Extended real Clifford - Dirac algebra and bosonic symmetries of the Dirac equation with nonzero mass
aa r X i v : . [ m a t h - ph ] A ug Extended real Clifford - Dirac algebra and bosonic symmetries ofthe Dirac equation with nonzero mass
V.M.Simulik and I.Yu.Krivsky
Institute of Electron Physics, National Academy of Sciences of Ukraine, 21 UniversitetskaStr., 88000 Uzhgorod, UkraineE-mail: [email protected]
The 64-dimensional extended real Clifford – Dirac (ERCD) algebra is introduced. On this basisthe new pure matrix symmetries of the Dirac equation in Foldy – Wouthuysen representation arefound: (i) the 32-dimensional A = SO(6) ⊕ ˆ ε · SO(6) ⊕ ˆ ε algebra, which is proved to be maximalpure matrix symmetry of this equation, (ii) two different realizations of the (1 / , ⊕ (0 , / , ⊕ (0 ,
0) and irreduciblevector (1/2,1/2) bosonic S(1,3)-symmetries. Finally, spin 1 Poincare symmetries both for theFoldy – Wouthuysen and standard Dirac equations with nonzero mass are found.
An interest in the problem of the relationship between the Dirac and Maxwell equations emergedimmediately after the creation of quantum mechanics [1-11]. One of our own results in thisfield is the proof of bosonic (spin 1) Poincare symmetry of the massless Dirac equation [12-15]and the relationship between the Dirac ( m =0) and slightly generalized Maxwell equations inthe field strengths terms [12-16]. We found the irreducible vector (1/2,1/2) and the reducibletensor-scalar (1 , ⊗ (0 ,
0) representations of the Lorentz group L and corresponding (generatedby these representations) representations of the Poincare group P , with respect to which themassless Dirac equation is invariant ( P is the universal covering of the proper ortochronousPoincare group P ↑ + = T(4) × )L ↑ + , L – of the Lorentz group L ↑ + , respectively). Now we are ableto present below the similar results for the general case, when the mass in the Dirac equationis nonzero.In order to derive this assertion we essentially use two new constructive ideas. (i) We starthere from the Foldy – Wouthuysen (FW) equation (i. e., from the Dirac equation in the FWrepresentation [17])( iγ µ ∂ µ − m ) ψ ( x ) = 0 V ↔ V − ( i∂ − γ ˆ ω ) φ ( x ) = 0 , ψ, φ ∈ S , , (1) V = −→ γ · −→ p + ˆ ω + m q ω (ˆ ω + m ) , V − = V ( −→ p → −−→ p ) , ˆ ω ≡ √− ∆ + m , S , ≡ S(R ) × C , R ⊂ M(1 , , (2)where (S(R ) is the Schwartz test function space, M(1,3) is the Minkowski space. (ii) Weintroduce here into consideration the 64-dimensional extended real Clifford – Dirac algebra(ERCD). We essentially apply it here as a constructive mathematics for our purposes.1xplanation of (ii) needs more details. For the physical purposes, when the parametersof the relativistic groups are real, it is enough to consider the standard Clifford - Dirac (CD)algebra as a real one (corresponding generators associated with real parameters are identifiedas primary). Thus, starting from the standard (Pauli - Dirac) representation of the γ -matriceswe are able to extend the complex 16-dimensional CD algebra to the 64-dimensional ERCDalgebra. For the definiteness it is useful to choose 16 independent (ind) generators of standard CDalgebra as { indCD } ≡ n I , s ˆ µ ˆ ν , ˆ µ ˆ ν = 0 , o = (cid:26) s ˇ µ ˇ ν ≡
14 [ γ ˇ µ , γ ˇ ν ] , ˇ µ ˇ ν = 0 , , s ˇ µ = − s µ = 12 γ ˇ µ (cid:27) , (3)where γ ≡ iγ stand = γ γ γ γ and the matrices s ˆ µ ˆ ν are the primary generators of the SO(1,5)group and satisfy the following commutation relations[ s ˆ µ ˆ ν , s ˆ ρ ˆ σ ] = − g ˆ µ ˆ ρ s ˆ ν ˆ σ − g ˆ ρ ˆ ν s ˆ σ ˆ µ − g ˆ ν ˆ σ s ˆ µ ˆ ρ − g ˆ σ ˆ µ s ˆ ρ ˆ ν , { g µν } = diag(+1 , − , − , − . (4)On this basis ERCD is constructed as { ERCD } = n indCD, i · indCD, ˆ C · indCD, i ˆ C · indCD o , (5)where ˆ C is the operator of complex conjugation in { φ } ∈ S , , the operator i is the ordinarycomplex unit. Thus, the ERCD algebra is the composition of standard CD algebra (3) andalgebra of Pauli – Gursey – Ibragimov [18, 19]], i. e. it is the maximal set of independentmatrices, which may be constructed from the elements i , ˆ C and (3).All the physically meaningful symmetries of the FW and Dirac equations, which are putinto consideration below, are constructed using the elements of ERCD algebra. The 32-dimensional subalgebra A = SO(6) ⊕ ˆ ε · SO(6) ⊕ ˆ ε of ERCD algebra (ˆ ε = iγ ), whichSO(6) generators have the form s AB = 14 [ γ A , γ B ] = − s BA , A, B = 1 , , γ ≡ γ γ ˆ C, γ ≡ iγ γ ˆ C (6)(our γ = γ stand ) and together with ˆ ε = iγ satisfy the commutation relations[ s AB , s CD ] = δ AC s BD + δ CB s DA + δ BD s AC + δ DA s CB , [ s AB , ˆ ε ] = 0; A, B, C, D = 1 , , (7)2f SO(6) ⊕ ˆ ε -algebra, is the maximal pure matrix algebra of invariance of the FW-equation from(1) (the commutation relations for the ˆ ε · SO(6) generators ˜ s AB = ˆ εs AB differ from (7) by thegeneral multiplier ˆ ε = iγ = − γ γ ··· γ , which is the Casimir operator of the whole A algebra.The proof of this assertion is fulfilled by straightforward calculations of the correspondingcommutation relations of this algebra and the commutators between the elements of A andthe operator ( i∂ − γ ˆ ω )of the FW-equation taken from (1). The maximality of A is theconsequence of the ERCD algebra maximality. The FW-equation from (1) is invariant with respect to the two different spin 1 representationsof the Lorentz group L (below s Vµν are the generators of the irreducible vector (1/2,1/2) and s T Sµν are the generators of the reducible tensor-scalar (1 , ⊕ (0 ,
0) representations of the SO(1,3)algebra). The explicit forms of corresponding pure matrix operators are following s T Sµν = { s T S k = s I k + s II k , s T Smn = s Imn + s IImn } , s Vµν = { s V k = − s I k + s II k , s Vmn = s T Smn } , (8)i.e. they are constructed as a fixed sum of two different subsets s Iµν = { s I k = i γ k γ , s Imk = 14 [ γ m , γ k ] } , γ ≡ γ γ γ γ , ( k, m = 1 , , (9) s IIµν = { s II = i γ ˆ C, s II = − γ ˆ C, s II = − γ , s II = i , s II = i γ γ ˆ C, s II = 12 γ γ ˆ C } , (10)of the (1 / , ⊕ (0 , /
2) generators of the L group (two different realizations (9), (10) ofthe (1 / , ⊕ (0 , /
2) representation of SO(1,3) algebra are the subalgebras of the A =SO(6) ⊕ ˆ ε · SO(6) ⊕ ˆ ε algebra). The validity of this assertion is evident after the transition s Boseµν = W s
V,T Sµν W − in Bose-representation of the γ -matrices, where the operator W has theform W = 1 √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − C i i ˆ C − C − − ˆ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , W − = 1 √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − − − − i C − ˆ C ˆ C i ˆ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , W W − = W − W = 1 . (11)We call this transition a new natural form of the supersymmetry transformation of theFW-equation. The FW-equation from (1) is invariant with respect to the canonical-type spin 1 representationof the Poincare group
P ⊃ L , i.e. with respect to the unitary, in the rigged Hilbert spaceS , ⊂ L (R ) × C ⊂ S , ∗ (12)3 representation, which is determined by the primary generators p = − iγ ˆ ω, p n = ∂ n , j ln = x l ∂ n − x n ∂ l + s Iln + s IIln , j k = x ∂ k + iγ { x k ˆ ω + ∂ k ω + [( −→ s I + −→ s II ) × −→ ∂ ] k ˆ ω + m } , (13)where ˆ ω is given in (2), s Iln and s IIln are given in (9), (10), respectively, and ~s I,II = ( s , s , s ) I,II .The proof is fulfilled by the straightforward calculations of (i) corresponding P -commutatorsbetween the generators, (ii) commutators between generators and operator ( i∂ − γ ˆ ω ), (iii) theCasimir operators of the Poincare group. It is easy to see that the Dirac equation from (1) has all above mentioned spin 1 symmetriesof the FW-equation. The corresponding explicit forms of the generators Q D in the manifold { ψ } ∈ S , are obtained from the corresponding formulae (6), (8) – (10), (12) for the FW-generators q F W with the help of the FW-operator V (2): Q D = V − q F W V . As a meaningfulexample we present here the explicit form for the spin 1 P -symmetries of the Dirac equationˆ p = − iγ ( ~γ~p + m ) = − iH D , p k = ∂ k , j kl = x k ∂ l − x l ∂ k + s kl +ˆ s kl , j k = x ∂ k − ix k ˆ p + s k + ˆ s l ∂ n ε kln ˆ ω + m , (14)where ε kln is the Levi-Chivitta tensor, and operators s µν , ˆ s µν = V − s IIµν V have the form s µν = 14 [ γ µ , γ ν ] , ˆ s µν = { ˆ s = 12 iγ ˆ C, ˆ s = − γ ˆ C, ˆ s = − γ −→ γ · −→ p + m ω = − H D ω , (15)ˆ s = i , ˆ s = iH D ω γ ˆ C, ˆ s = H D ω γ ˆ C } (a part of spin operators from (14) is not pure matrix because it depends on the pseudodiffer-encial operator (ˆ ω ≡ √− ∆ + m ) − well-defined in the space S , . The Dirac equation in theBose-representation has the form of corresponding generalized Maxwell equations in the termsof field strengths. Let us note that the generators (12) without the additional terms s IIln (10) and ~s II = ( s , s , s ) II directly coincide with the well-known [17] generators of standard Fermi (spin 1/2) P -symmetriesof the FW-equation (similar situation takes place for the generators (13) taken without termsincluding operators ˆ s µν from (15)). These well-known forms determine the Fermi-case while theoperators, which are suggested here, are associated with the Bose interpretation of equations(1), which is found here also to be possible. The only difference of our Fermi-case from the spin4 / / References [1] C.G. Darwin, Proc. Roy. Soc. Lon. A , 654 (1928).[2] R. Mignani, E. Recami, M. Badlo, Lett. Nuov. Cim. L , 572 (1974).[3] O. Laporte, G.E. Uhlenbeck, Phys. Rev. , 1380 (1931).[4] J.R. Oppenheimer, Phys. Rev. , 725 (1931).[5] R.H. Good, Phys. Rev. , 1914 (1957).[6] J.S. Lomont, Phys. Rev. , 1710 (1958).[7] H.E. Moses, Phys. Rev. , 1670 (1959).[8] C. Daviau, Ann. Found. L. de Brogl. , 273 (1989).[9] A. Campolattaro, Intern. Journ. Theor. Phys. , 141 (1990).[10] H. Sallhofer, Z. Naturforsch. A , 1361 (1990).[11] J. Keller, On the electron theory, In Proceedings of the International Conference “Thetheory of electron” - Mexico. 24-27 September 1995, Adv. Appl. Cliff. Alg. (Special), 3(1997).[12] V.M. Simulik, I.Yu. Krivsky, Adv. Appl. Cliff. Alg. , 69 (1998).[13] V.M. Simulik, I.Yu. Krivsky, Ann. Found. L. de Brogl. , 303 (2002).;[14] V.M. Simulik, I.Yu. Krivsky, Rep. Math. Phys. , 315 (2002).[15] Simulik V.M. The electron as a system of classical electromagnetic and scalar fields. Inbook: What is the electron? P.109-134. Edited by V.M. Simulik.
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