Adv. Studies Theor. Phys., Vol. 6, 2012, no. 26, 1289 1307 On Some Considerations of Mathematical Physics: May we Identify Clifford Algebra as a Common Algebraic Structure for Classical Diffusion and Schrödinger Equations? Elio Conte Department of Pharmacology and Human Physiology –TIRES –Center for Innovative Technologies for Signal Detection and Processing University of Bari, Bari, Italy and School of Advanced International Studies for Applied Theoretical and non Linear Methodologies of Physics-Bari-Italy Abstract We start from previous studies of G.N. Ord and A.S. Deakin showing that both the classical diffusion equation and Schrödinger equation of quantum mechanics have a common stump. Such result is obtained in rigorous terms since it is demonstrated that both diffusion and Schrödinger equations are manifestation of the same mathematical axiomatic set of the Clifford algebra. By using both such )( iSA and the 1,±iN algebra, it is evidenced, however, that possibly the two basic equations of the physics cannot be reconciled. 1. INTRODUCTION It is well known that several tools enable us to derive the governing equations of classical dynamics from Lagrange equations and variational principles or directly from classical Newton's laws of motion. In all the cases we arrive to consider differential equations that are admitted to describe the dynamical system under consideration. 1290 E. Conte Very often, however, it is tacitly admitted that they express the universal deterministic behavior of our reality. Instead we outline that determinism is not spontaneously exhibited from such equations. On the contrary, determinism is forced to be exhibited from such equations imposing from the outside that they must satisfy Lipschitz conditions. They are a mathematical restriction that we impose from outside to the set of considered differential equations, and such imposed restriction guarantees the uniqueness of solutions when fixed the initial conditions. However, there are some results that legitimate doubts on the validity of such conditions as universally admitted as it has been largely discussed by us elsewhere [1]. There are many cases in which Lipschitz conditions do not result compatible with the physical or the biological nature of the dynamics in consideration. The finding of Lipschitz violation in different systems regarding biological and physical dynamics, constitutes a promising acquisition in a current attempt to coherently and correctly describe the time dynamics of reality. The discovery of chaos contributed to better understanding the dynamics of evolution of systems as well as the interpretation and modeling of complex phenomena in physics and biology. The discovery of possible Lipschitz violation in a given set of differential equations cleared up on a new step in the framework of such chaotic studies. It follows from Lipschitz violation that a class of phenomena cannot be represented by deterministic chaos. In these cases the behavior of systems is governed by a new kind of dynamics that has been called discrete event dynamics. Here randomness appears as point events so that there is a sequence of random occurrences at fixed or random times but there are not additional components of uncertainty between these times [2-12]. In conclusion, the role of the mathematics is central in physics, and, in this particular case, it evidences in a substantial manner that are possible real conditions in which the determinism, generally assumed as an universal tenet regarding physical and biological dynamics, may be violated. In an excellent paper M. Zak [13-17] gave a non-Lipschitz approach to quantum mechanics. He made an effort to reconcile quantum mechanics with Newton's laws corrected this time by the non Lipschitz formalism. In this paper we should pursue a step one. We would attempt to investigate if we may acknowledge an algebraic structure from which both classical diffusion equation and Schrödinger equation may be derived. It is well known that diffusion equation and Schrödinger equation evidence some profound conceptual differences in physics. We should be able to derive their common algebraic structure, if existing and the mathematical point in which such two basic and fundamental equations of physics bifurcate. 2. INTRODUCTORY MATHEMATICS Let us fix the basic mathematical framework of our paper. The Clifford algebra Cl3 of R3 is the real associative algebra generated by the set of abstract mathematical objects (e1 , e2 ,e3) satisfying the following two basic axioms Clifford algebra as a common algebraic structure 1291 e12 = e22 = e32 = 1 (2.1) and e1 e2 = e2 e1 ; e1 e3 = e3 e1 ; e2 e3 = e3 e2 (2.2) The (2.1) and the (2.2) represent the two basic mathematical axioms of Clifford algebra. In this paper we will show that only such two mathematical axioms are required in order to derive both the diffusion and Schrödinger equations. The employed algebra is 8-dimensional with the following basis 1 the scalar e1, e2, e3 vectors (2.3) e1 e2, e1 e3, e2 e3 bivectors e123 = e1 e2 e3 a volume element An arbitrary element in Cl3 is the sum of a scalar, a vector, a bivector, and a volume element, and it must be written in the following manner q = γ + a + b e123 + μ e123 (2.4) with q ∈ Cl3; γ, μ ∈ R ,and a, b ∈ R3. As in particular we will discuss also in the following section, we are mainly concerned with the problem of a matrix representation of Cl3. Therefore, let us denote the set of 2 x 2 matrices with complex numbers as entries by Mat (2, C). This set may be regarded also as a real algebra with scalar multiplication taken over the real numbers in R also if the matrix entries are in the complex field C. Let us remember that the Pauli spin matrices σ1 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 01 10 , σ2 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 0 0 i i , σ3 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −10 01 (2.5) satisfy the multiplication rules σ12 = σ22 = σ32 = I and (2.6) σ1σ2 = -σ2 σ1 = i σ3 σ3 σ1 = σ1σ3 = i σ2 σ2 σ3 = σ3 σ2 = i σ1 The (2.5) and the (2.6) generate the real algebra Mat(2,C). The correspondence I ↔ 1 σ1,σ2,σ3 ↔ e1, e2, e3 (2.7) σ1σ2 ,σ2σ3 , σ1σ3 ↔ e1e2, e2e3, e1e3 establish an isomorphism between real algebras. We have that Cl3 ≅ Mat (2, C) (2.8) An essential difference between the Clifford algebra Cl3 and Mat (2, C) remains in the fact that in Cl3 we distinguish, by definition, a particular subspace, the vector space R3, while no distinguished subspace is signed in the definition of Mat (2, C). The theory of Clifford algebra includes basically the statement that each Clifford algebra is isomorphic to a matrix representation. 1292 E. Conte Idempotents are members of the previous algebra with the particular well known property that, when squared, give themselves. For Cl2, we have two basic members (basic elements) ei (i = 1,2) (e1e2 = e2e1), and one such idempotent involves only one basic element, i.e., ψ1 = 2 1 ( 1 + e1) , ψ1ψ1 = ψ1 (2.9) If the idempotent is multiplied by the other basic element, e2, other functions can be generated, as it follows: ψ2 = e2ψ1 = ( 2 1 – 2 1 e1) e2 ; ψ3 = ψ1 e2 = ( 2 1 + 2 1 e1) e2; ψ4 = e2ψ1e2 = 2 1 - 2 1 e1. (2.10) In addition, we have also that ψ1 e1ψ1 = 0 The important thing that we must retain for the following arguments is that the four functions ψi (i=1, 2, 3, 4) provide means to represent any member of the space. A general member q is given in terms of the basis members of the algebra in the following manner q = a0 + a1 e1 + a2 e2 + a3 e1e2 (2.11) and q may be represented by the series of terms of the idempotents q= a11 ψ1 + a21ψ2 + a12ψ3 + a22ψ4 (2.12) with a11+a22 = 2 a0 a12 + a21 = 2 a2 (2.13) a11 – a22 = 2 a1 a12 – a21 = 2 a3 On the other hand, calculating ψ1qψ1, ψ1qψ2, ψ3qψ1, ψ3qψ2, one finds that a matrix A may be defined A = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 2221 1211 aa aa (2.14) with a11 = a0 + a1, a22 = a0-a1, a12= a2+ a3 , a21= a2-a3 and (1 e2) A ψ1 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 2 1 e = q = a11ψ1 + a21ψ2+a12ψ3 + a22ψ4 (2.15) Thus, we may conclude that the (2.15) generates the general Clifford number q. When equating q with 1, e1, e2, e1e2, respectively, we obtain the final matrix expressions multiplied by the idempotent: q = 1, A ψ1 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 10 01 ψ1 ; q = e2 , Aψ1 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ 01 10 ψ1 q = e1, Aψ1= ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −10 01 ψ1 ; q = e1e2 , Aψ1 = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − 01 10 ψ1 (2.16) These are the usual basis matrices for Cl2. Clifford algebra as a common algebraic structure 1293 A set of basis matrices for Cl3 may be obtained following the same previous procedure. In this case one obtains that e0 1 0 0 1≡ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ; e1 0 1 1 0= ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ; e i i2 0 0= −⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ; e3 1 0 0 1= − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (2.17) With a proper choice of the idempotent, we may also arrive to express new sets of basis matrices at n = 4 , 8 , .... . They are expressed in the following manner E0i = I1 ⊗ e i ; Ei0 = ei ⊗ I2 (2.18) The notation ⊗ denotes direct product of matrices, and Ii is the ith 2x2 unit matrix. Thus, in the case of n= 4 we have two distinct sets of basis matrices, E0 i and Ei 0, with E i0 2 1= ; Ei0 2 1= ; i = 1, 2, 3; (2.19) E0i E0j = i E0k ; Ei0 Ej0 = i Ek0 ; j = 1, 2, 3; i ≠ j and Ei0 E0j =E0j Ei0 (2.20) with (i, j, k) cyclic permutation of (1, 2, 3). Let us examine now the following result (I1 ⊗ ei) (ej ⊗ I2) = E0i Ej0 =Eji (2.21) We have that E0i Ej0 = Eji with i = 1, 2, 3 and j =1, 2, 3, with E j i 2 = 1, Eij Ekm ≠ Ekm Eij, and Eij Ekm= Epq 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 matrices E0i , Ei0 and, in addition, basic sets of unities (Eij, Eip, E0m) with (j, p, m) basic permutation of (1, 2, 3). Similarly, we may realize other basic sets of basis matrices using (Eji, Epi, Em0). Note the basic role explained from cyclic permutations of (1, 2, 3) and their strong connection with non commutativity of the chosen basic elements. In the case of matrix representation at order n= 8, we have the possibility to introduce three sets of biquaternion basic unities. We will have E00i, E0i0, and Ei00, i = 1, 2, 3 and E00i = I1 ⊗ I1 ⊗ ei ; E0 i0 = I2 ⊗ ei ⊗ I2 ; Ei 0 0 = ei ⊗ I3 ⊗ I3 and (I1⊗I1⊗ei)(I2⊗ei ⊗I2)(ei ⊗I3⊗I3) = ei⊗ei⊗ei = E00iE0i0Ei00 = Eiii (2.22) Still we will have that E00i E0i0 = E0i0 Ei00; E00i Ei00 = Ei00 E00i ; E0i0 Ei00 = Ei00 E0i0 (2.23) In the case n = 8 we have three distinct basic unities, and, in addition, we have other sets of basis matrices, as example (Eijk, Eiji, Eijj). Other cases are obviously possible. Generally speaking, fixed the order n of the matrix representation of the set of basis matrices, we will have that Γ1 = Λ n Γ2m = Λ n-m ⊗ ⊗ ⊗ ⊗− + − +e I In m n m n2 1 2( ) ( ) ......... (2.24) Γ2m+1 = Λ n-m ⊗ ⊗ ⊗ ⊗− + − +e I In m n m n3 1 2( ) ( ) ......... Γ2n = e I I n2 2⊗ ⊗ ⊗( ) ......... 1294 E. Conte with Λn= e e e n1 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 I1, I2, ..., In in the distinct direct products. Basis matrices are determined by the number of dispositions possible for ei and I(n). In this manner we have established that matrix representations of Cl3 exist at different orders n = 2, 4, 8, ..... . We may now give the explicit expressions of E0i, Ei0, and Eij. 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 = − − ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ 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 = − − ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ; (2.25) 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 = − − ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ . Let us return to our Cl3 mathematical formulation and let us explore still some other important features of the idempotents. In order to semplify our elaboration we remain to consider matrix representation of Cl3 at order n=2. Consider that, with regard, as example, to the basic element e3 of Cl3, we may identify an idempotent ψ, and we write that e3 ψ = ψ and ψ e3 = ψ (2.26) Still, we may identify an idempotent φ so that e3 φ = φ and φ e3 = φ (2.27) We may also generalize such definitions. In fact, we may have that Clifford algebra as a common algebraic structure 1295 e3 ψ1 = ψ1 and ψ2 e3 = ψ2 (2.28) and e3 φ1 = φ1 and φ2 e3 = φ2 (2.29) Calculations show that in the case of the (2.25), we have that ψ = 2 1 3e+ (2.30) In the case of the (2.26), we have φ = 2 1 3e− (2.31) In the case of the (2.27), we have ψ1 = 2 1 3e+ + 2 21 iee + and ψ2 = 2 1 3e+ + 2 21 iee − (2.32) In the case of the (2.28), we have φ1 = 2 1 3e− + 2 21 iee − and φ2 = 2 1 3e− + 2 21 iee + (2.33) Consider again the (2.26) that now we rewrite as it follows (e3 1) ψ = 0 and ψ( e3 1) =0 (2.34) As rigorously admitted from our elaboration of the previous pages, let us calculate now the (2.34) with regard to ei, i = 1,2 . We obtain that ei ( e3 1) ψ = 0 and ψ ( e3 1) ei = 0 (2.35) Thus, we may conclude, with regard to ψ that, as example, we have that (i e1 e2 ) ψ = 0 and ψ ( i e1 + e2 ) = 0 (2.36) Considering the (26), we may rewrite that (e3 + 1) φ = 0 and φ (e3 + 1 ) = 0 (2.37) We may calculate the (2.36) with regard to ei, i = 1,2, obtaining ei ( e3 + 1) φ = 0 and φ ( e3 + 1) ei = 0 (2.38) Thus we may conclude that with regard to φ we have that (i e1 + e2 ) φ = 0 and φ ( i e1 e2 ) = 0 (2.39) In previous papers we have largely used Clifford algebra to realize a bare bone skeleton of quantum mechanics and therefore we will not discuss such basic features furterly here.[18-19] 3. THE DERIVATION OF DIFFUSION AND SCHRÖDINGER EQUATIONS Let us attempt to show that we may derive Diffusion and Schrödinger equations using only the two basic algebraic axioms (2.1) and (2.2) that we introduced in the previous section without assuming quantum mechanics physical foundations. In this elaboration we will utilize some results that were previously obtained by G.N. Ord and A.S. Deakin [20-22]. Consider the diffusion equation in 1+1 dimensions 1296 E. Conte 2 2 x uD t u δ δ = δ δ (3.1) According to A. Einstein in 1905 it goes with a microscopic model of Brownian motion with u(x,t) describing the ensemble average concentration of small particles undergoing such Brownian motion on a scale much less than the scale of observation. Schrödinger equation is written as 2 2 x iD t δ ψδ = δ δψ (3.2) It is seen that apparently one may obtain the (3.2) from the (3.1) in a mathematical way considering the time coordinate to be imaginary in the previous diffusion equation (3.1). This was correctly indicated to be a mathematical extension, a formal analytic continuation (FAC) and this indicated that we may compare the solutions of the two equations also if the solutions of diffusion equations result qualitatively very different from those of free particle Schrödinger's equation. G.N. Ord and A.S. Deakin [20-22] discussed in detail this problem. According to such authors, both diffusion and Schrödinger equation happen in the domain of classical statistical mechanics. The model they used is the standard lattice random walk model of Brownian motion: solutions of the diffusion equation appeared directly in a first order projection out of the considered space while solutions of Schrödinger's equation appeared directly in a second order projection. The conclusion was that two qualitatively different behaviors coexist in the same physical system since the two considered projections resulted to be orthogonal. Ord and Deakin's conclusion was that no FAC is required to be conceived in order to introduce Schrödinger equation from this classical model: both the equations delineate different views of the same system represented by an ensemble of Brownian particles. FAC indicated a link of the two equations, while, according to [20-22], in this model we have a "wave function" that is an observable property of the examined ensemble of real point particles. As basic framework the authors [20-22] considered a space-time lattice with spacings δ and ε. Particles were considered to hop on lattice a distance ± δ at each time step ε. The walks are symmetric and at each lattice site walks are equally likely to take either direction. Also we shall be interested in the statistics of the number of direction changes in trajectories on the lattice. G.N. Ord and Deakin considered that between lattice sites, each particle will be in one of two direction states, the right or the left movement, respectively with one of the two spin states. Ising spin variable was used with σ = ±1 in order to describe such two states. The direction state was considered to change with every collision, the direction change, while instead the spin was considered to change with every two collisions. In this manner, a particle starting off in state 1 (right moving and σ = +1) changes to state 2 having left moving and σ = +1 at the first collision, it will have state 3, right moving and σ = -1, at the second collision, while instead it will arrive at state 4 it with left moving and σ = -1 at the third collision, and back to state 1 at the fourth collision. States one and three will correspond both to right Clifford algebra as a common algebraic structure 1297 moving particles and states two and four will correspond to left moving. A particle starting in state 1 (2) and ending in the same direction state 3 (4) will result to have changed its spin from +1 to –1.Ising spin was not considered a new property added to the particle but a simple label helping to classify particle trajectories. Finally, Ord and Deakin considered a so called parity in order to account for differences between two states with identical directions. The authors [20-22] indicated by pμ (mδ, sε)δ (μ = 1,2,3,4) the probability that the particle, leaving the space-time point (mδ, sε), is in state μ (m=0, ± 1,........; s=0, ± 1,...........). The master equation expressed by the following set of difference equations was written p1(mδ,(s+1)ε) = 2 1 p1((m-1)δ,sε) + 2 1 p4((m+1)δ,sε) p2(mδ,(s+1)ε) = 2 1 p2((m+1)δ,sε) + 2 1 p1((m-1)δ,sε) (3.3) p3(mδ,(s+1)ε) = 2 1 p3((m-1)δ,sε) + 2 1 p2((m+1)δ,sε) p4(mδ,(s+1)ε) = 2 1 p4((m+1)δ,sε) + 2 1 p3((m-1)δ,sε) If we multiply the first equation by δ, we have that the probability for a particle leaving the node (mδ, (s+1)ε) in state 1, is equal to the sum of two probabilities, the first representing the probability that a particle leaves node ((m-1)δ, sε) in state 1 and remains in this state when it leaves node (mδ, (s+1)ε) and the second representing probability that particle leaves the node ((m+1)δ, sε) in state 4 and changes to state 1 when it leaves the node (mδ, (s+1)ε). The remaining equations may be interpreted in a similar way. The probability pμ is uniquely determined since the initial conditions are fixed. Fixed s 0≥ , we have also that ∑ ∑ =μ +∞ −∞= μ =δεδ 4 1 m 1)s,m(p so that the probability for particle to be somewhere on the lattice at a given time is equal to one. Assuming qi(s) (i=1,2,3,4) is the probability for a particle to be in the i-th state (i=1,2,3,4) at the s-th step on the lattice, one has that (q1(s+1),q2(s+1),q3(s+1), q4(s+1))T=T(1)(q1(s),q2(s),q3(s),q4(s))T (3.4) where the transition matrix T(1) defines a Markov chain. In this manner we have reached the central point of our paper. Generally speaking, the most general Markov chain with four states may be defined by the following matrix ⎟⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎝ ⎛ α =α 1100 0110 0011 1001 2 )(T (3.5) 1298 E. Conte where, with α = 1, Tij is the probability of a transition from state j to state i in one step. Ord and Deakin just started with (3.5) in their elaboration. The problem that at this point we pose is the following: have we a set of a purely mathematical axioms by which we may derive all the results that were previously obtained by these authors? Let us start observing that T(α), as indicated in the (3.5), is a general member of the Clifford algebra that we introduced in the previous section. In detail we may write that T(α) = [ ]211 ZeZ12 ⊗+⊗ α (3.6) where Ζ1 = 1 + 2 1 e1 2 i e2; Ζ2 = 2 1 e1 + 2 i e2 (3.7) In order to discuss other features of the Markov process, one considers a change of variables from qi(s) (i = 1, 2, 3, 4) to μi(s) (i = 1, 2) and ξi(s) (i = 1, 2): μ1 = q1 + q2 + q3 + q4 ; μ2 = (q1 + q3) – (q2 + q4) ξ1 = q1 q3 ξ2 = q2 q4 (3.8) where μ1 ≡ μ1(s) is the sum of all the occupation probabilities, μ2 ≡ μ2(s) is the difference of occupation probabilities by direction, ξ1 ≡ ξ1(s) and ξ2 ≡ ξ2(s) represent instead the differences of occupation of the 2 spin states for right and left moving particles, respectively. In matrix form it is obtained that ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ξ ξ μ μ 2 1 2 1 = R ⎟⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎝ ⎛ 4 3 2 1 q q q q with R = ⎟⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − −− 1010 0101 1111 1111 (3.9) In our mathematical framework the (3.9) becomes the following equation involving general members of Clifford algebra U = R P (3.10) where R is the Clifford general member R = Z3 ⊗ Z4 + Z5 ⊗ 1 (3.11) and Ζ3 = 2 1 + 2 1 e1 + 2 i e2 + 2 1 e3 ; Z4 = e1 + e3 ; Ζ5 = − 2 1 + 2 1 e1 − 2 i e2 + 2 1 e3 P is the following Clifford member P= Z6⊗Z7 + Z8⊗Z9 (3.12) with Z6 = 2 1 + 2 1 e3 ; Z7 = 311 e2 q 2 q + 321 2 eq 2 ie 2 q −+ ; Z8= 21 e2 ie 2 1 − ; And Clifford algebra as a common algebraic structure 1299 Z9 = 2414333 e2 iqe 2 qe 2 q 2 q −++ Finally, U is given in the following manner U = Z6⊗Z'7 + Z8⊗Z'9 (3.13) where Z'7= 2212311 e2 i e 2 e 22 μ − μ + μ + μ and Z'9= 2 2 1 2 3 11 e 2 ie 2 e 22 ξ − ξ + ξ + ξ At this stage a basic result may be evidenced from a theoretical as well as mathematical physics view points: the equations contained in Ord and Deakin's paper may be obtained in terms of Clifford algebra. The interesting conclusion is only one. The results obtained in [20-22] are expression of the basic set of axioms of Clifford algebra. In the calculations, Ord and Deakin normalized ξi(s) defining ξ'i(s) = ( 2 )s ξi(s) and introducing the (3.9) in the (3.5), they splitted the problem in two final equations that again are given in terms of Clifford members Ξ(s+1) = V Ξ(s) (3.14) with V= )ie1( 2 1 2− and Ξ(s)= 2212311 e2 )s(ie 2 )s(e 2 )s( 2 )s( ξ − ξ + ξ + ξ and similar Clifford member for Ξ(s+1). The equations (3.3) characterize the full random walk of the system. It remains to show that the solutions of these equations can be approximated in terms of solutions of diffusion and of Schroödinger equations. To this purpose in [20-22] it was used the usual diffusive scaling of random walks for small δ )(OD 2 2 δ+= ε δ or )(O D2 3 2 δ+ δ =ε (3.15) where D is the constant of diffusion. Let us consider the following shift operators, Ex±1 and Et, acting in the following manner Ex±1 pi(mδ,sε) = pi( (m ± 1)δ,sε) ; Et pi(mδ,sε) = pi(mδ,(s+1)ε) (3.16) The (3.3), the basic master equation of the paper, results to be still a basic equation expressed by Clifford members. In fact, we have that P(mδ,(s+1)ε) = L P(mδ,sε) (3.17) with L= )TeT1( 2 1 211 ⊗+⊗ 1300 E. Conte and (3.18) T1 = 3x 1 x 2 1 x 1 1 x 1 xx e) 2 EE(e 2 Eie 2 E 2 )EE( − +−+ + −−−− ; T2 = 2x1x e2 Eie 2 E + Instead P is given by the following Clifford member P = Z6⊗Q7+Z8⊗Q9 where Q7 = 3122121 e2 pe 2 pie 2 p 2 p +−+ and Q9 = 3324143 e2 pe 2 pie 2 p 2 p +−+ Thus, in conclusion, the master equation, given in the (3.3), is expression of the basic axiom set of the Clifford algebra. Ord and Deakin operated the following change of variables z1 = p1+p2+p3+p4 ; z2 = (p1+p3)-(p2+p4) ; (3.19) φ1 =p1-p3 ; φ2=p2-p4 and we may arrive to write the following and final Clifford equation χ1 = Nχ (3.20) with N = 2 1 [ ]21016 NZNZ ⊗+⊗ N1 = ( ) 2 EE 1xx −+ + ( ) 2 EE 1xx −+ e3 + ( ) 2 EE x 1 x − − e1 + i ( ) 2 EE x 1 x − − e2 ; Z10 = 2 1

e3 ; N2 = ( ) 2 EE x 1 x + − + ( ) 2 EE x 1 x − − e3 + ( ) 2 EE x 1 x − − e1 –i ( ) 2 EE 1xx −+ e2 and χ = 2816 BZBZ ⊗+⊗ where 12223 11 1 ez2 iez 2 ie 2 z 2 z B +−+= ; 22123112 e2 ie 2 e 22 B φ− φ + φ + φ = By using such mathematical procedure, we have arrived now to explain in mathematical terms the reason by which both the diffusion and the Schrödinger equations appear within the domain of classical statistical mechanics and the reason by which diffusion equations appears directly in a first order projection of the considered space while Schrödinger equation appears in a second order projection. We have two equations that express two qualitatively different behaviors to coexist in the same physical system. The reason of such two different projections resides in the intrinsic mathematical features of Clifford algebra. As said in the previous section, a significant feature of such algebra is that it admits primitive idempotents that here we may consider to be given in the following manner Clifford algebra as a common algebraic structure 1301 2 e1 i i + =ψ , i = 1, 2, 3 and whose important properties were outlined in the (2.26) – (2.39) of the previous section. We also outlined in other papers their importance in order to explain the actual origin of quantum mechanics [23-26]. Here we obtain that they have a decisive role in determining the two projections that give origin to diffusion and Schrödinger equations respectively. We have a basic mathematical set that from one hand determines both the two equations and at the same time is able to differentiate between diffusion and Schrödinger equations on the basis of its intrinsic mathematical features. Consider as example the idempotent 2 e1 3 3 + =ψ One has that e3 3ψ = 3ψ ; 3ψ e3 = 3ψ ; (e3 –1 ) 3ψ = 0 and 3ψ (e3-1) = 0 In the same manner one has that ej(e3-1) 3ψ = 0 and 3ψ (e3-1)ej = 0 ; j = 1, 2. Such idempotents have a decisive role in characterizing Clifford algebra since some their products, as seen in the previous trivial example, give zero. At this stage one may note that all the Clifford members that we calculated for N and χ of the (3.20) are just expressed always in terms of the idempotents that we have here indicated. Owing to the intrinsic mathematical features of such idempotents in the (2.20) one obtains that the general Clifford member N1 will act only on the basic Clifford element B1 of χ , depending only from z1 and z2, while the other Clifford member N2 will act only on the Clifford member B2 of χ , depending this time only from the completely different variables. As a result of such basic mathematical features, using N1 we will arrive to establish only the usual diffusion equation, while, using N2, we will arrive only to Schrödinger's equation, both formulated in the framework of the same model, and, mathematically speaking, on the basis of the same basic axiom set defining the employed algebra. In a previous paper we demonstrated two theorems in Clifford algebra showing the manner in which they arise giving algebraic solution to the well known quantum measurement problem in quantum mechanics. They arose showing that during a measurement we have a transition from the the standard Clifford A( iS ) algebra to the Dihedral algebra 1,±iN where new commutation rules appear and linking in some manner the present formulation [27]. In addition, using idempotents, interpreted, according to von Neumann, as logical statements, we also arrived to support the logical origin of quantum mechanics [23-26]. An intrinsic cognition principle appears in our formulation as this problem is investigated in detail in [28-37]. Here, we have given a clear Clifford algebraic explanation on the common mathematical structure that determines such two equations but at the same time we have given large evidence about the mathematical origin of their formal and conceptual differentiation in physics. 1302 E. Conte We may now conclude moving the calculations in the direction of the continuum limit as previously was performed in ref. [20-22]. One considers Ex )s,M(i εδγ = )s,M(i εδ+δγ expanding )s,M(i εδ+δγ in a power series of δ with Ex = 1 + L + 2 1 L2 + O ( )3δ (3.23) where L = xδ δ δ , one arrives to L )s,M(i εδγ = L iγ (x,t) (3.24) with the domain definition of iγ extended to all the (x, t) and iγ (x, t) considered to be continuously differentiable. Since the (3.23), one has that 1 xE − =1 – L + 2 L2 + O ( )3δ (3.25) and for N2 one has that ( ) )(OLee 2 L2iLe 2 L2 2 1N 332 2 1 2 2 δ+⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + −− + = (3.26) that may be written in the following manner )(O)ee(L 2 1)ie1( 2 1)L 2 11(N 3312 2 2 δ++−−+= (3.27) One may call V= )ie1( 2 1 2− as in the (3.14) and B=- )ee( 2 1 31 + (3.28) and thus obtains that N2 = V + BL + )(OVL 2 1 32 δ+ (3.29) Owing to the intrinsic features of Clifford algebra, one has that 2 2 ie1V2V −=−= ; BV=-e3; VB=e1; VB+BV = 2 B; B2 =1 (3.30) Therefore, it is obtained that N 22 = (1+L 2) V2 + L (VB+BV) +L2 B2 + O ( )3δ = 2 L2 B + 2 L B –ie2 (3.31) In the same manner one has that N24 = -1 –2ie2 L2 + O ( )3δ (3.32) since it is that V(-ie2)=(-ie2)V=V2 ; B (-ie2) =-(-ie2)B = 2 2 (e1-e3) (3.33) Finally, one obtains that Clifford algebra as a common algebraic structure 1303 N28 = 1 + 4ie2 L2 + O ( )3δ (3.34) In conclusion, remembering that in the (3.20) we have 1χ = Et χ with Et giving Et8 = 1 + 8 ε )(O t 2ε+ δ δ (3.35) and χ given in the following manner χ = 3122121 e2 e 2 ie 22 φ + φ − φ + φ (3.36) we have that (1+4ie2L2)( 3122121 e2 e 2 ie 22 φ + φ − φ + φ )= ( )L2 2 2 21 φ+ φ + +( 11 22 e)L2 2 φ− φ +( )iL2 2 i 1 22 φ+ φ− 2e +( 32 21 e)L2 2 φ+ φ (3.37) and calculating Et8 χ = (1+8 tδ δ ε )( )e 2 e 2 ie 22 3 1 2 2 1 21 φ+ φ − φ + φ (3.38) and equaling the (3.37) with the (3.38) as it is on the basis of the (3.20) and of the (3.3), one obtains that such two Clifford members coincide if and only if it happens that t 2 x 2 22 1 2 δ δφ δ ε −= δ φδ and t 2 x 1 22 2 2 δ δφ δ ε = δ φδ (3.39) Since )(OD 2 2 δ+= ε δ we have that 2 2 2 1 x D t δ φδ = δ δφ and 2 1 2 2 x D t δ φδ −= δ δφ (3.40) as they were also derived in ref.[20-22]. Again we may outline here that all the present matter seems to run about the important presence of new basic commutation rules that are involved in algebraic transition from the A( )iS to 1,±iN . One may also introduce the following functions )t,x(i)t,x()t,x( 12 φ+φ=ψ+ and )t,x(i)t,x()t,x( 12 φ−φ=ψ− and thus obtaining the final forms of Schrödinger's equations 2 2 x )t,x(D t )t,x(i δ ψδ −= δ δψ ++ and 2 2 x )t,x(D t )t,x(i δ ψδ = δ δψ −− (3.41) simultaneously admitted from Clifford algebra in complex and complex conjugate forms. At the same time, using this time the general Clifford member N1, given in the (3.20), following the same Clifford algebraic procedure one arrives to write the well-known diffusion equation 1304 E. Conte 2 2 x )t,x(zD t )t,x(z δ δ = δ δ (3.42) The important conclusion of such long derivation is thus clear: both diffusion and free Schrödinger equations are Clifford members. In detail, they are emanations of the same basic axiomatic set of the Clifford algebra, the A( )iS , in particular the (2.1) and the (2.2) that we enounced in the previous section, and, in case, the associated 1,±iN Dihedral algebra. By using both such algebras, it seems, however, that both such equations, also arising from the same axiomatic set, possibly they cannot be reconciled. REFERENCES [1] A more general discussion on Determinism as basic rule in time dynamics including also some features of the present paper, was held by us in E. Conte, A. Federici, A. Khrennikov, J.P Zbilut, Is determinism the basic rule in dynamics of biological matter? Conference Proceedings on Quantum Theory: Reconsideration of Foundations-2, Sweden,1-7 June 2003, Vaxjo University press, (2003), 639-675 [2] The relevance of singularities in physiological systems was prospected for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. 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Federici, A Preliminary Experimental Verification of Violation of Bell inequality in a Quantum Model of Jung Theory of Personality Formulated with Clifford Algebra. Journal of Consciousness Exploration & Research, 1 (2010), 831-849. E. Conte, O. Todarello, A.Y. Khrennikov, A. Federici, J.P. Zbilut, A preliminary experimental verification on the possibility of Bell inequality violation in mental states, Neuroquatology, 6 (3) (2008) 214-221. [36] E. Conte, A. Khrennikov, O. Todarello, A. Federici, J.P. Zbilut, On the Existence of Quantum Wave Function and Quantum Interference Effects in Mental States: An Experimental Confirmation During Perception and Cognition in Humans. Neuroquantology, 7 (2009), 204-212. [37] E. Conte, O. Todarello, A. Federici, F. Vitiello, M. Lopane, A. Khrennikov, J.P. Zbilut, Some remarks on an experiment suggesting quantum-like behavior of cognitive entities and formulation of an abstract quantum mechanical formalism to describe cognitive entity and its dynamics. Chaos, Solitons and Fractals, 31 (2007), 1076-1088. Received: September,