The Time Flow Manifesto Chapter 2. Time Symmetry in Physics. Andrew Holster © September 2014 Contents Chapter 2. Time Symmetry in Physics.......................................................................3 Probabilistic Laws ..................................................................................................5 The Physicists Reversal Fails .................................................................................9 The Correct Principle for Time Reversal ...............................................................9 The Fallacy in the Physicists' Principle. ..............................................................11 Quantum Mechanics is Time Asymmetric...........................................................12 The Error in Quantum Mechanics Textbooks. .....................................................14 Conclusion. Fallacies 1* 4*. ..............................................................................15 References ................................................................................................................16 Footnotes. .............................................................................................................17 Principles of physical time directionality and fallacies of the conventional view 3 Chapter 2. Time Symmetry in Physics. I start with a simple conventional presentation of the view of time reversal in physics, and then return to analyse it. There are two common ways of explaining time symmetry in physics. The first is to make a concrete visualization: imagine first a (normal) physical process that obeys the laws of physics. Then imagine the same process running in reversed temporal sequence – what we would see if we ran a film of it backwards. This is the time reversed process. The laws of physics are time symmetric just in case any time-reversed process also obeys the same laws of physics as the normal process. If this is true for the general laws of physics, then the laws do not support an 'intrinsic' (or law-like) directionality of time, or a 'preferred direction' for physical processes. The processes we see in real life of course do not appear to be reversible – we cannot make a river run uphill, or make a broken egg fall upwards from the floor and reassemble on the bench. These reversed processes do not appear to be physically possible. But this, we are told by the physicists, is an illusion. It is merely the result of the peculiar 'low entropy' state in which our universe began – not a matter of any intrinsic asymmetry in the fundamental laws of physics themselves. And this, we are told, is one of the most profound results in the history of science. It shows there is no scientific foundation for what we intuitively believe, viz. that there is an 'intrinsic flow of time', from past to future. The whole process of the universe could have happened in time-reversed order, as far as the laws of nature are concerned. And then we would all identify the opposite directions of time as 'past' and 'future'. This conclusion is the starting point for most modern writing on the naturalistic philosophy of time for the last 50 years or so. But how do the physicists prove this result? Well of course we can't examine every possible process individually and check if it is 'reversible'. There are infinitely many possible processes. Instead, we check the general laws of physics for the property of time symmetry. These laws tell us what processes are possible at a fundamental level (according to present physics). These laws are written as equations ('fundamental The following are rejected in this chapter as fallacies. 1. Conventional Fallacies of time symmetry a d QM reversibility. 1* False Analytic Principle 1. The time reversal of a deterministic causal law like: s1(t)  s2 (t+t) is a law like: Ts2(t)  Ts1(t+t). 2* False Analytic Principle 2. The time reversal of a probabilistic law like: prob(s2(t+t)| s1(t)) = p is a law like: prob(Ts1(t+t)| Ts2(t)) = p 3* False Analytic Principle 3. The condition for time symmetry of a probabilistic theory is that: prob(s2(t+t)|s1(t)) = prob(Ts1(t+t)|Ts2(t)) for all state transition laws. 4* False Analytic Claim About Physics. Quantum mechanics is time symmetric (reversible or symmetric under time reversal transformation.) I repeat what time symmetry means.  Time symmetry means invariance under the time reversal transformation, a symmetry transforma io based on the mapping: T: t  -t.  A symmetry transformation is based on a 1-1 mapping of a fundamental variable (like time, space, charge, etc) back onto itself. This must logically induce transformations on all other complex constructions involving this quantity. E.g. the mapping t  -t determines that dr/dt  dr/d(-t) (velocity reversal follows from time reversal).  Any kind of well-defined object or logical construction (e.g. variables, states, processes, laws, worlds) for which the time reversal transformation is defined may have the property of time symmetry, meaning that the object or c nstruction is identical to its time-revers d image.  The laws of physics are time symmetric (reversible) just in case they are identical to their image under the time reversal transformation. THE TIME FLOW MANIFESTO 4 equations'), and by doing some formal transformations on the equations, we can check whether they are time symmetric. This gives the second common method for explaining the meaning of time symmetry in physics. The time reversal transformation, we are told, is simple and straightforward. It simply consists in replacing the time variable, t, with its negative image, -t, throughout the equations of physics. Oh, and replacing any state description, s, with its time-reversed image, T(S). If the laws are time symmetric, then the time reversal TL of any law L is also a law of physics. This seems easy enough to understand with examples. The simplest example of a process is a particle travelling in a straight line at a constant velocity: time, t P t1+Dt t1 r1 r2 space, r t = 0 -t1 -t1 -Dt TP Figure 1. Space-time diagram illustrating a simple process (P) and its time reversal (TP). TP is the reflection of P through t = 0. In P, a particle moves from r1 to r2 in a period t. In TP, the particle moves from r2 back to r1 in a period t. But in TP, the velocity is reversed, because it is moving 'backwards'. Both of Principles of physical time directionality and fallacies of the conventional view 5 these are possible processes for an isolated particle according to most theories of physics. The intuitive line of thought goes like this. We take this first of all to be a deterministic process. For the process P to be physically possible (as in classical physics), there must be a law like: LD s1(t)  s2 (t+t) meaning that an (isolated) system in a state s1 at time t will develop, according to the laws of physics, into a later state s2 at time t+t. Note that laws are assumed to be time translation invariant where we choose to assign the coordinate value: t = 0 is merely conventional so this law applies to any time t. Logicians would say that the general laws have an implicit universal quantifier on t, meaning that "for all moments t, ...". We are interested in whether the reversed process TP is possible given that P is possible. Since P starts in state s1 and ends in state s2, the reversed process must start with Ts2 and end with Ts1. Given the law LD that governs the process P, it seems that we then need a time reversed law like the following to allow the reversed process: TLD* Ts2(t)  Ts1(t+t) I.e. an (isolated) system in a state Ts2 at time t will develop, according to the laws of physics, into a later state Ts1 at time t+t. This is assumed to be the time reversal of the law LD in the conventional analysis. I have labelled it with an asterix, TLD*, however, because actually it is not the time reversal of LD at all! I will let the reader puzzle over this for a few moments, and see if they work out what the real time reversal of LD is – it is obvious enough when you see it, but the conventional presentation, as above, conceals the correct answer under false intuition. Before revealing the answer I consider probabilistic laws. Probabilistic Laws THE TIME FLOW MANIFESTO 6 The serious problems arise when we move on to probabilistic laws. Quantum mechanics is widely believed to be the fundamental theory of particle physics, and to require irreducibly probabilistic laws, and these laws are claimed to be time symmetric. Physicists take the time reversal of a probabilistic transition law of the following form: L Prob( s1(t)  s2 (t+t) ) = p (The probability of a transition from s1 to s2 after a period t equals p, with p a real number from 0 to 1) to be a corresponding law of the form: TL* Prob(Ts2 (t)  Ts1(t+t)) = p (The probability of a transition from Ts2 to Ts1 after a period of t equals p) Again I have labelled TL* with an asterix because it is not really the time reversal of L. The proof that quantum mechanics is time symmetric in its probabilistic laws then amounts to the claim that the following symmetry principle holds: [QM cause-effect exchange symmetry] Prob( s1(t)  s2 (t+t) ) = Prob(Ts2 (t)  Ts1(t+t)) for all quantum state transitions – since this assures us that for every law L of the theory there is a corresponding law TL*. Note also that if we take the transition probability to be p=1, then this reduces to the deterministic case above. However although this principle is generally true in quantum mechanics (with the exception of some meson decay processes), we will see that it does not represent time symmetry at all. This is why I have called it QM cause-effect exchange symmetry, instead of QM time reversal symmetry as stated in all the textbooks. To illustrate let us consider another very simple example, of a clearly time symmetric probabilistic process. Imagine a system with just three possible states, call them s0, s1, s2, which 'jumps' from state to state after every interval of time, t, like this: Principles of physical time directionality and fallacies of the conventional view 7 s1 s1 s1 s0 s0 s0 s0 s2 s2 s2 t t+Dt t+2Dt t+3Dt t+4Dt t+5Dt t+6Dt Figure 2. A simple probabilistic process. From state s0 the system jumps randomly to either s1 or s2 , i.e. with probability 0.5 in each case. From state s1 or s2 the system always jumps back to s0, i.e. with probability 1 in each case. The underlying probabilities are indicated in black, a series of actual events (actualised probabilities) is indicated in blue: ...0201010... There are four simple laws for the dynamics of this system: L01 prob(s1(t+t)|s0(t)) = 0.5 L02 prob(s2(t+t)|s0(t)) = 0.5 L10 prob(s0(t+t)|s1(t)) = 1 L20 prob(s0(t+t)|s2(t)) = 1 To ensure the theory of this process as a whole is time symmetric, we also ensure that there is no start or end to the process, with an extra law that: L+ prob(s0(t) or s1(t) or s2(t)) = 1, for all times, t. L01 means that the probability of the state s1 at time t+t given the state s0 at time t equals 0.5, and so on. L+ entails that system at any time always has an earlier and a later state. We could imagine this as an infinite coin-tossing process, where s0 is the randomised state before each toss, s1 is the outcome state heads, s2 is the outcome state tails, and after each toss the coin is returned to its randomised state. (In quantum physics, this could be modelled as a series of spin-1/2 experiments, with 'up' and 'down' as outcomes, and the system returned to the superposition after each event.) THE TIME FLOW MANIFESTO 8 To keep the example simple, we define the states to be their own time reversals, i.e. Ts0 = s0, Ts1 = s1, Ts2 = s2. Hence when we play a sequence of states backwards, we see a sequence of the same kind of states again. E.g. suppose a process has a subsequence: P ...020101010201020201020201020201010102010202010... Then the time reversed process has the sub-sequence: TP ...010202010201010102020102020102020102010101020... Now it seems patently obvious that this process is time symmetric, and that the set of laws L01, L02, L10, L20, that govern it forms a time symmetric set of laws. It is impossible to tell a sequence and its time reversal apart statistically. Of course a directional pattern could occur, e.g.: ...010101010101020202020202... But any directional pattern in an actual sequence is merely the result of coincidence, with the same probability of the reversed directional pattern occurring by coincidence, and this still doesn't help us determine any direction of time for the process from the stochastic laws. Principles of physical time directionality and fallacies of the conventional view 9 The Physicists Reversal Fails Let us now examine this set of laws using the physicist's criterion for finding the time reversal of probabilistic laws. According to that, the reversals of the laws for this system are: Original Theory Physicists' Time Reversal L01 prob(s1(t+t)|s0(t)) = 0.5 TL01* prob(s0(t+t)|s1(t)) = 0.5 L02 prob(s2(t+t)|s0(t)) = 0.5 TL02* prob(s0(t+t)|s2(t)) = 0.5 L10 prob(s0(t+t)|s1(t)) = 1 TL10* prob(s1(t+t)|s0(t)) = 1 L20 prob(s0(t+t)|s2(t)) = 1 TL20* prob(s2(t+t)|s0(t)) = 1 But there is something wrong here the physicists' time reversal of the theory contradicts the original theory! E.g. in the original theory, prob(s1(t+t)|s0(t)) = 0.5, but in the physicists' time reversal, prob(s1(t+ t)|s0(t)) = 1. In fact the physicists' time reversal of the theory gives a self-contradictory theory, stating that both: prob(s1(t+t)|s0(t)) = 1 and prob(s2(t+t)|s0(t)) = 1. This requires that the state s0(t) develops deterministically to the state s1(t+ t) and to the state s2(t+t). So this analysis using the physicist's principles would tell us that the theory is not time symmetric! But we know intuitively that the theory is perfectly time symmetric. The time reversal of the theory, if derived correctly, must be identical to the original theory. There is a fallacy in the physicists' derivation of time reversal. The Correct Principle for Time Reversal I now state the correct principle for deriving time reversal. First, for our original example of a deterministic law like: LD s1(t)  s2 (t+t) The time reversal is actually: TLD Ts1(t)  Ts2 (t-t) THE TIME FLOW MANIFESTO 10 This means that the state Ts1 at t determines the earlier state, Ts2 at t-t. That is to say, the future-directed deterministic law, LD, becomes a past-directed deterministic law, TLD, when the law LD is reversed. More generally, the time reversal of a probabilistic law like: L prob(s2(t+t)| s1(t)) = p Is actually: TL prob(Ts2(t-t)| Ts1(t)) = p Again this is a past directed law. The requirement for time symmetry of a probabilistic theory, T, is then that: [T is time symmetric] T entails that: prob(s2(t+t)| s1(t)) = prob(Ts2(t-t)| Ts1(t)), for all state transition laws of the theory. Applying this to our example: Original Theory True Time Reversal L01 prob(s1(t+t)|s0(t)) = 0.5 TL01 prob(s1(t-t)|s0(t)) = 0.5 L02 prob(s2(t+t)|s0(t)) = 0.5 TL02 prob(s2(t-t)|s0(t)) = 0.5 L10 prob(s0(t+t)|s1(t)) = 1 TL10 prob(s0(t-t)|s1(t)) = 1 L20 prob(s0(t+t)|s2(t)) = 1 TL20 prob(s0(t-t)|s2(t)) = 1 L+ prob(s0(t) or s1(t) or s2(t)) = 1, for all times, t, is identical in both. And the time reversed theory indeed turns out to be exactly the same as the original theory. E.g. the original theory requires that the state s1 is always followed by s0 – and it equally entails that the state s1 is always preceded by s0. Similarly, the original theory requires that the state s0 is followed by s1 with 0.5 chance – and it equally entails that the state s0 is preceded by s1 with 0.5 chance. Without this symmetry between future-directed transition statistics and past-directed transition statistics the theory clearly could not be time symmetric, and this example matches all our intuitions. Principles of physical time directionality and fallacies of the conventional view 11 The simplest way to assure yourself that TL is the time reversed image of L is simply to follow the 'formal recipe' recommended by physicists, and substitute all time variables for their negatives in L (including substitution of Ts for each state s). This first gives us: prob(Ts2(-t-t)| Ts1(-t)) = p. Because t is universally quantified but -t is a specific constant, this is logically equivalent to: prob(Ts2(t- t)| Ts1(t)) = p, which is TL as stated. Not that hard! The Fallacy in the Physicists' Principle. How did the physicists make this error? I think by using unanalysed intuition to formulate their 'reversal' principle, and then failing to check it. To obtain the physicists' TL* we have to perform the substitution of –t for t, and then also exchange the causal order of states. This does not give the time reversed image of L at all – it sneaks in a 'double reversal', to satisfy our normal intuition that causal laws must go forward in time. In fact, this does not represent a symmetry transformation at all. A symmetry transformation is based on a 1-1 mapping of a fundamental variable (like time, space, charge) back onto itself. This must logically induce transformations on all other complex constructions involving this quantity. But TL* does not have any possible underlying transformation! A full proof of this is given in Holster 2003, where it is proved that the conventional criterion is neither a necessary nor sufficient condition for time symmetry. The physicists' time reversal principle is actually logically irrelevant to time symmetry! What physicist have called time reversal is best called cause-and-effect-reversal, or causal exchange for short, because it involves exchanging the order of cause and effect, along with the time reversal of states. This is already seen in the deterministic case. The law LD states that s1 at t will cause s2 at t+t. The physicists' reversal of this, TL*, states that Ts2 at t will cause Ts1 at t+ t. It may seem intuitive that this is time reversal, but that is a fallacy of intuition: it does not represent the time reversal transformation, as induced by the mapping: t  -t, and it does not have any of the implications of time reversal that are critical to the philosopher's interpretation of what this means. Equally, what is called time symmetry (or reversibility) of quantum THE TIME FLOW MANIFESTO 12 mechanics in textbooks should be called 'causal exchange symmetry of quantum mechanics'. I note that there is another problem with time reversal in both quantum theory and even classical electromagnetic theory, viz. the choice of the time reversal operator on states, i.e. the transformation: s  Ts. The literature on this reveals great confusion. Quantum Mechanics is Time Asymmetric. The famous result that quantum mechanics is time symmetric is based on the fallacious principle we have just seen, and it is completely wrong. It is wrong in its method: it uses the wrong principle to analyse time symmetry, identifying TL* instead of TL as the reversal of L. And it is wrong in its conclusion: when the analysis is done correctly, it is clear that quantum mechanics is time asymmetric (irreversible). The probabilistic laws of quantum mechanics simply do not hold of time-reversed quantum processes. This can be seen from a simple theorem to the effect that: Theorem of QM Equilibrium. Time symmetry and cause-effect exchange symmetry jointly entail thermodynamic equilibrium, where absolute probabilities of all micro-states are equally likely. This of course contradicts the observation of disequilibrium in our universe: Observation of Disequilibrium. The real universe is in a state of disequilibrium. A simple derivation of the previous theorem follows. Derivation of the Theorem of QM Equilibrium. The easiest way to demonstrate this is by combining the quantum principle of causal exchange: prob(s2(t+t)| s1(t)) = prob(Ts1(t+t)| Ts2(t)) With the requirement for true time symmetry: prob(s2(t+t)| s1(t)) = prob(Ts2(t-t)| Ts1(t)) If these both held generally, then equating the right hand sides: Principles of physical time directionality and fallacies of the conventional view 13 prob(Ts1(t+t)| Ts2(t))= prob(Ts2(t-t)| Ts1(t)) By substitution of Ts1 and Ts2 for s1 and s2 and using the identities: TTs1 = s1 and TTs2 = s2 and the general quantification of t, we then obtain: prob(s1(t+t)| s2(t))= prob(s2(t-t)| s1(t)) = prob(s2(t)| s1(t+t)) But this can only hold if the absolute probabilities for the two states, s2(t) and s1(t+t) are equal. This is seen by expanding into conditional probabilities: prob(s1(t+t)| s2(t))= prob(s1(t+t))/ prob(s1(t+t) and s2(t)) = prob(s2(t)| s1(t+t))= prob(s2(t))/ prob(s2(t) and s1(t+t)) Hence equating the right hand sides: prob(s2(t)) = prob(s1(t+t)) (absolute probability law). And since the laws are universalised w.r.t. time, this requires that: prob(s2(t)) = prob(s1(t)) This states that the absolute probabilities of any two micro-states, s1(t) and s2(t), are equal. But this is a condition for thermodynamic equilibrium. It is absolutely not a condition that is met by the real universe. See Holster (2003) for more detailed proofs. In summary:  Time symmetry and cause-effect exchange symmetry can both hold only in a thermodynamic equilibrium. Our universe is not in equilibrium. Hence at least one symmetry must fail. Since cause-effect exchange symmetry holds in quantum mechanics, time symmetry must fail in quantum mechanics. This shows that it is quite impossible for quantum theory to be time symmetric. As a result, quantum mechanics implies an intrinsic time direction. This is the direction of actualisation of quantum probabilities. In Holster, 1990 [PhD Thesis], I adapted McCall 1976 [9], in interpreting this as the direction of time flow. THE TIME FLOW MANIFESTO 14 The Error in Quantum Mechanics Textbooks. This shows that the claims 1* to 4* are fallacies. This fallacy is perpetuated in philosophical accounts in a deeply misleading way, but also advanced in textbooks on quantum mechanics, in a relatively more harmless way, but needing correction. E.g. "A system is said to exhibit symmetry under time reversal if, at least in principle, its time development may be reversed and all physical processes run backwards, with initial and final states interchanged. Symmetry between the two directions of motion in time implies that to every state  there corresponds a time-reversed state  and that the transformation  preserves the values of all probabilities, thus leaving invariant the absolute value of any scalar product between the two states." Merzbacher, 1970, p.406-407. [10]. To correct the fallacy, this might be modified to read (with alterations underlined): "A system is said to exhibit symmetry under causal exchange if, at least in principle, its time development may be reversed and all physical processes run backwards, with initial and final states interchanged. This symmetry implies that to every state  there corresponds a time-reversed state T  and that the transformation  preserves the values of all probabilities, thus leaving invariant the absolute value of any scalar product between the two states. In quantum mechanics we normally identify the time reversal transformation, T, with the antiunitary operator, . Note that this causal exchange symmetry is identified in older texts as time reversal symmetry, but it has been shown that it does not represent time reversal symmetry. True time reversal symmetry is not physically valid in quantum mechanics, and consequently of no interest in the technical development of the theory here. Implications of true time reversal symmetry cannot be inferred from the causal exchange symmetry which is explained here. There are currently no reliable textbooks treating time symmetry in quantum mechanics." Along with similar replacement of the term time reversal symmetry with causal exchange symmetry at a few other places, this corrects the error represented in general physics textbooks. Of course this now leaves the concept of time symmetry unexplained, and leaves the rationale for choosing  instead of T unclear, and leaves the implications of CPT theorems for time symmetry unclear, but that goes beyond correcting the explicit error. The subsequent mathematical derivations in physics Principles of physical time directionality and fallacies of the conventional view 15 textbooks are usually reliable, the initial interpretation of what it means is incorrect. We can correct this by calling the symmetries by their proper names. To forestall a common objection, I insist that this is not just a 'semantic issue' or 'playing with definitions'. The meaning of the term 'time reversal symmetry' is not being conventionally defined or changed to our convenience – on the contrary we are insisting on using it with its correct meaning. The term has an objective meaning in physics. It means symmetry under the time reversal transformation. What is being corrected is a false identification, viz. of causal exchange symmetry as time reversal symmetry. Conclusion. Fallacies 1* 4*. The fallacies of 1*4* have been demonstrated. This removes the present case for the conventional conclusion that the conventional reversibility of physics means time has no intrinsic temporal direction. THE TIME FLOW MANIFESTO 16 References 1. David Z Albert. 2000. Time and Chance. Harvard University Press. 2. Davies, P.C.W. 1974. The Physics of Time Asymmetry. Surrey University Press. 3. de Beauregard, Olivia Costa. 1987. Time, The Physical Magnitude. Reidel. 4. de Beauregard, Olivia Costa. 1980. "CPT Invariance and Interpretation of Quantum Mechanics". Found.Phys. 10 7/8, pp. 513-531. 5. Healey, R. 1981. "Statistical Theories, Quantum Mechanics and the Directedness of Time". pp. 99-127. In Reduction, Time and Reality, ed. R. Healey. Cambridge. 6. Holster, A.T. 2003. "The criterion for time symmetry of probabilistic theories and the reversibility of quantum mechanics", New Journal of Physics, (www.njp.org) http://stacks.iop.org/1367-2630/5/130. (Oct. 2003). 7. Holster, A.T. 2003. "The quantum mechanical time reversal operator." 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Watanabe, Satosi. 1965. "Conditional Probability in Physics". Suppl.Prog.Theor.Phys. (Kyoto) Extra Number, pp. 135-167. 21. Zeh, H.D. 1989. The Physical Basis of the Direction of Time. Springer-Verlag. 22. Prigogine, Ilya; Stenger, Isabelle. 1985. Order out of Chaos. Flamingo. Principles of physical time directionality and fallacies of the conventional view 17 Footnotes.