Quantum mechanics in terms of realism ∗ Arthur Jabs Alumnus, Technical University Berlin. Vossstr. 9, 10117 Berlin, Germany arthur.jabs@alumni.tu-berlin.de (5 July 2017) Abstract. We expound an alternative to the Copenhagen interpretation of the formalism of nonrelativistic quantum mechanics. The basic difference is that the new interpretation is formulated in the language of epistemological realism. It involves a change in some basic physical concepts. The ψ function is no longer interpreted as a probability amplitude of the observed behaviour of elementary particles but as an objective physical field representing the particles themselves. The particles are thus extended objects whose extension varies in time according to the variation of ψ. They are considered as fundamental regions of space with some kind of nonlocality. Special consideration is given to the Heisenberg relations, the Einstein-PodolskyRosen correlations, the reduction process, the problem of measurement, and the quantum-statistical distributions. Key words: foundations of quantum mechanics, interpretation, realism, wavepackets, measurement, reduction, entanglement, nonlocality, Einstein-PodolskyRosen problem, Bell inequality ∗The 1996 version was also paper-published, in Physics Essays 9, 36-95, 354 (1996). The present version is a thorough revision, taking experimental and theoretical progress into account and omitting some more speculative sections. 2 CONTENTS 1. Introduction: General Principles 3 1.1 Difficulties in Present-Day Quantum Theory 3 1.2 Realism 5 1.3 Quantum Wavepackets 7 2. Specific Postulates: Single Particles 13 2.1 The Double-Slit Experiment 13 2.2 The Action Probability 15 2.3 The One-Particle Nonlocality of the Quantum Wavepackets 16 2.4 The Heisenberg Relations 20 3. Specific Postulates: Systems of Particles 23 3.1 Entangled Wavepackets and Multi-Particle Nonlocality 23 3.2 Similar (Identical) Wavepackets 25 3.3 Condensed Wavepackets 27 4. Measurements and Reduction 28 4.1 The Copenhagen Measurement Postulates 28 4.2 The Underlying Physical Problem 30 4.3 The Stern-Gerlach Experiment 31 4.4 Schrödinger's cat and Wigner's Friend 34 4.5 Determinism and Indeterminism 36 5. The Einstein-Podolsky-Rosen Problem 39 5.1 The EPR Problem and Nonlocality 39 5.2 Relativity and Causality 42 5.3 Bell's Inequality 44 5.4 Experiments 47 6. Quantum Statistics with Wavepackets 50 6.1 Field Quantization 50 6.2 The Many Aspects of the Condensed Wavepackets 51 6.3 The Balance Relation 54 6.4 The Bose and Fermi Distributions 58 6.5 Quantum Count Fluctuations 61 Acknowledgement 66 Appendix A: Collection of Wavepacket Spreading Formulas 67 Appendix B: Proof of the Bell Inequality 75 Appendix C: EPR Joint Probability Formulas 77 Appendix D: EPR Probabilities in Different Systems of Eigenfunctions 80 Notes and References 83 100 3 The environment as we perceive it is our invention. Heinz von Foerster 1. INTRODUCTION: GENERAL PRINCIPLES 1.1 Difficulties in Present-Day Quantum Theory There is no doubt that quantum theory is one of the most successful physical theories. Yet there is also no doubt that it contains serious difficulties. These difficulties are nowadays felt more and more strongly by those concerned with the unification of quantum theory and relativity and the future basis of physics. The difficulties may be divided into two kinds: conceptual and mathematical. The conceptual difficulties are related to the so-called Copenhagen interpretation. Any physical theory consists of a mathematical formalism, that is, a set of mathematical symbols and the rules for connecting these among themselves, and a set of interpretation rules, connecting the symbols of the mathematical formalism with the concepts of our sensory experience. The Copenhagen interpretation represents that set of interpretation rules that is presented more or less explicitly in the present textbooks on quantum mechanics. Actually, it is difficult to say who exactly constitutes the "Copenhagen school", supporting the Copenhagen interpretation; certainly Bohr and Heisenberg, but also Dirac, Pauli and von Neumann [1], [2]. Also, many versions of "the Copenhagen interpretation", from orthodox to liberal, can be found when different authors or textbooks are consulted. The difficulties of the Copenhagen interpretation may be characterized in the following way: 1) The wave function ψ(x, t) is not taken as an objective physical field like the electromagnetic field, but as a probability amplitude. And the probabilities to which it refers are not the probabilities that something is true or something will happen, whether it is observed or not (as in statistical mechanics) but the probabilities of specified outcomes of measurements or observations. Moreover, and most importantly, the observer is not just another physical object but the linguistic ego, something which appears nowhere as a mathematical symbol in the formalism. An electron does not have an exact location as long as we do not observe it, but it does have one when we observe it. "The 'trajectory' arises only by our observing it" [3]. In this way the Copenhagen interpretation speaks of position and of momentum, angular momentum, etc. only as "observables", not as real properties which objects have regardless of whether we observe them or not. The observer and the measurement are therefore indispensable elements in defining the theory. In all other physical theories the observer's only function is to test and apply the theory, not to define it. The Copenhagen interpretation thus rejects the language of epistemological realism. In my opinion, this is the most serious difficulty with that interpretation. 2) According to the Copenhagen interpretation it is impossible, in principle, to explain the probabilistic behavior in quantum physics as the result of some 4 underlying deterministic processes and variables that specify the physical situations in more detail than is done by the ψ function. This means that the Copenhagen interpretation rejects determinism. This point will be examined further in Sec. 4.5. 3) In the Copenhagen view concepts that refer to pointlike particles (single, sharp position, etc.) are applied to microscopic objects, whereas the ψ function is the solution of a field equation, namely a partial differential equation, like Schrödinger's, and may show up wave-like behavior. This has been called the "wave-particle duality", and it has been asserted that a unifying picture of microscopic objects cannot exist. As is well known Einstein, von Laue, Schrödinger, Planck, and de Broglie never accepted the Copenhagen interpretation. In fact, although the Copenhagen interpretation seems to be without apparent logical inconsistencies, any really thoughtful scholar finds it difficult to digest. Inquiring students are usually silenced by authoritarian statements such as that they do not yet know enough and will understand later, or that their questions are not relevant to physics. In particular, the defenders of the Copenhagen interpretation generally assert that a unifying picture is really unnecessary. They say that the formalism plus some working rules for its application give us the correct prescriptions for calculating the probabilities of the outcomes of any experiment, and that that is all we want. But as if some tectonic tensions were felt, discussions concerning the foundations of quantum mechanics continue. Moreover, it is noticeable that in the course of historical development as well as in the mind of any particular scientist the final mathematical formalism describing a set of physical phenomena emerges from a more or less pictorial view, conception or model. A good picture is very helpful since it has the same logical structure as the region of reality which it aims to represent, and it leads to a correct mathematical formulation of this reality. An example is Faraday's intuitive picture of lines of force and their subsequent mathematical formulation by Maxwell. "It is mainly with the hope of making these [Faraday's] ideas the basis of a mathematical method that I have undertaken this treatise", Maxwell writes [5]. A bad picture leads to no or to an only partially correct formalism. In this latter case it may happen that the emerging formalism describes the known phenomena correctly in its initial stage, but when it is developed further to include more and more experimental facts it sooner or later comes off the track. This is what I think has happened to quantum theory. I think that the lack of a good picture is responsible for the mathematical difficulties, and that their solution will emerge only from a solution of the conceptual difficulties. The mathematical difficulties of present-day quantum theory arise with the attempt to extend nonrelativistic quantum mechanics into the relativistic domain, that is, into quantum electrodynamics and relativistic quantum field theory. Here, divergent integrals have shown up in the perturbation expansions as solutions of the basic equations. Even if these integrals are made finite by means of renormalization procedures or are avoided by means of Epstein-Glaser methods [6], [7] nobody knows whether the expansions converge, and nobody has found an exact solution of the equations including interactions in the real world of 3+1 dimensions, although enormous efforts have been undertaken [8], [9, Sec. 11.1]. Thus Dirac [10], [11] writes: 5 I feel pretty sure that the changes which will be needed to get over the present difficulties facing quantum theory and appearing as a resistance between the quantum theory and relativity will be very drastic just as drastic as the change from Bohr orbits to the quantum mechanics of Heisenberg and Schrödinger and therefore one should not become too much attached to the present quantum mechanics. One shouldn't build up ones whole philosophy as though this present quantum mechanics were the last word. If one does that, one is on very uncertain ground and one will in some future time have to change one's standpoint entirely. 1.2 Realism While Dirac says nothing about the nature of the expected changes in quantum mechanics, Einstein's critique [12] - [14, p. 667] - [16] is more specific. Einstein [15, p. 6] writes: But in any case my conception starts from a thesis which is strongly rejected by most present-day theoreticians: There is something like the "real state" of a physical system, which independent of any observation or measurement exists objectively and which can in principle be described by means of physical terms [Which adequate terms or basic concepts have to be employed for this is in my opinion unknown at the present moment (material points? field? concepts that have still to be invented?)]. Because of its "metaphysical" nature, this thesis of reality does not have the purpose of providing a statement of fact: it has really only a programmatic character. However, everybody, including the quantum theoreticians, sticks consistently to this thesis of reality so long as he does not discuss the foundations of quantum theory. Nobody doubts, for example, that there has been at a certain time a certain position of the moon's center of gravity even if no real or potential observer existed. This thesis of Einstein's is what we mean by epistemological realism. We do not attempt to give a fool-proof definition of realism. We emphasize, however, that the type of realism adopted here does not mean that physical objects with their properties exist independent of whether or not we observe them (this would be naive realism); it only means that the laws of nature can be formulated as if that were the case. Realism, as it is meant here, is not considered as a matter to be proved or disproved, it is a way of speaking, a language. It is in fact the manner in which our language works in normal (not philosophical) use; to quote Wittgenstein [17]: For this is what disputes between Idealists, Solipsists and Realists look like. The one party attack the normal form of expression as if they were attacking a statement, the others defend it, as if they were stating facts recognized by every reasonable human being. 6 If we were to distinguish this type of realism from naive realism we would call it epistemological or linguistic realism. Von Laue [18], Schrödinger [19] - [21], and Planck [22], [23] have always shared a realist attitude with Einstein. And in the course of the years the number of physicists who openly advocate realism in quantum theory has continually increased. Jammer's book [24] already quotes Bohm, Bunge, de Broglie, Jaynes, Ludwig, Popper and Renninger. And we want to add the papers by Janossy [23], de Broglie [25], Bunge and Kalnay [26] - [28], Bell [29] - [31], Rayski [32], Lévy-Leblond [33], Stapp [34], [35], Roberts [36], Maxwell [37], Burgos [38], Popper [39], [40], Pearle [41], Bohm, Hiley and Kaloyerou [42], Rohrlich [43], Dorling [44], and Dieks [45]. Actually, it is difficult to do justice to everybody because there are several types of realism, because statements in favor of realism range from very outspoken to rather casual, and because the problem of a realist interpretation is often mixed up with a change in the formalism to be interpreted. Scientists I found particularly outspoken in favor of realism are Popper, Bunge, and Bell. Bell [30, p. 40] in particular postulated "beables" to replace the "observables", and his work will concern us in Sec. 5 when we discuss the Einstein-Podolsky-Rosen (EPR) problem. Thus in the present work we propose to overcome the conceptual difficulties of quantum mechanics by interpreting the formalism in terms of realism. We replace the interpretation rules of the Copenhagen school by interpretation rules formulated in the language of epistemological realism. What is interpreted is the same, namely the standard formalism of nonrelativistic quantum mechanics, as for example presented in the textbooks by Messiah [46] and by Cohen-Tannoudji, Diu, and Laloë [47]. In other words, we separate the problem of changing the interpretation from the problem of changing the formalism and restrict ourselves to the interpretation problem. This certainly cannot solve all problems. Nevertheless it can solve a number of them, as I intend to show in this article. In other words, this work is an attempt to see how far one can get with a realist interpretation without proposing new formulas. Our main result is that a realist interpretation is possible, in contradiction to what is asserted by the Copenhagen school. Other interpretations of the unmodified standard formalism in terms of realism might also be possible, differing from the present one in the details of the elaboration, although to my knowledge in 1996 there was none that had been expounded to a comparable extent. Since then a number of ontological or realist interpretations have appeared, but according to them they have nothing to do with my work. It is mainly for convenience of presentation that the nonrelativistic formalism is chosen, with the familiar Schrödinger equation and wave function as the basis of the reinterpretation. We think that the concepts developed will prove fruitful in the relativistic domain as well; at least we do not know of any argument that would point to the contrary. Everything that can be described by the Schrödinger equation can also be described by the Klein-Gordon, Dirac etc. equation. Thus we might as well have used the Klein-Gordon, Dirac, etc. equation, and used a Lorentz scalar, spinor, vector, etc. instead of the Schrödinger scalar ψ(x, t). Moreover, we include photons in our considerations, that is, we treat classical electromagnetic radiation 7 pulses on the same footing as pulses of Schrödinger etc. waves. The essential point is that the wave function ψ(x, t) is taken as an objective physical field, comparable in this respect to the function Fμν(x, t) as the sourcefree electromagnetic field. This implies that ψ is not merely a device for calculating the probabilities of specified outcomes of observations; it does not merely describe "knowledge" [48]. There are no longer two different elements, namely the particle and the field associated with it; rather these two are the same. There is no wave-particle duality. The function ψ(x, t) describes the particle like a region of excessive water density ρ(x, t) in the sky describes a cloud. And the particle has its properties ("beables") even when they are not observed. What these properties are is another question. In fact we shall see that the new properties are different from the old ones. Thus, for example, single sharp values of position or momentum are no longer among the properties. The properties that we shall have to attribute to the particles in order to carry through the realist programme at first sight might appear rather strange. This is the price one has to pay. Nowadays we are perhaps willing to pay more than physicists in Schrödinger's time when the emphasis was on the successes rather than on the the difficulties of quantum mechanics. Also, it is not to be expected that the difficulties that have beset quantum theory for more than 90 years can be overcome by some cheap trick. On the other hand, we try to show that the price is not too high in that the new postulates, together with [49], [50], lead to a coherent picture. The conceptual mist dissolves and the view opens for a fresh outlook. Quantum mechanics is formulated in the same language as classical mechanics and any other physical theory, and the power that lies in realist language is available for quantum mechanics too. 1.3 Quantum Wavepackets The identification of an elementary particle with a field means that quantum mechanics becomes a field theory, albeit a special one. The Schrödinger equation, or any of the quantum equations of motion, in any case is a field equation, that is, a partial differential equation, with the solution ψ depending on the four independent variables x, y, z and t. The equations of motion of the point particles of classical mechanics, on the contrary, are ordinary differential equations for the three functions x(t), y(t) and z(t). On this point Einstein [51] writes: The most difficult point for such a field theory at present is how to include the atomic structure of matter and energy. For the theory in its basic principles is not an atomic one in so far as it operates exclusively with continuous functions of space, in contrast to classical mechanics whose most important feature, the material point, squares with the atomistic structure of matter. 8 The modern quantum theory, as associated with the names of de Broglie, Schrödinger, and Dirac, which of course operates with continuous functions, has overcome this difficulty by means of a daring interpretation, first given in a clear form by Max Born: the space functions which appear in the equations make no claim to be a mathematical model of atomic objects. These functions are only supposed to determine in a mathematical way the probabilities of encountering those objects in a particular place or in a particular state of motion, if we make a measurement. This conception is logically unexceptionable, and has led to important successes. But unfortunately it forces us to employ a continuum of which the number of dimensions is not that of previous physics, namely 4, but which has dimensions increasing without limit as the number of the particles constituting the system under examination increases. I cannot help confessing that I myself accord to this interpretation no more than a transitory significance. I still believe in the possibility of giving a model of reality, a theory, that is to say, which shall represent events themselves and not merely the probability of their occurrence. On the other hand, it seems to me certain that we have to give up the notion of an absolute localization of the particles in a theoretical model. This seems to me to be the correct theoretical interpretation of Heisenberg's indeterminacy relation. And yet a theory may perfectly well exist, which is in a genuine sense an atomistic one (and not merely on the basis of a particular interpretation), in which there is no localizing of the particles in a mathematical model. For example, in order to include the atomistic character of electricity, the field equations only need to involve that a three-dimensional volume of space on whose boundary the electrical density vanishes everywhere, contains a total electrical charge of an integral amount. Thus in a continuum theory, the atomistic character could be satisfactorily expressed by integral propositions without localizing the particles which constitute the atomistic system. Only if this sort of representation of the atomistic structure be obtained could I regard the quantum problem within the framework of a continuum theory as solved. The idea that elementary particles are extended objects has repeatedly appeared in the literature. However, the size has always been considered to be fixed, for example equal to the Compton length of the electron −λC = h/mc, the classical electron radius rcl = e 2/mc2, or the Planck length lP = (hG/c 3)1/2. In the interpretation presented here the size of any individual particle is variable, namely equal to the size of the ψ function traditionally associated with it. Mathematically, the ψ function need not have a sharp boundary but for our purposes it may be considered to have the extension given by the usual standard deviation ∆x := 〈(x − 〈x〉)2〉1/2, which varies in time according to the variation of ψ(x, t). In order to emphasize that we consider the elementary particles not as pointlike 9 but as extended objects we call them wavepackets, always written in one word. The function ψ(x, t) is the mathematical representation of a wavepacket. Sometimes we shall neglect the difference between the wavepacket and its mathematical representation and just call ψ a wavepacket. The term 'wavepacket' does not, however, in any way mean a restriction to a linear superposition of plane waves, and even when it is mathematically expressed as such it does not mean that plane waves are physical constituents of the particles. It just means the region(s) of non-vanishing ψ. A particle thus may be regarded as a "matter pulse", on an equal footing with an electromagnetic radiation pulse. Indeed, in our interpretation the radiation pulse, under certain conditions, is also a particle, namely a photon. For each kind of particle the "matter" field is specified by additional parameters like mass, charge, etc. Note that considering the elementary particles as quanta of some fields, in particular considering the Schrödinger function ψ(x, t) as representing one elementary particle, goes beyond what is supported by the Schrödinger equation alone. This equation as it stands is still a classical field equation [52]. The appearance of the typical quantum constant h in it only indicates the kind of fields to which the equation refers, namely to Schrödinger, de Broglie or matter fields whose characteristic wave quantities ω and λ are related to the particle quantities E and p by the relations E = hω and p = h/λ. Even the special boundary conditions introduced in order to get "quantized" solutions, which can be distinguished from one another by parameters that take on discrete (eigen)values, are not enough to provide a genuine quantum character. The energy eigenvalues in the hydrogen atom, for example, follow from the normalizability postulate I < ∞. Eigenvalues already appear in classical macroscopic physics, and it is significant that the title of Schrödinger's famous papers was "quantization as a problem of eigenvalues". However, taking a one-particle wavepacket as a quantum of some field means to attribute to it a special property: in the process of reduction (Section 4.2) it can never contract to more than one place at a time. Time taken as that in its centre system, say. In other words the one-particle wavepacket can never induce more than one effect at a time (Sections 2.2 and 2.3). At the time when Schrödinger developed his wave mechanics he already tried to identify the electron with a wavepacket of small but finite extension constructed by superposition of particularly simple solutions of the Schrödinger equation [53]. The physicists of his time did not become convinced of this idea because in general the wavepackets spread out in time and for other reasons. Though Schrödinger was aware of the difficulties [54] and although he admitted not knowing how to overcome them, he never gave up the belief that a realist wavepacket picture should be possible [55] - [57]. In order to get such a picture one must be willing to accept a rather drastic change in the basic concepts. Let us first consider three popular objections against the general idea that wavepackets can be models of elementary particles. The first objection is that ψ can take on complex values and hence cannot represent a physical object. The objection is very weak and has been dealt with conclusively by Bunge [58]: it dissolves as soon as one realizes that a complex function 10 is nothing but a couple of real functions united in a convenient way, and as soon as one remembers other situations in physics where complex quantities are used; for example, the impedance of an electric circuit Z = R + iX. Notice that we never directly measure "the electron", but only its charge, velocity, etc., that is its properties. Thus "the electron" may be represented by a complex function in the mathematical formalism, provided only that the values of its directly measurable properties are represented by real numbers. This is related to the old philosophical distinction between substance and attributes. Actually, it is the very phase of the complex wave function ψ that has recently opened a way to introduce determinism into quantum mechanics [49]. The second objection is that a free wavepacket of Schrödinger waves inevitably spreads out in time. A corresponding difficulty already exists in ordinary quantum mechanics where the wavepacket gives the probability that the position of the alleged particle in it will show up. If, for example, the wavepacket of an electron which moves perpendicularly through successive registering screens were to spread out rapidly, the "positions" of the electron on the successive screens would be distributed erratically in a rapidly widening cone, and would not exhibit anything like a straight track. The actually observed straight track is usually ascribed to the facts that (1) the spreading of the wavepacket numerically is negligibly small because the packet need not have a smaller extension that that of an observed spot on a screen, which has at least the dimension of a grain in a photographic emulsion (10−7 m), and (2) the wavepacket is reduced in size whenever its "position" is observed. This has been worked out in [49] (cf. Chapter 4). Nothing prevents us from taking over these arguments, suitably adapted, when we consider the wavepacket as the electron itself. Notice also that there is hardly any individual free elementary particle that is experimentally controlled for more than a fraction of a second. The assertion that it keeps together after it has left the apparatus goes far beyond any real experience. The wavepacket model may clash with some of our favored ideas on what an elementary particle is, but the decisive counter-argument could only be a clash with observed facts. We have not detected any such clash. It is instructive to consider also non-free wavepackets which spread out. Consider an electron in an impenetrable macroscopic box. We may construct a wavepacket whose initial dimensions ∆x(0) and ∆p(0) are rather small, in particular ∆x(0) very small compared to the dimension of the box. This rather well localized packet may move back and forth between the walls, and it is a good model of a classical Newtonian particle. In the course of its oscillations, however, it will spread out and its width will become of the order of the dimension of the box, so that it ceases to be a good model of a classical particle; it is then a model of a classical wave swashing back and forth between the walls. Pictures of this behavior are shown by Brandt and Dahmen [59]. Another example is a small electron wavepacket orbiting around the nucleus in an atom, as in the planetary (Rutherford) model. Such a packet may be constructed by the superposition of some stationary eigenpackets, as suggested by Schrödinger [53]. However, as Heisenberg [3] soon pointed out, in the course of time the packet inevitably spreads out all around the nucleus although it remains 11 within a finite radius [60], [61]. It follows from the Schrödinger equation with a Hermitean Hamilton operator that the quantity I = ∫ +∞ −∞ ρ(x, t)d3x with (1.1) ρ = |ψ(x, t)|2 is independent of time. In the Copenhagen interpretation the integral is the probability of obtaining any value for the position of the particle whose wave function is ψ(x, t) in an appropriate position measurement. The probability assumes a 100% efficiency of the apparatus, hence the value of the integral has to be 1. What is the conserved quantity I = ∫ ρd3x if not a probability? The question what the field ψ(x, t) is, is easy: the field is the field. It is an irreducible entity, a "primitive notion", in the same sense that the electromagnetic field is. Originally the electromagnetic field had been "explained" by the ether, but for well known reasons the ether has been dismissed and the field has been established as a quantity in its own right (Einstein [62]). We become familiar with the field by studying how it behaves, that is, by finding rules that connect it with well-known phenomena. ψ is just a "matter" field (electron field, proton field etc.) . As Jammer puts it [63]: "physical reality is what it is because it does what it does." Likewise, the meaning of the integral I = ∫ ρd3x depends on what role I , or ρ, plays in the formulas describing familiar phenomena. In quantum field theory the integral I , with some more general ρ than just (1.1), is identified with the operator for particle number, or for particle number minus antiparticle number, or for total charge, either electric or baryonic or leptonic, etc. Leaving the meaning of I open does not pose a problem for a realist interpretation. The only thing that matters is that I a priori is not a probability. In addition to the three objections discussed above, there are other, more serious ones. These include the objection that what is observed is always "a whole particle", never parts of it (dealt with in Sec. 2.3), and that the wave-function reduction, taken to occur instantaneously, allegedly cannot be a physical process (Sec. 4). The subsequent sections are organized as follows: In Secs. 2 and 3 we develop the specific features of our interpretation, which go beyond the general principles of realism and of extended wavepackets discussed in Sec. 1, and which are different from those of the Copenhagen interpretation. In doing so we adopt an inductive approach, selecting special physical situations to motivate and illustrate the postulates to be introduced. Then, in Secs. 4 and 5, we turn to the application of these postulates to special problems. These sections show how the conceptual problems of the Copenhagen interpretation can be avoided, and by contrasting the two interpretations, they lead to a deeper understanding and a clarification of some additional points. 12 Finally, in Sec. 6, we show how the concepts of our interpretation applied to Schrödinger's configuration space with symmetrized functions lead us to the quantum statistical distribution and fluctuation formulas. The appendices contain the mathematical derivations of some important formulas in order to make the work self-contained. 13 2. SPECIFIC POSTULATES: SINGLE PARTICLES 2.1 The Double-Slit Experiment We begin by considering the interpretation of the expression |ψ(x, t)|2d3x . Since in our interpretation the wavepacket is an extended object, we can no longer speak of the position of the wavepacket, and the above expression can no longer be the probability of observing, at time t, the position within d3x about x, as in the Copenhagen interpretation. We shall introduce the new interpretation of |ψ|2d3x by considering the well-known double-slit experiment. Figure 1: The double-slit experiment. Consider a beam of electrons with average momentum p and little spread about this value. The beam is directed towards a wall, as shown in Fig. 1. The wall contains two parallel slits, which can be opened and closed. Behind the wall there is a detecting screen which registers the intensity I (x ) of the beam (number of electron counts per second) as a function of the distance x from the center O. If only slit 1 is open the intensity function I (x ) will look like I1; if only slit 2 is open it will look like I2. If both slits are open the intensity function is not, however, the sum I1 + I2 but will look like I12. The shape of I12 is obtained simply by regarding the beam of electrons as a plane wave with a wavelength λ = h/p and calculating the interference of waves originating from slit 1 and slit 2. These interference effects constitute the difference between I12 and I1 + I2. The important point is that no matter how low the intensity of the incoming beam, the intensity function on the screen, when both slits are open, is always given by I12, provided we compensate for the lower incoming intensity by a longer exposure time in order to have the same total amount of energy (or total number of electrons) deposited on the screen. We may adjust the incident intensity until it is so low that it corresponds to one incoming electron per day. Thus, even a single electron must correspond to a number of wave trains capable of interference with one another. The Copenhagen interpretation holds to the idea of a pointlike position and at the same time has to account for what is observed in the double-slit experiment. If the particle at every instant of time had a pointlike position, this would mean 14 that any single particle goes either through slit 1 or through slit 2 so that the intensity function on the screen would be I1 + I2 but never the observed I12. Of course, one might invent a special type of motion of the particle, for instance some orbiting through both slits, before it finally arrives at the registering screen. It is, however, clear that this would have to be a quite complicated motion in order to explain the distribution I12, and it would compete rather unfavorably with the simple explanation of interference of waves. Thus, if such constructions are left out of consideration, the observed intensity distribution I12 leads to the conclusion that the particle does not have a sharp trajectory. In spite of this, the Copenhagen school wants to continue, in some way or other, with the concept of a sharp position. The material point had already been a basic concept in classical mechanics [62], [64], and it had remained so in the first steps into the microworld, namely in Rutherford's and Bohr's planetary models of the atom. Now, since the double-slit results make it impossible for the particle to have a (sharp) position at all times, that position is ascribed to them only at the moment of measurement. Thus, "position" is no longer a permanent objective property of a particle but an "observable", which comes into existence only in the act of observation. If there is no observation, a position must not be ascribed to the particle. This is not to be understood in the sense that the particle is extended when it is unobserved, and contracts to the point of observation when we observe it. In the Copenhagen interpretation the contraction happens only to the wave function, which is, however, not the particle itself. As with position, other quantities like the components of momentum and angular momentum must not be ascribed to the particle, except at the moment of the respective measurement. All these quantities are merely "observables". Now turn to the realist interpretation. Here, the electron is a wavepacket, and its extension covers both slits, so that in any single passage both slits are involved and determine the final interference pattern. Only if the wavepacket has a transverse width (normal to the direction of its centre) that is smaller than the distance between the slits will no double-slit interference effects be observed. Let us have a closer look at how the intensity function I (x ) on the screen is obtained. The screen might be a photographic emulsion. The incoming electron beam causes black spots (after development) at certain positions on the screen. The intensity function I (x ) is then proportional to the spatial density of these spots. These spots are not points but have some extension. A spot is the result of a cascade of processes which is initiated by the ionization of an atom of the emulsion by the incoming wavepacket. The localization is thus limited in practice by the extension of the black spot and in principle by the extension of the initiating atom or of the smallest emulsion wavepacket with which the incoming packet interacts. This does not mean that any distances that are smaller than the smallest wavepacket would have no meaning. We are only discussing more or less direct position measurements. There are indirect measurements: a distance l may appear in a complicated equation, and by measuring all the other quantities in that equation the value of l may be calculated. In this way the formulas of quantum electrodynamics for leptons for 15 instance, have been verified down to distances very much smaller than the diameter of any atom. The limitation in direct localizability does not present any basic difficulty, and we may assume in the present considerations that the black spots are small enough compared to the incoming wavepackets to be considered practically as points. Then, of course, we can no longer say that a black spot at a certain position indicates that the electron is (or was) at this place. Rather, in our interpretation the packet, when traversing the special physical environment represented by the photographic emulsion, has induced a black spot at the observed position. The packet is like a cloud moving along while triggering thunder and lightning here and there. 2.2 The Action Probability In the present interpretation the expression |ψ|2d3x in some physical situations is also a probability, but conceptually different from the Copenhagen localization probability, and may be called an action probability. It is the probability that the wavepacket ψ(x, t) when placed into an appropriate physical environment, at time t induces an observable effect within d3x about x. 'At time t' in the Copenhagen interpretation is the moment when the observer chooses to take notice of the result. In the realist interpretation it means a moment somewhere in the time interval in which the incoming wavepacket is given the possibility of inducing an effect, roughly speaking during the interval the wavepacket sweeps over the apparatus [49]. It is presupposed that the wavepacket during that interval with certainty induces an observable effect somewhere in space. This is what is meant by an appropriate physical environment. If the actual environment meets this requirement only partly, the raw data must be corrected for this, and |ψ|2d3x refers to the corrected data. Recall that the Born rules already presuppose perfect apparatus since ∫+∞ −∞ |ψ(x, t)|2d3x = 1. In the Copenhagen interpretation the expression |ψ|2d3x is defined as a probability, and accordingly the function ψ is defined as a probability amplitude. In our interpretation ψ is defined as a physical field, but the expression |ψ|2d3x is not defined as a probability. Rather |ψ|2d3x in certain physical situations can be put numerically equal to an action probability [49]. The point might appear subtle, but it is important. Compare it with the following picture: Smoke comes out of a funnel. Let ρ(x, t) be the density of smoke, and let p(x, t) be the probability that a test body at x and t will gather on its surface a certain amount of smoke particles in a certain time interval. This probability may be calculated on the basis of the physical laws involved in the process and may turn out to be proportional to the density ρ. Then, by referring ρ to some standard density and properly adjusting the chosen amount of gathered smoke particles, we can always arrange that the density ρ(x, t) is numerically equal to the probability p(x, t). Nevertheless, it is still smoke, and not probability, that comes out of the funnel. The expression |ψ|2 in our interpretation plays the logical role of the smoke density ρ, not of the probability p, whereas in the Copenhagen interpretation |ψ|2 16 plays the role of the probability p, and there is no expression in the formalism of quantum mechanics that in the Copenhagen interpretation would play the role of the smoke density ρ. In principle, the effect induced by the wavepacket ψ need not be observed. In realism, if an effect occurs, it occurs independently of our observing it. And it may be any effect. It is only because we want to refer to the same physical situation as the Copenhagen interpretation that it must be an observable effect, observable according to the criteria of the Copenhagen interpretation. A stable macroscopic fixation, memory or record of the effect will be required, but how to achieve this is not a specific problem of quantum mechanics and does not concern us here. Although in our interpretation the black spot on the screen does not mean that the particle "has been there", it still means that it has been somewhere around, in the sense that it has covered the position of the spot. For example, suppose we know the shape of the wavepacket but not its absolute location. We can then calculate, by maximum-likelihood methods, the probability that the region of the black spots includes the centre of the packet. Or suppose the other case, that we know the position of the centre of the packet but not its shape. We may then measure the shape in the following manner: let the direction of the wavepacket's centre be the y direction, put a screen in the x-z plane in its way, and register the black spot appearing on the screen. Repeat this under identical initial conditions. The distribution of spots on the screen then images the shape of the wavepacket, or, more precisely, its projection on the x-z plane. And the width ∆x of the distribution in x direction is the width of the wavepacket. 2.3 The One-Particle Nonlocality of the Quantum Wavepackets We will now consider more closely how the black spot in an emulsion screen is brought about. This will lead us to the perhaps most remarkable property that we have to ascribe to the quantum wavepacket. Consider a beam of electromagnetic radiation falling onto a screen. Let us first treat the beam classically as a continuous field with the time-averaged energy flux density S = εocE2 in W/m 2, say. The registering screen, on the other hand, is conceived to consist of atoms in the sense of quantum mechanics. The incoming radiation will then cause black spots at certain places on the screen, and their density classically is proportional to the energy flux density of the radiation at these places. The resulting pattern will exhibit a granular structure but this does not demonstrate that the incoming radiation has particle or quantum properties; it only shows that the screen has, which we have presupposed anyway. The wind is not made of particles only because it causes an integral number of trees to fall in the forest, as Marshall and Santos put it [65]. To see what may be called the quantum or particle aspect of the incoming radiation we recall that the black spot is the result of a cascade of processes which is initiated by the ionization of an atom. This atom, in order to become ionized, needs some threshold energy Ethr. Of course, knowing quantum mechanics we assume that hν ≥ Ethr, but this is not sufficient for ionization in the classical picture. Imagine that the atom gets 17 this energy by absorbing and accumulating all the energy of the incoming classical radiation which arrives on its area σ, like a dust collector that is put in a stream of polluted air. With the energy flux density of the radiation S the energy accumulated by the atom during the time t is E = Sσt . When the atom has accumulated the energy Ethr it becomes ionized and initiates the formation of the black spot. The accumulation time needed for this is tacc = Ethr/(Sσ). Now, the value for tacc according to the above formula turns out to be of the order of hours or years in situations where effects are actually observed immediately after the arrival of the radiation. As an example consider the interference experiments of Reynolds et al. [66], [67]. Light of wavelength λ = 4.358× 10−7 m and ∆λ = 10−12 m passing a Fabry-Perot interferometer produced an interference pattern on the multi alkali cathode of an image intensifier tube. The entire interference pattern had an area of 1.5×3.1 mm2= 47 × 10−6 m2. From the size of the coherence length of the light (cf. Appendix A, (A25)f.) and the density of the excited atoms in the source it could be concluded that the light intensity was so low that only 15 photon wavepackets per second passed through it and there was only one photon in the apparatus at a time. The wavelength of 4.358×10−7 m means photons of energy 2.85 eV. So, a light energy of 15×2.85 eV = 43 eV/s (S = 1.5×10−13 W/m2) passed over the area of the pattern. Each photon wavepacket covers the whole interference pattern. With a minimum linear size of the photon-absorbing molecule of 8× 10−10 m, the energy passing per second over the area of a molecule located in an interference maximum (≈ 2× average energy) may be approximately 2 × 15 × (64 × 10−20/47 × 10−6) × 2.85 eV/s. With a threshold energy of 1.36 eV (λ = 0.9 × 10−6 m) necessary for the emission of an electron [68], the accumulation time is 1.36/(11.6×10−13) s = 1.17×1012 s = 37500 years! The interference pattern would thus only appear after centuries, but then it would appear fully in one flash. Actually the interference patterns were obtained in 15 seconds, and in various runs with reduced exposure time the authors verified that the pattern is built up gradually as time proceeds. The first black spots are induced immediately after the arrival of the radiation. In fact, upper limits of the time lags between the arrival of the radiation and the ionization of the molecule as short as 3 × 10−9 s [69] and 10−10 s [70] have been reported. Of course the total energy absorbed by all atoms during the whole accumulation time is the same in both cases. The difference is that classical theory would have it absorbed all in the last moment (at tacc), whereas experiment shows that it is absorbed in many small portions distributed over the accumulation time. Even with somewhat different assumptions one arrives at the same conclusion, as already shown by Campbell [71], Planck [72], Mandel [73], Paul [74], and others [75]. It is thus safe to conclude that the energy for ionizing the atom is not the energy contained in the cylinder that the atom has cut out of the field up to the moment of its ionization. 18 The traditional way of interpreting this situation was that the energy which ionized the atom was concentrated in a pointlike particle (the quantum) and that this particle happened to hit the atom. But in order to account for the interference effects observed, the particle was not allowed to exist until a measurement was made. This is the "wave-particle duality". Alternatively, we suggest: the field quanta are not pointlike objects but spatially extended wavepackets of mean energy 〈E〉 = h〈ν〉, and in the act of measurement they contract to an effectively pointlike region. And what we have here concluded for the photon wavepackets we conceive to hold for any wavepackets. This contraction is different from the shrinking (or spreading) of the wavepacket governed by the Schrödinger equation (Appendix A). In order to account for the situations described above, the contraction must occur with superluminal velocity in the reference system of the measurement apparatus: With the dimension of the interference pattern (≤ lateral dimension of the wavepacket) in [66], [67] of 3.1 cm and a time lag between the arrival of the radiation and the ionization of the atom of 10−10 s [70] the contraction velocity would be 3.1×108 m/s. This is only slightly larger than light velocity, but wavepackets with larger dimensions are easily met: In fact, the one-particle wavepacket may consist of several non-overlapping spatially well separated parts. In [76], for example, each photon of a low intensity radiation beam was split by a beam splitter into two separate parts, and either part was directed into a different detector. No coincidence counts between the two detectors could be observed. That is, if the photon is detected in detector 1 it immediately contracts to a small spot in that detector, so that there is no longer any part of the wavepacket at detector 2. The detectors were separated by 20 m. With a time lag between arrival and detection of 10−10 s [77] (0.074× 10−10 s [78]) the contraction had to occur at a velocity of 20 c (270 c). In other words, the count in one detector effects that there is no count in the other detector. In view of the multi-particle nonlocality to be considered in Sec. 3.1 we regard this as one-particle nonlocality [79]. And we ascribe this oneparticle nonlocality also to massive wavepackets, for example to the atoms in the Stern-Gerlach apparatus (Sec. 4.3) and the neutrons in the single-crystal neutron interferometer [80]. There are no parts of an electron wavepacket, say, which could dynamically interact with each other. We want to call this internal structurelessness of the wavepackets. In fact, we may regard the success of the Schrödinger equation as a confirmation of the absence of dynamic self-interactions (i.e. those that are explicitly represented by interaction terms in the Schrödinger equation) [83]. Consider the Schrödinger equation for an electron in an electrical potential V(x, t) (2.1) ih ∂ψ(x, t) ∂t = − h 2 2m ∆ψ(x, t)− eV(x, t)ψ(x, t). Let us for the moment regard the quantity ρ(x, t) := − e|ψ(x, t)|2 as the charge 19 density of the electron, and let us write the potential V(x, t) as the sum of two terms (2.2) V(x, t) = Vo(x) + Ve(x, t), where Vo(x) is the potential produced by the atomic nucleus (proton) plus some outside charges, and Ve(x, t) is the potential produced by the charge distribution of the electron itself. This can be written as (2.3) Ve(x, t) = 1 4πεo ∫ ρ(x′, t) |x− x′| d3x′. Ve(x, t) represents some action of the electron on itself. Inserting (2.2) and (2.3) into (2.1) leads us to the nonlinear integro-differential equation ih ∂ψ(x, t) ∂t = − h 2 2m ∆ψ(x, t)− eVo(x)ψ(x, t) (2.4) + ψ(x, t) e2 4πεo ∫ |ψ(x′, t)|2 |x− x′| d3x′, which differs from the familiar Schrödinger equation by the last, self-interaction term. On the other hand, it is the familiar Schrödinger equation (2.1), and not Eq. (2.4), that gives the correct results, for example for the hydrogen eigenfunctions. Of course, the absence of dynamic interactions between spatial parts of a wavepacket does not exclude the existence of recoil effects of the emitted radiation on the emitting wavepacket as a whole, as it is considered by Barut and collaborators in their approach to quantum electrodynamics without canonical quantization [84]. In this context we may quote Lorentz [85]: In speculating on the structure of these minute particles we must not forget that there may be many possibilities not dreamt of at present; it may very well be that other internal forces serve to ensure the stability of the system, and perhaps, after all, we are wholly on the wrong track when we apply to the parts of an electron our ordinary notion of force. and also Dirac [86]: it is possible for a signal to be transmitted faster than light through the interior of an electron. The finite size of the electron now reappears in a new sense, the interior of the electron being a region of failure, not of the field equations of electromagnetic theory, but of some of the elementary properties of space-time. The contraction in any case shows that the quantum wavepacket must be a special object of its own kind. We have already mentioned in Section 1.3 that a one-quantum wavepackets can never induce more than one effect at a time. This is just another aspect of the here described contraction to one single place. Does the 20 contraction occur even with infinite velocity? In which reference system? [50]. In any case a velocity inside the wavepacket has no direct physical meaning: there cannot be observers sitting inside the wavepacket at determined positions and reading off synchronized clocks. Might the failure of the elementary properties of spacetime mentioned by Dirac go so far that there is no space at all (i.e. no distance to travel) inside the wavepacket, as speculated in the 1996 version of this article? Note also Bell's remark in [87]: Behind the apparent Lorentz invariance of the phenomena, there is a deeper level which is not Lorentz invariant. 2.4 The Heisenberg Relations The Heisenberg relations (2.5) ∆x∆px ≥ h/2 play a central role in the continuing discussions on the physical meaning of quantum theory. Many different interpretations have been advanced [24], and it is not our intention to review them here. Rather, we shall pick out a few points that serve to clarify the present interpretation. In the realist interpretation the ranges ∆x and ∆px are simply the extensions of the wavepacket. ∆x is the extension (in x direction) in ordinary space, ∆x = 〈(x−〈x〉)2〉1/2, for example, and ∆px is the extension in momentum space. Both ∆x and ∆px are simultaneously ascribed to a wavepacket. The Heisenberg relations (2.5) express a relation between these ranges. This is just like the relation ∆x∆ν ≥ c/4π and its interpretation for a pulse of classical electromagnetic radiation, known long before the advent of Heisenberg's relation. The importance of the Heisenberg relations stems from the importance of the wavepacket nature of the elementary particles. In fact, one may characterize quantum mechanics as the theory of those phenomena where the concept of a point particle or point localization irreparably breaks down. In the regime of classical mechanics only the first moment of the function |ψ(x, t)|2, that is, the centre value 〈x〉 plays a role (Ehrenfest theorem), whereas in the proper quantum regime the higher moments come into play. Already the first step, the inclusion of only the second moments ∆x, marks the essential differences from classical mechanics. This is one reason why the wavepacket concept captures more characteristics of the elementary particle than does the point particle concept. Consider a one-dimensional wavepacket with a momentum width ∆p such that ∆p/〈p〉 = const 1. A beam made up of such wavepackets would be quasimonochromatic, and any beam to which an energy is at all ascribable ("a 45 keV electron beam") must be of this type. In this case the Heisenberg relation for Gaussian minimum packets reads ∆x = r/(4π const) × h/〈p〉 with r = 1, and, if other things remain constant, the "classical limit" h → 0 is indeed equivalent to ∆x→ 0. For other than Gaussian packets or other definitions of the width ∆x only different values of the factor r appear in the relation between ∆x and h/〈p〉. 21 Since the Copenhagen interpretation insists that a sharp position, or a sharp momentum of the particle can be obtained (at least in principle) as a result of a measurement, the Heisenberg relations are called uncertainty, inaccuracy or indeterminacy relations. In the realist interpretation presented here, there is no sharp position to be uncertain about. Asking for any sharp value of position or momentum of a particle is asking the wrong question. Once we are given the shape of the wavepacket (which includes its ranges), no uncertainty is left concerning this packet. We therefore do not speak of uncertainty relations; rather we would speak of complementarity, or just Heisenberg or Born-Heisenberg relations. As ∆px is the width of the wavepacket ψ(px) in momentum space, where ψ(px) is the Fourier transform of ψ(x) (cf. Appendix A), 'Fourier reciprocity relation' would appear the most fitting denomination. One may say that there is still the uncertainty about the place where the wavepacket will induce an effect. This uncertainty depends, however, on the environment and is not a characteristic of the packet alone. While in a realist interpretation the Heisenberg relations a priori have nothing to do with any measurement (cf. Planck [88] and Bunge and Kalnay [28], [26]), the opposite is the case in the Copenhagen interpretation. Thus, Bohm writes [89, p. 99]: If a measurement of position is made with accuracy ∆x, and if a measurement of momentum is made simultaneously with accuracy ∆p, then the product of the two errors can never be smaller than a number of order h. This suggests that the widths ∆x and ∆px were primarily those of the measurement apparatus and not those of the wavepacket, which is also suggested in the example given by Messiah [46, p. 142, 146], where the ∆'s are widths of slits in diaphragms. Also, in the framework of the Copenhagen interpretation efforts have been made to find a physical mechanism by which the Heisenberg relations are brought about, and consideration has been given to the influence of the measuring devices on the object measured. Bohr describes physical situations where the experimental arrangement suited for measuring the exact positions excludes the experimental arrangement suited for measuring the exact momenta of some particles, and he writes [14, p. 233]: As repeatedly stressed, the principal point is here that such measurements demand mutually exclusive experimental arrangements. However, in classical physics such situations exist as well. Consider a sugar cube and the two properties of being soluble and being burnable. Evidently, measurements of these properties demand mutually exclusive experimental arrangements. In normal language simultaneous existence has nothing to do with simultaneous measurability. The justification for calling the individual cube both burnable and soluble rests on the fact that one has at one's disposal many cubes that are presupposed to be equal with respect to the properties considered. And even though the two different 22 properties can only be verified on separate cubes nobody would have any objection to saying that every single cube is burnable and soluble. It is easy to imagine cubes made up of a mixture of several substances so that, depending on the mixing ratio, the cubes are easier to burn but more difficult to dissolve, and the degrees of solubility and combustibility are in a certain reciprocal or complementary relation to each other, in analogy to ∆x and ∆px. Similar examples have been described by Janossy [23] and by Koopman [90]. Indeed, any procedure of destructive testing of materials can provide more examples. The relations (2.5) may be generalized to any two quantities which are represented by Hermitean operators A and B [55], [91] ∆A∆B ≥ 12 |〈[A,B]〉|, and our realist interpretation is again that these relations and the noncommutativity of the operators express correlations between properties (the ranges ∆A and ∆B) of wavepackets. This interpretation is independent of whether the operators have discrete or continuous eigenvalues. For example, the relation between the components of angular momentum ∆lx∆ly ≥ 12 h|〈lz〉|, means that the particle which is an eigenpacket of lz( 6= 0) cannot at the same time be an eigenpacket of lx and ly, rather it has the finite ranges ∆lx and ∆ly. When the same packet in an appropriate physical situation changes into an eigenpacket of lx, say, then its range ∆x shrinks to zero while it acquires the finite ranges ∆ly and ∆lz. 23 3. SPECIFIC POSTULATES: SYSTEMS OF PARTICLES 3.1 Entangled Wavepackets and Multi-Particle Nonlocality In the preceding sections the concepts of a realist interpretation concerning single elementary particles or single field quanta were set out. Now we shall extend our considerations to include systems of several particles. Some new concepts will be encountered here. Multi-particle systems in quantum mechanics are described by configuration space wave functions Ψ(x1,x2, . . . ,xN , t). This does not prevent us from maintaining a realist interpretation: the variable x1 refers to wavepacket number 1, x2 to wavepacket number 2 and so on. The wavepackets have their properties even when not observed, and the mathematical expression |Ψ(x1,x2, . . . ,xN , t)|2d3x1d3x2 . . . d3xN , for example, which in the Copenhagen interpretation is the probability that particle 1 is observed at x1, particle 2 at x2, and so on, in the realist interpretation is the probability that wavepacket 1 induces an effect at x1, wavepacket 2 at x2, and so on. Essentially, what the configuration-space formalism effects is to introduce correlations between the spatial one-particle wavepackets, as we shall see below (and in Sec. 5). In quantum mechanics it is of particular importance to distinguish between multiparticle wave functions that can be written (perhaps after some transformation to a different system of eigenfunctions) as a product of one-particle functions and those that cannot. In those that can, the particles are independent of each other. In those that cannot, the particles, viz. the wavepackets representing them, in a sense are dependent on each other and are called entangled, a term coined by Schrödinger [92], [93], and they form a system of entangled wavepackets. We will consider these entangled wavepackets more closely. A two-particle entangled wavepacket, for example, may be written in the form (3.1) Ψ = a1 φ1(x1;u1,m1)φ2(x2;u2,m1) + a2 φ1(x2;u2,m2)φ2(x1;u1,m2). The function φ1 (in both parts) represents one particle, and φ2 the other. The parameters u1, u2 determine the spatial shapes of the wavepackets, which includes their centre position and their width. The parameters m1, m2 are additional properties, for example spin or polarization components [94] or energy and arrival time [96]. The time variable t is the same for all, and is omitted. For simplicity only one spatial dimension x is considered. It may happen that the wavepackets φ1 and φ2 develop so as to occupy disconnected regions R,L of space, that is, the distance between their centers being large compared with their widths. Let us write this as (3.2) Ψ = a1φ1(R,m1)φ2(L,m2) + a2φ1(R,m2)φ2(L,m1). 24 The two wavepackets are still entangled because both components m1 and m2 appear in each paket. In the realist interpretation Eq. (3.2) means that particle 1 neither has the property m1 nor the property m2, and the same holds for particle 2. Considering the experiments where such processes occur [97] - [100] it seems that a necessary condition for independent particles getting entangled is that the wavepackets must come close to each other or even overlap to some degree (3.3) φ1(x, t) φ2(x, t) 6= 0 at some instant of time. It does not seem that dynamic interactions are sufficient for establishing entanglement. An electron and a proton, for example, in principle interact with each other via the Coulomb force even if the proton is in the Andromeda galaxy and the electron on Earth. I do not think that there is any physicist who would assume that the two are entangled, with the properties of entanglement described below, if they never satisfied condition (3.3). An exact mathematical specification of the entangling condition is not the concern of the present article. Its concern is only to emphasize that it is a real physical process, occurring with real physical wavepackets, not just a loss of 'information' or 'knowledge'. How do entangled wavepackets become disentangled? This also happens in a real physical process namely in the reduction or collapse. A criterion for it to occur has been conjectured in [49], where it is taken to be a spatial contraction. It occurs independently of being observed, but in a measurement it is always present [101]. This will be discussed more thoroughly in Sec. 4.2 We now come to the most remarkable feature of entangled wavepackets. As early as 1932 [102] Ehrenfest had emphasized that the mere use of a non-product configuration space wave function implies some kind of sinister action at a distance. Indeed, the conditional probability that particle 1 acts in d3x1 about x1 (or: that the result of a position measurement is a position about x1), if particle 2 acts in d3x2 about x2, is (3.4) P (x1|x2, t)d3x1 = |Ψ(x1,x2, t)|2d3x1d3x2 d3x2 ∫ |Ψ(x1,x2, t)|2d3x1 , and this depends on x2, that is, on the result of a simultaneous action of (observation of) the second particle. On the other hand, the distance between the particles in ordinary space |x1 − x2| may be many kilometers long, and the correlations are independent of whether there are dynamic interactions between the two particles or not. With a product form of Ψ(1, 2) the probability (3.4) would be independent of x2, and there would be no correlations. Indeed, the two events may occur at spacelike intervals of spacetime, that is, if a signal from one event to the other were to connect the two, this signal would have to proceed with superluminal speed. There are many experimental confirmations of this. In [103] a lower bound of that speed was found to be four orders of magnitude larger than the speed of light. As in the case of one-particle nonlocality an interesting question is: can there be a reference system where the speed is infinite [50]? 25 What has been said here concerning the case of two wavepackets holds also for N entangled one-particle wavepackets [104]. A system of N entangled elementary wavepackets can induce effects effects at up to N places at a time in the reduction process (Sec. 4.2). A macroscopic body, which in the classical Newton-Euler description is a system of point particles, in the present description is a system of wavepackets, where in addition to the dynamic interactions between them there are entanglement and disentangling contractions between them. Why are the correlations between spacelike separated events (spacelike correlations, for short) such a remarkable feature? Could it not be that the spacelike distance is caused by so far unknown common causes in the overlap of the past light cones of the events, like the consecutive illumination of a series of places on the Moon by searchlight pulses emitted from a place on the earth? In Section 5 it will be proven that no such causes can exist in principle because the observed spacelike correlations can lead to to a violation of the Bell inequality ([106], with one exception, described in the next section). In anticipation of that proof we will call these special spacelike correlations multi-particle nonlocality. They can be shown to be ascribable to the entanglement between wavepackets. Therefore the range of the nonlocality in our theory is limited to the extension of the system of entangled wavepackets. And one-particle and multi-particle nonlocality are conceived to be basically of the same nature, and we will just speak of nonlocality. As we also shall see in Sec. 5 nonlocality does not lead to a superluminal connection between cause and effect and does not allow superluminal signaling. 3.2 Similar (Identical) Wavepackets The case of identical particles deserves special consideration. Following Dirac's book [107] we call the particles 'similar' rather than 'identical'. One reason for this is the ambiguity in the meaning of the word identical. We may say "Lord Kelvin and William Thompson are identical", which means that the two are one and the same person; but we may also speak of identical twins, which means two different persons. In quantum mechanics the effects of similarity go beyond those met in classical mechanics [108]. In classical mechanics similarity or indistinguishability always means essentially the indistinguishability of equal billiard balls. Imagine one billiard ball in your right hand and the other in your left hand. You are blindfolded and somebody else takes the balls out of your hands, then puts them back. If you look at them again you cannot tell whether or not they have been interchanged. However, if you were not blindfolded, you could follow their paths and decide which ball was initially in your right hand and which was in your left hand. Or, imagine a situation where the balls are in rapid movement around each other so that you can see nothing but a fuzzy cloud of whirling balls. Nevertheless, when you are allowed to use more refined methods of observation, you will always be able to follow the paths of the balls individually. Clearly, in classical physics with its mass points representing the centers of mass 26 of impenetrable bodies, there is no indistinguishability that could not be resolved in principle. In quantum physics this is no longer true. There are situations, associated with wave function overlap, where the observer is unable, in principle, to distinguish the particles, in the sense that he is unable to follow the path of a given particle unmistakably through all processes. In other words, for him the particles lose their individuality. In the realist programme this is conceived not as any incapability on the part of the observer but as an objective physical fact. The wave function representing a system of similar particles must be symmetric or antisymmetric under the exchange of function parameters [108]. Consider the product function (3.5) Ψ = φ1(R ↑)φ2(L ↓) representing one particle with spin up (↑) in the spatial region R and another particle with spin down (↓) in the separate region L. Considering the case where there has been the possibility of spin flip in a previous overlap of the two wavepackets so that they can no longer be identified by their spin components, function (3.5) can be superposed with the exchange function (3.6) Ψ = φ1(L ↓)φ2(R ↑) , and we obtain the (anti)symmetric function (3.7) Ψ = φ1(R ↑)φ2(L ↓)± φ1(L ↓)φ2(R ↑) . However, as emphasized by Ghirardi et al. [109], though this is no product state, it does not mean entanglement, with the property of nonlocality. The single particles all have definite properties of their own. True, in function (3.7) neither particle 1 nor particle 2 have definite spin values; nevertheless there is one particle in R, whichever of the two it is, with definite spin up and another particle in L with definite spin down. This does not suffice to violate the Bell inequality (Appendix C, Eq. (C12) and following). An entangled wavepacket is (3.8) Ψ = φ1(R ↑)φ2(L ↓)± φ1(L ↑)φ2(R ↓) . This function indeed means that the particle in region R, whichever of the two it is, has neither the property ↑ nor the property ↓, and the same holds for the particle in region L. Eq. (3.8) is the type of function which usually is the base of the discussions of the experiments designed to verify a violation of the Bell inequality (e.g. [96], [110] - [116]). In fact, Eq. (3.8) represents 2 of the 4 "Bell states", which can lead to maximal violation (cf. Sec. 5). The 2 others are (3.9) Ψ = φ1(R ↑)φ2(L ↑)± φ1(R ↓)φ2(L ↓) . Thus, (anti)symmetrizing a product of two similar one-quantum wavepackets by itself does not guarantee entanglement with its nonlocality. 27 3.3 Condensed Wavepackets Another interesting special case is when the similar wavepackets, such as φ1 and φ2 in Eqs. (3.8) or (3.9), are equal in all respects. Any configuration space function, we recall, can be expanded in terms of a complete set of one-particle functions in ordinary space φri(x, t) [46, Sec. VII.6]. (3.10) ΨSA(x1, . . . ,xN , t) = ∑ r1,...,rN c(r1, . . . , rN , t) φr1(x1, t) * * *φrN (xN , t). The xi may include the spin components. The functions φri may be taken as functions of time too, as in the Dirac or interaction picture. c(r1, . . . , rN , t) is then the transformed wave function. In the case of similar wavepackets it is completely determined if we specify the number of times each of the arguments r1, r2, r3, . . . occurs in it. These numbers are the occupation numbers nri , and the set |n1, n2, n3, . . . , t〉 is another representation of the wave function ΨSA(x1, . . . ,xN , t). The set |n1, n2, n3, . . . , t〉 is the state (vector, wave function) in the occupation-number, N , or Fock representation [117] - [120]. The change in occupation numbers in the course of time, due to interactions, is described by the creation and annihilation operators a† and a. This works for bosons (nri = 0, . . . ,∞) and, with minor additions, for fermions (nri = 0, 1). The occupation-number representation is well suited for our interpretation because numbering of quanta within one wavepacket φri(xi, t) is not even mentioned in it. The term "occupation number" is, however, likely to mislead one to think that the quanta and the states filled with them are two different things. In our interpretation there are only wavepackets and therefore we choose a different formulation: the oneparticle basis functions are wavepackets, and occupation numbers of 2 or more mean that two or more wavepackets have condensed to form one single wavepacket, even though this is still normalized to 1 in the current formalism. Thus boson wavepackets can condense, but fermi packets are excluded from doing so. This is our formulation of the Pauli exclusion principle. The inverse process we call decondensation. The change in occupation numbers will then be described as condensation and decondensation of wavepackets. Condensation and decondensation occur only between Bose but not between Fermi packets. In Chap. 6 we shall show how these processes lead to a new derivation of the quantum-statistical Bose and Fermi distributions and to the corresponding fluctuations in the wavepacket picture. Another manifestation of the multi-quantum condensed wavepackets is the 'photon bunching' in thermal radiation; that is, the observation that the photon coincidence rate in small temporal coincidence windows is higher than can be explained by random coincidences [121], [122], [123]. By now, multi-quantum wavepackets have also been isolated experimentally: in the Bose-Einstein condensates of atoms [124], [125] and of photons [126]. In Sec. 6.2 we shall also see that there are many other objects, conceived long ago in radiation theory, that are very similar to the condensed packets and may be taken to be just other aspects of them. 28 4. MEASUREMENTS AND REDUCTION 4.1 The Copenhagen Measurement Postulates The preceding section concluded the presentation of the basic features of our interpretation. In the following sections we shall apply them to specific problems. As a first application, the problem of measurement in quantum mechanics will be discussed. This problem is indicative of the basic difficulties of the Copenhagen interpretation. The basic postulates concerning measurements in Copenhagen quantum mechanics may be expressed in the following way: 1) Let ψ(x, t) be the normalized wave function in x space. The expression |ψ(x, t)|2d3x is then the probability that the result of a position measurement at time t lies in the interval d3x . Or, more generally, let c(n, t) = ∫ ψ(x, t)φ∗n(x, t)d 3x be the wave function in A space where the quantity A is represented by a self-adjoined operator with discrete nondegenerate eigenvalues an and eigenfunctions φn. Then the expression |c(n, t)|2 is the probability that the result of a measurement of A is the value an. 2) The only values that the quantity A may assume in the measurement are those of the eigenvalue spectrum of the operator associated with A. This is the usual, somewhat sloppy formulation. It is empty in the case of continuous eigenvalues which cover the whole real axis. In the case of a discrete spectrum we may say, more precisely, that the only values that can be found will lie in arbitrarily small intervals around the discrete eigenvalues. Notice that if the result of the measurement is the eigenvalue an, it is not said that the system considered had the value an before the measurement; rather the value an comes into existence by the act of measurement, and only immediately after the measurement is the system said to have the value an. 3) The third postulate introduces an assumption regarding the shape of the wavepacket after the measurement. This has been worked out very clearly by von Neumann [127] and is therefore often called von Neumann's axiom. It assumes that immediately after the measurement the wavepacket is a superposition of eigenfunctions of the respective operator which belong to the interval of eigenvalues specified by the measurement [46, p. 298], [47, p. 221]. This statement covers the case of continuous as well as discrete eigenvalues. In the case of continuous eigenvalues the measurement interval can be identified with the above interval d3x . In the extreme case of a discrete spectrum, a non-degenerate eigenvalue and sufficient measurement accuracy, the wavepacket immediately after the measurement will be a completely specified eigenfunction of the respective operator. The proviso "immediately after the measurement" emphasizes the fact that if we wait too long, the wavepacket will already have changed in accordance with the time-dependent Schrödinger equation. All this refers to "ideal" measurements, i.e. where any change in the wavepacket measured, due to effects other than those mentioned above, can be neglected. For 29 example, the components of the momentum of a charged particle are changed when they are measured by means of the deflection of the particle in a magnetic field. This is a calculable effect. It is assumed that such effects can always be completely reduced or compensated for. All this also applies only to specific quantum measurements, by which we mean those where the wavepacket nature of the elementary particles cannot be neglected, that is, where a particular quantity is to be measured with an error interval that is smaller than, or of the order of, the corresponding width of the wavepacket in the respective space. In other measurements the above postulates lose their relevance and the elementary particles may be treated like classical objects, for example when the electron's e/m ratio is measured. This is in accordance with the classical limit mentioned in Sec. 2.4. According to postulate 3 the measurement effects a reduction of the initial wavepacket: the initial packet may always be expanded mathematically in a series (or an integral) of eigenfunctions φn of the self-adjoined operator belonging to the measured quantity (nondegenerate eigenvalues) (4.1) ψ = +∞∑ n=−∞ c(n)φn, where n ranges over all values that specify the complete set of eigenfunctions φn, symbolized in (4.1) by n = −∞ . . .+∞. The measurement reduces the wavepacket to only a part of the sum ψ = +∞∑ n=−∞ c(n)φn → n2∑ n=n1 c(n)φn. That is, it narrows down the range of values of n. In the extreme case reduction leads to one single discrete eigenfunction φ0 ψ = +∞∑ n=−∞ c(n)φn → c(0)φ0, where the normalization of φ has to be re-adjusted so that ∫+∞ −∞ |c(0)φ0|2d3x = 1. The reduction is therefore also called a collapse, preparation, filtering, or projection of the wavepacket. It clearly means a nonlinear evolution of the wave function: if φm and φn are two normalized eigenfunctions belonging to different nondegenerate eigenvalues, reduction of φm leads to φm, reduction of φn to φn, but reduction of (φm + φn)/ √ 2 does not lead to (φm + φn)/ √ 2 but to either φm or φn [128]. It is thus a change in the wavepacket that is not described by the Schrödinger equation. Thus, a wavepacket may vary with time in two ways: sometimes deterministically (as determined by the Schrödinger equation) and at other times with a random element in its behavior (in a 'measurement"). The question is: what is the feature of the measurement that makes the measurement interaction so different from the Schrödinger evolution? This is the "measurement problem" of traditional quantum mechanics. 30 In the realist interpretation the above postulates assume different forms: Regarding postulate 1, we have already explained in Sec. 2.2 that the expression |ψ(x, t)|2d3x can be taken to be numerically equal to the probability that the wavepacket ψ(x, t) induces an effect within d3x and that from this we cannot, without further steps, draw conclusions regarding the properties of the wavepacket. Regarding postulate 2, in a realist formulation a measurement measures what already exists. This implies that many operations which in the Copenhagen interpretation are called measurements in a realist interpretation are not. Thus, neither is the observation of the position of the black spot produced by an electron in a photographic emulsion a measurement of the electron's position, as we have seen in Secs. 2.1 and 2.2, nor is the registration of an atomic electron in the spin-up path of a Stern-Gerlach apparatus a measurement of the electron's spin component, as we shall see in Sec. 4.3. The above operations may indeed be used for a measurement of properties that already exist, but not without further steps. In any case an ensemble of wavepackets must be given that are equal with respect to the property considered, as we have already seen in Sec. 2.2 and in the sugar-cube example in Sec. 2.4. Regarding postulate 3, we emphasize that von Neumann's axiom does not just mean that the reduction process occurs; it means that it occurs in and only in a measurement or observation. We do not do away with the reduction process, but we do away with the assertion that it occurs only in a measurement or observation. This is the main point. In the realist interpretation the reduction processes are objective physical processes that occur in some physical situations, whether these involve a measurement or not. Nevertheless reduction is involved in measurement [133]. This will be outlined in the next section, after we have described the reduction process itself. 4.2 The Underlying Physical Problem Thus, the problem of the reduction process is no longer a "measurement problem". It remains a problem, but it assumes a different form: what feature of the physical environment is it that makes the reduction process, rather than the Schrödinger evolution, occur? The problem now is how to physically (and then mathematically) characterize the reduction process. A physical characterization would be one in terms of physical conditions and processes that are not already taken into account in the Schrödinger (or Dirac etc.) equation for a particle in an external field. Recall that reduction in a measurement implies nonlinearity. In this respect traditional quantum mechanics is incomplete, although this is a different incompleteness than that pointed out by Einstein, Podolsky and Rosen (Sec. 5). The incompleteness revealed here is a serious drawback, and the wish to avoid fully acknowledging this incompleteness presumably was another reason why the Copenhagen school ascribed those changes of the ψ function that cannot be described by a Schrödinger equation to the acts of His or Her Majesty, the Observer. In a realist interpretation this is no longer permitted. Actually, a quantitative physical criterion for the occurrence of the reduction pro- 31 cess has been formulated in [49], where also determinism is discussed. The traditional formalism of quantum mechanics is not changed, but a gap is filled. We here only want to sketch the features which are concerned with reduction. Reduction occurs when two wavepackets, ψ1 and ψ2, in ordinary space for the first time overlap and satisfy a certain criterion. Both then contract suddenly to the place of overlap. The criterion depends on the parameters of both wavepackets in a symmetric way: (4.2) |α1 − α2| ≤ 12 αs (4.3) [ ∫ R3 |ψ1(r, t)| |ψ2(r, t)|d3r ]2 ≥ α/2π. αs is Sommerfeld's fine-structure constant. α1 and α2 are the absolute phase constants of wavepacket 1 and 2, respectively. They are nonlocal 'hidden' variables. They are physical because there are situations where they can be determined, when, for example, two independent weak quasi-monochromatic laser or maser beams (photon wavepackets) are superposed [74, Sec. 7.4], [136]. In the superposition the absolute phases become relative and determine the position of the interference fringes. The same can be said of the wave functions of Bose-Einstein condensates of atoms [137] and photons [126]. The phase constant α in (4.3) is the smaller one of α1 and α2. Now, as every physicist can verify, in any quantum mechanical measurement the measurement apparatus fans out the incoming wavepacket in ordinary space into spatially separated eigenpackets of the chosen observable (self-adjoint operator). When one of the eigenpackets and some wavepacket in the sensitive region of the apparatus satisfy the criterion, the reduction associates the place of contraction with an eigenvalue of the observable. There must be such wavepackets in the sensitive region which under contraction initiate an avalanche of effects that result in an observable spot. This is the step from an unobservable microworld into the directly observable macroworld. An example is the well known Stern-Gerlach experiment. 4.3 The Stern-Gerlach Experiment This experiment is well suited to demonstrate the new concept of measurement outlined in the preceding section. The Stern-Gerlach experiment has traditionally been regarded as the prototype of a measurement in quantum mechanics. First let us briefly recall the facts. We consider hydrogen atoms in the ground state which move in the y direction with velocity ν through an inhomogeneous magnetic field B produced by a Stern-Gerlach magnet [138] - [141], [46, Sec. III.10]. The magnet is positioned so that along the path of the atoms both B and gradBz point in the z direction. This is then the "spin-reference axis", or simply the "axis" of the apparatus. The hydrogen atom has a permanent magnetic moment ~μ which comes from the magnetic dipole moment of the electron, the contribution of the 32 proton being negligible. Therefore it is mainly the electron that interacts with the magnetic field, and it is the spin of the electron (4.5) s = −(m/e)~μ that determines the precession in the magnetic field. Accordingly, in (4.5) m is the electron mass. We might thus just speak of electrons moving through the SternGerlach magnet, and we shall occasionally do so. We note, however, that the SternGerlach magnet does not work for free electrons. This is due to the Lorentz force and to the spreading of the electron wavepacket [142] - [144, p. 214]. If we want to perform the Stern-Gerlach experiment for free electrons we may scatter electrons by atoms [144, Chap. IX], [99]. We call such devices Stern-Gerlach-type apparatuses. Under the conditions of the experiments performed the laws of classical mechanics and electrodynamics predict that the atom when it has spent the time ∆t in the Stern-Gerlach magnet will be deflected along the z direction by the angle (4.6) αz = pz/py = μz(∂Bz/∂z)∆t/py, where py and pz are the momentum components of the atom, and μz is the z component of ~μ. When a beam of atoms goes through the Stern-Gerlach magnet with the spins of the atomic electrons initially oriented at random, μz can take on all values between +μ and −μ, and the deflection angles can take on all values between the corresponding extreme values ±μ(∂Bz/∂z)(∆t/py). On the screen behind the magnet, the sensitive region, one would therefore observe one single spot elongated along the z direction. What is actually observed, however, is two separate spots corresponding to the above two extreme values of αz, with μ = eh 2m = ec 4π λC corresponding to the electron spin value s = h/2 in formula (4.5). The upper spot on the screen thus corresponds to spin-up electrons and the lower spot to spin-down electrons with respect to the axis of the apparatus. So far our considerations of the Stern-Gerlach experiment have been independent of our interpretation since they have been formulated with beams consisting of many atomic electrons. Difficulties arise when the behavior of the individual electron wavepackets of the beam is considered. Any single wavepacket is fanned out into two coherent parts, one corresponding to spin up and the other corresponding to spin down, and it covers both the upper and the lower path in portions that can be calculated by the standard formulas. According to the orthodox version of the Copenhagen interpretation it is only in a subsequent measurement, for example, when a black spot at the proper "up" position on the screen is observed, that the packet contracts and is reduced to a pure spin-up eigenpacket. When we choose not to look at the measurement device, no reduction can occur [129]. In the Copenhagen interpretation the Stern-Gerlach experiment is called a measurement of the initial z component of the spin of the atomic electron [145], [89, p. 593] In the present interpretation this is different. Consider an atomic electron 33 which is initially described by a wave function that is a product of a spatial and a spin function. Such a wavepacket never has a definite pointlike position, but it always has a definite spin component, in the sense that the wave function can always be written as a spin-up eigenfunction of the spin-component operator sz′ , with some axis z ′, which of course need not coincide with the axis z of the Stern-Gerlach apparatus. This is connected with the fact that the group SU(2) is locally isomorphic to O(3). We may call the direction of the axis z′ the spin direction of the electron before it entered the apparatus. Now the inhomogeneous magnetic field accomplishes that the eigenfunctions of the spin component of the incoming electron become spatially separated: the spinup component eigenfunction goes upward (say) and the spin-down eigenfunction downward. Both functions then enter the sensitive screen in which contraction can occur. It is only when the function contracts at a cluster in the upper region of the screen, say, that the electron has become a pure spin-up electron with respect to the apparatus axis z, whatever the electron's initial spin direction z′. Because of total angular momentum conservation, the angular momentum of the apparatus is thereby also changed. This has been verified experimentally in the case of photon polarization apparatuses, which in principle function like Stern-Gerlach apparatuses [134]. In the present interpretation we do not call the operation of the Stern-Gerlach apparatus on an individual incoming electron and the observation of its respective final position on the screen a measurement of the electron's initial spin component. We may indeed use the Stern-Gerlach apparatus for such a measurement, but not without further steps: a large number of equal electrons must be given. Let the electrons enter the magnet one after the other. Then the direction of the axis of the apparatus must be varied, and from the abundance ratios of the up and down spots for the various chosen directions the original spin direction can be derived, using the standard formulas of quantum mechanics. For example, let the spin of the incoming electrons be restricted to lie in the x-z plane perpendicular to the direction of motion (y axis). The probability of inducing a spot in the "up" or "down" position, respectively, is then given by (4.7) Pup/down = 1 2(1± cosθ) where θ denotes the angle between the spin axis of the incoming electron and the axis of the apparatus. The ratio of the corresponding abundances will give (Pup)/(Pdown) = tan −2(θ/2), hence θ and the spin direction of the incoming electrons becomes known. Alternatively, we may rotate the apparatus round the y axis until a position is obtained where only spots at the up position are observed. This signifies θ = 0 in (4.7), and the apparatus axis coincides with the spin direction of the incoming electrons. For particles with higher spin the situation is more complicated. These particles need not have a definite spin component since the expression of the general spin state as a superposition of spin-component eigenstates can no longer always be reduced to one term by a spatial rotation of the spinreference axis. Still, the coefficients in 34 the superposition can be determined in essentially the same way as before, and it may now be the set of these coefficients that represents what already existed before the particles entered the apparatus. In the present interpretation the superposition principle does not mean "that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states . . . in a way that cannot be conceived on classical ideas" ([107, p. 12]). In the situation of the Stern-Gerlach experiment this would mean that the electron even before it has entered the apparatus, when it still is in an eigenstate of sz′ (z ′ = arbitrary axis), is already partly in each of the eigenstates of sz (z = apparatus axis). We feel that this peculiar notion of Dirac's arises from the attempt to adjust the usual view that a measurement measures what already exists to the postulate that the Stern-Gerlach apparatus operating on an individual electron is such a measurement. Indeed, if the electron were a classical gyroscope, the measurement of the place where it hits the screen behind the magnet could be considered as a measurement of its initial spin component (formula (4.6)). However, the electron is not a classical gyroscope, and to describe the quantum situation in the same way as the classical situation is misleading. In our interpretation, expressing a wave function as a superposition of certain eigenfunctions in general is no more than a mathematical procedure. Only in special physical situations like the one within the Stern-Gerlach magnet are these eigenfunctions physical parts of a wavepacket. 4.4 Wigner's Friend and Schrödinger's Cat A difficulty with von Neumann's axiom, namely that reduction occurs in and only in a measurement, is pointed out in the "Wigner's Friend" example [129]: Usually the whole measurement apparatus consists of a long chain of sub-apparatuses (amplifier, channel analyzer, transmitter etc.) between the micro-object considered and me as the final observer. And in the most orthodox version of the Copenhagen interpretation it is my being conscious of the result that completes the measurement and effects the reduction. Now, a friend of mine may form a sub-apparatus in that chain in that he, for example, reads off the pointer position on a display and then telephones it to me. The difficulty arises as soon as I credit my friend with the same capabilities as I have because this implies that the reduction has already taken place in the apparatus due to his being conscious of it. This is essentially the conflict between the Copenhagen description where the observer is the linguistic ego, and any realist description, where the observer is just another physical object and the reduction is a physical process that occurs whether we notice it or not. One may ask why von Neumann's axiom has been introduced at all. What were von Neumann's reasons, for example? In his book [127, Sec. III.3] von Neumann refers to an experiment carried out by Compton and Simon [146] in which photons were scattered by electrons at rest and where the azimuthal directions of both the scattered photon and the scattered electron from the same scattering event were measured. The situation was such that, if the validity of the classical kinematical conservation laws was assumed, the azimuthal angle φ of the scattering plane could 35 be determined by measuring either the azimuthal direction of the scattered electron or that of the scattered photon. These two measurements may be made in succession, and the experiment confirms that the second measurement always gives the same result for φ as the first one. In other words: initially the system is in a state in which φ cannot be predicted with certainty, then a measurement transforms it into a state in which φ is definitely predictable. One may accept this as an argument in favor of the reduction process; however, there is nothing in the argument that would force one to conclude that the reduction depends on the presence of an observer. The connection with an observation was perhaps conceived under the influence of some of Bohr's ideas [147], quoted by von Neumann [127, Footnote 207] (cf. also Jammer [1, p. 370]). These ideas do not, however, amount to a conclusive argument against realism. Another difficulty with von Neumann's axiom is described in the example of Schrödinger's cat [19, p. 812]. Consider a closed box containing a cat, a certain amount of radioactive nuclei, a Geiger counter and a cat-killing device, all protected against the cat. Circumstances are arranged so that the probability that the Geiger counter discharges at the decay of at least one nucleus within one hour is just 1/2. If the counter discharges it triggers the cat-killing device, which consists of a hammer and a flask of prussic acid. The flask is smashed, the acid is released, and the cat is poisoned. The probability that after one hour the cat is dead is 1/2. Since the box is closed we cannot know after an hour whether the cat is dead or alive, unless we cautiously open the box and look into it. In orthodox quantum mechanics, where the wave function represents our knowledge, there is one wave function ψL that represents our knowledge that there is a living cat in the box and another function ψD that there is a dead cat, and the situation is not described by the sum of the probabilities but by the superposition of the probability amplitudes (4.8) cLψL + cDψD in the two-dimensional Hilbert space of dead and living cats. (4.8) is then interpreted as the wave function of neither a dead nor a living cat but a superposition of both. Only when we look into the box a reduction occurs and the cat's wave function becomes either ψL or ψD. The cat is a macroscopic object, and in the realm of macro-objects the language of realism is spoken: the cat is either alive or dead even if we do not observe it. The radioactive nucleus is a microscopic object, and in the realm of micro-objects the language of realism is forbidden by the verdict of Copenhagen. In the example there is a chain of reactions beginning in the microworld with the decay of the unstable nucleus and ending in the macroworld with the death of the cat. If both the microand the macro-object are described by a ψ function, the character of the ψ function must change when the chain of reactions crosses the borderline between the two realms. This is another difficulty. In our interpretation the language of realism is spoken in the microworld as well as in the macroworld, and the character of the ψ function is always that of 36 a real physical field representing real physical objects, microor macroscopic. The point is however that these objects must exist at the same time [148]. Probabilities may refer to different times, but wavepackets not. The wavepackets ψL and ψD, on the contrary, exist at different times, ψL before and ψD after the decay of the radioactive nucleus. No such superposition is met elsewhere in the standard formalism of quantum mechanics, not even for microscopic objects. In fact I do think that Schrödinger considered the cat example in order to point out to what incredible features the superposition (4.8) would lead. Thus, although the particular superposition (4.8) for the cat is is not allowed in realism, this does not mean that there is no superposition at all of wavepackets representing macroscopic objects. The restriction is that these wavepackets must represent something that really exists at the same time. Examples are the recent experiments with large molecules and clusters [149], [150], if one accepts these to be already macroscopic objects. 4.5 Determinism and Indeterminism It is well known that Einstein favored determinism [151] - [154]: "At any rate I am convinced that he is not playing dice" [152]. And this has often been considered as Einstein's main criticism of quantum theory. However, as indicated by Einstein [155] and emphasized by Pauli in a letter to Born [156]: Einstein (as he explicitly repeated to me) does not consider the concept of "determinism" to be as fundamental as it is frequently held to be . . . Einstein's point of departure is "realistic" rather than "deterministic." Both determinism and indeterminism are compatible with realism. Determinism means a programme. It means the expectation that as science advances we will be able to make more and more phenomena predictable, by means of laws of nature, and that this process is infinite. The Copenhagen interpretation takes the stand of strict indeterminism and decrees a definite limit to the described process of the deterministic programme. It maintains that the probabilities in quantum mechanics are unlike those in statistical mechanics and can never be explained by underlying determining processes that would specify the physical situation in more detail. The probability statements in quantum mechanics, according to this interpretation, are the last word. Even if we knew all the laws and all the wavepackets in the world, we would not be able, in principle, to calculate the exact future result of an individual measurement. Only in some degenerate cases can we obtain probabilities that reach the value one and thus give certainty. In general, identical initial conditions do not lead to identical results. Why does the Copenhagen interpretation assume such an extreme position? Admittedly, so long as no theory is found that can specify the hypothetical underlying processes postulated in the deterministic attitude, it might seem reasonable, from the viewpoint of economy of concepts, to eliminate the concept of these processes altogether. This would give the additional bonus that the indeterminacy no longer 37 points at an inability of the quantum theorists to build a complete theory but is a property of nature. It seems that the attitude of the Copenhagen school received additional support from von Neumann's demonstration [127, Secs. IV.1 and IV.2] that some basic features of quantum mechanical states are incompatible with the introduction of additional hidden variables besides ψ in order to further specify the physical situations and to restore determinism. This statement seems to have been taken to mean that no deterministic theory at all is possible. When Bell examined the case [157], [158] he found that those basic features of the quantum mechanical states which von Neumann postulated also for the states in a hidden-variable theory, are actually more than can reasonably be postulated in such a theory. Thus, von Neumann's proof, although mathematically correct, leaves the real question untouched and does not exclude deterministic hidden-variable theories. The same conclusion had been reached by Grete Hermann in 1935 [159]. On the other hand, Bell's investigations revealed that any hidden-variable theory which after averaging over the hidden variables reproduces the formulas of quantum mechanics must have a grossly nonlocal structure. A more detailed account of this specific aspect will be given in Sec. 5. Here, the essential lesson is that von Neumann's proof does not exclude deterministic theories, and the apodictic exclusion of determinism in the Copenhagen interpretation is unjustified. Let us therefore consider what a deterministic programme might look like in a realist interpretation. In fact, in [49] a deterministic programme in a realist interpretation is established. The point is that in an ensemble of quantum wavepackets their overall phase constants α in the reduction criterion (4.2), (4.3) are taken as pseudorandom numbers, determined by certain initial conditions, in the spirit of the theory of deterministic chaos. The criterion (4.2), (4.3) then reproduces the Born probability rules in measurements. Nevertheless, although everything may be determined, we may not always be able to completely predict the future (or retrodict the past) because we do not have all relevant data. We must distinguish between determinism and predictability. Compare it with throwing dice: the result is unpredictable, though it is conceived to be determined. Complete predictability is, in fact, impossible: Imagine a physicist at O in the spacetime of Fig. 2 who wants to predict exactly what will happen at A [160]. He can do this only on the basis of what he knows, and he can at most have knowledge of the events within his past light cone. However, the event at A can obviously be influenced by events lying outside that cone, for instance at B. Hence, as a matter of principle, future events are not completely predictable. One may restrict oneself to isolated systems and their future development. An isolated system would be one where such outside influences as those of B on A are negligible. Such a restriction may be reasonable for macroscopic bodies since appreciable influences are only to be expected from other macroscopic bodies which usually move at very low velocities and hence reach the observer's future light cone only after a very long time. For microscopic bodies, however, the isolation is much more difficult, if not impossible, to maintain. 38 Figure 2: The event at A can be influenced by an event at B lying outside the observer's cone of knowledge. One might try to regain predictability by extending the past light cone far backwards in time. If one assumes an appropriate cosmological model of the world the spacetime diagram will be deformed as a result of non-zero spacetime curvature, and the past light cone (horizon) might eventually include all of the world, and no events such as event B in Fig. 2 will be left outside it. But even so one cannot obtain complete predictability, for one has to bear in mind that all things consist of wavepackets. If one wants to fully describe one specific wavepacket at one particular time, one can do this only with the help of macroscopic stable bodies such as books, microfilms, disks, etc. All of these bodies in turn consist of a large number of wavepackets. So, for noticing and registering one wavepacket many other wavepackets are needed. Only a part of the world can thus be "known", the other part is used in representing the "knowledge". On the other hand, some wavepacket from the registering body may escape from it without any perceivable effect, but with a large effect on the wavepacket registered. Macroscopic objects are like icebergs floating in the sea, and the microscopic particles are like the waves in this sea. Imagine the difficulties you would meet trying to influence an individual wave by handling icebergs. And the size of an iceberg does not exceed the size of a wave by as much as the size of a man exceeds the size of an atom. This last size ratio (1010) is almost equal to that between the galaxy and the Sun. Seen from the microworld the macroworld of human beings and their apparatuses is of cosmic dimensions. 39 5. THE EINSTEIN-PODOLSKY-ROSEN PROBLEM 5.1 The EPR Problem and Nonlocality In this section we consider the problem of "simultaneous elements of reality" and of action at a distance formulated by Einstein, Podolsky and Rosen (EPR) in 1935 [161]. The title of the EPR paper is "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" The authors wanted to demonstrate that the answer is no. For a physical theory to be complete it is necessary that "every element of the physical reality must have a counterpart in the physical theory", and reality is characterized by the following sufficient criterion: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." EPR consider two systems, 1 and 2 (imagine two protons) that have interacted from time t = 0 to t = T , after which time they are well separated, so, EPR assume, there is no longer any interaction between them. Let the (exactly calculable) wave function of the combined system 1+2 after T be Ψ(1, 2). The number 1 stands for all variables used to describe the first system and 2 for those of the second system. In general the function Ψ(1, 2) cannot be written as a product (or a more general function) of one function φ1(1) of the variables 1 and one function φ2(2) of the variables 2, and hence we cannot describe the state in which either one of the two systems is left after the interaction. This state, according to the Copenhagen interpretation, can only be known by a subsequent measurement: Let m1,m2,m3, . . . be the eigenvalues of some physical quantity M pertaining to system 1 and u1(1), u2(1), u3(1), . . . the corresponding orthonormal eigenfunctions; then Ψ(1, 2) can be expanded into a series of these eigenfunctions with coefficients that are functions of the variables 2 (5.1) Ψ(1, 2) = ∞∑ r=1 ζr(2)ur(1). Although not necessary for the argument, we assume for simplicity of presentation, that the eigenvalues are discrete. The functions ζr(2) are not normalized and in general are not orthogonal to each other, but this is not relevant here. Suppose that the quantity M is measured on system 1 and that the value m7 is found. According to von Neumann's reduction axiom, the first system after the measurement is left in the state u7(1) [i.e. the first wavepacket assumes the form u7(1)]. Hence the sum (5.1) is reduced to the single term ζ7(2)u7(1), and due to the simple product form of this term the second system is left in the state ζ7(2), apart from normalization. The set of functions un(1) is determined by the choice of the physical quantity M . If, instead of M , we had chosen a different quantity N , with eigenvalues n1, n2, n3, . . . and orthonormal eigenfunctions υ1(1), υ2(1), υ3(1), . . . we would have obtained a different expansion (5.2) Ψ(1, 2) = ∞∑ s=1 ηs(2) υs(1), 40 where the ηs(2) are the new coefficient functions. If the quantity N is now measured and the value n5 is found, then system 1 is left in the state υ5(1) and system 2 in the state η5(2). Therefore, as a consequence of two different measurements performed on the first system, the second system may be left in states with two essentially different wave functions. On the other hand, at the time of measurement the two systems, according to EPR, no longer interact, that is, no real change can take place in the second system as a result of anything that may be done to the first system. Thus it is possible to assign two different types of wave functions, ζ and η, to the same physical reality, namely to system 2 after the interaction. Figure 3: Arrangement for a proton spin correlation experiment. It is even possible to choose noncommuting operators M and N , operating in system 1, in such a way that the two sets of wave functions ζr, ηs of system 2 are discrete eigenfunctions of two noncommuting operators, for example of the two operators sx and sz of the spin component of a proton in x direction and in z direction respectively. Such a case was first considered by Bohm [89, p. 614]. Let the two protons interact at O (Fig. 3) and let the scattering proceed through an intermediate state of zero total spin (singlet spin state). The general expression (5.1) in this particular case becomes [46, p. 562] (5.3) Ψ(1, 2) = 1√ 2 ( |+〉(1)|−〉(2) − |−〉(1)|+〉(2) ) A(1, 2), where the spin projections up |+〉 and down |−〉 refer to an arbitrary axis. A(1,2) is the spatial part and the bracket is the spin part of the wave function. Notice that the spin part follows solely from spin algebra (Clebsch-Gordon coefficients) and happens to be antisymmetric in the particle labels, regardless of whether the particles are similar. In the case of similar particles the spatial part may take care of the correct symmetry. After the interaction the protons propagate with opposite momentum |p| towards the observers A and B respectively. Each observer is equipped with a Stern-Gerlachtype apparatus (e.g. a scattering device with counters; a Stern-Gerlach magnet with registering screen would not do in this case, see Sec. 4.3). The (spin-reference) axes of the apparatuses can be oriented in any direction. Observer A may thus put the axis of his apparatus either in x direction or in z direction, thereby obtaining either 41 the x or the z spin component of the first proton. He is then in a position to predict with certainty, and without in any way disturbing the second proton, either the value of the x or the value of the z component of the spin of the second proton. According to the above criterion of reality both components are elements of physical reality. Therefore, the values of both must enter into the complete description of reality. On the other hand, in quantum mechanics no wave function can contain both an eigenvalue of some operator M and an eigenvalue of an operator N that does not commute with M . Therefore EPR conclude that the quantum-mechanical description of reality by the wave function is not complete. As EPR remark, one would not arrive at this conclusion if one regarded the given criterion of reality as not sufficiently restrictive, that is, if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this. The burnable and soluble sugar cube in Sec. 2.4 is another illustration of this point. In the disputes following the publication of the EPR paper the adherents of orthodox quantum mechanics pointed out that the conclusions of EPR were only valid provided the two systems after the interaction are truly independent of each other in every respect. Present quantum mechanics, however, does not say this. Instead, it conceives the two particles to be entangled, that is, inseparably incorporated into the single wave function (5.1), so that we cannot operate on the one particle "without in any way disturbing" the other, and only a reduction of the sum to one of its terms by means of an observation (von Neumann's axiom) can achieve a separation of the two particles. This is independent of whether ζ, u, η and υ in expressions (5.1) and (5.2) represent Schrödinger scalar wave functions or relativistic Dirac spinors or other tensors. We may consider the functions ζ, u in (5.1) or η, υ in (5.2) as functions not only of the space coordinates but of time as well, with the same time variable t in all functions (cf. Sec. 3.1). Thus, the moment of the observation at the one place achieves the simultaneous reduction at the other place. Since the time of this subsequent observation is at the observer's disposal, he or she may perform it with an arbitrarily long delay after the interaction, so that the wave functions of the two systems can be taken to be separated from one another by an arbitrarily large distance. On the other hand, things may be arranged so that observer A operates on system 1 such a short time before B operates on system 2 that no light signals could connect these two events. What happens at B then depends on what A has done 42 in a region that is separated from B by a spacelike distance in spacetime: there are spacelike correlations. Before the appearance of the EPR paper these correlations had not been fully noticed. Thus in regarding the two systems as independent, EPR are not in accordance with quantum mechanics; hence they cannot maintain that it provides only an incomplete description of physical reality. But then the original question "Can quantum-mechanical description of physical reality be considered complete?" is replaced by another question: "Are these spacelike correlations a feature of physical reality?". Einstein agreed that the EPR conclusion rests on the assumption of complete independence of the two systems after the interaction, but the assumption of non-independence in the form suggested by orthodox quantum mechanics appeared to him an unacceptable "spooky action at a distance" [162], [12], [13], [14, p. 84, 682, 683]. The question is so important for our conception of nature that, in spite of the fact that the spacelike correlations mentioned are predicted by quantum mechanics and that quantum mechanics has been confirmed in innumerable situations, one would wish that this particular prediction be tested in specific experiments. We shall come to these experiments in Sec. 5.3. 5.2 Relativity and Causality Here we want to consider the question whether the spacelike correlations permit the transmission of signals or messages with superluminal speed from one person to another. Such a transmission would mean a drastic violation of relativistic causality because we may consider A's sending a message the cause and B's receiving it the effect, and with superluminal transmission this cause and effect could appear in reverse order of time in different Lorentz systems. Since we use the same formulas as orthodox quantum mechanics, the discussion is relevant for our interpretation too. In contrast to the preceding section, where we argued in terms of wave functions and eigenvalues, we now want to argue in terms of results, or probabilities of results, of experiments. Let us try to construct an early-warning system. Consider Fig. 3 of Sec. 5.1. Imagine B to be the Earth and O and A two space stations. Invaders (the Borg) from a distant star are expected to approach the Earth from the direction of A. The task of A is to inform the Earth as soon as the invaders have been seen (emergency case). For this purpose the auxiliary space station O continually emits pairs of scattered protons, say at a rate of 1 pair per second, and the protons are to pass through Stern-Gerlach-type apparatuses on station A and on the Earth respectively. The distance between O and the Earth is made only a little larger than the distance between O and A, so that A receives its proton such a short time (which still may amount to some hours) before the Earth receives its proton that no light signal could have informed the Earth of A's operation. In routine cases, the apparatuses at both A and B have their axes in +z direction, and in the emergency case A turns the axis of its apparatus into the +x direction. One might think that this changes the 43 probability of an up or down result in apparatus B on Earth, and from the changes in the ratio of up and down results the physicists on Earth would soon learn (before any light signal could be sent from A to B) that the invaders had been seen. Now, the joined probability that proton 1 in apparatus A becomes an rA-proton (= up proton if rA = +1, down proton if rA = −1) and that proton 2 in apparatus B becomes an rB-proton, according to the formulas of quantum mechanics, is (Appendix C) (5.4) P (rA, rB|a,b) = 14(1− rArB cosθ), where the unit vector a specifies the axis of apparatus A, b that of apparatus B, and θ (0 ≤ θ ≤ π) is the angle between a and b. Hence the probability that B observes the result rB, whatever the result rA, is just P (rB|a,b) = ∑ rA P (rA, rB|a,b) = 12 , regardless of the axis a (as well as of b), and in this way the early-warning system will not work. In fact, the above arrangement cannot transmit any message, superluminal or subluminal; the superluminal case is only the most interesting aspect of this general incapability. We may try to exploit the fact that the change of A's axis a, if it does not change B's probabilities, will at least change the correlations between A's and B's results rA and rB respectively. For example, when both apparatuses have their axes in +z-direction, formula (5.4) gives P (+,−|z, z) = P (−,+|z, z) = 12 , that is, if A's proton is deflected in +z direction (rA = +) B's proton will be deflected in −z direction (rB = −) and vice versa, and we obtain a series of correlated results such as indicated in Fig. 4, before the emergency case. In the emergency case, when A turns the axis of its apparatus in the +x direction, formula (5.4) gives P (+,+|x, z) = P (−,−|x, z) = P (+,−|x, z) = P (−,+|x, z) = 14 , so that there is no longer any correlation between the up and down results of A and of B, and the series indicated in Fig. 4 continue with uncorrelated results. However, the physicists on Earth do not know this. They only know the results of their own apparatus, that is, the lower line of Fig. 4, but not the upper line. Thus, they do not know the correlation of their results with those of A, still less can they realize any change in those correlations. Either of the two lines of Fig. 4 is just a random series; the probability of an up result is equal to that of a down result, before and after the emergency case. So, the early-warning system does not work this way either. One may try more general apparatuses than just Stern-Gerlach-type ones. These also will not work. It can be shown quite generally that no faster-than-light warning system can be built with devices obeying the formulas of quantum mechanics. The proof is shown in Appendix D. 44 Figure 4: Records of results of observers A and B before and after the emergency case. On the other hand we note that the formulas of quantum mechanics only give probabilities for the various possible results. If the physicist at A could arrange with certainty that his proton always goes into the up state (say) with respect to his axis, superluminal messages would be possible. For then, with A's and B's axes parallel, the physicist at B would register only down protons. In the emergency case let A turn his apparatus upside down. From then on B would obtain only up protons, and the first of these would tell B that A has seen the invaders. For the construction of an early-warning system one might therefore try situations in the grey zone between quantum and classical physics, hoping that here the probability features of quantum mechanics have already sufficiently approached classical deterministic behavior while the superluminal features persist. An attempt has been made in that direction by Herbert [163], who used the amplification of a weak beam of light. But it was soon shown that the proposal would not work because the amplification of arbitrary states by one and the same apparatus even is at variance with the linearity of the quantum mechanical operators [164] - [167]. Thus, the very theory, quantum mechanics, that predicts superluminal features also predicts that these features cannot be used for transmitting superluminal messages from one person to another. What, then, is the remarkable feature of the quantum mechanical formula (5.4) expressing superluminal features, i.e. spacelike correlations? To see this we have to consider Bell's inequality. 5.3 Bell's Inequality A priori, spacelike correlations, i.e. corelations between spacelike separated events, may be thought of as caused by arrangements in the past, for example by the searchlight spots on the Moon mentioned at the end of Section 3.1, or the letters running over the lights of a billboard. Such effects are brought about by common causes in the past, i.e. by events in the overlap of the past light-cones of the correlated events. The remarkable feature of formula (5.4) is that the correlations described by 45 it cannot be accomplished with the above mentioned arrangements. This is shown by means of the Bell inequality. Spacelike correlations with no common causes in the past exhibit what we have called multiparticle nonlocality (Section 3.1) or just nonlocality. The Bell test of whether spacelike correlations exhibit nonlocality is that the joint-probability formula leads to expectation values of the product rArB of the dichotomic variables rA and rB E(a, b) := P (+,+|a, b) + P (−,−|a, b)− P (+,−|a, b)− P (−,+|a, b) that violate Bell's inequality (5.5) K := |E(a, b) + E(a, b′) + E(a′, b)− E(a′, b′)| ≤ 2 for some choice of the parameters a, b, a′, b′. (For convenience we write a and b instead of a and b). Actually, there are many versions of the Bell inequality; the particular version (5.5) was first written down in [168]. It is easy to see that formula (5.4) leads to E(a, b) = − cosθ, and this may violate (5.5). For example, choose vectors a and b that lie in planes normal to the direction of propagation of the protons, and let a form the angle 0◦,b 45◦,a′ 90◦, and b′ − 45◦ relative to some standard direction. This choice results in K = 2 √ 2 = 2.83 > 2. Thus the quantum mechanical formula (5.4) for spacelike correlations implies nonlocality. In order to see why Bell's inequality accomplishes this we must look at the assumptions that are made in its derivation. In Appendix B it is shown that Bell's inequality is satisfied if the joint probability can be written in the form (5.6) P (rA, rB|a, b) = ∫ P1(rA|a, λ)P2(rB|b, λ) f(λ) dλ. This form means that we first go to a more detailed level of description by introducing the additional variable λ into the joint probability in the integrand, and that on this level we write the joint probability in the special form [169] (5.7) P (rA, rB|a, b, λ) = P1(rA|a, λ)P2(rB|b, λ). The variable λ fluctuates with the probability density (5.8) f(λ) ≥ 0, ∫ f(λ) dλ = 1, and f(λ) and the range of λ do not depend on a, b, rA, and rB. Actually, λ stands for any set of variables that might be relevant in determining the probabilities. The product form (5.7) is more than just the separability P (rA, rB|a, b, λ) = P1(rA|a, b, λ)× P2(rB|a, b, λ) since in (5.7) the first factor does not depend on b nor the second on a. Thus, not only are the events rA and rB statistically independent for given λ, but the probability that A obtains the result rA [i.e., ∑ rB P (rA, rB|a, b, λ) = P1(rA|a, λ)] is also independent of B's parameter b; and similarly P2 is independent of a. There is however still a link between A and B, namely the common variable λ 46 and after the integration the probability (5.6) need no longer have the product form P (rA, rB|a, b) = P1(rA|a)P2(rB|b), and correlations between rA and rB may arise that depend on a and b. Nevertheless, in order to calculate the probabilities P1(rA|a, λ) at A it is sufficient to take into account simply the common parameter λ and the local parameter a, but not the remote parameter b; and analogously for B. This is why the correlations based on the probability formula (5.6), satisfying the Bell inequality, are called locally explicable [170]. Compare formula (5.4) with the formula (5.9) P (rA, rB|a,b) = 14(1− 1 3rArB cosθ). All considerations in Section 5.2 would remain unaltered if instead of formula (5.4) we used formula (5.9). Formula (5.9) results if we assume, in a "semiclassical" model, that the two protons in Fig. 3 were completely independent after their interaction at O (cf. [171], [172], [173]), one having the spin ~σ and the other −~σ (total spin zero) already at O, and that the probability formulas for the single protons were still those of standard quantum mechanics ((C.9) of Appendix C) P (rA|a, ~σ) = 12(1 + rÃσa) = 1 2(1 + rA cosα). Then proton 1 would arrive at A with ~σ, proton 2 at B with −~σ, and the conditional joint probability would be PSC(rA, rB|a,b, ~σ) = 12(1 + rÃσa) 1 2(1− rB~σb). The integration over an isotropic distribution of ~σ would then result in (5.10) PSC(rA, rB|a,b) = 1 4π ∫ dφ ∫ sinαdα12(1 + rÃσa) 1 2(1− rB~σb) = 1 4π ∫ dφ ∫ sinαdα14(1 + rA cosα) (1− rB[sinθ cosφ sinα+ cosθ cosα]) = 14(1− 1 3rArB cosθ). Expression (5.10) differs from the quantum-mechanical formula (5.4) only by the factor 1/3 inside the last bracket. This has however the consequence that the expectation is now E(a, b) = −13 cosθ and the Bell inequality (5.5) |E(a, b) + E(a, b′) + E(a′, b)− E(a′, b′)| ≤ 4/3 < 2 is always satisfied. The variable ~σ in the semiclassical model corresponds to the λ introduced in formula (5.6) because it is independent of a and b. In the quantummechanical case it is not so, because there the value of ~σ for proton 2 before it enters B's apparatus is influenced by A's variable a, ~σ for proton 2 being either +a or −a. 47 Thus correlations described by formulas such as (5.4), which do not satisfy the Bell inequality, mean local inexplicability, i.e. nonlocality [170]. It is always assumed that the choice of the parameters a and b can be made at any time at will by the experimenters. If we accept a strictly deterministic view, where free will is an illusion, it is possible to assume that all correlations arise from common causes in the past. Nevertheless, it is also possible to assume that there are particular correlations which do not arise in this way. Some support of this view is seen in the superluminal contraction of the one-particle wavepackets discussed in Sec. 2.3 and in the arguments given in [49] (cf. also [50]). Finally we note that in the above considerations about the Bell inequality the question of determinism is not involved because the parameters a, b and λ only determine the probability of an outcome, not the outcome itself. Whether this probability is reducible to some underlying constellations of additional variables is left open. – Neither is the question of realism involved because it is left open whether the outcomes come into existence by our observation or arise independently of the observer. The Bell inequality is just about nonlocality. 5.4 Experiments Are the formulas of quantum mechanics that lead to a violation of the Bell inequality confirmed in specific experiments? Many experiments have been performed [98], [115], [116], [174], [175] and the result is that they generally confirm quantum mechanics. Most experiments were concerned with the Bell inequality in its different but essentially equivalent forms. There are also other experiments confirming the nonlocal features [176], [177], but the experiments related to the Bell inequality seem to be the most stringent ones and have been subjected to the closest scrutiny. In all of them, except in two early cases which are now considered unreliable, a violation of the respective variant of the Bell inequality has been found. Moreover, the particular type of violation was exactly that predicted by the formulas of quantum mechanics. The probability P (rA, rB|a, b) in formula (5.6) or the average E(a, b) are measured by means of normalized coincidence rates and appropriate average values. Not all experiments were absolutely conclusive because simplifying though very plausible assumptions had to be made. These assumptions were necessary because the filters and detectors employed were not ideal, because the two photons in the atomic-cascade experiments are not strictly antiparallel, and because of other reasons. Clauser and Horne [169], for example, introduced the "no-enhancement assumption", which means that the photon detection probabilities [P1 and P2 in (5.7)], for every value of the variable λ, can at most be reduced but not enhanced by a polarization filter placed in front of the detector ("detection loophole"). In subsequent experiments most of the simplifying assumptions have been gradually eliminated or reduced in their influence. For example, in all pre-1982 experiments the spin-(polarization)-reference axes were fixed before the individual measurements were done, hence the measurements were not separated by spacelike intervals ("locality loophole"). Therefore the result 48 of A could, in principle, have been transmitted to B with light (or even under-light) velocity before the measurement by B had taken place and so could have influenced B's result. Of course, in the actual experiments any mechanism that might, according to current knowledge, have permitted this was excluded; still the possibility was only excluded technically, not in principle. In 1982 Aspect et al. [178] performed an experiment in which this was excluded in principle. They used variable polarizers that jumped between two orientations in a time that was short compared with the photon transit time. In this experiment, too, a violation of Bell's inequality and a confirmation of the quantum mechanical formulas was found (see also [179], [180], [181], [182]). Thus, the experiments provide overwhelming evidence that nonlocality is indeed a feature of physical reality. Finally, let us have a look at the spatial separations of the wavepackets between which spacelike correlations have been observed in the experiments. (1) In the proton-proton scattering experiment of Lamehi-Rachti and Mittig [99] the distance OA in Fig. 3 was about 5 cm. After scattering at O the protons had a kinetic energy of 6 MeV, and the length of the proton packets was calculated from the lifetime of the intermediate singlet s-wave state to be 4 × 10−15 m. A proton packet of the above energy, for which 4×10−15 m is the minimum width, spreads out to an extension of 2.3 cm while its centre traverses the distance of 5 cm [Appendix A, Eq. (A23)]. Thus, the separation between the two proton packets is about 4 times their width as measured by the standard deviation. (2) In some experiments for testing Bell's inequality photons from a cascade decay of excited atoms are employed. The length of the photon packets is estimated from the mean lives of the decaying levels, which gives values of the order of 1.5 – 3 m. This is comparable to or even larger than the dimensions (OA) of the apparatuses used up to 1980. In the subsequent experiments [178], [183], [184] the apparatuses A and B are separated by about 13 m. This is 8 times the estimated length of the photon packets. (3) In correlation experiments with photon pairs from e+e− annihilation [185] - [187] the individual photon packets are usually assumed to have lengths of the order of 7− 15 cm while the distance between O and A was up to 2.5 m. This is 16 to 35 times the packet length. Admittedly, the lengths ascribed to the individual wavepackets may be larger than assumed, in particular they may be larger than the usually adopted standard deviation ∆y . The value of ∆y is often calculated from ∆py by means of the Heisenberg relation with the equality sign ∆y∆py = h/2. However, the equality sign can only hold for a Gaussian form of the wavepacket, and even for a Gaussian form it holds only at one instant of time; at other times the length may have spread out to values considerably larger than the minimum value. Also, the length ∆y of a wavepacket is often taken to be the coherence length of the beam in which it takes part [99], [185], [186]. Actually, the coherence length of the beam is of the order of a lower bound for the length of the constituent wavepackets (Appendix A). Brendel et al. [188] used pairs of parametrically down-converted photons and measured the correlations in coincidence counts over distances of 55 cm. At the same 49 time they measured the length of the individual photon wavepackets and obtained values of less than 10 cm. There is thus very little overlap between the photon wavepackets. But at the same time they also obtained the high value of 87% for visibility in the coincidence rate as a function of wavepacket separation. This cannot be explained as interference of wavepackets in ordinary space. And there are by now many other photon-correlation experiments that point in the same direction [94], [189], [190]. In fact, spacelike correlations between photons (length of the order of 30μm) entangled over more than 10 km [191], [192], 16 km [103], and 143 km [193] have been observed [194]. 50 6. QUANTUM STATISTICS WITH WAVEPACKETS 6.1 Field Quantization The Schrödinger equation in ordinary (3+1)-dimensional space is a classical field equation, and the discrete eigenfunctions following from imposing the usual normalizability and uniqueness conditions are no more than the standing waves of classical physics. This is just a consequence of de Broglie's idea of matter waves. We would thus not call that appearance of discrete eigenvalues and eigenfunctions real quantum effects but would reserve this denomination to effects that cannot be explained in the mentioned way. At first, Schrödinger seems to have believed that there are no such other effects, but he was opposed by Heisenberg [195], [196] who pointed, among other things, to the photoelectric effect and to the Planck radiation law. Schrödinger's equation in its general form is not an equation in ordinary space but in (3N + 1)-dimensional configuration space, and this goes beyond de Broglie's conception. The Schrödinger or de Broglie function ψ(x, t) in ordinary space in itself does not tell us whether it refers to one or to more particles or to particles at all. It is just a field and expresses only the wave aspect. With the introduction of the configuration-space function Ψ(x1,x2, . . . ,xN , t), however, the number N of particles or quanta is explicitly introduced. In 1927 in his fundamental paper on the quantum theory of the emission and absorption of radiation Dirac [197] derived, among other things, the Einstein A and B coefficients and hence Planck's law by means of a new procedure, which had first been introduced by Born and Jordan [198] and in the famous three-man work on matrix mechanics by Born, Heisenberg and Jordan [199]. The procedure consisted in turning some canonically conjugate variables of the Hamilton formalism into operators satisfying the canonical commutation relation, such as between position and momentum. Here the now familiar creation and annihilation operators a† and a (in present notation) showed up for the first time. In the same paper Dirac, and later Tomonaga [200], showed that this procedure was equivalent with Schrödinger's configuration space treatment with symmetrical wave functions. The equivalence was subsequently elaborated and extended by Jordan, Klein and Wigner [201] - [205] to include antisymmetric wave functions, i.e. fermions, in which case anticommutators were to replace the commutators. In 1932 Fock [117] gave a lucid summary of these developments, and he showed that the restriction to a configuration space of fixed dimension can easily be overcome. This he made particularly clear by casting Schrödinger's configuration-space formalism into the form of the Fock, occupationnumber, or N representation [117], [119], which we have already discussed in Sec. 3.3. In this representation the total number of particles no longer appears explicitly, and this makes it possible to apply the formalism to systems in which the total number of particles is not conserved, as for example to the photons in a cavity. The method of commutation relations was then further developed into a comprehensive scheme by Heisenberg and Pauli [206], [207]. They no longer derived the commutation relations from configuration space but set out from ordinary-space 51 fields ψ(x, t) and introduced the commutation relations by way of postulate. In this way the quanta of the fields (i.e. the particles) arise from interpreting the operator a†a, which has only non-negative integer eigenvalues, as a particle-number operator. Moreover, and most important, they extended the formalism to include Lorentz invariant interactions and hence retardation between the similar particles. Retardation effects cannot be taken into account in Schrödinger's configuration space, so the two schemes are no longer equivalent. We call Heisenberg's and Pauli's scheme canonical quantization. In it the well known difficulties with the diverging integrals, irreparable by simple normal ordering, began. Thus, in my opinion, this is where something went wrong with the relativistic formulation of quantum theory, and I suspect that this is related to the general negative attitude towards nonlocality at that time, as reflected, for example, by Pauli's classification [208] of Landau's and Peierls' nonlocal density as "unnatural". In fact, some features, which had been thought to be explicable only in the formalism of canonical quantization, could by now be shown to be explicable within the formalism of quantum mechanics, reduction and entanglement being duly taken into account [95], [209]. Thus here we stop, and we conclude our treatise with the re-interpretation of the nonrelativistic field-quantization formalism as far as it is equivalent with Schrödinger's configuration-space formalism, including symmetrization and particle non-conservation. Our treatment still includes quantization of the electromagnetic radiation field, that is photons in the spirit of Einstein. There are no retardation effects between the photons, because there are no direct interactions between them. In the following sections we shall derive the Bose, Planck and Fermi distributions as well as the corresponding fluctuations, by speaking of alteration, condensation and decondensation of the realist quantum wavepackets, rather than of distributing particles over phase-space cells. 6.2 The Many Aspects of the Condensed Wavepackets We consider similar particles of mass m in a cavity of volume V at temperature T , and we write the general quantum statistical distribution function in the well-known form (6.1) N(p, T )dp = 4πV p2dp h3 × { exp [(√ p2c2 +m2c4 − μ ) 1 kT ] ± 1 }−1 where N(p, T )dp means, in the usual interpretation, the time averaged number of particles in V whose absolute value of momentum lies in the interval dp about p. The plus sign refers to fermions and the minus sign to bosons. μ is the chemical potential [fugacity z = exp(μ/kT )]. In the special case of photons we have m = 0, p = hν/c, μ = 0, and formula (6.1) with the minus sign reduces to the Planck distribution for polarized radiation. We have written the distribution (6.1) as the product of two factors. The first factor is (6.2) gp = 4πV h3 p2dp = 4πV ε √ ε2 − (mc2)2 h3c3 dε 52 where ε = √ p2c2 + (mc2)2 is the total energy of a particle. In this section we will consider only this factor; the second factor will be considered in the next section. In the special case of photons (6.2) becomes (6.3) gν = 4πV ν 2dν/c3, and in this case it has a long history: In 1899 it was calculated by Planck [210] as the proportionality factor between the mean energy of electromagnetic radiation in V and dν and the mean energy of a charged oscillator with radiation damping. In 1900 and 1905 Rayleigh [211] and Jeans [212] considered the factor as the number of degrees of freedom of the ether inside the cavity, this number in turn being considered equal to the easily calculable number of eigenvibrations (modes of vibration) within dν of the ether. In 1914 von Laue [213] decomposed the cavity radiation into mutually independent radiation bundles, each converging to its focal region and then diverging. To these bundles he attributed degrees of freedom and obtained (6.3) as the sum of the degrees of all these bundles (see below). Bose, in his famous paper of 1924 [214], considered the factor (6.3) as the number of phase-space cells of size h3. Such cells had already been considered by Planck in 1906 [215] in the special case of harmonic oscillators. In 1925 Landé [216] called those of von Laue's radiation bundles that had just one degree of freedom elementary light-quantum bundles or just quantum bundles, and he proposed to identify these with Bose's quantum phase-space cells. In present-day quantum mechanics (6.2) is the number of eigenvalues of the Hamilton operator for a free particle in V that fall into the energy interval dε which corresponds to dp. Each eigenvalue is multiply counted according to its order of degeneracy. In other words, (6.2) is the number of eigenstates in V and dp. For photons we are thus effectively back at Rayleigh's and Jeans' determination. In quantum mechanics (6.2) holds, however, for any kind of particle, not just photons, because de Broglie waves are associated with each kind of particle. Finally, in canonically quantized radiation theory (6.3) is the number of oscillators. But in contrast to Planck's oscillators, which represent atoms ('resonators") interacting with the radiation field, these oscillators are to represent the field itself, a point of view that had already been indicated by Ehrenfest in 1906 [217]. Now we want to show that (6.2) or (6.3) can also be taken as the number of (condensed) wavepackets in the cavity covering the momentum interval ∆p. For this purpose we employ the fact that (6.3) is the total number of degrees of freedom of von Laue's radiation bundles and that a bundle of F degrees of freedom may be taken to consist of F wavepackets. Von Laue defines the number F of degrees of freedom of a bundle of length l (from wall to wall of the cavity), spectral range dν (equal to the spectral range of the radiation considered), convergence half angle α and focal cross section A with the help of the theory of optical resolving power and the counting of Fourier coefficients. He arrives at the expression (6.4) F = Aldν ac 53 where a is the minimum focal area that is possible for a bundle of convergence angle α (cf. Appendix A formula (A16), with α = υsx∞/c). Now imagine that the bundle of F degrees of freedom consists of a stream of wavepackets, all moving parallel to the axis of the bundle and going side by side through its focal area. The convergence of the bundle to the focal plane and its subsequent divergence comes about by the contraction and subsequent spreading in the transverse direction of each one of these wavepackets, assuming that all packets have their minimum transverse extension in the focal plane. In front of and behind the focal plane the wavepackets may overlap in the lateral direction. We then write von Laue's degrees of freedom F as the product of three factors: F = N1N2N3 where N1 = dν/∆ν,N2 = A/a, and N3 = l/(∆y). Each factor is the ratio of some quantity relating to the bundle divided by the corresponding quantity relating to the packets. ∆ν is the frequency range of a wavepacket of total length ∆y. ∆ν is related to ∆y by ∆ν = rc/(4π∆y), (r ≥ 1), which follows from the Fourier reciprocity (Heisenberg) relation ∆y∆py = rh/2 with ∆py = (h/c)∆ν. With this relation we obtain N1N2N3 ≤ (4π/r)F . The number r may be set equal to 4π because the packets in the cavity do not all have their minimum phase-space extension (r = 1). Moreover, there is always some degree of arbitrariness in the exact definition of the widths ∆y,∆py etc., resulting in some arbitrariness in r . Thus we take r = 4π to mean the average extension of the wavepackets in the cavity with an appropriate definition of the widths, and with this we obtain (6.5) N1N2N3 = F. Now, by the definitions given aboveN1 is the number of spectral types of wavepackets in the bundle, as defined by their individual frequency ranges ∆ν (colors), N2 is the number of wavepackets of a particular spectral type that go side by side and N3 those that go one after the other through the focal area of the bundle. It follows that the product N1N2N3 is equal to the total number of wavepackets that make up the bundle, and the relation (6.5) means that this number is equal to the number of degrees of freedom of the bundle. Thus the total number of wavepackets can be identified with the total number (6.3) of degrees of freedom, or of von Laue's bundles, if we imagine the radiation to consist only of elementary bundles. This not only holds for photon packets but for packets of any kind. The considerations by von Laue can easily be extended to matter waves. One just has to replace ν/c by 1/λ in (6.4) and the ensuing text, and then to replace 1/λ by p/h, and with r = 4π the same result obtains. Now, the discrete energy values nhν may be attributed to each of Planck's oscillators or Jeans' degrees of freedom [218], and n quanta may occupy each of Bose's cells. These are then our condensed wavepackets representing n quanta. Empty wavepackets, without a quantum, are also included in formula (6.2). This is a convenient means of indicating that there is space left for more wavepackets to show up in V and dp (cf. Bose [214] and Schrödinger [219]) Equivalently, one may say that (6.2) is the maximal possible number of existing (non-empty) wavepackets. 54 The condensed wavepackets also resemble the degenerate light pulses of Mandel [220]. Mandel introduced the degeneracy parameter δ, meaning the average number of photons in a light beam that are to be found in the same cell of phase space. He expressed the phase space volume with the help of a certain coherence volume defined in the theory of optical coherence. We obtain the same formulas when we take the phase-space volume to be the product of the ranges of the wavepacket at the moment of its minimum extension (Appendix A). Finally we want to point out to the "light molecules", "quantum multiples", "n-quantum rays" and "radiation bundles" first mentioned parenthetically by Joffé (1911) [221] and considered more closely by Ishiwara (1912) [222] , Wolfke (1921) [223], de Broglie (1922) [224], Bothe (1923,1924) [225] and especially by Schrödinger (1924) [219] and Bothe (1927) [226]. These authors noticed that the Planck distribution can be written in the form of a sum and that the n-th term can be interpreted as a contribution from objects that are composed of n light quanta (see formula (6.16) below). It seems that these ideas retreated under the blow of Dirac's quantization of radiation in 1927 [197], but it is seen that they also strongly resemble the condensed wavepackets. The wavepackets, not the single quanta (if we were to take these for a moment as entities of their own) are the statistically independent objects, and condensation of wavepackets is our means of expressing the "mutual influence of the molecules [i.e. quanta] which for the time being is of a quite mysterious nature" mentioned by Einstein in 1925 [227]. 6.3 The Balance Relation Now we turn to the second factor in (6.1) { exp [(√ p2c2 +m2c4 − μ ) 1 kT ] ± 1 }−1 . We want to derive this factor by means of Einstein's method of balance relations between transition rates [228], formulated in terms of wavepackets. Einstein's original treatment of 1917 did not explicitly take into account that photons are bosons and not fermions; it would give the same distribution function in both cases. Of course, the Fermi distribution was published only in 1926 [229]. The fact that photons are bosons can only be taken into account when phase-space regions or energy intervals are subdivided into those fundamental units that are counted by formula (6.2). It is not enough to consider the number of photons in a given energy interval, as in Einstein's procedure of 1917, but one must further specify how the photons are distributed over the various fundamental units within this interval. This was done by Einstein later in 1924 in Bose's quite different phase-space cell approach. The fact that Einstein did obtain the Planck distribution by his 1917 method, in spite of not accounting for the subdivision into fundamental units, is due to the special way he formulated the balancing equations. We shall return to this point below. 55 With the subdivision into fundamental units we shall be able to derive both the Bose and the Fermi distribution on an equal footing by means of the method of balancing equations between wavepackets. The Fermi distribution has already been obtained in the framework of this method by several authors, though in a different way [118], [230] - [235]. Our procedure is inspired by the comprehensive treatment by Oster [232] and the remarkable paper by Bothe [226]. The general mechanism in all these works is exchange of quanta between fundamental units. These units have sometimes been taken to be the discrete energy states of atoms or oscillators which emit and absorb photons. We emphasize, however, that the fundamental units are not restricted to discrete energy states. That they may well be small but finite energy intervals, centred about any energy values, had already been pointed out by Pauli [236] and by Einstein and Ehrenfest [237] when considering photons scattered by free electrons in the Compton effect. In our interpretation the fundamental units are the wavepackets and we are going to consider processes taking place between these. As we have discussed in Sec. 3.3 this is our interpretation of the change in "occupation numbers", which in the occupation-number representation is described by means of the creation and annihilation operators. The one-particle basis functions in the expansion (3.10) now are energy eigenfunctions or narrow superpositions of these. Specifically, we consider two types of elementary processes. Either of them is decomposed in alteration, condensation and decondensation. Alteration changes the energy of a wavepacket, whereas condensation and decondensation change the number of quanta it represents. Thus the first type is: (1) An s-quantum wavepacket of kind 1 which represents s quanta in the energy interval εi1 . . . ε i 1 + dε i 1 (an s-packet in dε i 1, for short) decondenses into an (s − n)packet and an n-packet in dεi1. The n-packet exchanges energy and momentum with an n′-packet of kind 2 in dεi2 whereby it is altered and goes into the energy interval dεf1 and then condenses with an r-packet in dε f 1 to form an (r + n)-packet in dε f 1. Simultaneously an s′-packet of kind 2 in dεi2 decondenses into an (s ′ − n′)-packet plus an n′-packet in dεi2. The n ′-packet is altered in an interaction with the n-packet in dεi1 whereby it goes into the interval dε f 2 and then condenses with an r ′-packet in dεf2 to form an (r ′ + n′)-packet in dεf2. A graphical scheme is presented in Fig. 5. Conservation of energy requires (6.6) n (εi1 − εf1) = n′ (εf2 − εi2). The energy intervals dε are chosen so that they correspond to the respective intervals dp in absolute value of momentum used in formula (6.1), i.e. dε = (dε/dp)dp. Effectively, if we may use here the picture of quanta as standard portions of water, an s-packet in dεi1 gives n quanta to an r-packet in dε f 1, and an s ′-packet in dεi2 gives n′ quanta to an r′-packet in dεf2. Thus, process (1) leads from the state |a〉 characterized by the 4 packets which represent s, r, s′ and r′ quanta respectively, to some state |b〉 characterized by the 4 packets that represent s− n, r+ n, s′ − n′ and r′ + n′ quanta respectively. The probability of such a transition is denoted by W1. 56 Figure 5: Scheme of the considered processes (1) and (2) between the wavepackets of kind 1 and kind 2. The process comprises most particular physical situations that lead to Fermi or Bose distributions as special cases. There is no interaction between photons, so here we need the second kind of wavepacket (atoms, electrons etc.). Electrons interact with each other, and kind 1 and kind 2 may be the same. In the Boltzmann case there is no change in the number of quanta but only an alteration in the energies of the wavepackets. This can be described by putting r = r′ = 0, n = s, n′ = s′ in in the scheme of Fig. 5 and dropping p(0, ε) and q(0, ε) from the Eqs. (6.7) and (6.8) below, which then lead to the Boltzmann distribution for the wavepackets (of energy εs). (2) The second type of processes is this: an (r+n)-packet of kind 1 in dεf1 (which need not be the same packet as that at the end of process (1)) decondenses into an r-packet and an n-packet. The n-packet exchanges energy and momentum with an n′-packet of kind 2 whereby it goes into the energy interval dεi1 and then condenses with an (s− n)-packet of that interval. Simultaneously an (r′ + n′)-packet of kind 2 in dεf2 decondenses into an r ′-packet and an n′-packet. The n′-packet is altered in an interaction with the n-packet of kind 1 whereby it goes into dεi2 and then condenses with an (s′ − n′)-packet. Conservation of energy is again guaranteed by Eq. (6.6). 57 The initial (final) s, r, s′ and r′-packets of process (2) have the same momenta etc. as the final (initial) s, r, s′ and r′-packets of process (1) and differ from those only by spatial translations. Thus, process (2) goes back from state |b〉 to state |a〉. The probability of this transition is denoted by W2. Process (2) is not the time reversed ("converse") process to process (1), but may be called the "reverse" process, after Dirac [238]. Only the reverse process can lead to statistical equilibrium [237], but only the converse process is suggested to exist and to occur at the same rate as the original process on account of the general principle of time-reversal invariance of basic processes. Now, in an isotropic medium the reverse process can be obtained from the converse one by successive reflections in three mutually perpendicular mirrors at rest relative to the system as a whole, and their frequencies of occurrence must be equal [238]. Thus, in statistical equilibrium the two processes (1) and (2) also occur at equal rates. We now consider these rates. Rate 1 is the mean number of processes (1) that occur per second in the volume V . According to the above-given description it should be equal to (6.7) p(s, εi1)dε i 1 p(r, ε f 1)dε f 1 q(s ′, εi2)dε i 2 q(r ′, εf2)dε f 2 W1 where p(s, εi1)dε i 1 is the mean (time averaged) number of s-packets of kind 1 in V that represent quanta in dεi1, q(s ′, εi2)dε i 2 is the mean number of s ′-packets of kind 2 in V and dεi2, and so on. Analogously, for process (2) the rate is (6.8) p(s− n, εi1)dεi1 p(r + n, εf1)dεf1 q(s′ − n′, εi2)dεi2 q(r′ + n′, εf2)dεf2 W2, and the two rates (6.7) and (6.8) have to be equated. Now, according to the preceding section our wavepackets and hence the states |a〉 and |b〉 mean pure states of quantum mechanics, and since the probability of a transition in quantum mechanics is the same for a process that goes from |a〉 to |b〉 as for a process that goes from |b〉 to |a〉 (Hermitean operators), the probabilities W1 and W2 are equal and disappear from the balancing equation. We shall thus obtain the statistical distribution functions without using any special property of the transition probabilities. The differentials dε also cancel, and our balance relation acquires the simple and symmetric form (6.9) p(s, εi1) p(r, ε f 1) q(s ′, εi2) q(r ′, εf2) = p(s− n, εi1) p(r + n, εf1) q(s′ − n′, εi2) q(r′ + n′, εf2). The relation is reminiscent of the relation for chemical equilibrium between several kinds of molecules. It still comprises the Bose and Fermi cases. There is no spontaneous emission term, i.e. one that would be independent of the number of wavepackets. A general solution is (6.10) p(s, ε) = a(ε) exp[−(bε− c)s] q(s, ε) = a′(ε) exp[−(bε− c′)s]. 58 - [Insert and use (6.6)]. Notice that only the parameter b (which shortly will be identified with 1/kT ) is the same for the two kinds of packets. In any other respect the distribution function for kind-1 packets is independent of the distribution function for kind-2 packets. The parameters c and b are obtained via the thermodynamic relations ∂S/∂Etot = 1/T and ∂S/∂Ntot = −μ/T where (6.11) S = k ln ∏ {dεi} gp! [p(0, εi)dεi]! [p(1, εi)dεi]! * * * is Bose's or Natanson's [239] formula (in our notation) for the entropy of the total system, with Ntot = ∑ {dεi}Ndp and Etot = ∑ {dεi} εiNdp . The total energy here is thought to be subdivided into a set of intervals {dεi}. The entropy (6.11) does not depend explicitly on the numbers s. Formula (6.11) also implies that the wavepackets (as units representing s quanta) are thermodynamically independent [223], [240]. The number of all wavepackets in V and dε, including the empty ones, is given by (6.2), so we have (6.12) ∑ {s} p(s, ε)dε = gp. The total number of quanta in V and dε is N dp, so (6.13) ∑ {s} s p(s, ε)dε = Ndp. With (6.10), (6.13) and Stirling's approximation ln p! ≈ p ln p the entropy in thermal equilibrium may be written as S = k ( bEtot − cNtot − ∑ {dεi} gp ln [a(εi)dεi/gp] ) hence the above-mentioned thermodynamic relations lead to b = 1/kT, c = μ/kT. So far all mathematical operations could be carried out even if the numbers s, s′, r, r′ were not integers. 6.4 The Bose and Fermi Distributions Now we go to special cases. First we take the kind-1 packets to be boson packets. In this case the numbers s, r and n are non-negative integers and (6.12) becomes (6.14) ∞∑ s=0 p(s, ε) dε = gp = adε 1− exp[−(ε− μ)/kT ] . From this we obtain adε = gp(1− exp[−(ε− μ)/kT ]) and (6.15) pB(s, ε)dε = gp (1− exp[−(ε− μ)/kT ]) exp[−s(ε− μ)/kT ]. 59 This formula coincides with Bose's expression for the number of phase space cells occupied with s quanta. The total number of quanta (6.13) becomes (6.16) Ndp = ∞∑ s=0 s p(s, ε) dε = gp exp[(ε− μ)/kT ]− 1 and we identify these quanta, not the wavepackets, with the particles in the usual interpretation of formula (6.1). Thus we have arrived at the desired Bose-Einstein distribution function, Eq. (6.1) with the minus sign. Let us further consider photons, as a special kind of bosons, and let us consider the processes where photons are absorbed and emitted by atoms. In this case we take the packets of kind 1 to be the photons and the packets of kind 2 to be the atoms. In one respect the situation goes beyond the scheme of Fig. 5, in that the number of photons is no longer conserved. In its interaction with the atom the photon is absorbed and exists no longer. Thus, in Fig. 5 the arrow that points from the n-packet in dεi1 (second line) to the (r + n)-packet in dε f 1 (third line) no longer exists, and the (r + n)-packet remains an r-packet. Equivalently, one may say that the n-packet turns into an empty packet (n = 0). Thus, in the balance relation (6.9) the function p(r + n, εf1) on the right-hand side becomes equal to the function p(r, εf1) on the left-hand side and disappears from the equation. In the energy-conservation relation (6.6) we have to put εf1=0. What the atom does beyond satisfying the energy conservation in absorbing and emitting a photon is irrelevant. Likewise, in the reverse process the arrow that points from the (r + n)-packet in dεf1 (fourth line) to the n-packet in dε f 1 (fifth line) no longer exists, or equivalently, refers to an empty packet. This does not, however, affect the balance equation, which thus is (6.17) p(s, εi1) q(s ′, εi2) q(r ′, εf2) = p(s− 1, εi1) q(s′ − n′, εi2) q(r′ + n′, εf2). The solution is again given by (6.10), but only if c = 0 in p(s, ε) there, so we have obtained Planck's law. There is no spontaneous emission in our treatment. Let us compare this with Einstein's treatment. Einstein's balancing equation (Sec. 3 in his 1917 paper [228]) is (6.18) exp(−εn/kT ) ρ = exp(−εm/kT ) (ρ+Anm/Bnm) where ρ = hνN dp/(V dν) is hν times the (time averaged) number of photons per unit volume and per unit frequency interval in the cavity. This equation, unlike our Eq. (6.9) or (6.17), is concerned with the number of quanta, not with the number of wavepackets. The second term in the bracket, Anm/B n m = 4πhν 3/c3 (polarized radiation), is independent of ρ and is the spontaneous emission term. One may obtain Einstein's Eq. (6.18) from our Eq. (6.17) if one puts back the transition probabilities W1 and W2 into this equation and uses a special property of them. The left-hand side of (6.17) effectively means a process where an atom absorbs a photon from an s-photon packet, and the right-hand side a process where an atom 60 emits a photon into an (s − 1)-packet. In order to compare with Einstein's 1917 treatment, which disregards the wavepacket structure of the radiation, one has to sum Eq. (6.17), with W1 and W2 restituted, over all photon packets, i.e. over all values of s (with n=1): q(s′, εi2) q(r ′, εf2) ∞∑ s=0 p(s, εi1) W1(s, α) = (6.19) = q(s′ − n′, εi2) q(r′ + n′, εf2) ∞∑ s=0 p(s− 1, εi1) W2(s− 1, β). α and β are the other arguments in W1 and W2, which do not depend on s. Now we use the special property (6.20) W1(s, α) = W2(s− 1, β) = f * s where f may depend on anything but s. This fits with Dirac's statement that the probability of a transition in which a boson is absorbed from (emitted into) state x is proportional to the number of bosons originally in state x (in state x, plus one) [197]. With Eq. (6.20) one may write (6.19) in the form q(s′, εi2) q(s′ − n′, εi2)} {{ } C1 dεi1 ∞∑ s=1 s p(s, εi1) = q(r′ + n′, εf2) q(r′, εf2)} {{ } C2 dεi1 ∞∑ s=1 s p(s− 1, εi1) = C2 [ dεi1 ∞∑ s=1 (s− 1) p(s− 1, εi1) + dεi1 ∞∑ s=1 p(s− 1, εi1) ] , and with (6.14) and (6.16) one obtains C1N dp = C2(N dp+gp). If one multiplies this by hν/(V dν) and observes that C1/C2 = exp[n ′(εf2− εi2)/kT ] and n′ = 1 one obtains Einstein's balancing equation (6.18). We note that it is only the special form (6.18) of the balancing equation that requires the special property (6.20) of the transition probabilities in order to arrive at the Planck distribution. Second, we take the kind-1 packets to be fermion packets. In this case we have only zeroand one-quantum packets, and the sums in (6.14) and (6.16) range only from 0 to 1. Thus 1∑ s=0 p(s, ε)dε = adε(1 + exp[−(ε− μ)/kT ]) = gp. Hence (6.21) pF(s, ε)dε = gp (1 + exp[−(ε− μ)/kT ])−1 exp[−s(ε− μ)/kT ] 61 and the total number of quanta is Ndp = 1∑ s=0 s p(s, ε)dε = gp exp[(ε− μ)/kT ] + 1 , which is the desired Fermi-Dirac distribution, i.e. formula (6.1) with the plus sign. The balance relation (6.9) in the Fermi case may be specified to read (s = n = 1, r = 0) p(1, εi1) p(0, ε f 1) q(s ′, εi2) q(r ′, εf2) = p(0, ε i 1) p(1, ε f 1) q(s ′ − n′, εi2) q(r′ + n′, εf2) and may also be given a specific interpretation: a fermion 1-packet in dεi1 is altered in an interaction with a wavepacket of kind 2 whereby it goes into dεf1, conservation of energy requiring εi1 − εf1 = n′(εf2 − εi2). Then it goes into a region of phase space within dεf1 that is not yet occupied by a (non-empty) wavepacket. The transition rate is proportional to the size of this region, expressed in fundamental units, that is, to the number of empty packets in dεf1, p(0, ε f 1)dε f 1. 6.5 Quantum Count Fluctuations Finally we extend our considerations on wavepackets in a cavity to include fluctuations. To be definite, we consider a small subvolume υ of the total cavity volume V . We imagine that the subvolume is homogeneously filled with detectors (groups of sensitive atoms). The detectors are sensitive only within the interval p . . . p + dp of the absolute value of momentum or the corresponding energy interval ε . . . ε+ dε, and we assume that within this interval the sensitivity is constant. These detectors are switched on during the interval ∆t, and the number of counts is registered. This procedure is repeated a great many times, where the time intervals between the repetitions are large compared with ∆t . We then ask for the mean square deviation, or variance, (∆m)2 of the number of counts. Again we shall treat both boson and fermion wavepackets on an equal footing. The subvolume υ together with the interval dp define a certain volume of phase space and with this a certain number gυ of (empty plus non-empty) wavepackets, given by formula (6.2) with V replaced by υ. The number of wavepackets with which the counter can interact during ∆t is larger than gυ because (1) the switch-on time ∆t may be so long that many sets of wavepackets, each set filling the counter volume at one time, may pass through the counter during ∆t, and (2) the counter can also interact with wavepackets that only partially extend into it. The (integer) number of wavepackets that partially and/or totally cover the phase-space region of the counter during ∆t is denoted by g, where g ≥ gυ and g ≥ 1. A count is always an interaction of the counter with a wavepacket, not with a quantum. The number of quanta represented by the g packets fluctuates because between two measurements some few-quantum packets may have replaced some many-quantum packets and vice versa. This is the only source of fluctuations. Fluctuations that arise from a nonempty wavepacket leaving the region without another non-empty packet entering 62 it are already accounted for because our number of wavepackets includes empty packets, so that a non-empty packet leaving the region is equivalent with an empty packet entering it. First we want to consider a special property of a condensed wavepacket which we shall use below. As the condensed packet arises from the process where some of the one-particle functions in the expansion of ΨS (3.10) become equal we may describe it by means of the product (6.22) ΨS = φ(x1, s1, t)φ(x2, s2, t) * * *φ(xN , sN , t) where the si signify the spin variables, and the same φ is used in all factors. φ(x, s, t) here does not necessarily mean the lowest-energy one-particle state, as it does in Bose-Einstein condensation proper [241]. We may mention that in the treatment of laser coherence [242], [243] the wave function of a stationary N -photon state can also be written as a product of the type (6.22). The condensed packet is thus effectively described by the one function φ(x, s, t) in ordinary space. Consider the expression (6.23) P1 = ∫ x1∈D3 ∫ xi∈R3 (i≥2) * * * ∫ |ΨS(x1,x2, . . . ,xN , t)|2d3x1d3x2 . . . d3xN = ∫ x∈D3 |φ(x, t)|2d3x. x ∈ D3 means xa ≤ x ≤ xb, ya ≤ y ≤ yb, za ≤ z ≤ zb. R3 means total space. If necessary x, D3 and R3 are to include the spin variables. For D3 = d3x the expression reduces to P1 = |φ(x, t)|2d3x . In the Copenhagen interpretation expression (6.23) means the probability that particle 1 of a system of N similar particles is found in the spatial region D3, irrespective of where the other N − 1 particles are found. In our interpretation it is the probability that wavepacket 1 acts in D3, irrespective of where the other N − 1 wavepackets act. Actually, the wavepackets are equal, so what we really want is an expression for the probability that any one packet acts in D3, irrespective of where the other N−1 packets act, which is N *P1. In the same way the standard formalism gives the probability P2 that any m wavepackets of the N act in D 3 while the other N −m do not act in D3 (6.24) P2 = (N m ) ∫ x1∈D3 * * * ∫ xm∈D3 ∫ ′ xm+1∈R3 * * * ∫ ′ xN∈R3 |ΨS(x1,x2, . . . ,xN , t)|2 ×d3x1d3x2 * * * d3xN where a prime at the integral sign means that in the integration over the variables xi the region D3 has to be excluded. The combinatorial factor (N m ) is just the number of ways m billiard balls (or particle labels) can be chosen from N . In the case of the condensed packet of the product form (6.22) P2 reduces to (6.25) P2 = (N m ) ηm(1− η)N−m 63 with (6.26) η = ∫ x∈D3 |φ(x, t)|2d3x. This is the binomial distribution. It is just the probability that N independent trials with probabilities η for success and 1−η for failure result in m successes and N −m failures. We may thus say that P2 is the probability that the condensed packet acts with m of its N quanta in D3. Now, of all the quanta of a wavepacket only some fraction will be counted. The probability of m counts from an n-quantum wavepacket is formula (6.25) with n instead of N : (6.27) b(m;n, η) = (n m ) ηm(1− η)n−m where η = m/n (0 ≤ η ≤ 1, (∆m)2 = m(1 − η)) now is the average fraction of quanta of the packet that are counted during the interval ∆t . In (6.26) we would have η = 1 if D3 = R3. Now, the detector volume υ may happen to be much larger than a wavepacket, and in this case υ is effectively equivalent to R3, leading to η = 1 in Eq. (6.26). Eq. (6.27) would then always give zero for m 6= n and be useless. It is, nevertheless, possible to maintain formula (6.27) even in this situation if we take into account that the efficiency η is not only limited by the finite counter volume, as accounted for in Eq. (6.26), but also by the intrinsic counter efficiency and short interval ∆t, so that even when D3 → R3 the efficiency η may be less than unity, and we can maintain formulas (6.25) and (6.27), independently of whether the counter volume covers the wavepacket totally or partially. The difference between these two cases is absorbed in the numerical value of η, that value suffering an additional decrease when we go from the case of total to that of partial spatial covering. What, then, is the probability W (m; g) of counting m quanta from g wavepackets? To answer this question we first evaluate the probability w(n; g) that the g packets represent n quanta and then the probability B(m;n) that of these n quanta m are counted, and we write W (m; g) = ∑∞ n=mw(n; g) B(m;n). We first consider boson packets. The probability that a randomly chosen boson packet is an s-quantum packet is given by (6.15). From this we obtain the average number s of quanta represented by one packet averaged over all packets in V and dp (or the corresponding dε) (6.28) s = ∞∑ s=0 s p(s)dε/gp = exp[−(ε− μ)/kT ](1− exp[−(ε− μ)/kT ])−1 so that exp[−(ε− μ)/kT ] = s/(1 + s) and (6.29) p(s)dε/gp = 1 (1 + s) 1 (1 + 1/s)s . The probability that of the g packets in υ the first one represents s1, the second s2, . . . and the g-th sg quanta, with ∑g i=1 si = n, is the product (6.30) g∏ i=1 p(si)dε/gp = 1 (1 + s)g 1 (1 + 1/s)n . 64 We are, however, not interested in the particular distribution (which quanta are represented by which packets), so we have to form a sum of expressions (6.30), one for each distribution. Since (6.30) is the same for any distribution we need only to multiply (6.30) by the number of possible distributions, given by the well-known combinatorial expression (6.31) ( g + n− 1 n ) ≡ (g + n− 1)! (g − 1)! n! . Thus the probability that the g packets represent n quanta is (6.32) w(n; g) = (g+n−1 n ) 1 (1 + s)g 1 (1 + 1/s)n . This formula was already obtained by Mandel [244] in a related context. Next we evaluate the probability B(m;n) that if n quanta are present m are counted. Here we take advantage of the fact that this probability is independent of how these quanta are represented by the different wavepackets. This is seen in the following way: assume that all the n quanta are from one and the same wavepacket. Then the probability of m counts is given by the binominal distribution (6.27). Next assume that the quanta are from two wavepackets, one with n1 and one with n2 = n− n1 quanta. The probability of m counts would then be∑ m1 ∑ m2} {{ } m1+m2=m (n1 m1 ) ηm1(1− η)n1−m1 (n2 m2 ) ηm2(1− η)n2−m2 . But due to a special folding property of the binominal distribution [245, p. 173, 268], this is equal to b(m;n, η) of (6.27). And this would remain so if the quanta were from any number of packets. We may thus assume that all n quanta are from one and the same wavepacket whence B(m;n) = b(m;n, η) of (6.27). In deriving this result we have used the same value of η for all wavepackets. This requires a justification because when the phase-space volume of a wavepacket ∆3x∆3p and that of the counter 4πυp2dp with which it interacts are comparable, their overlap and with this the value of η may vary appreciably from one wavepacket to the next, even if, as we assume, all wavepackets in the cavity have very nearly the same size in phase space. In this case the above folding theorem in fact no longer holds generally, although still in special cases (η 1, η ≈ 1, Poisson approximation, normal approximation [245, Chap. 6]). When, however, the phase-space volume of the counter is large (in each direction) compared with that of a wavepacket the counter covers almost each wavepacket completely, and since we consider a counter with constant sensitivity over its whole phase-space volume, η is still the same for all packets. When, in the opposite limit, the counter is small compared with the wavepacket, it is true that η may vary considerably because the counter may cover regions of the wavepacket with varying |ψ|2 (in x space or in p space) since we have not assumed that |ψ|2 is constant over the wavepacket. But now, whichever region of a wavepacket is covered 65 by the counter, η will always be small, and under this condition the folding theorem still holds in the form∑ m1 ∑ m2} {{ } m1+m2=m b(m1;n1, η1) b(m2;n2, η2)− b ( m;n1 + n2, n1η1 + n2η2 n1 + n2 ) ∝ (η1 − η2)2 + terms of higher order in η1 and η2 , so that B(m;n) = b(m;n, η) still holds with η representing some average over different regions of the wavepacket. Therefore we consider B(m;n) = b(m;n, η) as an acceptable approximation. With this the probability W (m; g) of counting m quanta when g packets are in 4πυp2dp is (6.33) W (m; g) = ∞∑ n=m w(n; g) b(n;m, η) = ∞∑ n=m w(n; g) (n m ) ηm(1− η)n−m = ∞∑ n=m (g+n−1 n ) 1 (1 + s)g 1 (1 + 1/s)n (n m ) ηm(1− η)n−m. Formula (6.33) is the same as that obtained via the standard quantization formalism [246], as it should be. Substituting l = n − m and using the binomial identity(l+g−1+m l ) = (−1)l (−g−m l ) and Newton's binomial formula ∑∞ l=0 (−g−m l ) ( − 1−η1+1/s )l = ( 1− 1−η1+1/s )−g−m we obtain (6.34) W (m; g) = (g+m−1 m ) 1 (1 + ηs)g 1 (1 + 1/(ηs))m . From this we can calculate the variance (∆m)2 := ∑∞ m=0(m − m)2 W (m; g) = ηsg(1 + ηs) [use the generating function [245, p. 266] of W (m; g)]. Observing that ηsg = m is the average number of counted quanta from g packets we may finally write the variance of the number m of counts as (6.35) (∆m)2 = m ( 1 + m g ) . Comparing (6.34) with (6.32) we see that the distribution, and hence variance, of the counted quanta has the same form as that of the existing quanta. Our procedure is easily carried over to fermion packets. We only have to observe that there are only zeroand one-quantum packets. Of course (6.22) no longer holds, but (6.27) still does since for m ≤ n ≤ 1 it reduces to three trivial expressions. Further, instead of (6.29) we have to take (6.21) which we write in the form (cf. (6.28)) (6.36) p(s)dε/gp = (1− s) 1 (1/s− 1)s , 66 and instead of (6.31) we have to take( g n ) = g! (g − n)!n! . (6.34) is then replaced by (6.37) W (m; g) = ( g m ) (1− ηs)g 1 (1/(ηs)− 1)m and (6.35) by (∆m)2 = m ( 1− m g ) . For g = 1 Formula (6.34) becomes (m = ηsg) (6.38) WB(m; 1) = mm (1 + m)m+1 (Bose) and (6.37) WF(m; 1) = mm (1− m)m−1 (Fermi). For g →∞ both (6.34) and (6.37) approach the Poisson distribution (6.39) WBM(m;∞) = mm m! e−m (Boltzmann). It seems that our results not only hold for cavities but also for beams, at least in some situations. This we infer from the observation that the distributions (6.38) and (6.39) are also obtained for stationary beams of chaotic light in standard quantized radiation theory [247] and are confirmed by experiment [248]. Formula (6.38) obtains when the counting time ∆t is short compared with the coherence time τc, ∆t τc, and (6.39) obtains in the opposite limit, ∆t τc. These limits can be compared with the limits g = 1 and g 1 of our treatment because the wavepacket structure of the radiation field reflects its coherence properties, in that the spatial size of a photon wavepacket is a measure of the size of the coherence region with length Lc = cτc (Appendix A). The limit ∆t τc means that many wavepackets can interact with the counter during ∆t, either consecutively, if the packets are large, or simultaneously, if they are small compared with the counter volume υ. Thus g 1 in this limit. In the opposite limit, ∆t τc, the wavepackets can interact with the counter only during such a short time interval that their movement is negligible. Whether we have g 1 or g = 1 now depends on further specifications. When the packets are large and pass over the counter one after the other we have g = 1. This was in fact implicitly assumed in the (plane-wave) calculations referred to by Loudon [247] and was explicitly stated in the experimental verification [248] of Formula (6.38). So, here ∆t τc means g 1, and ∆t τc means g = 1, and the situations are those covered by the above formulas. Acknowledgement I thank my wife Marianne for her understanding and support over all these years. 67 APPENDIX A: Collection of Wavepacket Spreading Formulas The wavepackets considered here are the usual free packets of Schrödinger or de Broglie waves. Since the formulas are rather spread out in the literature and are often restricted to special cases we have collected here some more general results for easy reference. The general wavepacket is written in the form (A1) ψ(x, t) = (2π)−3/2 ∫ +∞ −∞ ψ(k) exp[i(kx− ω(k)t)]d3k where ψ(k), and hence ψ(x, t), is normalized∫ +∞ −∞ |ψ(k)|2d3k = 1. The (three-dimensional) Fourier transform of ψ(x, t) is ψ(k, t) = ψ(k) exp[−iω(k)t]. This is the wavepacket in k (momentum) space. Mathematically the wavepacket need not have a sharp boundary in x space (or in k space), but for practical purposes it may be considered to have a finite extension given, for example, by the standard deviation (A2) ∆x := 〈(x− 〈x〉)2〉1/2 with 〈x〉(t) = ∫ +∞ −∞ ψ∗(x, t) x ψ(x, t) d3x, and with analogous expressions for the widths ∆y and ∆z . In the special case of the Gaussian form (and t = 0) |ψ(x)|2 = (2πσ2)−1/2 exp[−(x− 〈x〉)2/(2σ2)] ∆x of (A2) is equal to the parameter σ and is the distance between the position of the maximum and the point where the distribution has fallen off to exp(−1/2) = 0.61 of the maximum value. Sometimes the form of ψ(x) or of ψ(k) is such that the integrals in (A2) etc. diverge, for example for ψ(k) ∝ sin(ak)/k and for ψ(k) ∝ (1 + a2k2)−1/2. In such cases one uses other definitions of the width, for instance the distance between the maximum and the first zero, the half width, the equivalent width [249] or the overall width [250]. The dispersion law ω(k) is determined by the relation between the energy E and the momentum p of the object the wavepacket is to represent provided we make use of the Einstein-Planck relation ν = E/h or ω = E/h 68 and the de Broglie relation p = hk where k ≡ |k| = 2π/λ. Thus, the relativistic relation for a free particle E = Etot = ±(p 2c2 +m2c4)1/2 leads to (A3) ω(k) = ±c(k2 + κ2)1/2 where −λC = 1/κ = h/(mc) is the Compton length belonging to the rest-mass parameter m. The most general solution of a relativistic wave equation would be a function of the type (A1) in which ω(k) from (A3) has the positive sign plus a function (A1) in which ω(k) has the negative sign. That is, the most general form would be a superposition of waves with positive as well as those with negative frequencies viz. energies. Although we shall consider the general relativistic formula (A3) we will restrict ourselves to positive-energy wave functions in order to have a simple connection with the results of nonrelativistic quantum mechanics. Let us consider the time dependence of the wavepacket's "centre" 〈x〉 and "width" σ(t) = ([∆x(t)]2 + [∆y(t)]2 + [∆z(t)]2)1/2 where [∆x(t)]2 := ∫ (x− 〈x〉)2|ψ(x, t)|2d3x and analogously for the y and z components. A very general calculation of these quantities has been given by Bradford [251]. His treatment works in three dimensions and does not require a special form either for ψ(k) or for ω(k), except for the usual convergence requirements of the integrals that appear in the averaging procedures. In particular, the treatment is valid in the nonrelativistic as well as in the relativistic domain. The result for the centre is (A4) 〈x〉(t) = 〈x〉(0) + 〈~υg〉t with 〈x〉(0) = − ∫ Im { ψ∗∇kψ } d3k (A5) 〈~υg〉 = ∫ |ψ|2∇kω d3k 69 ψ = ψ(k). According to (A4) and (A5) the centre moves at constant velocity 〈~υg〉 which is the mean group velocity. The result for the width is (A6) σ2(t) = σ2(t0) + [∆υg] 2(t− t0)2 where σ2(t0) = σ 2(0)− [∆υg]2t20 (A.7) σ2(0) = − ∫ Re { ψ∗∇2kψ } d3k− [〈x〉(0)]2 t0 = (∫ Im { ψ∗∇kψ } ∇kω d3k + 〈x〉(0)〈~υg〉 ) [∆υg] −2 and [∆υg] 2 := ∫ |ψ|2(∇kω − 〈~υg〉)2 d3k. Formula (A6) shows that the width varies hyperbolically with time. This type of variation is even independent of the form of the dispersion law ω(k), except when ∆υg is zero: there is either a hyperbolic dependence or none. The minimum extension of the wavepacket occurs at t = t0; before t0 the wavepacket shrinks, after t0 it spreads out. When we write ψ(k) in the polar form ψ(k) = ρ(k) exp[iα(k)] we are led to (A8) 〈x〉(0) = −〈∇kα〉 (A9) t0 = (〈∇kα ∇kω〉 − 〈∇kα〉〈∇kω〉) [∆υg]−2. We thus can fix the time and the place of the minimum extension by an appropriate choice of the phase α(k), that is, by 〈∇kα〉 and 〈∇kα ∇kω〉. Resolving the two Eqs. (A8) and (A9) for these two quantities we obtain (A10) 〈∇kα〉 = 〈∇kω〉 t0 − 〈x〉(t0) 〈∇kα∇kω〉 = 〈[∇kω]2〉 t0 − 〈∇kω〉 〈x〉(t0). For example, the especially simple form with three parameters ~ξ0, τ0 and η0 α(k) = −k~ξ0 + ω(k)τ0 + η0 70 leads to 〈∇kα〉 = −~ξ0 + τ0〈∇kω〉 which, by comparison with (A10), shows that the parameters ~ξ0 and τ0 coincide with the initial values ~ξ0 = 〈x〉(t0), τ0 = t0, and η0 is an arbitrary constant. In what follows we will always assume 〈∇kα〉 = 〈∇kα∇kω〉 = 0, so that the minimum extension occurs at t = t0 = 0, and the centre of the packet at that time is 〈x〉(0) = 0. Formulas (A6) and (A7) then simplify to σ2(t) = σ2(0) + [∆υg] 2t2 and σ2(0) = − ∫ Re { ψ∗∇2kψ } d3k. Let us now consider the spreading velocity (A11) υs := ∂σ(t)/∂t = [∆υg] 2 t/σ(t) which for large t tends to the asymptotic spreading velocity υs∞ := lim t→∞ υs = ∆υg. The packet spreads out to double its initial (t = 0) extension in a time τ2 =√ 3σ(0)/∆υg. After this time the spreading velocity is υs = ( √ 3/2)∆υg = 0.87υs∞. That is, 87% of the asymptotic spreading velocity is already reached when the packet has doubled its initial extension. The asymptotic value may thus be used in all practical estimates. At the time t1 when the asymptotic spreading velocity has (practically) been reached, the time for the wavepacket to further double its extension is σ(t1)/∆υg which is somewhat smaller than τ2 provided we identify σ(t1) with σ(0) in this comparison. To proceed further we must make specific assumptions about ψ(k). We shall assume a nearly unidirectional and quasimonochromatic packet, that is, a narrow packet in k space; that is, ψ(k) is assumed to be appreciably different from zero only in a narrow region concentrated about the point k0 = (0, k0, 0) so that (A12) ∆kx,∆ky,∆kz |k0| ≡ k0 = 2π/λ0. With the help of the Fourier reciprocity (Heisenberg) relations for the considered wavepackets (A13) ∆x(0)∆kx = 1 2 r, etc. with r ≥ 1. 71 where r depends on the form of the wavepacket, and r=1 can only occur for a Gaussian form for ψ(k), the condition (A12) can be written as a condition in ordinary space (A14) ∆x(0),∆y(0),∆z(0) r/(2k0). It is then possible to express ∆υg as a function of ∆kx and ∆ky. In the case where k0 6= 0 it is reasonable to consider separately the longitudinal spreading, along the direction y of the centre, and the transverse spreading, normal to that direction, say in the x direction. We then expand ∇kω = c2k/ω(k) in a three-dimensional Taylor series about k0 and break the series off after the quadratic terms. After a straightforward but tedious calculation one arrives at υsx∞ := lim t→∞ ∂∆x(t)/∂t = ∆υgx := (∫ |ψ|2(∂ω/∂kx − 〈υgx〉)2d3k )1/2 (A15) = c2 ω0 ∆kx = c (k20 + κ 2)1/2 ∆kx (A16) = r c 2(k20 + κ 2)1/2∆x(0) , where (A13) has been used for obtaining (A16). Likewise we obtain ~υ0 ≡ 〈~υg〉 = (0, k0c2/ω0, 0) υ0 = k0c 2/ω0, ω0 = c(k 2 0 + κ 2)1/2, and with this we may write (A15) as (A17) υsx∞ = c κ ∆kx ( 1− (υ0/c)2 )1/2 (A18) ≥ c ( 1− (υ0/c)2 )1/2 2κ∆x(0) . In the same way we arrive at the longitudinal asymptotic spreading velocity (A19) υsy∞ = ( cκ ω0 )2 c2 ω0 ∆ky = c κ ∆ky ( 1− (υ0/c)2 )3/2 (A20) ≥ c ( 1− (υ0/c)2 )3/2 2κ∆y(0) and (A21) υsy∞/υ0 = ∆ky/k0 × ( 1 + (k0/κ) 2 )−1 = ∆ky/k0 × ( 1− (υ0/c)2 ) . 72 Figure 6: Relation between minimum extension and spreading angle. We thus have the interesting result that the spreading velocities of a narrow packet in k space do not depend on the detailed form of the wavepacket over and above its second central moments ∆kx,∆ky. In particular, formulas (A15) to (A21) hold in the nonrelativistic as well as in the relativistic domain. Fig. 6 pictures σ(t) according to (A6), and according to (A16) shows that the smaller the minimum extension ∆x(0), at the waist of the packet, the larger is the spreading angle α: (A22) tanα := υsx∞/υ0 = r c 2(k20 + κ 2)1/2∆x(0)υ0 , where r ≥ 1 (Heisenberg relation) depending on the form of the wavepacket (r = 1 for Gaussian form). In the nonrelativistic domain it is k20 κ2 and (A22) becomes (A23) tanα = r λdeBr 4π∆x(0) , λdeBr = h mv0 . For photons it is κ = m = 0, v0 = c, and (A22) becomes (A24) tanα = r λ0 4π∆x(0) , where λ0 is the centre wavelength of the photon packet. Relation (A24) coincides, except for some numerical factors ≈ 1, with Verdet's condition for the cone of coherence [252], [253], and also with the angular distance between the maximum and the first zero in diffraction at a slit of width 2∆x(0), though the above formulas were obtained without any use of holes, slits, or microscopes. From (A20) it might appear that υsx∞ → ∞ if ∆x(0) → 0. This is not true because by (A13) ∆x(0) → 0 would imply ∆kx → ∞, and our assumption of a 73 narrow wavepacket would no longer hold. In fact, closer inspection shows that the spreading velocity (A11) is always limited when ∆x(0)→ 0 [254]. For equal widths ∆kx = ∆ky the transverse spreading is always larger than the longitudinal one since by (A17) and (A19) it is then υsx∞ υsy∞ = 1 1− (υ0/c)2 ≥ 1. In the nonrelativistic domain we have k20 κ2, and condition (A12) now takes the form ∆kx,∆ky,∆kz κ. The integrals involved in the averaging procedures may then be restricted to regions where the dispersion law (A3) can be approximated by ω(k) = mc2/h + hk2/(2m) from which one obtains the nonrelativistic relation ∆~υg = h∆k/m = ∆p/m. With (A13) the last condition in ordinary space reads ∆x(0),∆y(0),∆z(0) 1/(2κ) = 12 −λC. In the relativistic domain the spreading becomes slower as the velocity υ0 of the packet (centre) approaches c. In the zero-mass limit, κ → 0, ω0 → ck0, formulas (A15) and (A21) lead to υsx∞ → c k0 ∆kx υsy∞ → 0. Thus for photons there is a finite transverse spreading but no longitudinal spreading, in accordance with the well known absence of spreading of one-dimensional electromagnetic pulses composed of unidirectional waves. Let us finally consider another length, the coherence length ∆cy of a wavepacket (in the y direction). We want to show that this length is equal to the coherence length of the beam in which the wavepackets take part and that ∆cy does not spread out in time. The coherence length of a wavepacket is defined by means of the autocorrelation function (A25) γ(b) = ∫ ψ∗(x, y, z, t)ψ(x, y + b, z, t) d3x. The function |γ(b)| is maximal (=1) at b = 0, and ∆cy is defined as that value of b where |γ(b)| has decayed for the first time to exp(−1/2). ∆cy is closely related to the "mean peak width" of Hilgevoord and Uffink [250], [255], [256]. 74 The coherence length Lc of a beam is defined by means of the "contrast" or "visibility" V (b) = (Imax − Imin)/(Imax + Imin) where Imax (Imin) is the maximal (minimal) intensity in the interference pattern obtained by dividing the beam into two sub-beams, delaying the one sub-beam by the distance b, and then reuniting the two. The length Lc is defined as that value of b where V (b) has decayed to exp(−1/2). It is a quantity that is easy to measure. In order to see that ∆cy equals Lc suppose that the beam is a stream of equal wavepackets and that the subdivision of the beam means the subdivision of each single wavepacket. The reunited beam then means reunited wavepackets and for each wavepacket we have ψf = ψ(x, y + b, z, t) + ψ(x, y, z, t). The probability that it will cause a count in the final counter, at any moment of time and at any place within the large counter (assuming 100% detection efficiency) is proportional to (A26) W = ∫ +∞ −∞ |ψf |2 d3x = ∫ |ψ(x, y, z, t)|2d3x+ ∫ |ψ(x, y + b, z, t)|2d3x + 2Re ∫ ψ∗(x, y, z, t)ψ(x, y + b, z, t) d3x. The function ψf need not be normalized because the original beam may be divided into more than the two sub-beams considered. The integrals in the second line of (A26) are equal and we denote them by A. The third line may then be written as 2AReγ(b) when we use (A25) and observe that ψ in (A25) is normalized but in (A26) perhaps not. Writing Reγ = |γ| cosα we have W (b) = 2A+ 2A|γ(b)| cos(α(b)). Now, the detection probability W (b) of the wavepacket is proportional to the registered time averaged intensity I(b) of the final beam, and when one assumes that cos(α(b)) in typical cases varies much more rapidly than does |γ(b)| one obtains V (b) = |γ(b)| and with this ∆cy = Lc. γ(b) and with it ∆cy is independent of time because γ(b) is just the mean value of the y-translation operator in the state ψ(x, y, z, t), and this operator commutes with the Hamilton operator for free packets [257]. Thus, whereas the length ∆y(t) of the packet may spread out with time, the coherence length ∆cy does not and is usually proportional to the minimum length ∆ymin = ∆y(0) (assumed to occur at t = 0). For a Gaussian packet, for example, one obtains ∆cy = 2∆ymin [258]. There is no strict relation between the coherence length ∆cy and the length ∆y(t) of the wavepacket. Any beam, with long or with short coherence length, may be considered to be one very long wavepacket, simply by superposing the shorter wavepackets that originally were conceived as its constituents. All that can be said 75 is that the coherence length is of the order of a lower bound for the length of the constituent wavepackets. The difference between ∆y(t) and ∆cy may always be small in the case of photon wavepackets which propagate in y direction, since photon packets do not spread out in longitudinal but only in transverse directions. In fact, it is often found that for photon packets the coherence length as measured by means of the interference pattern extends over almost the whole length of the packet as measured by means of the distance the light travels during the mean life of the decaying state. In these cases the packet length is always close to its lower bound. In the case of wavepackets describing massive particles, however, there is considerable spreading even in the direction of propagation. In the neutroninterference experiment of Kaiser et al. [259], for example, the length ∆y(t) of the neutron packet at the time of its registration may be larger than the measured coherence length ∆cy = 20 Å by more than a factor of 10 5. APPENDIX B: Proof of the Bell Inequality In this appendix we derive the inequality (5.5) (B1) K := |E(a, b) + E(a, b′) + E(a′, b)− E(a′, b′)| ≤ 2 of Sec. 5.3 following Bell [260]. E(a, b) is the expectation of the product rArB of two dichotomic variables, rA and rB, either of which can take on the values +1 and −1 only: (B2) E(a, b) := P (+,+|a, b) + P (−,−|a, b)− P (+,−|a, b)− P (−,+|a, b) and P (rA, rB|a, b) = ∫ P (rA, rB|a, b, λ) f(λ) dλ rA, rB ∈ {−1,+1} (B3), (58) P (rA, rB|a, b, λ) = P1(rA|a, λ) P2(rB|b, λ) (B4), (59) f(λ) ≥ 0, ∫ f(λ) dλ = 1. Since f(λ) does not depend on rA and rB it is possible to write (B2) as E(a, b) = ∫ dλ f(λ) ×{P1(+|a, λ) P2(+|b, λ) + P1(−|a, λ) P2(−|b, λ) − P1(+|a, λ) P2(−|b, λ)− P1(−|a, λ) P2(+|b, λ)}, and since the first factor on the right-hand side of condition (B3) does not depend on rB, nor the second on rA (outcome independence), we may factorize the integrand (B5) E(a, b) = ∫ dλ f(λ) {P1(+|a, λ)− P1(−|a, λ)} {P2(+|b, λ)− P2(−|b, λ)} 76 - = ∫ dλ f(λ) Ā(a, λ) B(b, λ) where Ā(a, λ) = P1(+|a, λ)− P1(−|a, λ) B(b, λ) = P2(+|b, λ)− P2(−|b, λ). The probability nature of P1 and P2 means that 0 ≤ P1 ≤ 1, 0 ≤ P2 ≤ 1, hence (B6) |Ā(a, λ)| ≤ 1, |B(b, λ)| ≤ 1. Using (B5) and the fact that Ā(a, λ) does not depend on b nor B(b, λ) on a (parameter independence) we have E(a, b)± E(a, b′) = ∫ dλ f(λ) Ā(a, λ) [B(b, λ)± B(b′, λ)], and using (B6) for Ā(a, λ), (B4) for f(λ) and the fact that f(λ) does not depend on a and b we obtain (B7) |E(a, b)± E(a, b′)| ≤ ∫ dλ f(λ) |B(b, λ)± B(b′, λ)|. Likewise (B8) |E(a′, b)∓ E(a′, b′)| ≤ ∫ dλ f(λ) |B(b, λ)∓ B(b′, λ)|. Now, we have (B9) |B(b, λ)± B(b′, λ)|+ |B(b, λ)∓ B(b′, λ)| = 2 max(|B(b, λ)|, |B(b′, λ)|). This can be seen by observing that |x± y|+ |x∓ y| = 2 max(|x|, |y|), which in turn may be obtained by considering separately the various possible cases of positive and negative x and y. For example, for x positive and y negative it is x = |x|, y = −|y| and |x+ y| = | |x| − |y| | = max(|x| − |y|, |y| − |x|), and |x− y| = | |x|+ |y| | = |x|+ |y|, so that |x+ y|+ |x− y| = max(|x| − |y|+ |x|+ |y|, |y| − |x|+ |x|+ |y|) = max(2|x|, 2|y|). Then (B6) means |x| ≤ 1, |y| ≤ 1 so that (B9) with (B6) can be written as |B(b, λ)± B(b′, λ)|+ |B(b, λ)∓ B(b′, λ)| ≤ 2. With the normalization (B4) in (B7) and (B8) we arrive at |E(a, b)± E(a, b′)|+ |E(a′, b)∓ E(a′, b′)| ≤ 2 77 and this includes the desired Bell inequality (B1), (5.5). APPENDIX C: EPR Joint Probability Formulas Here we derive formula (5.4) of Sec. 5.2 (5.4) P (rA, rB|a, b) = 14(1− rArB cosθ) from the rules of quantum mechanics. We have two spin-12 particles (similar or not) in a state of zero total spin (spin singlet state). The particles move in opposite directions and each enters a Stern-Gerlach-type apparatus where it is deflected upwards or downwards with respect to the axis of its respective apparatus (cf. Fig. 3 in Sec. 5.1). Particle 1 enters apparatus A, which has its axis in the direction of the unit vector a, and particle 2 enters apparatus B with axis b. We first calculate the joint probability P (+,−|a, b) that A obtains an up deflection and B a down deflection. The wave function of the two-particle system in the spin singlet state is (5.3) (C1) Ψ(1, 2) = 1√ 2 [ |+〉(1)|−〉(2) − |−〉(1)|+〉(2) ] A(1, 2), where the up |+〉 and down |−〉 eigenfunctions of the spin component refer to a fixed but arbitrary axis. This is an entangled wave function. Only the spin part needs to be considered here. It happens to be antisymmetric, independent of whether it refers to similar or dissimilar particles. Let the two eigenfunctions of the one-particle spin component operator of apparatus A be |a+〉 and |a−〉, and those of apparatus B |b+〉 and |b−〉. The operation of a Stern-Gerlach-type apparatus on either of the two particles leads to either |a+〉(1)|b−〉(2) or |b−〉(1)|a+〉(2) with equal probability. Hence (C2) P (+,−|a, b) = 12 ∣∣∣(|a+〉(1)|b−〉(2),Ψ(1, 2))∣∣∣2 + 12 ∣∣∣(|b−〉(1)|a+〉(2),Ψ(1, 2))∣∣∣2, whether the particles are similar or not. In the case of dissimilar particles the factors 1/2 reflect the fact that we are not interested in distinguishing the particles but only in the average result. In the case of similar particles the probability expressions should be invariant under particle-label permutation (Sec. 3.2), and this is now the reason for the factors 1/2. To be able to evaluate expression (C2) we need the eigenfunctions |a+〉, |b−〉 etc. in terms of the eigenfunctions |+〉, |−〉, which refer to the fixed but arbitrary axis employed in (C1). We let this axis coincide with A's axis a, so that (C3) |a+〉 = |+〉, |a−〉 = |−〉. 78 Let the axis of B form an angle θ with the axis of A We then have to express the eigenfunctions |b+〉, |b−〉 in the rotated system B in terms of the eigenfunctions of system A [261], [46, p. 1073] |b+〉 = cos θ 2 exp[i(β + γ)/2] |+〉 + i sin θ 2 exp[−i(β − γ)/2] |−〉 (C4) |b−〉 = i sin θ 2 exp[i(β − γ)/2] |+〉 + cos θ 2 exp[−i(β + γ)/2] |−〉. The angles β and γ define possible rotations of the other axes but will not appear in the final formulas. Inserting (C1), (C3) and (C4) into (C2) and observing the orthonormality of the functions |+〉, |−〉 for the respective particles we obtain (C5) P (+,−|a, b) = 1 2 ( cos θ 2 )2 = 14(1 + cosθ). Proceeding in the same way in the other cases (A up, B up; A down, B up; A down, B down) we obtain (C6) P (−,+|a, b) = P (+,−|a, b), (C7) P (+,+|a, b) = P (−,−|a, b) = 1 2 ( sin θ 2 )2 = 14(1− cosθ). (C5), (C6) and (C7) may be summarized in the form (C8) P (rA, rB|a, b) = 14(1− rArB cosθ) where rA, rB ∈ {−1,+1}. This is formula (5.4) of Sec. 5.2. It is amusing to notice that we can also obtain formula (5.4) by proceeding as if the following situation were to hold: after any single interaction either of the two protons of Fig. 3 has a definite direction of spin (i.e. is a spin-up eigenfunction of some sz′ , cf. Sec. 4.3), say ~σ and −~σ, respectively, where ~σ is a unit vector, and the total spin is zero. The direction of ~σ varies from one interaction to the other in such a way that there is spherical symmetry on the average. The Stern-Gerlach-type apparatus which obtains its proton first, say A, turns the spin of its proton into either up or down direction with respect to its axis, say into direction +a, and at the same time turns the spin of the other proton into the opposite direction −a. Thus here angular momentum is conserved within the system of the two protons, and the apparatuses are not involved in angular-momentum conservation. Then B's apparatus turns the spin of its proton from direction −a into either up or down direction with respect to B's axis b, without, however, influencing the spin of proton 1 any more. Here, angular-momentum conservation involves proton 2 and apparatus B, as mentioned in Sec. 4.3 on the Stern-Gerlach experiment. The spin direction ~σ plays the role of the parameter λ in the general consideration of Sec. 5.3. From the point of view of quantum mechanics the parameter ~σ, 79 interpreted in the above way, is hidden. Of course, in the one-particle states |+〉 etc. ~σ is not hidden but explicitly specifies the spin direction, but the two-particle state Ψ(1, 2) of (C1), which is the only state existing after the interaction at O, has spherical symmetry, and there can be no parameter specifying any direction in such a state in quantum mechanics. This is why we used the proviso as if. In order to prove our above assertion regarding the as-if derivation of (5.4) we observe that the conditional probability of obtaining the result rA (i.e. either + or −) with respect to the axis a, given that the spin of the proton before it enters the apparatus points in the direction b, is, by (C3) and (C4), P (rA|a,b) = |〈arA|b+〉|2 = |〈rA|b+〉|2 = 12(1 + rAba), and if we replace the axis b by the axis ~σ and the angle θ between a and b by the angle α between a and ~σ we get the probability that A obtains the result rA given that the spin of the proton before it entered the apparatus pointed in the direction ~σ (C9) P (rA|a, ~σ) = 12(1 + rÃσa) = 1 2(1 + rA cosα). However, the probability of B obtaining the result rB is not the analogous formula (1 + rB(−~σ)b)/2, because A has turned not only the spin of proton 1 into the direction rAa but also the spin of proton 2 into direction −rAa and then separated the two protons. Thus here we have to replace −~σ by −rAa, and B's probability is (C10) P (rB|b,−rAa) = 12(1 + rB(−rAa)b). The conditional joint probability of A obtaining rA and B obtaining rB is given by the product of (C9) with (C10) P (rA, rB|a,b, ~σ) = 12(1 + rÃσa) 1 2(1− rArBab). If we integrate over all directions of ~σ, assuming an isotropic distribution, we obtain P (rA, rB|a,b) = 1 4 π ∫ +π −π dφ ∫ π 0 sinαdα 1 4 (1 + rA cosα) (1− rArB cosθ) (C11) = 1 8 (1− rArB cosθ) ∫ π 0 sinα (1 + rA cosα) dα where the system of coordinates (x, y, z) for the integration is chosen such that the z axis is in the direction a, and b lies in the x-z plane: a = (0, 0, 1), b = (sinθ, 0, cosθ), so that ~σ = (sinα cosφ, sinα sinφ, cosα), ab = cosθ, ~σa = cosα, ~σb = sinθ cosφ sinα+ cosθ cosα. 80 It is not difficult to verify that (C11) leads to formula (C8) or (5.4). Such an as-if procedure is in fact possible in any EPR situation, not only in that of two spin-12 particles in the singlet state. APPENDIX D: EPR Probabilities in Different Systems of Eigenfunctions We shall show here that no faster-than-light warning system can be built with devices that obey the formulas of quantum mechanics [262] - [266]. The proof uses the fact that in quantum mechanics the apparatuses are represented by operators. Different apparatuses mean different operators, and different operators in general mean different systems of eigenfunctions, and these can be transformed into one another. The normalized quantum-mechanical wave function for a system of two similar particles can be written in the form (D1) ΨSA(1, 2) = C ∞∑ k=1 [ ζk(1) uk(2) ± ζk(2) uk(1) ] , which is formula (5.1) from Sec. 5.1 with a properly symmetrized function. The uk(x) form a complete set of orthonormal eigenfunctions of some operator representing the apparatus of experimenter B. C is a real overall normalization constant, which need not be equal to 1/ √ 2 because the functions ζk(x), which describe the particle at A, are not presupposed to be orthonormal. With the expansion (D2) ζk(x) = ∑ l alk wl(x), where the wl(x) form a complete set of orthonormal eigenfunctions of some operator representing the apparatus of experimenter A, (D1) can be written as (D3) ΨSA(1, 2) = C ∑ lk alk [wl(1)uk(2)± wl(2)uk(1)] . We first calculate the probability P1(un, wm) of a transition where the state (D3) changes into either the state wm(1)un(2) or the state wm(2)un(1) with equal probability. This is the probability that in B's apparatus there will be a particle ("whichever of the two it is") with state un and in A's apparatus a particle with state wm. It is [see the remark on (C2) in Appendix C] P1(un, wm) = 1 2 ∣∣∣∣(C∑ kl alk (wl(1)uk(2)± wl(2)uk(1)) , wm(1)un(2) )∣∣∣∣2 + 1 2 ∣∣∣∣(C∑ kl alk (wl(1)uk(2)± wl(2)uk(1)) , wm(2)un(1) )∣∣∣∣2 = C2 2 ∣∣∣∣∑ kl a∗lk [( wl(1)uk(2), wm(1)un(2) ) ± ( wl(2)uk(1), wm(1)un(2) )]∣∣∣∣2 81 - + C2 2 ∣∣∣∣∑ kl a∗lk [( wl(2)uk(1), wm(2)un(1) ) ± ( wl(1)uk(2), wm(2)un(1) )]∣∣∣∣2. With ( wl(1)uk(2), wm(1)un(2) ) = ( wl(2)uk(1), wm(2)un(1) ) = δlmδkn this becomes (D4) P1(un, wm) = C2 2 ∣∣∣∣a∗mn ±∑ kl a∗lk ( wl(2), un(2) )( uk(1), wm(1) )∣∣∣∣2 + C2 2 ∣∣∣∣a∗mn ±∑ kl a∗lk ( wl(1), un(1) )( uk(2), wm(2) )∣∣∣∣2. As the scalar products in (D4) are zero the formula reduces to P1(un, wm) = C 2|amn|2. The scalar products are zero because the final wave function ul of the particle in apparatus B and the final wave function wl of the particle in apparatus A are well separated from each other and do not overlap. The interaction of a wavepacket from the entangled system (D1) with that apparatus that operated first, had led to reduction and to disentanglement of the system. Second we consider the probability of B observing that his particle assumes the state un irrespective of the state of A's particle. This is obtained by summing the probability P1(un, wm) over all states of A's particle (D5) P2(un) = C 2 ∑ m |amn|2, and we want to show that this probability is unchanged when A uses a different apparatus. Let the eigenfunctions of the new operator corresponding to the new apparatus be w′m(x). They are related to the eigenfunctions wm(x) of the original operator by (D6) wm(x) = ∑ k Umkw ′ k(x). Umk is a unitary matrix ( ∑ m U ∗ jmUkm = δjk), and the index k may even be continuous and the sum an integral. Actually, the transformation (D6) need not even be unitary [266], [267], but we will not pursue this here. By inserting (D6) into (D3) we can write ΨSA(1, 2) in the form ΨSA(1, 2) = C ∑ mn amn [ un(2) ∑ k Umkw ′ k(1)± un(1) ∑ k Umkw ′ k(2) ] = C ∑ lk ∑ j ajkUjl} {{ } blk [ w′l(1)uk(2)± w′l(2)uk(1) ] . 82 The probability that this changes into either w′m(1)un(2) or w ′ m(2)un(1) is C 2|bmn|2, by analogy with (D3) and (D5). Hence P2(un) = C 2 ∑ m |bmn|2 = C2 ∑ m b∗mnbmn = C 2 ∑ mjk a∗jnU ∗ jmaknUkm = C2 ∑ jk a∗jnakn ∑ m U∗jmUkm = C 2 ∑ k |akn|2, and this coincides with (D5), concluding the proof of our assertion. Finally we want to show that even if A chooses not to use his apparatus and to do nothing this will make no difference. In this case, when B's particle assumes the state un, A's particle will assume some correlated state ζn. The probability of B's particle assuming the state un is the probability of the transition where the state ΨSA(1, 2) changes either into the state ζn(1)un(2) or into the state ζn(2)un(1). When ζn is expressed in terms of the wm, according to formula (D2), the two states become ∑ m amnwm(1)un(2) and ∑ m amnwm(2)un(1), respectively, and the transition probability becomes (D7) P ′2(un) = 1 2 ∣∣∣∣(ΨSA(1, 2),∑ m amnwm(1)un(2) )∣∣∣∣2 × ∣∣∣∣∑ m amnwm(1)un(2) ∣∣∣∣−2 + 12 ∣∣∣∣(ΨSA(1, 2),∑ m amnwm(2)un(1) )∣∣∣∣2 × ∣∣∣∣∑ m amnwm(2)un(1) ∣∣∣∣−2. The denominators are different from 1 because the amn come in via the ζn(x) in formula (D2), and the ζ's are not normalized. With (D3) the first term of expression (D7) becomes T1 = ∣∣∣∣∑ klm a∗lkamn [( wl(1), wm(1) )( uk(2), un(2) ) ± ( wl(2), un(2) )( uk(1), wm(1) )]∣∣∣∣2 × ∣∣∣∣∑ mk a∗mnakn ( wm(1)un(2), wk(1)un(2) )∣∣∣∣−1 × C22 = C2 2 ∣∣∣∣∑ m a∗mnamn ∣∣∣∣2 × ∣∣∣∣∑ m a∗mnamn ∣∣∣∣−1 = C22 ∑m |amn|2. The second term of (D7) leads to the same expression, so P ′2(un) = C 2 ∑ m |amn|2 = P2(un), which is what we wanted to show. 83 Notes and References [1] Jammer, M.: The Conceptual Development of Quantum Mechanics (McGrawHill, New York, 1966) Chap. 7. [2] Remarkably, Bohr is also reported to have said: "There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature." Petersen, A.: The philosophy of Niels Bohr, Bulletin of the Atomic Scientists 19 (7), 8-14 (1963), the quotation is from p. 12. [3] Heisenberg, W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43, 172 198 (1927). "Die 'Bahn' entsteht erst dadurch, dass wir sie beobachten", this quotation is from p. 185. My translation. Another English translation in [4, p. 73]. [4] Wheeler, J.A. and Zurek, W.H. (eds.): Quantum Theory and Measurement (Princeton University Press, Princeton, 1983). [5] Maxwell, J.C.: A Treatise on Electricity and Magnetism (Clarendon Press, Oxford, 1891), Vol 2, Sec. 528. Republished by Dover, New York, 1954, p. 176. [6] Epstein, H. and Glaser, V.: The role of locality in perturbation theory, Ann. Inst. Poincaré A 19, 211 295 (1973). [7] Scharf, G.: Finite Quantum Electrodynamics, 2nd ed. (Springer-Verlag, Berlin, 1995). [8] Jauch, J.M. and Rohrlich, F.: The Theory of Photons and Electrons, 2nd ed. (Springer-Verlag, New York, 1976), Sec. 16-5. [9] Bjorken, J.D. and Drell, S.D.: Relativistic Quantum Fields (McGraw-Hill, New York, 1965). [10] Dirac, P.A.M.: Relativity and quantum mechanics, Fields and Quanta 3, 139 164 (1972). The quotation is from p. 163, 164. [11] Dirac, P.A.M.: Does renormalization make sense? in Duke, D.W. and Owens, J.F. (eds.): Perturbative Quantum Chromodynamics, (American Institute of Physics, NY, 1981), p. 129 130. 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