A Reformulation of von Neumann's postulates on quantum measurement by using two theorems in Clifford algebra Elio Conte Department of Pharmacology and Human Physiology – TIRES – Department of Physics: Center for Innovative Technologies for Signal Detection and Processing, University of BariItaly; School of Advanced International Studies for Applied Theoretical and Non Linear Methodologies of Physics, Bari, Italy Abstract: According to a procedure previously introduced from Y. Ilamed and N. Salingaros, we start giving proof of two existing Clifford algebras, the Si that has isomorphism with that one of Pauli matrices and the 1,±iN where iN stands for the dihedral Clifford algebra. The salient feature is that we show that the 1,±iN may be obtained from the Si algebra when we attribute a numerical value (+1 or –1) to one of the basic elements ( 321 ,, eee ) of the Si. We utilize such result to advance a criterium under which the Si algebra has as counterpart the description of quantum systems that in standard quantum mechanics are considered in absence of observation and quantum measurement while the 1,±iN attend when a quantum measurement is performed on such system with advent of wave function collapse. The physical content of the criterium is that the quantum measurement with wave function collapse induces the passage in the considered quantum system from the Si to 1,+iN or to the 1,−iN algebras, where each algebra has of course its proper rules of commutation. After a proper discussion on the difference between decoherence and wave function collapse, we re-examine the von Neumann postulate on quantum measurement, and we give a proper justification of such postulate by using the Si algebra. Soon after we study some applications of the above mentioned criterium to some cases of interest in standard quantum mechanics, analyzing in particular a two state quantum system , the case of time dependent interaction of such system with a measuring apparatus and finally the case of a quantum system plus measuring apparatus developed at the order n=4 of the considered Clifford algebras and of the corresponding density matrix in standard quantum mechanics. In each of such cases examined , we find that the passage from the algebra Si to 1,±iN , considered during the quantum measurement of the system, actually describes the collapse of the wave function. Therefore we conclude that the actual quantum measurement has as counterpart in the Clifford algebraic description, the passage from the Si to the 1,±iN Clifford algebras, reaching in this manner the objective to reformulate von Neumann postulate on quantum measurement and proposing a self-consistent formulation of quantum theory. PACS .03.65.Ta. INTRODUCTION Quantum mechanics has had a so great success to leave very little reasons to doubt its intrinsic validity. It has never been found in disagreement with experimental data, and in explaining a very large variety of physical processes and in predicting basic results also in other fields. Nevertheless, we cannot ignore that some questions concerning fundamental features of this theory remained unsolved, and some historic debates among scientists deeply influenced the early development of the theory. These basic issues were and often continue to be prevalently discussed mainly in philosophical contexts 1 . They will not receive here our direct consideration. In our opinion the object of direct investigation is to understand where the foundations of the theory lie, and why so many deep questions are still unanswered. The first important question concerns the problem of the wave-function collapse by measurement. Its solution would be of relevant significance because it would provide us with a self-consistent formulation of the quantum-mechanical formalism. This result might be of importance also to foresee the way to be followed in order to understand and to explain also the other basic issues that remained often understandable in the story of this theory. As we know , they gave origin to a profound debate. On this basis the completeness of quantum mechanics as a physical theory was discussed, and the very validity of quantum mechanics was often questioned. The aim of the present paper is to reformulate the basic von Neumann's postulate on quantum measurement on the basis of two theorems that we proof in the framework of the Clifford algebra. The results that we obtain seem to be of some relevance for the problem of wave function collapse since, based on two algebraic theorems, we are supported from the asepsis language of an algebraic framework and thus without resource to philosophical or to epistemological indications. SOME FEATURES ABOUT THE COLLAPSE OF THE WAVE FUNCTION In quantum mechanics we have the well known phenomenon of quantum interference. We consider a quantum-mechanical particle to be a "physical entity" represented by a quantum wave function ),( txψ , which depends on the space coordinate x and the time variable t of this particle. Consider the well known interference experiment of the Young type, in which a beam of particles hits a target with two open slits . Also the theoretical description of this experiment is well known. It holds about two basic postulates: (a) The total outgoing wave function ),( txψ behind the slits is written as 21 ψψψ += (1) where 1ψ and 2ψ are the two waves originating from slits 1 and 2, respectively. This is the so-called superposition principle. (b) The intensity of the wave function ψ is proportional to 2ψ . We understand the above experimental facts by means of a purely probabilistic interpretation of the wave function. It is assumed that 2 ),( txP ψ= (2) is proportional to the probability of finding a particle at a space-point x at time t, when it is in a state represented by ψ . The intensity observed at the screen is proportional to )Re(2)Re(2 212121 2 2 2 1 2 ψψψψψψψ ∗∗ ++=++== PPP (3) where there is the presence of the characteristic interference term )Re(2 21ψψ ∗ which is responsible for the observed interference pattern. Suppose we find a particle at point X on the screen .On the basis of the probabilistic interpretation, we can state that the particle state immediately after the observation must be represented by a wave function )(tXψ , distributed only around X so that we conclude that the measurement has caused the change Xψψ → (4) of the wave function. We call this change the wave-function collapse by measurement. The wave-function collapse is not a causal wave motion, continuously shrinking from ψ to Xψ or to 'Xψ , but it is an acausal and purely probabilistic event. Quantum mechanics only gives the probabilistic prediction that the probability of finding each event is proportional to 2 Xψ or to 2 'X ψ . Of course, the wave-function collapse cannot be described by the Schrödinger equation which gives only deterministic changes. Consequently, quantum mechanics becomes a non self-contained theory since the measuring process cannot be described by quantum mechanics itself. These are only some preliminary features regarding the more articulated problem of measurement in quantum mechanics. For a complete examination of the actual problems that are involved, we refer to the several reviews that may be found in literature 2 . In particular, we intend to hint here only at some recent developments as the theory of quantum decoherence, a term that was used for the first time by Bohr, while an articulated elaboration was introduced more recently by Zurek 3 . It considers the mechanism by which quantum systems interact with their environments giving the appearance of wave function collapse. Still we mention here the theory of Ghirardi, Rimini and Webber (GRW) 4 who claim that particles undergo spontaneous wave-function collapses. The leading idea of the theory is to eradicate observers from the picture and view state reduction as a process that occurs as a consequence of the basic laws of nature. The theory achieves this by adding to the fundamental equation of quantum mechanics, the Schrödinger equation, a stochastic term which describes the state reduction occurring in the system. After such preliminary remarks, we may now set the basis for our discussion. As previously stated, we consider the measurement of a given observable F on a quantum-mechanical system S in a normalized superposed state i i ic φψ ∑= ; ),( ψφiic = ; 1 2 =∑ i ic (5) where iφ is a normalized eigenstate of F , relative to an eigenvalue iλ ,so that iiiF φλφ = and ijji δφφ =),( . The probabilistic interpretation means that the probability of finding the eigenvalue iλ (i.e. the corresponding eigenstate iφ ) in the measurement of F on a state ψ is equal to 2 ic . The wave-function collapse is expressed in this case as iφψ → (6) The previous equation still does not describe the wave-function collapse by measurement, intended as an acausal and purely probabilistic event. A complete expression for the wave-function collapse must be formulated in terms of density matrix as it was initiated by von Neumann 5 kk k kFSji j ji i S ccc φφρφφψψρ ><=→><=><= ∑∑∑ ∗ 2 , (7) The above expression describes rather well a process in which all the phase correlations among different eigenstates are erased. We obtain a sum of exclusive probabilities of finding each eigenstate. However, also such formulation may still give origin to contradictions. In order to avoid such possible difficulties, we have to modify the previous expression for the wavefunction collapse, by introducing the states of a given measurement apparatus system A obtaining in this case tkAtkk k ktASAjij j i i AS ccc ),( 2 ,, ρφφρρφφρρρ ⊗><=→⊗><=⊗= ∑∑∑ ∗ (8) Let us see in more detail the von Neumann's postulate about quantum measurement. If a quantum system is in an eigenstate of the operator corresponding to the observable being measured, the outcome of the measurement will be the eigenvalue associated with that eigenstate. However, if the system is in a superposition of such eigenstates, the outcome will be unpredictable, and all that quantum theory can give, are the probabilities for the different outcomes. If the system is not destroyed by the measurement, and if the interaction fits into the so called 'measurement of the first kind', then the quantum state after the measurement will be the eigenstate associated to the measurement outcome, or more generally (to include degenerancies), the normalized projection of the original state onto the eigensubspace associated with the outcome. This rule is known as the projection postulate. It originated with Dirac and von Neumann, and was later formalized in degenerate cases by Luders and Ludwig 6 . According to such projection postulate the complete phase-damping way for a two state system may be written 11110000)( ><><+><><= ρρρD (9) where the effect of this mapping is to zero-out the off-diagonal entries of a density matrix:       =      δ α δγ βα 0 0 D (10) If we have a set of mutually orthogonal projection operators ( ),.....,, 21 mPPP which complete to identity, i.e., jijji PPP δ= and 1=∑ i iP when a measurement is carried out on a system with state >ψ then (1) The result i is obtained with probability >=< ψψ ii Pp (2) The state collapses to >ψi i P p 1 In detail, note that von Neumann's projection postulate only relates the vanishing of interference terms or decoherence. It does not explain the collapse of a pure state to another pure state associated to individual object systems. The formal distinction between decoherence and collapse is substantial as authors as Sussmann 7 in 1957 and Bell 8 more recently outlined. They distinguished clearly between what we call 'division' (decoherence) and 'reading' (the collapse which follows decoherence). Decoherence is a statistical concept, involving the transition from a pure state to a 'mixture', and the disappearance of interference terms. Collapse refers to an individual system, and it describes a transition from a pure state to another pure state. For a single quantum object, we may therefore write: k i iia φφ >→∑ (11) with probability 2 ka For an ensemble of measurements of the same observable performed on the same initial pure state (that is, each measurement being performed on a different single object, all prepared in the same pure state), one may represent the statistical transition described by the projection postulate as )( 2 k k k i ii PaaP φφ ∑∑ → (12) In brief, an explanation for collapse implies an explanation for decoherence, but an explanation for decoherence doesn't imply an explanation for collapse 9 . TWO THEOREMS IN CLIFFORD ALGEBRA Let us start with a proper definition of the 3-D space Clifford (geometric) algebra 3Cl . It is an associative algebra generated by three vectors ,, 21 ee and 3e that satisfy the orthonormality relation jkjkkj eeee δ2=+ (13) for [ ]3,2,1,, ∈λkj . That is, 12 =λe and jkkj eeee −= for kj ≠ Let a and b be two vectors spanned by the three unit spatial vectors in 0,3Cl . By the orthonormality relation the product of these two vectors is given by the well known identity: )( baibaab ×+⋅= where 321 eeei = is a Clifford algebraic representation of the imaginary unity that commutes with vectors. To give proofs, let us follow the approach that, starting with 1981, was developed by Y. Ilamed and N. Salingaros 10 . Let us consider the three abstract basic elements, ie , with 3,2,1=i , and let us admit the following two postulates: a) it exists the scalar square for each basic element: 111 kee = , 222 kee = , 333 kee = with R∈ik . (14) In particular we have also the unit element, 0e , such that that 100 =ee . b) The basic elements ie are anticommuting elements, that is to say: 1221 eeee −= , 2332 eeee −= , 3113 eeee −= . (15) It is iii eeeee == 00 . Theorem n.1. Assuming the two postulates given in (a) and (b) with 1=ik , the following commutation relations hold for such algebra : 31221 ieeeee =−= ; 12332 ieeeee =−= ; 23113 ieeeee =−= ; 321 eeei = , ( 1 2 3 2 2 2 1 === eee ) (16) They characterize the Clifford Si algebra . We will call it the algebra A(Si). Proof. Consider the general multiplication of the three basic elements ,,, 321 eee using scalar coefficients kkk γλω ,, pertaining to some field: 33221121 eeeee ωωω ++= ; 33221132 eeeee λλλ ++= ; 33221113 eeeee γγγ ++= . (17) Let us introduce left and right alternation: for any ),( ji , associativity exists jiijii eeeeee )(= and )( jjijji eeeeee = that is to say 211211 )( eeeeee = ; )( 221221 eeeeee = ; 322322 )( eeeeee = ; )( 332332 eeeeee = ; 133133 )( eeeeee = ; )( 113113 eeeeee = . (18) Using the (15) in the (18) it is obtained that 3132121121 eeeekek ωωω ++= ; 2332221112 eekeeek ωωω ++= ; 3232212132 eekeeek λλλ ++= ; 3332231123 keeeeek λλλ ++= ; 3323213113 keeeeek γγγ ++= ; 1331221131 eeeekek γγγ ++= . (19) From the (19), using the assumption (b), we obtain that 332 3 2 13 3 1 32 2 3 221 2 1 γ γγω ω ω +−=−+ ee k ee k ee k ee k ; 332 3 2 13 3 1 13 1 3 21 1 2 1 λ λλωω ω ++−=−+ ee k ee k ee k ee k ; 32 2 3 221 2 1 13 1 3 21 1 2 1 ee k ee k ee k ee k λ λ λγγ γ ++−=+− (20) By the principle of identity , we have that it must be 0313221 ====== γγλλωω (21) and 02211 =+− kk γλ 03322 =− kk ωγ 03311 =− kk ωλ (22) The (22) is an homogeneous algebraic system admitting non trivial solutions since its determinant 0=Λ , and the following set of solutions is given: ,321 ωγ−=k 312 ωλ−=k , 213 γλ−=k (23) Admitting 1321 +=== kkk , it is obtained that i=== 213 γλω (24) In this manner, using the (14) and the (15), as a theorem, the existence of such algebra is proven. The basic features of this algebra are given in the following manner 123 2 2 2 1 === eee ; 31221 ieeeee =−= ; 12332 ieeeee =−= ; 23113 ieeeee =−= ; 321 eeei = (25) The content of the theorem n.1 is thus established: given three abstract basic elements as defined in (a) and (b) ( )1=ik , an algebraic structure is established with four generators ( ).,,, 3210 eeee Let us go on now to give proof of theorem n.2. Before let us note that the algebra A(Si), now given, admits idempotents. Let us consider two of such idempotents: 2 1 3 1 e+ =ψ and 2 1 3 2 e− =ψ (26) It is easy to verify that 1 2 1 ψψ = and 2 2 2 ψψ = . Let us examine now the following algebraic relations: 13113 ψψψ == ee (27) 23223 ψψψ −== ee (28) Similar relations hold in the case of 1e or 2e . From a conceptual point of view , looking at the (27) and (28) we reach only a conclusion. With reference to the idempotent 1ψ , the algebra A(Si) (see the (27)), attributes to 3e the numerical value of 1+ while, with reference to the idempotent 2ψ , the algebra A(Si) attributes to 3e (see the (28)), the numerical value of -1 . However, assuming the attribution 3e → +1, from the (25) we have that new commutation relations should hold in a new Clifford algebra given in the following manner : 122 2 1 == ee , 1 2 −=i ; iee =21 , iee −=12 , 12 eie −= , 12 eie = , 21 eie = , 21 eie −= (29) with three new basic elements ( ),, 21 iee instead of ( ),, 321 eee . In other terms, in the case in which we attribute to 3e the numerical value +1, a new algebraic structure should arise with new generators whose rules should be given in (29) instead of in (25). Therefore, the arising central problem is that we should be able to proof the real existence of such new algebraic structure with rules given in the (29). We repeat: in the case of the starting algebraic structure, the algebra A(Si), we showed by theorem n.1 that it exists in the following manner 123 2 2 2 1 === eee ; 31221 ieeeee =−= ; 12332 ieeeee =−= ; 23113 ieeeee =−= ; 321 eeei = (30) In the present case in which we attribute to 3e the numerical value +1 , we should show that it exists a new algebra given in the following manner 122 2 1 == ee ; 1 2 −=i ; iee =21 , iee −=12 , 12 eie −= , 12 eie = , 21 eie = , 21 eie −= (31) So we arrive to give proof of the theorem n.2. Theorem n.2 . Assuming the postulates given in (a) and (b) with 11 =k , 12 =k , 13 −=k , the following commutation rules hold for such new algebra: 122 2 1 == ee ; 1 2 −=i ; iee =21 , iee −=12 , 12 eie −= , 12 eie = , 21 eie = , 21 eie −= (32) They characterize the Clifford Ni algebra. We will call it the algebra 1,+iN Proof To give proof, rewrite the (17) in our case, and performing step by step the same calculations of the previous proof, we arrive to the solutions of the corresponding homogeneous algebraic system that in this new case are given in the following manner: 321 ωγ−=k ; 312 ωλ−=k ; 213 γλ−=k (33) where this time it must be 121 +== kk and 13 −=k . It results 11 −=λ ; 12 −=γ ; 13 +=ω (34) and the proof is given. The content of the theorem n.2 is thus established. When we attribute to 3e the numerical value +1 we pass from the Clifford algebra Si (algebra A) to a new Clifford algebra 1,+iN whose algebraic structure is no more given from the (30) of the algebra A but from the following new basic rules: 122 2 1 == ee ; 1 2 −=i ; iee =21 , iee −=12 , 12 eie −= , 12 eie = , 21 eie = , 21 eie −= (35) that are totally different from the basic commutation rules that we have in the case of the algebra A(Si). The theorem n.2 also holds in the case in which we attribute to 3e the numerical value of 1− . Assuming the postulates given in (a) and (b) with 11 =k , 12 =k , 13 −=k , the following commutation rules hold for such new algebra 122 2 1 == ee ; 1 2 −=i ; iee −=21 , iee =12 , 12 eie = , 12 eie −= , 21 eie −= , 21 eie = (36) They characterize the Clifford Ni algebra. We will call it the algebra 1,−iN To give proof , consider the solutions of the (33) that are given in this new case by 11 +=λ ; 12 +=γ ; 13 −=ω (37) and the proof is given. The content of the theorem n.2 is thus established. When we attribute to 3e the numerical value –1, we pass from the Clifford algebra Si (algebra A) to a new Clifford algebra 1,−iN whose algebraic structure is not given from the (30) of the algebra A and not even from the (35) but from the following new basic rules: 122 2 1 == ee ; 1 2 −=i ; iee −=21 , iee =12 , 12 eie = , 12 eie −= , 21 eie −= , 21 eie = (38) In a similar way, proofs may be obtained when we consider the cases attributing numerical values ( )1± to 1e or to 2e . Of course, the Clifford algebra, 1,1 ±N , given in the (35 ) and in the (36) are well known . They are the dihedral Clifford algebra iN (for details, see ref.10 page 2093 Table II). In conclusion, in this section, using a Clifford algebraic framework, we have shown two basic theorems, the theorem n.1 and the theorem n.2. As any mathematical theorem they have maximum rigour, and an aseptic mathematical content that cannot be questioned. The basic statement that we reach by the proof of such two theorems is that in Clifford algebraic framework, we have the Clifford algebra A(Si) and inter-related Clifford algebras 1,±iN . When we consider ( ),, 321 eee as the three abstract elements with rules given in (30) , we are in the Clifford algebra A(Si) .When we attribute to 3e the numerical value +1, we pass from the algebra A ( the Clifford algebra Si, with basic features given in (30)), to the algebra B, the Clifford 1,+iN , with basic algebraic rules given in the (31). Instead, when we pass from the Clifford algebra A, (the Clifford algebra Si) to the Clifford algebra 1,−iN , the basic features are given in the (38) and we attribute to 3e the numerical value –1 . The same conceptual facts hold when we reason for Clifford basic elements 1e or to 2e , attributing in this case a possible numerical value ( 1± ) or to 1e or to 2e , respectively. A POSSIBLE IMPLICATION FOR QUANTUM MECHANICS If one looks at the algebraic rules and commutation relations given in the (30), the algebra A(Si) shown by theorem n.1, immediately acknowledges that they are universally valid in quantum mechanics. We called the algebra A as the Si Clifford algebra because it links the Pauli matrices that are sovereign in quantum mechanics. Still the isomorphism between Pauli matrices and Clifford algebras is well established at any order. Passing from the algebra A(Si) to 1,±iN it happens an interesting feature. Consider the case , as example, of 3e . While in A(Si) 3e is an abstract algebraic element that has the potentiality to assume or the value +1 or the value –1( in correspondence, in quantum mechanics it is an operator with possible eigenvalues 1± ) , when we pass in the algebra 1,±iN , 3e is no more an abstract element in this algebra , it becomes a parameter to which we may attribute the numerical value +1 , and we have 1,+iN whose three abstract element now are ( ),, 21 iee with commutation rules given in the (35) . If we attribute to 3e the numerical value –1 , we are in 1,−iN whose three abstract elements are still ( ),, 21 iee , and the commutation rules are given in (38). Reading this statement in the language and in the logic of quantum mechanical measurement , it means that if we are measuring the given quantum system S with a measuring apparatus and , as result of the actualized and performed measurement, we read the result +1 , we are in the corresponding algebraic case , in the algebra 1,+iN . If instead , performing the measurement , we read the result –1, in this case we are in the algebra 1,−iN . In each of the two cases this means that a collapse of the wave function has happened. During a process of quantum measurement , speaking in terms of Clifford algebraic framework, we could have the passage from the Clifford algebra A,(Si), having such fundamental basic commutation rules : 31221 ieeeee =−= ; 12332 ieeeee =−= ; 23113 ieeeee =−= ; 321 eeei = 123 2 2 2 1 === eee ; (39) to the new 1,+iN Clifford algebra having the following and totally new commutation rules: iee =21 , iee −=12 , 12 eie −= , 12 eie = , 21 eie = , 21 eie −= 122 2 1 == ee ; 1 2 −=i ; (40) in the case in which the result of the measurement of 3e is +1 (read on the instrument), and instead we could have the passage to the new 1,−iN Clifford algebra , having the following and totally new commutation rules: iee −=21 , iee =12 , 12 eie = , 12 eie −= , 21 eie −= , 21 eie = 122 2 1 == ee ; 1 2 −=i (41) in the case in which the result of the quantum measurement of 3e gives value –1 (read on the instrument). In such way it seems that a reformulation of von Neumann's projection postulate may be suggested. The reformulation is that, during a quantum measurement (wave-function collapse), we have the passage from the Clifford algebra A(Si), to the new Clifford algebra 1,±iN . In other terms Quantum Measurement (wave-function collapse) = passage from algebra A ( Si) to B ( 1,±iN ). In conclusion we think that the two previously shown theorems in Clifford algebraic framework give justification of the von Neumann's projection postulate and they seem to suggest , in addition , that we may use the passage from the algebra A(Si) to 1,±iN to describe actually performed quantum measurements. APPLICATIONS OF THE PREVIOUS CRITERIUM TO SOME CASES OF QUANTUM MECHANICAL INTEREST Let us start discussing a trivial application. It is important only to illustrate better the sense in which we must intend the present formulation. Assume a two –level microscopic quantum system S with two states +u , −u corresponding to energy eigenvalues +ε , −ε . The Hamiltonian operator SH can be written 333 )( 2 1 )( 2 1 )1( 2 1 )1( 2 1 eeeH S −+−+−+ −++=−++= εεεεεε (42) The standard quantum methodological approach is also well known . We have that       =+ 0 1 u ,       =− 1 0 u , and iiiS uuH ε= . (43) We may also choose εε =+ and 0=−ε simplifying the (42) to ε)1( 2 1 3eH S += (44) Indicate an arbitrary state of such quantum microsystem as −−++ += ucucSψ (45) where, according to Born's rule, we have 1δiepc ++ = , 2 δi epc −− = (46) with jp ( −+= ,j ) (47) corresponding probabilities with 1=+ −+ pp . This is the standard quantum mechanical formulation of the system. Let us admit now that we want to measure the energy of S using a proper apparatus . The rules of quantum mechanics tell us that we will obtain the value ε with probability +p , and the value zero with probability −p . After the measurement the state of S will be either +u or −u according to the measured value of the energy. The experiment will enable us also to estimate +p as well as −p . In such simple quantum mechanical example we have , as known, the (42), 3e , the (44) that are linear Hermitean operators with quantum states acting on the proper Hilbert space. Let us see instead the question from a different point of view. The 3e , and SH given in the (42) or in the (44) are members of the Clifford algebra. They are Clifford algebraic members of what we have called the algebra A( Si), with basic rules given in the following manner: 123 2 2 2 1 === eee 31221 ieeeee =−= ; 12332 ieeeee =−= ; 12332 ieeeee =−= ; 321 eeei = (48) However, on the basis of theorems n.1 and n.2 shown in the previous sections, starting with the Clifford algebra A(Si), we must use the existing Clifford, dihedral algebra B, 1,±iN when we arrive to attribute ( by a measurement) as example to 3e in one case the numerical value +1 and, in the other case, the numerical value –1. In the first case we have a dihedral Clifford iN algebra that is given in the following manner: 122 2 1 == ee 1 2 −=i iee =21 , iee −=12 , 12 eie −= , 12 eie −= , 12 eie = , 21 eie = , 21 eie −= (49) that holds when we are attributing to 3e the numerical value +1 ( in analogy with quantum mechanics: the quantum measurement process has given as result +1). In the second case, we have instead that 122 2 1 == ee ; 1 2 −=i ; iee −=21 , iee =12 , 12 eie = , 12 eie −= , 21 eie −= , 21 eie = (50) that holds when we have arrived to attribute to 3e the numerical value –1 by a direct measurement Reasoning in terms of a Clifford algebraic framework, we are authorized to apply the passage from algebra A(Si) to algebra B in the (42) . From it , we obtain: +− = ε)( elementCliffordSH (51) if the instrument has given as result of the measurement , the value +1 to 3e (Clifford algebraic parameter of dihedral 1,+iN algebra ), and −− = ε)( elementCliffordSH (52) if the instrument has given as result of the measurement , the value -1 to 3e . During the measurement we have had the passage from algebra A(Si) to the dihedral 1,±iN algebra in which , with given probabilities, 3e has assumed or the +1 or the value –1 , respectively. In the first case, we have ε=− )( elementCliffordSH and in the second case, we have 0)( =−elementCliffordSH Consider now the second application . Let us introduce a two state quantum system S with connected quantum observable 3σ ( )3e . We have 2211 φφψ cc += ,       = 0 1 1φ ,       = 1 0 2φ (53) and 1 2 2 2 1 =+ cc As we know, the density matrix of such system is easily written 321 decebea +++=ρ (54) with 2 2 2 2 1 cc a + = , 2 * 212 * 1 ccccb + = , 2 )( 2 * 1 * 21 ccccic − = , 2 2 2 2 1 cc d − = (55) where in matrix notation, 1e , 2e , and 3e are the well known Pauli matrices       = 01 10 1e ,       − = 0 0 2 i i e ,       − = 10 01 3e (56) Of course , the analogy still holds. The (54) is still an element of the Clifford algebra, and precisely of Clifford algebra A(Si). As Clifford algebraic member, the (54) satisfies the requirement to be ρρ =2 and Tr( 1) =ρ under the conditions 2/1=a and 02222 =−−− dcba as shown in detail elsewhere in ref11. In the algebraic framework previously outlined, let us admit that we attribute to 3e the value +1 (that is to say ... the quantum observable 3σ assumes the value +1 during quantum measurement ) or to 3e the numerical value –1 (that is to say... the quantum observable 3σ assumes the value –1 during the quantum measurement). As previously shown, in such two cases the algebra A, (Si) no more holds, and it will be replaced from the Clifford 1,±iN . To examine the consequences , starting with the algebraic element (54), write it in the two equivalent algebraic forms that are obviously still in the algebra A(Si). 3 2 2 2 1212 * 121 * 21 2 2 2 1 )( 2 1 ))(( 2 1 ))(( 2 1 )( 2 1 eccieeccieecccc −+−++++=ρ (57) and 3 2 2 2 1212 * 121 * 21 2 2 2 1 )( 2 1 ))(( 2 1 ))(( 2 1 )( 2 1 eccieeccieecccc −+−++++=ρ (58) Both such expressions contain the following interference terms. ))(( 2 1 ))(( 2 1 212 * 121 * 21 ieeccieecc −++ (59) and ))(( 2 1 ))(( 2 1 212 * 121 * 21 ieeccieecc −++ (60) Let us consider now that the quantum measurement gives as result +1 for 3e . In this case there are the (57) and the (59) that we take in consideration. On the basis of our principle, we know that the previous Clifford algebra A(Si) no more holds, but instead it is valid the 1,1 +N that has the following new commutation rules: iee =21 , iee −=12 , 12 eie −= , 12 eie = , 21 eie = , 21 eie −= (61) Inserting such new commutation rules in the (57) and the (59), remembering that here 3e is now a parameter that has value +1, one sees that the interference terms are erased and the density matrix, given in the (57), now becomes 2 1cM =→ ρρ (62) The collapse has happened. In the same manner let us consider instead that the quantum measurement gives as result -1 for 3e . In this case there are the (58) and the (60) that we take in consideration On the basis of our principle, we know that the previous Clifford algebra A(Si) no more holds, but instead it is valid the 1,1 −N that has the following new commutation rules iee −=21 , iee =12 , 12 eie = , 12 eie −= , 21 eie −= , 21 eie = (63) Inserting such new commutation rules in the (58) and (60), remembering that the parameter 3e now assumes value –1, one sees that the interference terms are erased and the density matrix, given in the (54) or in the (58), now becomes 2 2cM =→ ρρ (64) The collapse has happened. Let us examine now von Neumann results. In order to formulate in detail von Neumann's projection postulate , consider the spinor basis given in (53). Outer products give projection operators that are the idempotents in the A(si) Clifford algebra as explicitly given in (26). Consider again the (9). Reasoning in terms of Clifford algebra 00 >< and (65) 11 >< (66) are respectively the idempotents 2 1 3e+ and (67) 2 1 3e− (68) Considering the first term on the right in the (9) one has that ( 2 1 3e+ ) ρ ( 2 1 3e+ ) (69) that , in terms of the matrix given in the (10) , gives ( 2 1 3e+ ) ρ ( 2 1 3e+ ) =α ( 2 1 3e+ ) and explicitly (70)       00 0α (71) Applying the same procedure in the case of 2 1 3e− (72) (the second term in the (9)) , one obtains as result (β ) 2 1 3e− (73) and explicitly       δ0 00 (74) The sum , as indicated in the (9), gives       δ α 0 0 (75) In conclusion we have given full justification of von Neumann's projection postulate in Clifford A(Si) algebra. As expected, there is total equivalence between von Neumann postulate and corresponding A)Si) formulation . It is important to reaffirm here that it has been obtained using only the framework of A(Si) algebra. In accord with von Neumann we obtain α ( 2 1 3e+ ) (76) and (β ) 2 1 3e− (77) Note now that , in application of our criterium, quantum measurement is obtained passing from algebra A(Si) to 1, ±i N . In this case we no more obtain the (76) and the (77) as it happens remaining in the framework of the A(Si) algebra, but we obtain respectively the (62) or the (64), that is to say, 2 1cM =ρ (78) or 2 2cM =ρ (79) as it must be when the collapse has happened. The nature of such result obviously does not change if we explore a time dependent situation. The reader is advised that we will use a lightly modified formalism that however does not alter the significance of our application. Consider the quantum system S and indicate by 0ψ the state at the initial time 0. The state at any time t will be given by 0)()( ψψ tUt = and )0(0 == tψψ (80) An Hamiltonian H must be constructed such that the evolution operator U(t), that must be unitary, gives iHtetU −=)( . It is well known that, given a finite N-level quantum system described by the state ψ , its evolution is regulated according to the time dependent Schrödinger equation )()( )( ttH dt td i ψ ψ =η with 0)0( ψψ = . (81) Let us introduce a model for the hamiltonian H(t). Details of this formalism may be found in reff.12 and 13. We express by H0 the hamiltonian of the system S, and we add to H0 an external time varying hamiltonian, H1(t), representing the perturbation to which the system S is subjected by action of the measuring apparatus. In conclusion we write the total hamiltonian as H(t) = H0 + H1(t) (82) so that the time evolution will be given by the following Schrödinger equation [ ] )()()( 10 ttHH dt td i ψ ψ +=η (83) and 0)0( ψψ = . We have that 0)()( ψψ tUt = (84) where U(t) pertains to the special group SU(N). We will write that [ ] )()()()()( 10 tUtHHtUtH dt tdU i +==η and U(0)=I (85) Let A1,A2,........,An , (n=N 2 -1), are skew-hermitean matrices forming a basis of Lie algebra SU(N). In this manner one arrives to write the explicit expression of the hamiltonian H(t). It is given in the following manner [ ] j n j jj n j j AbAatHHitiH ∑∑ == +=+−=− 11 10 )()( (86) where aj and bj = bj(t) are respectively the constant components of the free hamiltonian and the time-varying control parameters characterizing the action of the measuring apparatus. If we introduce T, the time ordering parameter (for details see reff. 12 and 13), we arrive also to express U(t) that will be given in the following manner )))((exp())(exp()( 0 0 ττττ dAbaiTdHiTtU jj t t j +−=−= ∫ ∫ (87) that is the well known Magnus expansion. Locally U(t) may be expressed by exponential terms as it follows )........exp()( 2211 nnAAAtU γγγ +++= (88) on the basis of the Wein-Norman formula               + + + =               Ξ nnn n ba ba ba ...... ),......,,( 22 11 2 1 21 γ γ γ γγγ & & & (89) with Ξ n x n matrix, analytic in the variables iγ . We have 0)0( =iγ and I=Ξ )0( , and thus it is invertible. We obtain               + + + Ξ=               − nnn ba ba ba ...... 22 11 12 1 γ γ γ & & & (90) Consider a simple case based on the superposition of only two states. We have [ ]Tyy 21,=ψ and 1 2 2 2 1 =+ yy (91) As previously said, we have here an SU(2) unitary transformation, selecting the skew symmetric basis for SU(2). We will have that       = 01 10 1e ,       − = 0 0 2 i i e ,       − = 10 01 3e (92) Now we consider the following matrices jj e i A 2 = , j = 1,2,3 The reader may now ascertain that the previously developed formalism is moving in direct correspondence with our Clifford algebra A(Si). We are now in the condition to express H(t) and U(t) in our case of interest. The most simple situation we may examine is that one of fixed and constant control parameters bj. The hamiltonian H will become fully linear time invariant and its exponential solution will take the following form         ++= ∑ = +∑ = 3 1 ))(( )() 2 ( 2 ) 2 cos( 3 1 j jjj Abat Abat k sen k It k e j jjj (93) with 233 2 22 2 11 )()()( bababak +++++= . In matrix form it will result [ ] [ ] ( )           +−++−− +++++ = 331122 112233 22 cos)( 2 1 )( 2 1 )( 22 cos )( bat k sen k i t k baibat k sen k baibat k sen k bat k sen k i t k tU (94) and, obviously, it will result to be unimodular as required. Starting with this matrix representation of time evolution operator U(t), we may deduce promptly the dynamic time evolution of quantum state at any time t writing 0)()( ψψ tUt = (95) assuming that we have for 0ψ the following expression         = false true c c 0ψ (96) having adopted for the true and false states (or yes-not states, +1 and –1 corresponding eigenvalues of such states ) the following matrix expressions       = 0 1 trueφ and       = 1 0 falseφ (97) Finally, one obtains the expression of the state )(tψ at any time [ ] [ ] falsefalsetrue truefalsetrue bat k sen k i t k cbabait k sen k c baibat k sen k cbat k sen k i t k ct φ φψ           +−+    +−+ +          ++++    ++= )( 22 cos)()( 2 1 )()( 2 1 ( 22 cos)( 332211 1122)33 (98) As consequence, the two probabilities Ptrue(t) and Pfalse(t) , will be given at any time t by the following expressions )()( 2 1 2 cos)()( 222 2 222 BQAP k senkt QPt k sen k t k BAtPtrue +++++= and (99) )()( 2 1 2 cos)()( 222 2 222 DSRC k senkt RSt k sen k t k DCtPfalse +++++= where A= Re ctrue ,B=Im ctrue, C=Re cfalse , D=Im cfalse , P=-D(a1+b1)+C(a2+b2)-B(a3+b3), Q=C(a1+b1)+D(a2+b2)+A(a3+b3), (100) R=-B(a1+b1)-A(a2+b2)+D(a3+b3), S=A(a1+b1)-B(a2+b2)-C(a3+b3) Until here we have developed only standard quantum mechanics. The reason to have developed here such formalism has been to evidence that at each step it has its corresponding counterpart in Clifford algebraic framework A(Si) , and thus we may apply to it the two theorems developed in the previous section and the previously introduced criterium, passing from the algebra Si to 1,±iN . In fact , to this purpose , it is sufficient to multiply the (94) by the (96) to obtain the final forms of )(tctrue and )(tc false In the final state we have that         = )( )( tc tc false true tψ (101) We may now write the density matrix that will result to have the same structure of the previously case given in the (54) but obviously with explicit evidence of time dependence. . In the Clifford algebraic framework it will pertain still to the Clifford algebra A( Si). In order to describe the wave-function collapse we have to repeat the same procedure that we developed previously from the (54) to the (64) , considering that , in accord to our criterium ,we have to pass from the algebra A(Si) to 1,±iN , and obtaining 2 )(tctrueM =→ ρρ (102) in the case 1,+iN and 2 )(tc falseM =→ ρρ (103) in the case 1,−iN , as required in the collapse. Let us examine now the fourth application of our criterium. Until here we considered only examples of two state quantum systems. Let us expand our formulation at any order n. First consider Clifford Si algebra at order n=4 (for details see ref.14) . One has E0 i = I 1 ⊗ e i ; Ei 0 = e i ⊗ I 2 (104) The notation ⊗ denotes direct product of matrices, and I i is the ith 2x2 unit matrix. Thus, in the case of n= 4 we have two distinct sets of Clifford basic unities, E0 i and Ei 0, with 120 =iE ; 1 2 0 =iE , i = 1, 2, 3; (105) E0 i E0 j = i E0 k ; Ei 0 Ej 0 = i Ek 0 , j = 1, 2, 3; i ≠ j and Ei0 E0 j =E0 j Ei 0 (106) with (i, j, k) cyclic permutation of (1, 2, 3). Let us examine now the following result (I 1 ⊗ ei) (ej ⊗ I 2 ) = E0 i Ej 0 =Ej i (107) It is obtained according to our basic rule on cyclic permutation required for Clifford basic unities. We have that E0 i Ej0 = Ej i with i = 1, 2, 3 and j=1, 2, 3, with E j i 2 = 1, Ei j Ek m ≠ Ek m Ei j, and Ei j Ek m = Ep q where p results from the cyclic permutation (i, k, p) of (1, 2, 3) and q results from the cyclic permutation (j, m, q) of (1, 2, 3). In the case n = 4 we have two distinct basic set of unities E0 i , Ei 0 and, in addition, basic sets of unities (Ei j , Ei p , E0 m) with ( j, p, m) basic permutation of (1, 2, 3). This is the Clifford algebra A at order n=4. In the other more general cases we have E0 0 i, E0 i 0, and Ei 0 0, i = 1, 2, 3 and E0 0 i = I 1 ⊗ I 1 ⊗ ei ; E0 i0 = I 2 ⊗ ei ⊗ I 2 ; Ei 0 0 = ei ⊗ I 3 ⊗ I 3 and (I 1 ⊗ I 1 ⊗ ei ) . (I 2 ⊗ ei ⊗ I 2 ) . (ei ⊗ I 3 ⊗ I 3 ) = ei ⊗ ei ⊗ ei = = E0 0 i E0 i 0 Ei 0 0 = E i i i (108) Still we will have that E0 0 i E0 i 0 = E0 i 0 Ei 0 0 ; E0 0 i Ei 0 0 = Ei 0 0 E0 0 i ; E0 i 0 Ei 0 0 = Ei 0 0 E0 i0 (109) Generally speaking, fixed the order n of the Si Clifford algebra in consideration , we will have that Γ1 = Λ n Γ2m = Λ n-m ⊗ ⊗ ⊗ ⊗ − + − +e I In m n m n2 1 2( ) ( ) ......... (110) Γ2m+1 = Λ n-m ⊗ ⊗ ⊗ ⊗− + − +e I In m n m n3 1 2( ) ( ) ......... Γ2n = e I I n 2 2⊗ ⊗ ⊗( ) ......... with Λn= e e e n 1 1 1 2 1 ( ) ( ) ( ).....⊗ ⊗ ⊗ = ( e I I n1 1⊗ ⊗ ⊗( ) ..... ).(........).( I I I en( ) ( ) ( )...1 2 1⊗ ⊗ ⊗ ); m = 1, ....., n 1 according to the n-possible dispositions of e1 and I 1 , I 2 , ..., I n in the distinct direct products. We may now give the explicit expressions of E0 i, Ei 0, and Ei j at the order n=4. E01 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 =             ; E i i i i 02 0 0 0 0 0 0 0 0 0 0 0 0 = − −             ; E03 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 = − −             (111) E10 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 =             ; E i i i i 20 0 0 0 0 0 0 0 0 0 0 0 0 = − −             ; E30 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 = − −             ; E11 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 =             ; E22 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 = − −             ; E33 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 = − −             ; E i i i i 12 0 0 0 0 0 0 0 0 0 0 0 0 = − −             ; E13 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 = − −             ; E i i i i 21 0 0 0 0 0 0 0 0 0 0 0 0 = − −             ; E31 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 = − −             ; E i i i i 23 0 0 0 0 0 0 0 0 0 0 0 0 = − −             ; E i i i i 32 0 0 0 0 0 0 0 0 0 0 0 0 = − −             . Note the following basic feature: we have now some different sets of Clifford algebras A, (Si). In detail, we have the following sets of basic Si Clifford algebras: ),,( 131201 EEE , ),,( 232201 EEE ,( ),, 333201 EEE , ),,( 131102 EEE , ),,( 232102 EEE ,( ),, 333102 EEE , ),,( 121103 EEE , ( ),, 222103 EEE , ),,( 323103 EEE ,( ),, 332310 EEE , 322210 ,,( EEE ), ),,( 312110 EEE ,( ),, 331320 EEE , ),,( 321220 EEE ,( ),, 311120 EEE , ( ),, 231330 EEE ,( ),, 221230 EEE ,( ),, 211130 EEE (112) All these are the sets of Clifford algebras A (Si) that we have at order n=4. To each of these sets we may apply the theorems n.1 and n.2 previously shown and we may apply the criterium of the passage from the Si to the 1,1 ±N that we have just used in the other previous cases of application. Fixed such algebraic features, we may now consider the problem that we formulated in the introduction of the present paper. It is that, in order to avoid possible contradictions, we should still modify the previous expression for the wavefunction collapse, by introducing the states of a given measurement apparatus system A obtaining in this case tkAtkk k ktASAjij j i i AS ccc ),( 2 ,, ρφφρρφφρρρ ⊗><=→⊗><=⊗= ∑∑∑ ∗ (113) See the previous discussion that we introduced by the (8). We may refer the algebraic sets iE0 to the quantum system S to be measured , and consider the algebraic sets 0iE to the measuring apparatus A. Still we have the basic algebraic set ijE that relates the coupling of S with A. Let us write the density matrix ρ at such order n=4. To simplify, we may write it in the following general form             −−− +−− ++− +++ = sittiqqidd itthifficc iqqiffeibb iddiccibba 212121 212121 212121 212121 ρ (114) Obviously, the correspondence between Clifford algebra and quantum mechanics still holds also at the present order. The ρ of the (114) is still a member of the Clifford algebra A(Si) that in fact, on the basis of the (111) may be written in the following manner ) 4 () 4 ( 3303300033300300 EEEE e EEEE a −−+ + +++ =ρ + ) 4 ( 33300300 EEEE h −−+ + ) 4 ( 33300300 EEEE s +−− +     + − + ) 2 () 2 ( 32022 3101 1 EE b EE b +     + − + 2 () 2 ( 20232 1310 1 EE c EE c +     + − − ) 2 () 2 ( 21122 2211 1 EE d EE d +     − + + ) 2 () 2 ( 21122 2211 1 EE f EE f +             − + − 2 ) 2 ( 20232 1310 1 EE q EE q +     − + − ) 2 () 2 ( 02322 3101 1 EE t EE t (115) It is in A(Si). Applying the previous criterium we must now pass from A(Si) to 1,±iN . Let us start considering for 33E the numerical value +1 and this is to say that or 13003 +== EE or 13003 −== EE . On the basis of such condition of the measuring instrument, by inspection of the (115) it is seen that the terms by e and h go to zero. It remains the term by a for 13003 +== EE and the term in s for 13003 −== EE . All the other terms containing ib , ic , id , if , iq , it ( 2,1=i ) go to zero and the wave function collapse has happened. Let us explain as example as the term 2 3202 EE + (116) pertaining to 2b , goes to zero. Remember that we have attributed to 33E the value +1. By inspection of the (112) ,one sees that the basic algebraic A (Si) set in which 33E enters is ( ),, 333201 EEE . Passing from the algebra A to the algebra 1,+iN (in fact we have attributed to 33E the numerical value +1) we obtain the new commutation rule that iEE =3201 . (117) On the other hand, considering the basic algebraic A(Si) set ( 030201 ,, EEE ) of the (112) with attribution to 03E the numerical value -1, we have the new commutation rule that iEE −=0201 (118) In conclusion we have that iEE 0132 = (119) and 2 3202 EE + = 0 22 01010102 = +− = + iEiEiEE (120) Following the same procedure , one obtains that also the other interference terms are erased and in conclusion, passing from the algebra A (Si) to the 1,±iN , one obtains that the wave-function collapse has happened. If 13003 +== EE ( )133 +=E from the (115) we obtain aM =→ ρρ (121) If 13003 −== EE ( )133 +=E from the (115) we obtain sM =→ ρρ (122) and the collapse has happened. CONCLUSION In section three , following Y. Ilamed and N. Salingaros 10 , we have given proof of two theorems (n.1 and n.2) on two existing Clifford algebras , the Si and the iN . Such two algebras are of course well known in Clifford algebraic framework 10 , the first holding with isomorphism with Pauli matrices , the second representing the well known dihedral Clifford algebra iN . We also gave previously a very preliminary proof of such theorems by exposition at the conference on Reconsideration of Quantum mechanics Foundations in Vaxjio – Sweeden 15 . The substance of the results that we obtain in the present paper is that we may pass from the algebra Si to 1,±iN attributing to one of the abstract elements ( 321 ,, eee ) a direct numerical value (as example , consider 3e attributing to it the value +1 and thus passing from Si to 1,+iN or attributing to 3e the value –1 and thus passing from the algebra Si to 1,−iN . The algebra Si has its commuation rules based on the abstract elements ( 321 ,, eee ), the algebra 1,+iN has its three abstract elements ( );, 21 iee and its basic commutation rules while the algebra 1,−iN has its three abstract elements ( );, 21 iee and its basic commutation rules . We foresee the possibility of a profound implication for the quantum measurement problem based on existence of such two Clifford algebras Si and 1,±iN , and , in particular, on the basic feature that has been shown in section three, that 1,±iN , may be obtained from Si by direct attribution , as example to 3e , of a direct numerical value (+1 or –1) . The reason is that when , given a quantum system S, we arrive to attribute to S a definite numerical value for some selected quantum observable , say 3e , actually this happens because we measure S with a proper measuring apparatus "reading" the numerical value +1 or –1 , respectively. This reason has motivated us to introduce a criterium. A quantum system without direct observation and actualization , induced from a proper measuring apparatus, has its Clifford algebraic counterpart in the Clifford algebraic structure Si while the collapse , happening on the considered system during the proper actualization by an instrument apparatus, may be described passing from the algebra Si to 1,±iN . We have given three cases of application of such criterium showing in detail that it holds . On the other hand, there are still other basic considerations that in some manner legitimate the choice of such criterium. In section three, in the (65)- (77) we have re-obtained , as expected, the results of von Neumann projection postulate in quantum measurement. It is important to observe that we have re-obtained von Neumann projection postulate using only the Si algebra. Therefore, we have given a justification of von Neumann projection postulate showing that it is articulated only in the Si Clifford algebra. On the other hand , by using only the Si Clifford algebra one shows (see Appendix A) that one may obtain a rough scheme of quantum mechanics as shown in detail elsewhere 16 . Finally, in order to confirm still that , passing from Si to 1,±iN , we have a description of quantum wave function collapse, we may also add two final considerations. The first is that remaining in a geometric interpretation of Clifford algebra one has that 1s 1v D = 1 1s 2v 1b D = 2 1s 3v 3b 1t D = 3 where s means scalar, v means vector, b means bivector, t means trivector. To describe standard Si one needs D=3 that is 1 scalar, 3 vectors , 3 bivectors and 1 trivectors that is the imaginary unity i of complex numbers . This is a classical two state quantum system with quantum dimension d=2, Hilbert space. When we pas to 1,±iN , we have 1s 2v 1b D = 2 one needs 1 scalar, 2 vectors , 1 bivector . The dimension has been decreased to D= 2 . In correspondence the dimension d of the quantum system has become d=1, Hilbert space. The system has collapsed. The second consideration is based on the following reasoning. In the Si , )3(Cl Clifford algebra , we have two elements )1( 2 1 3e±=±ε (123) that are idempotent, better they are primitive idempotents , as we outlined in the (26). The sets )3(Cl ±ε and )3(Cl±ε are left and right ideals in )3(Cl in Si. They are vector spaces of complex dimension 2 and the identification of i with complex imaginary unity makes each of them identical to 2C . A spinor is precisely an element of a two dimensional representation space for the group SL(2;C), which is 2C . Let us first consider +ε)3(Cl . If we chose an arbitrary frame : +=      ε 0 1 and       =+ 1 0 1εe (124) we may decompose any arbitrary element, that is to say +∈∀ εφ )3(Cl , 2 2 1 C∈        = φ φ φ (125) A similar procedure applies to )3(Cl+ε , choosing the basis (1 0) = +ε and (0 1) = 1e+ε . We may also look at ref. 17 for further details. This is in Si. Now , if we calculate 2 ) 2 1 ( 21311 ieee ee − = + =+ε (126) that holds in Si, it gives       =+ 1 0 1εe (127) When instead we pass to 1,1 +N , since we have in 1,1 +N that 21eei = , we obtain that 2 ) 2 1 ( 21311 ieee ee − = + =+ε = 0 (128) that is to say       =+ 1 0 1εe 0≡ (129) The collapse has happened. APPENDIX A We may now derive a rough scheme of quantum mechanics using the Si Clifford algebraic framework 18,16 . Consider in Si the three abstract basic elements, ie , with 3,2,1=i that , as we know, are submitted to the following basic postulate: 1 2 1 =e , 1 2 2 =e , 1 2 3 =e (A.1) If we consider the ie ( 3,2,1=i ) as abstract quantum entities, we may conclude that they have an intrinsic randomness that is their essential irreducible nature. This of course happens also for quantum events. In the algebra A (Si) the ie ( 3,2,1=i ) have the intrinsic potentiality that we may attribute them or the numerical value +1 or the numerical value –1 . A generic member of our algebra A(Si) is given by i i iexx ∑ = = 4 0 (A.2) with ix pertaining to some field R or C . Since the ie are abstract quantum entities, having the potentiality that we may attribute them the numerical values, or 1± , and they have an intrinsic and irreducible randomness , we may admit to be )1(1 +p the probability that 1e assumes the value )1(+ and )1(1 −p the probability that it assumes the value 1− , so that we have its mean value that is given by )1()1()1()1( 111 −−+++>=< ppe (A.3) Considering the same corresponding notation for the two remaining basic elements, we may introduce the following mean values: )1()1()1()1( 222 −−+++>=< ppe , (A.4) ).1()1()1()1( 333 −−+++>=< ppe We have 11 +≤>≤<− ie )3,2,1(=i (A.5) Selected the following generic element of the algebra A(Si): i i iexx ∑ = = 3 1 R∈ix (A.6) Note that 2 3 2 2 2 1 2 xxxx ++= (A.7) Its mean value results to be ><+><+><>=< 332211 exexexx (A.8) Let us call 2/12 3 2 2 2 1 )( xxxa ++= (A.9) so that we may attribute to x the value a+ or a− We have that aexexexa ≤><+><+><≤− 332211 (A.10) The (A.8) must hold for any real number ix , and, in particular, for >=< ii ex so that we have that axxx ≤++ 23 2 2 2 1 that is to say aa ≤2 1≤→ a so that we have the fundamental relation 123 2 2 2 1 ≤><+><+>< eee (A.11) This is the basic relation we are writing in our Clifford algebraic quantum like scheme of quantum theory. Let us observe some important things: (a) First of all it links the Clifford algebra A(Si) with the 1,1 ±N . In absence of measurement, that is to say in absence of direct observation of one quantum entity ie the (A.11) holds. (b) If we attribute instead a definite numerical value to one of the three quantum entities, as example we attribute to 3e the numerical value +1 , we have 13 >=< e , the (A.11) operates now in the 1,+iN algebra , reduced to 022 2 1 =><+>< ee , 021 >=>=<< ee , (A.12) and we have complete, irreducible, indetermination for 1e and for 2e . This is an excellent example of the profound link existing between quantum phenomenology with and without direct observation expressed in a pure algebraic framework. ( c ) Finally, the (A.11) affirms that we never can attribute simultaneously definite numerical values to two basic non commutative elements ie Still let us examine another important consequence of our rough quantum mechanical scheme. As previously evidenced, in Clifford algebra A we have idempotents. Let us consider again two of such idempotents: 2 1 3 1 e+ =ψ and 2 1 3 2 e− =ψ (A.13) Let us consider the mean values of (A.13). We have that ><+>=< 31 12 eψ and ><−>=< 32 12 eψ (A.14) Using the last equation in (A.4) we obtain that 2 1 )1( 33 ><+ =+ e p and 2 1 )1( 33 ><− =− e p (A.15) Therefore, we have that >=<+ 13 )1( ψp and >=<− 23 )1( ψp (A.16) This is to say that probabilities 1,1,3 −+p are the mean values of the idempotents. The same result holds obviously when considering the basic elements 1e or 2e . Considering that in quantum mechanics (Born probability rule), given wave functions −+,φ , we have −+−+ = , 2 , pφ (A.17) we conclude that ( ) 113 θψφ ie><=+ and 223 )( θψφ ie><=− (A.18) and we have given proof that our rough scheme of quantum mechanics foresees the existence of wave functions as exactly traditional quantum mechanics makes. ACKNOWLEDGMENT I am deeply indebted with prof. 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