EJTP 8, No. 25 (2011) 109–126 Electronic Journal of Theoretical Physics On the Logical Origins of Quantum Mechanics Demonstrated By Using Clifford Algebra: A Proof that Quantum Interference Arises in a Clifford Algebraic Formulation of Quantum Mechanics Elio Conte∗ Department of Pharmacology and Human Physiology – TIRES – 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 Received 6 July 2010, Accepted 10 February 2011, Published 25 May 2011 Abstract: We review a rough scheme of quantum mechanics using the Clifford algebra. Following the steps previously published in a paper by another author [31], we demonstrate that quantum interference arises in a Clifford algebraic formulation of quantum mechanics. In 1932 J. von Neumann showed that projection operators and, in particular, quantum density matrices can be interpreted as logical statements. In accord with a previously obtained result by V. F Orlov , in this paper we invert von Neumann's result. Instead of constructing logic from quantum mechanics , we construct quantum mechanics from an extended classical logic. It follows that the origins of the two most fundamental quantum phenomena , the indeterminism and the interference of probabilities, lie not in the traditional physics by itself but in the logical structure as realized here by the Clifford algebra. c© Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Quantum Mechanics; Quantum Interference; Clifford Algebra PACS (2010): 03.65.-w; 42.50.-p; 03.65.Fd 1. Introduction It is well known that in 1936 Birkhoff and von Neumann extended conventional quantum mechanics by using quaternions in place of complex numbers in order to represent the wave-functions and probability amplitudes in such a theory [1]. Starting with 1972 [2], we began to apply Clifford algebra to a quantum mechanical framework, and our effort culminated in 2000 [3], when we were able to reformulate the whole standard framework ∗ elio.conte@fastwebnet.it 110 Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 of quantum mechanics by a rough algebraic scheme, showing in detail that, by this rough formulation, we may re-obtain all the standard quantum theory, and in particular we may give proof of the existing quantization of systems, of basic features of harmonic oscillator, of orbital angular momentum, of hydrogen atom, as well as of time evolution of quantum systems, arriving to consider also the well known EPR problem of quantum mechanics [4,5], the non locality, the Kochen and Specker theorem [5] as well as the more recent Bell approach [7]. We covered all the standard features of quantum mechanics. Clifford algebra gives an unifying framework of physical knowledge here including quantum mechanics, relativity, electromagnetism and other physical matter. When we introduce a Clifford rough scheme of quantum mechanics, we cannot ignore the emerging salient feature of this formulation. It is that in this case we obtain a quantum mechanical theoretical framework invoking only an algebraic structure that does not contain any further specific requirement. This is a very important and salient feature of this algebraic structure. Under a restricted and more methodological profile, it must be outlined with clearness that such our approach to quantum mechanics does not give a formulation with alternative and entirely new ideas in quantum mechanics. In fact, recalling as example the contributions given in [1], we outline that our elaboration results at least consistent with old ideas that were known from the earliest days of quantum theory, although it contains some, few, but very interesting new features that possibly deserve careful consideration. In our effort there is, first of all, a research finality that is evident. There is also a didactic finality. Starting with 2003 [8,910,11,12,13,16,17,20,26], we have performed a number of experiments showing that the well known quantum interference effect may be observed in perceptive-cognitive processes of human subjects, and, in particular, during their perception and cognition of ambiguous figures. We will not enter here in the theoretical and experimental details of such new obtained results [7], but their importance under the perspective to establish if or not the psychological functions of human subjects involve also the domain of quantum mechanics, results rather evident. In a recent contribution we tested the possible Bell quantum violation in mental states, and we introduced for the first time Clifford algebraic elements as basic quantum mental observables [17]. Such new results in psychological and neurophysiological studies open interesting perspectives, and lead as consequence that a whole set of researchers as psychologists and neurologists could be interested to approach the whole scheme of quantum mechanics in order to fully understand the possible potentialities of such theory when we attempt to explain by it some basic features of our mind and of our human thinking. In brief, a rough approach to quantum mechanics is required since it is well known that such researchers in some cases may have not the necessary mathematical competence to approach the sophisticated and standard formalism of the traditional quantum mechanics. In these cases, our simplified, rough , Clifford scheme of quantum mechanics, without violating the requirements of the scientific rigour that is required, but allowing at the same time some derogation under the strict formal profile, enables as counterpart to acquire the basic conceptual foundations of a theory, and thus to apply such new knowledge in the proper but distant sphere of competence. This is the basic objective that the effort, given in [3], may substantially reach Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 111 at the previously mentioned didactic profile. Clifford algebra uses few basic rules and algebraic elements. Therefore, the advantage to use such so restricted algebraic framework in a didactic perspective to acquire knowledge of the basic foundations of quantum mechanics, may be of importance for the researchers who unfortunately may have not direct competence in the standard language of quantum mechanics that, as it is well known, holds about the abstract field of Hilbert space and of acting linear operators. Of course, the finality to introduce such alternative didactic patterns is not new here. We remember, as example, the excellent book of T.F. Jordan [18], that reaches the same objective using the simple matrix form, and that inspired so much our elaboration. In any manner, our principal objective is to evidence the profound existing link between quantum mechanics and Clifford algebra since this specific link opens some basic questions that are of relevant and basic interest for the same widening of the basic foundations, nature and meaning of quantum mechanics in the whole complex of unifying physical knowledge. It offers also didactic opportunities. Therefore, our selected objective remains confined to the analysis and examination of such existing link. To this purpose, it may be of interest the result that we obtain in the present paper. We proof that quantum interference leads necessarily to a Clifford algebraic formulation of quantum mechanics. Consequently, it adds still rigour to the Clifford algebraic formulation of quantum mechanics . 2. A Clifford Algebraic Rough Scheme of Quantum Mechanics Let us explain briefly the basic framework of our approach [3]. Let us give a proper definition of Clifford algebra. By using the Clifford algebra in our rough quantum mechanical scheme it is intended that, specifically, a Clifford algebra is a unital associative algebra which contains and is generated by a vector space V equipped with a quadratic form Q. The Clifford algebra C(V ,Q) is the algebra generated by V subjected to the condition v2 = Q(v) for all v ∈ V where uv + vu = 1/2(Q(u, v)−Q(u)−Q(v)) is the symmetric bilinear form associated to Q. We will utilize and follow the work that, starting with 1981, was developed by Y. Ilamed and N. Salingaros [19], using sometimes the same technique that these authors introduced in their work. Let us anticipate that only two basic assumptions, quoted as (a) and (b), are required in order to formulate such rough scheme of quantum mechanics. Let us consider three abstract basic elements, ei, with i = 1, 2, 3, and let us admit the following two assumptions : (1) it exists the scalar square for each basic element: e1e1 = k1 , e2e2 = k2, e3e3 = k3 with ki ∈  (1) 112 Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 In particular we have also that e0e0 = 1. (2) The basic elements ei are anticommuting elements, that is to say: e1e2 = −e2e1 , e2e3 = −e3e2, e3e1 = −e1e3. (2) In particular it is eie0 = e0ei = ei. Note that, owing to the axioms (a) and (b), we consider the given basic elements ei(i = 1, 2, 3) as abstract entities that we call potentialities, given in a numerical field since do not exist actual numerical entities satisfying both the (1) and the (2) simultaneously. In detail, by the (1), the ei have the potentiality to simultaneously assume the numerical values ±k1/2i . According to [34], let us introduce the necessary and the sufficient conditions to derive all the basic features of the algebra that we have just introduced. To give proof, let us consider the general multiplication of the three basic elements e1, e2, e3, using scalar coefficients ωk, λk, γkpertaining to some field: e1e2 = ω1e1 + ω2e2 + ω3e3; e2e3 = λ1e1 + λ2e2 + λ3e3; e3e1 = γ1e1 + γ2e2 + γ3e3. (3) Let us introduce left and right alternation: e1e1e2 = (e1e1)e2; e1e2e2 = e1(e2e2); e2e2e3 = (e2e2)e3; e2e3e3 = e2(e3e3); e3e3e1 = (e3e3)e1; e3e1e1 = e3(e1e1). (4) Using the (4) in the (3) it is obtained that k1e2 = ω1k1 + ω2e1e2 + ω3e1e3; k2e1 = ω1e1e2 + ω2k2 + ω3e3e2; k2e3 = λ1e2e1 + λ2k2 + λ3e2e3; k3e2 = λ1e1e3 + λ2e2e3 + λ3k3; k3e1 = γ1e3e1 + γ2e3e2 + γ3k3; k1e3 = γ1k1 + γ2e2e1 + γ3e3e1. (5) From the (5), using the assumption (b), we obtain that ω1 k2 e1e2 + ω2 − ω3 k2 e2e3 = γ1 k3 e3e1 − γ2 k3 e2e3 + γ3; ω1 + ω2 k1 e1e2 − ω3 k1 e3e1 = −λ1 k3 e3e1 + λ2 k3 e2e3 + λ3; γ1 − γ2 k1 e1e2 + γ3 k1 e3e1 = −λ1 k2 e1e2 + λ2 + λ3 k2 e2e3 (6) For the principle of identity , we have that it must be ω1 = ω2 = λ2 = λ3 = γ1 = γ3 = 0 (7) Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 113 and −λ1k1 + γ2k2 = 0 γ2k2 − ω3k3 = 0 λ1k1 − ω3k3 = 0 (8) The (8) is an homogeneous system admitting non trivial solutions since its determinant Λ = 0, and the following set of solutions is given: k1 = −γ2ω3, k2 = −λ1ω3, k3 = −λ1γ2. (9) Admitting k1 = k2 = k3 = +1, it is obtained that ω3 = λ1 = γ2 = i (10) In this manner, using the (3), a theorem, showing the existence of such algebra, is proven. The basic features of this algebra are given in the following manner e1e2 = −e2e1 = ie3; e2e3 = −e3e2 = ie1; e3e1 = −e1e3 = ie2; i = e1e2e3 (11) The content of this theorem is thus established: given three abstract basic elements as defined in (a) and (b), an algebraic structure is established with four generators (e0, e1, e2, e3). Of course, as counterpart, the (11) are well known also in quantum mechanics and the isomorphism with Pauli's matrices at various orders is well known and discussed in detail in [34]. Here, they have been derived only on the basis of two algebraic assumptions, given respectively in (a) and (b). We may now add some comments to the previous formulation. Let us attempt to identify the phenomenological counterpart of the algebraic structure given in (1), (2), and (11) with e21 = 1, e 2 2 = 1, e 2 3 = 1 (12) A generic member of our algebra is given by x = 3∑ i=0 xiei (13) with xi pertaining to some field or C. The (12) evidences that the ei are abstract potential entities, having the potentiality that we may attribute them the numerical values, or ±1. Admitting to be p1(+1)the probability to attribute the value (+1)to e1 and p1(−1) the probability to attribute (−1), considering the same corresponding notation for the two remaining basic elements, we may introduce the following mean values: < e1 >= (+1)p1(+1) + (−1)p1(−1), < e2 >= (+1)p2(+1) + (−1)p2(−1), < e3 >= (+1)p3(+1) + (−1)p3(−1). (14) Selected the generic element of the algebra, given in (13), its mean value results < x >= x1 < e1 > +x2 < e2 > +x3 < e3 > (15) 114 Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 Let us call a = x21 + x 2 2 + x 2 3 (16) so that −a ≤ x1 < e1 > +x2 < e2 > +x3 < e3 > ≤ a (17) and −1 ≤< ei > ≤ +1 i = (1, 2, 3) (18) The (17) must hold for any real number xi, and, in particular, for xi =< ei > so that we have the fundamental relation < e1 > 2 + < e2 > 2 + < e3 > 2≤ 1 (19) See details of this proof in ref. [32] for quantum mechanics in simple matrix form and for its extension in Clifford algebra in ref. [3] Let us observe some important things: (1) The (19), owing to the (14), says that probabilities for basic elements ei are not independent and this is of basic importance to acknowledge the essential features of a rough quantum mechanical scheme. (2) The (19) still says that also mean values of ei are not independent. In detail, the (19) may be considered to represent a general principle of ontic potentialities. We have here a formulation of a basic, irreducible, ontic randomness. In particular, it affirms that we never can attribute simultaneously, definite numerical values to two basic elements ei. Let us consider, as example, < e3 >= +1, that is to say that e3 → +1, we have consequently that < e1 >=< e2 >= 0, that is to say that e1 and e2 are both in a complete condition of randomness. The values are equally probable, there is full indetermination. We have a condition of ontic potentiality. In conclusion, by using only the axioms (a) and (b), by the (11), the (14) and the (19), we have delineated a rough scheme of quantum theory using only an algebraic structure. Let us observe that the elective role in our formulation is performed in particular from the axiom (b) that relates non commutativity of the basic elements. In this algebraic scheme some principles of the basic quantum theoretical framework result to be represented. In particular, this algebraic structure reflects the intrinsic indetermination and the ontic potentiality that are basic components of quantum mechanics. This means that, in absence of a direct numerical attribution, such basic elements are abstract entities that act having an intrinsic, irreducible, indetermination, an ontic randomness, an ontic potentiality. Therefore, by using such rough quantum mechanical scheme, we may explore what is the actual role of potentiality in nature, what is its manner to combine with actual elements of our reality and what is the manner in which potentiality may contribute to the general dynamics of systems in Nature. Let us add still some other feature of the scheme that we have in consideration. Let us consider two generic elements of our algebra, given as in the (13), and let us indicate Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 115 them by x and y. Owing to the (11), they will result in general not commutative, that is to say xy = yx (20) However, under suitable conditions, non-commutativity may fail and such abstract entities return to have the actual and traditional numerical role in some selected field. In this condition we have that xy = yx Starting with 1974, [2] we introduced a theorem showing that necessary and sufficient condition for two given algebraic elements, x and y, to be commutative is that xy = yx↔ xj = λ yj , ∀λ(j = 1, 2, 3) (21) This theorem regulates the passage from potentiality of abstract elements in this algebra to actual numerical values relating instead any numerical field of our direct experience. In quantum mechanics this passage from potentiality to actualisation is called the collapse of wave function. An important feature of the theorem given in (21) is that the algebraic structure given in (1), (2), (11), and (19) admits idempotents. Let us consider two of such idempotents: ψ1 = 1 + e3 2 and ψ2 = 1− e3 2 (22) It is easy to verify that ψ21 = ψ1 and ψ 2 2 = ψ2. Let us examine now the following algebraic relations: e3ψ1 = ψ1e3 = ψ1 (23) e3ψ2 = ψ2e3 = −ψ2 (24) Similar relations hold in the case of e1or e2. The relevant result is that the (23) establishes that the given algebraic structure, with reference to the idempotent ψ1, attributes to e3 the numerical value of +1while the (24) establishes that, with reference toψ2, the numerical value of -1 is attributed to e3. The conclusion is very important: the conceptual counter part of the (23) and (24) is that we are in presence of a self-referential process. On the basis of such self-referential process, as given in (23) and in (24), this algebraic structure is able to attribute a precise numerical value to its basic elements. Each of the three basic elements may " transitate" from the condition of pure potentiality to a condition of actualization, that is to say, in mathematical terms, from the pure, symbolic representation of their being abstract elements to that one of a real number. Let us remember that, on the basis of the (19), this self-referential process may regard each time one and only one of the three basic elements. It is well known that self-referential processes relate the basic phenomenology of our mind and consciousness. In conclusion, for the first time we have an algebraic structure that represents a rough quantum mechanical scheme and that, at the same time, evidences, on the basis of a selfreferential process, that it is possible a transition from potentiality to actualization. Other features of our formulation are given in [2,3,11]. It remains to evidence that a profound 116 Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 link exists between the idempotents prospected as example in the (22) and the traditional wave function that is introduced in standard quantum mechanics. Let us consider the mean values of (22). We have that 2 < ψ1 >= 1+ < e3 > and 2 < ψ2 >= 1− < e3 > (25) Using the last equation in (14) we obtain that p3(+1) = 1+ < e3 > 2 and p3(−1) = 1− < e3 > 2 (26) Therefore, considering the (22), we have that p3(+1) =< ψ1 > and p3(−1) =< ψ2 > (27) The same result holds obviously when considering the basic elements e1 or e2. Considering that in quantum mechanics (Born probability rule), given the wave function φ+,−, we have |φ+,−|2 = p+,− (28) we conclude that φ3 (+) = √ < ψ1 >e iθ1 and φ3(−) = √ < ψ2 >e iθ2 (29) and we have given proof that our rough scheme of quantum mechanics foresees the existence of wave functions as exactly traditional quantum mechanics makes. We need here to make an important digression. Quantum mechanics runs usually about some fixed axioms. States of physical systems are represented by vectors in Hilbert spaces : historically, theoretical physicists as Planck, Bohr, Heisenberg, Pauli, Born, Dirac, established the rather general and consistent quantum mechanics in the form that is presently known to day. The question on the manner in which systems behave sometimes like particles and sometimes like waves as well as the question about the exact meaning of the complex wave functions are usually retained to represent examples of open question in the theory. In our opinion there is often no matter for such questions , and this is evidenced in our formulation about the rough quantum mechanical scheme by Clifford algebra. We consider the quantum wave function as the first evidence of the strong link existing between cognitive performance and linked physical description at some stages of our reality. Of course, we retain that superposition and interference effects by wave functions play a key role. We support that wave intensities and probability densities are not a matter of simple interpretation, that is added to quantum mechanics as it may be established evaluating that the Born probability rule was in fact introduced and thus added to quantum mechanics for purposes of probabilistic interpretation of quantum theory. It is no matter of a so simple Born interpretation. There is instead a precise theorem, proved and published well before quantum mechanics, that shows the fundamental role of the superposition principle and the profound link existing between quantum wave functions and probability densities. The theorem was published in 1915 Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 117 by Fejer and by Riesz [21,22]. There is an excellent paper by F.H. Frohner that, time ago, properly evidenced the profound existing link between probability theory and quantum mechanics [23]. For any purpose, we retain of importance to report here this theorem that states 0 ≤ ρ(x) ≡ n∑ l=−n cle ilx ≡ ∣∣∣∣∣ n∑ k=0 ake ikx ∣∣∣∣∣ 2 ≡ |ψ(x)|2 where the complex Fourier polynomial ψ(x) has not restrictions, where instead to the Fourier polynomial ρ(x) is imposed the requirement of its reality and non-negativity. So, in conclusion, such required link exists and it is mathematically established. This is the matter in spite of the continuous claims that in quantum mechanics such link holds only on the basis of a given Born's interpretation. Let us look now to another link existing between standard quantum mechanics and our rough quantum mechanical scheme. It is well known the central role that is developed in traditional quantum mechanics from density matrix operator . In our scheme of quantum mechanics, we have the corresponding algebraic member that is given in the following manner ρ = a+ be1 + ce2 + de3 (30) with a = |c1| 2 2 + |c2| 2 2 , b = c∗1c2 + c1c ∗ 2 2 , c = i(c1c ∗ 2 − c∗1c2) 2 , d = |c1|2 − |c2|2 2 (31) where in matrix notation, e1,e2, ande3 are the well known Pauli matrices. The complex coefficients ci(i = 1, 2)are the well known probability amplitudes for the considered quantum state ψ = ⎛ ⎜⎝ c1 c2 ⎞ ⎟⎠ and |c1|2 + |c2|2 = 1 (32) For a pure state in quantum mechanics it is ρ2 = ρ. In our scheme a theorem may be demonstrated that ρ2 = ρ↔ a = 1 2 and a2 = b2 + c2 + d2 (33) The details of this our theorem are given in [24,25]. Written in matrix form we have also Tr(ρ) = 2a = 1. In this manner we have the necessary and sufficient conditions for ρ to represent a potential state or, in traditional quantum mechanics, to have a superposition of states. We have to examine now quantum time evolution. It is clear that the quantum like scheme we are discussing is based on the two dimensional abelian subalgebra of the four dimensional Clifford algebra. Of course, generally speaking, we are considering our quantum rough scheme using quantum like operators acting on vectors of a given Hilbert space. 118 Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 For time evolution, we consider Heisenberg description. Given the operator α connected to some observable A, the mean value at a given time t will be given as < αt >= (ψ0, U −1αUψ0) with U time evolution operator. It is well known that we have ih d < α > dt = ih < ∂α ∂t + 1 ih [α,H] > (34) and dα dt = ∂α ∂t + 1 ih [α,H] (35) where His the Hamiltonian of the system. The manner in which such Hamiltonian may be constructed for psychological states in the Clifford algebra framework is given in [8]. It is well known that members of Clifford algebra transform according to e ′ i = U +eiU, U +U = 1 (36) In [3] we give a rigorous proof of the (34) and the (35) using the Clifford algebra. Still we have to remember here that in the past there were attempts to go beyond the linear Schrodinger equation [14,15,27,28,29,30], but, as well as we know, nobody tried to do the same thing in the Heisenberg's picture. It is very important to outline here that in the non linear case, such two, Heisenberg and Schrodinger, representations, no more result to be equivalent. We have in fact that U = exp(−ihHt) = 1− i h Ht+ ( iH h )2 t2 2! − (iH h )3 t3 3! + ........................ (37) and dU dt = − iH h [ 1− i h Ht+ ( iH h )2 t2 2! + ........... ] (38) By using the Clifford rough scheme of quantum mechanics we are in the condition to take account also for such possible non linear processes in Heisenberg like quantum representation. 3. Proof that Quantum Interference Arises in a Clifford Algebraic Formulation of Quantum Mechanics and the Irreducible, Ontic Randomness of Basic Clifford Algebraic Elements Consider a beam of particles impinging on a beam splitter A so that randomly may be either reflected to proceed a path L1 or transmitted to proceed along the path L2(Fig.1). At the end of L1, the particles impinge on the upper side of a second beam splitter , Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 119 B, and it may be either reflected and detected by the detector D1 or transmitted and detected by the detector D2.The particles arriving from path L2, impinge on the opposite side of to be either transmitted reaching the detector D1 or reflected to reach the counter D2. As it is well known we are considering here the interference pattern of a beam of particles passing through a Mach Zender interferometer. The considered random variable A assumes the value a = +1in the case of reflection and the value a = −1 in the case of transmission. The random variable Bassumes the value b = +1in the case of reflection and the value b = −1 in the case of transmission. We have a third variable C = ABthat is determined by the product of the values of A and B. In analogy with the rough quantum scheme previously developed we call still write the mean value of A by < A > and < A >= (a = +1)pab + (a = −1)pab (39) the mean value of Bby < B > and < B >= (b = +1)pab + (b = −1)pab (40) and the mean value of C by < C > and < C >= (ab; a = +1, b = +1)pab + (ab; a = +1, b = −1)pab +(ab; a = −1, b = +1)pab + (ab; a = −1, b = −1)pab (41) Let us follow directly the argument as it was recently developed in [31]. According to this interesting paper , we may write easily the expression of the probability for the corresponding four alternatives (a = ±1, b = ±1) in the following manner pab = 1 4 (1 + ax+ by + abz) (42) 120 Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 where x ≡< A >, y ≡< B >, z ≡< C > . (43) Still according to ref. [31] let us calculate the probability for counting the detector D1. We have that p++ = 1 4 (1 + x+ y + z) and p−− = 1 4 (1− x− y + z) (44) so that in the detector D1 we have pD1 = p++ + p−− = 1 2 (1 + z) = 1 2 (1+ < C >) (45) In the case of the detector D2, we have p+− = 1 4 (1 + x− y − z) and p−+ = 1 4 (1− x+ y − z) (46) and pD2 = p+− + p−+ = 1 2 (1− z) = 1 2 (1− < C >) (47) This is of course the classical statistical argument holding on an epistemic interpretation of randomness . In order to introduce the quantum like elaboration the author in ref.[31] correctly introduced three new variables: U = αA+ β B + γ C with α2 + β2 + γ2 = 1; (48) V = λA+ μB + ν C , with λ2 + μ2 + ν2 = 1, αλ+ βμ+ γν = 0 (49) and W = δ A+ ωB + θC with δ = βν − γμ;ω = γλ− αν, θ = αμ− βλ, (50) and considered < U >= u (51) plus < V >=< W >= 0 (52) in order to take into account a complete indetermination in the case of variables V and W . Following this argument one obtains α < A > +β < B > +γ < C >= u ; (53) λ < A > +μ < B > +ν < C >= 0; δ < A > +ω < B > +θ < C >= 0 that admits solutions < A >= αu , < B >= β u, < C >= γ u. (54) Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 121 Inserting the (54) in the (42), one obtains pab = 1 4 [1 + (α a+ β b+ γ ab)u] (55) and pD1 = p++ + p−− = 1 2 (1 + γ u) (56) and pD2 = p+− + p−+ = 1 2 (1− γ u) (57) Let us comment the obtained results. First consider the classical case. Probability given in (42) must be between the well known limits 0 ≤ pab ≤ 1 (58) Consequently, still according to the findings in ref.[31], < A >,< B >,< C >, are the coordinates of a point inside the equilateral octahedron having the vertices < A >= ±1, < B >= 0, < C >= 0; < B >= ±1, < A >= 0, < C >= 0; < C >= ±1, < A >= 0, < C >= 0 The author in ref. [31] correctly argues that the first limiting values correspond to the case of pure reflection (transmission) by A and equally probable reflection and transmission by B and zero correlation. The second limiting values correspond to equally probable reflection and transmission by A followed by pure reflection by B and zero correlation, and the third limiting values correspond to the case of complete correlation between the two splitters with equally probable transmission and reflection by A and B. These are the limiting cases while for the other possible conditions we have average values of the considered random variables having values less than one. This means that always particles with both the values (±1) are present. We have that −1 ≤< A > + < B > + < C >≤ +1 (59) According to the (54) in the case of the (48)-(50) and (51)-(52) , we have that −1 ≤ (α + β + γ) u ≤ +1 (60) which implies that the absolute value of uis always smaller than one. Particles with both values (±1) of A,B,C are always present, [31]. We may now explore the quantum case. Instead of the (45) and the (47) , of the (56) and the (57) , the correct probabilities in quantum theory result to be pD1 = p++ + p−− = 1 2 (1 + γ ) (61) and pD2 = p+− + p−+ = 1 2 (1− γ ) (62) 122 Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 that result to be pD1 = p++ + p−− = 1 2 (1 + cosφ ) (63) and pD2 = p+− + p−+ = 1 2 (1− cosφ ) (64) This is to say that we must have u = 1(u = −1), and < A >= α, < B >= β, < C >= γ (65) with < A >2 + < B >2 + < C >2= 1 (66) and γ = cosφ as polar angle of the unit vector on the sphere given in (66). This is to say that it must be < U2 >= 1 (67) and < V >=< W >= 0 (68) to assure complete indetermination. Let us consider again the variable U as given in the (48). It results that U2 = (αA+ β B + γ C)(αA+ β B + γ C) = α2 + β2 + γ2 + αβ(AB +BA) + αγ(AC + CA) + βγ(BC + CB) = 1 + αβ(AB + BA) + αγ(AC + CA) + βγ(BC + CB). (69) It is < U2 >= 1+ < αβ(AB + BA) + αγ(AC + CA) + βγ(BC + CB) > . (70) The only way to obtain the (67) is that AB = −BA , AC = −CA, BC = −CB (71) and this is to say that the variables A,B,C must be the basic elements of the Clifford algebra, the ei(i = 1, 2, 3) basic elements that we introduced in the previous section in the (1),(2),(11). It is A ≡ e1 , B ≡ e2, AB ≡ e1 e2 = ie3 (72) Therefore, A,B,C, as given in the (48),(49), and the (50) are members of the Clifford algebra. So we reach the following conclusion. Quantum mechanics holds about the basic phenomenon of quantum interference. We may realize it using the basic elements, and the structure of the Clifford algebra. The Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 123 author in [31] concluded that typical objects of the required kind are Hermitean matrices with eigenvalues (±1). We may now take a step on. Rather recently [32] we gave proof of two theorems on existing two Clifford algebras, the A(Si) that has isomorphism with that one of Pauli matrices, and the Ni,±1 where Ni stands for the dihedral Clifford algebra. The salient feature was that by using such two theorems, we showed that the Ni,±1 algebra may be obtained from the A(Si) algebra when we attribute a numerical value (+1 or –1) to one of the basic elements (e1, e2, e3) of the A(Si). The arising physical model was that the A(Si)−Clifford algebra refers to the representation of the general situation in quantum mechanics where the observer has no right to decide on the state of a two-state system while instead, through the operation represented by Ni,±1 algebra, he finally specifies which state is the one that will be or is being observed. The A(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 Ni,±1 attend when a quantum measurement is performed on such system with advent of wave function collapse. There is another salient feature that needs to be outlined here. As said, under a Clifford algebraic profile, the quantum measurement with wave function collapse induces the passage in the considered quantum system from the A(Si) to Ni,+1or to the Ni,−1 algebras: it is of interest from a mathematical and physical view points to observe that in the passage from A(Si) to N1,±1, each N1,±1 algebra has now its proper rules of commutation that are new and different respect to standard ones calculated in A(Si). Under the profile of a quantum measurement, wave function collapse is thus characterized, at least from an algebraic view point, just from such transition from standard to new commutation rules for the basic algebraic elements. This is an important feature that deserves careful physical consideration. In [32] we re-examined also the well known von Neumann postulate on quantum measurement, and we gave a proper justification of such postulate by using such two theorems. In detail, we studied some application of the above mentioned theorems 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, we found that the passage from the algebra A(Si) to Ni,±1 actually describes the collapse of the wave function. We concluded that the actual quantum measurement has as counterpart in the Clifford algebraic description, the passage from the A(Si) to the Ni,±1, reaching in this manner the objective to reformulate von Neumann postulate on quantum measurement and proposing at the same time a self-consistent formulation of quantum theory. The aim of the present paper has been to propose a step on. As it is well known, quantum mechanics runs about two basic foundations that are the indeterminism and the quantum interference. It is also well known that in 1932 J. 124 Electronic Journal of Theoretical Physics 8, No. 25 (2011) 109–126 von Neumann [34] gave proof that the projection operators and, in particular, quantum density matrices represent logical statements. We may say that he constructed a matrix logic on the basis of quantum mechanics. In the present paper we have re-constructed the two basic foundations of quantum mechanics starting from logic and thus arriving to explain that quantum mechanics has logical origins. Not logic deriving from quantum mechanics as in von Neumann but quantum mechanics having logical origins. This is to say that the two basic foundations of quantum mechanics, the indeterminism and the quantum interference, may be explained on a purely logical basis. In the development of the paper we have used our Clifford rough scheme of quantum mechanics including the theorems shown in [32]. No element of physics has been evoked by us but only the idempotents of Clifford algebra , given in the (22) once again one has admitted the necessary existence of the Clifford basic elements given in the (72). Such idempotents represent of course in quantum mechanics the projection operators that were introduced by von Neumann as logical statements. Therefore , a conclusion seems to be unavoidable. We have to consider the basic foundations of quantum mechanics as basic framework representing conceptual entities [33]. Acknowledgment The author is indebted with the friend and colleague Alessandro Giuliani (Istituto Superiore di Sanità – Rome) for the continuous and stimulating discussions held during the elaboration of the present paper about the fundamental theme on the possibility to represent cognitive processes of mind by the classical and quantum profiles of the physics. 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