A Clifford Algebraic Analysis and Explanation of Wave Function Reduction in Quantum Mechanics Elio Conte School of Advanced International Studies for Applied Theoretical and Non Linear Methodologies of Physics, Bari, Italy. Department of Pharmacology and Human Physiology, University of BariItaly. Abstract : proof is given of a theorem on Clifford algebra . Its implications are examined in order to give a Clifford algebraic formulation and explanation of the process of wave function reduction in quantum mechanics. PACS .03.65.Ta. Let us state a proper definition of Clifford algebra. The Clifford (geometric) algebra 0,3Cl is an associative algebra generated by three vectors ,, 21 ee and 3e that satisfy the orthonormality relation jkjkkj eeee δ2=+ (1.1) for [ ]3,2,1, ∈kj . That is, 12 =je 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 an imaginary number that commutes with vectors. To give proof, let us follow the approach that, starting with 1981, was developed by Y. Ilamed and N. Salingaros [1]. Let us consider three abstract basic elements, ie , with 3,2,1=i , and let us admit the following two assumptions: a) it exists the scalar square for each basic element: 111 kee = , 222 kee = , 333 kee = with R∈ik . (1.2) In particular we have also that 100 =ee . b) The basic elements ie are anticommuting elements, that is to say: 1221 eeee −= , 2332 eeee −= , 3113 eeee −= . (1.3) In particular it is iii eeeee == 00 . 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 ++= . (1.4) Let us introduce left and right alternation: 211211 )( eeeeee = ; )( 221221 eeeeee = ; 322322 )( eeeeee = ; )( 332332 eeeeee = ; 133133 )( eeeeee = ; )( 113113 eeeeee = . (1.5) Using the (1.3) in the (1.5) it is obtained that 3132121121 eeeekek ωωω ++= ; 2332221112 eekeeek ωωω ++= ; 3232212132 eekeeek λλλ ++= ; 3332231123 keeeeek λλλ ++= ; 3323213113 keeeeek γγγ ++= ; 1331221131 eeeekek γγγ ++= . (1.6) From the (1.6), 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 ++−=+− (1.7) By the principle of identity , we have that it must be 0313221 ====== γγλλωω (1.8) and 02211 =+− kk γλ 03322 =− kk ωγ 03311 =− kk ωλ (1.9) The (1.9) 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 (1.10). Admitting 1321 +=== kkk , it is obtained that i=== 213 γλω (1.11) In this manner, using the (1.2) and the (1.3), 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 = (1.12). The content of this statement is thus established: given three abstract basic elements as defined in (a) and (b), an algebraic structure is established as in (1.12) with four generators ( ).,,, 3210 eeee The previous Clifford (geometric) algebra 0,3Cl admits idempotents. Let us consider two of such idempotents: 2 1 3 1 e+ =ψ and 2 1 3 2 e− =ψ (1.13) 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 (1.14) 23223 ψψψ −== ee (1.15) Similar relations hold in the case of 1e or 2e . The given algebraic structure 0,3Cl , with reference to the idempotent 1ψ , see the (1.14), relates to 3e the numerical value of 1+ while the (1.15), with reference to 2ψ , relates to 3e the numerical value of -1 . With relation to 3e → +1 , from the (1.12) we have that 122 2 1 == ee , 1 2 −=i ; iee =21 , iee −=12 , 12 eie −= , 12 eie = , 21 eie = , 21 eie −= (1.16) with three new basic elements ( ),, 21 iee instead of ( ),, 321 eee . In other terms, in the case 13 +→e , a new algebraic structure arises with new generators whose rules are given in (1.16) instead of in (1.11). Therefore, the arising central problem is to proof the real existence of such new algebraic structure. Note that, in the case of the starting algebraic structure ,we showed that it exists in the following manner 123 2 2 2 1 === eee ; 31221 ieeeee =−= ; 12332 ieeeee =−= ; 23113 ieeeee =−= ; 321 eeei = (1.17). In the present case, ( )13 +→e , we have to show that it exists in the following manner 122 2 1 == ee ; 1 2 −=i ; iee =21 , iee −=12 , 12 eie −= , 12 eie = , 21 eie = , 21 eie −= (1.18) In this manner we arrive to proof a theorem that, given the algebraic structure A, fixed as in the (1.17), under the condition, 13 +→e , it exists an algebraic structure B with basic elements (generators) given in (1.18). To proof, rewrite the (1.4) in our case, and performing calculations 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 (1.19) where this time it must be 121 +== kk and 13 −=k . It results 11 −=λ ; 12 −=γ ; 13 +=ω (1.20) and the proof is given. The theorem also holds in the case in which we relate to 3e the numerical value of 1− . It is 13 −→e and 122 2 1 == ee ; 1 2 −=i ; iee −=21 , iee =12 , 12 eie = , 12 eie −= , 21 eie −= , 21 eie = (1.21) The solutions of the (1.19) are given in this case by 11 +=λ ; 12 +=γ ; 13 −=ω (1.22). In a similar way it is obtained the proof when considering the cases of 1e or of 2e . Of course, the Clifford algebra given in the (1.18) and in the (1.21) are well known. They are the dihedral Clifford algebra iN (for details, see ref.2 page 2093 Table II). The Possible Implications for Quantum Mechanics. For the problem under consideration, we consider the representation, called the density operator formulation, in which a quantum system is represented by a positive definite Hermitean operator of unit trace known as the density operator. The density operator ρ of a system described by the state vector ψ is simply the projection operator ψψ . In general, a density operator has the form iip i i∑=ρ (1.23) in some particular basis { i }. In this basis, it is immediately evident that the eigenstates of ρ are just the states i and the probabilities ip are the corresponding eigenvalues. To sketch the problem. Consider the most general form of the state vector of an arbitrary two state quantum system −++= δψ ibea (1.24) where, without loss of generality, δ,,ba are considered here to be all real. The density operator of this system is       = − 2 2 babe abea i i δ δ ρ (1.25) There is no nontrivial choice of a , b , and δ that could cause ρ to be diagonal in this basis and still satisfy the requirements of a density operator, and in particular the unit trace requirement. The only way to obtain zeros on the off diagonals is to specify that δ is completely undetermined, and that we must therefore average over all possible values of δ . By this requirement, the averaging turns the complex exponential to zero, giving a diagonal matrix. However, there is no way to accurately specify δ as a completely undetermined quantity in a manner that allows for rigorous calculations. Consider a two state quantum system S with connected quantum observable 3σ . We have 2211 φφψ cc += with       = 0 1 1φ ,       = 1 0 2φ (1.26) and 1 2 2 2 1 =+ cc (1.27) Let us represent the state of such system by a density matrix ρ given in the following terms 321 decebea +++=ρ (1.28) with 2 2 2 2 1 cc a + = , 2 * 212 * 1 ccccb + = , 2 )( 2 * 1 * 21 ccccic − = , 2 2 2 2 1 cc d − = (1.29) 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 (1.30) Of course, the (1.28) is an element of the Clifford algebra as given in the (1.17). As Clifford algebraic element the (1.28) satisfies the requirement to be ρρ =2 and Tr( 1) =ρ under the conditions 2/1=a and 02222 =−−− dcba (null norm of (1.28) algebraic element) as shown in detail in [2]. In the algebraic framework previously outlined, let us admit that we relate 13 +→e (that is to say that the quantum observable 3σ assumes the value +1) or 13 −→e (that is to say that the quantum observable 3σ assumes the value –1). As previously shown, the algebra given in (1.18) and the (1.21)will now hold, respectively. To examine the consequences, starting with the algebraic element (1.28) , write the two equivalent algebraic forms 3 2 2 2 1212 * 121 * 21 2 2 2 1 )( 2 1 ))(( 2 1 ))(( 2 1 )( 2 1 eccieeccieecccc −+−++++=ρ (1.31) and 3 2 2 2 1212 * 121 * 21 2 2 2 1 )( 2 1 ))(( 2 1 ))(( 2 1 )( 2 1 eccieeccieecccc −+−++++=ρ (1.32) Let us consider now when we relate 13 +→e . The (1.18) now hold in the (1.31) that becomes IcM ×= 2 1ρ Let us consider now when we relate 13 −→e . The (1.21) now hold in the (1.32) that becomes IcM ×= 2 2ρ being I the unity matrix. The quantum interference terms now disappear. References [1] Y. Ilamed, N. Salingaros, J. Math. Phys. 22(10), 2091, (1981) [2] E. Conte, Physics Essays, 6, 4, (1994)