Chisholm-Caianiello-Fubini Identities for S = 1 Barut-Muzinich-Williams Matrices

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1 Chisholm-Caianiello-Fubini Identities for S = 1 Barut-Muzinich-Williams Matrices M. de G. Caldera Cabral and V. V. Dvoeglazov UAF Universidad Autónoma de Zacatecas Apartado Postal 636 Suc. 3 Cruces Zacatecas Zac. México valeri@fisica.uaz.edu.mx Abstract The formulae of the relativistic products are found S = 1 Barut- Muzinich-Williams matrices. They are analogs of the well-known Chisholm- Caianiello-Fubini identities. The obtained results can be useful in the higher-order calculations of the high-energy processes with S = 1 particles in the framework of the 2(2S +1) Weinberg formalism which recently attracted attention again. PACS numbers: p t Ds This the modernized version of a EF-UAZ FT unpublished preprint of 1995 of the second author. The modifications are due to the Thesis of the first author. 1

2 The attractive Weinberg 2(2S + 1) component formalism for description of higher spin particles [1] is based on the same principles as the Dirac formalism for spin-1/2 Ref. [2]. Further developments [ ] showed that many interesting things can be found therein. For instance the connections with the modified Bargmann-Wigner formalism (BWW) [7] or the connections with the so-called Bargmann-Wightman-Wigner formalism [ ]. On the basis of the analysis of the (S 0) (0 S) representation space it was found there that the intrinsic parities of boson and its antiboson can be opposite see also [10]. If a neutrino is identified with the self/anti-self charge-conjugate representation space then it may be coupled with the BWW bosons to generate physics beyond the present day gauge theories. see the above-cited references. One more hint at the possible future application of these formalisms is the tentative experimental evidence for a tensor coupling in the π e + ν e + γ decay for instance [11]. There exist experimental opportunities to check the existence of the unconventional bosons and fermions and different types of interactions as well beyond the Standard Model e. g. Ref. [12]. The principal equation in this formalism is that of the 2s -order in the momentum operators. The analogs of the Dirac γ-matrices have also 2s vectorial indices: [γ µ1µ 2...µ 2s µ1 µ2... µ2s + m 2s ]Ψ(x) = 0. (1) The covariant-defined Γ- matrices for any spin have been introduced by Barut Muzinich and Williams [13] see also [14 15]. For the case of spin S = 1 they have the following form: 1 ( ) Γ (1) I 0 I = 0 I ( ) Γ (2) I 0 γ 5 = where Γ (3) αβ γ αβ = 0 I ( 0 S αβ S αβ 0 ) Γ (4) αβ γ 4αβ = iγ 5 γ αβ (2) Γ (5) αβ γ 5αβ = i [γ αλ γ βλ ] Γ (6) αβµν γ 6αβµν = [γ αµ γ βν ] + + 2δ αµ δ βν [γ αν γ βµ ] + 2δ αν δ βµ = = 1 12 [γ 5αβ γ 5µν ] (δ αµ δ βν δ αν δ βµ ) 4ɛ αβµν γ 5 1 The Eiclidean metric is used. S 44 = I Si4 = S 4i = is i S ij = S ij δ ij = S i S j + S j S i δ ij. (3) 2

3 S i are the spin-1 matrices and ɛ 1234 = 1. They have the simmetry properties [14]: γ αβ = γ βα γ αα = 0 γ 4αβ = γ 4βα α γ 4αα = 0 α γ 5αβ = γ 5βα (4) γ 6αβµν = γ 6βαµν γ 6αβµν = γ 6µναβ γ 6αβµν + γ 6αµνβ + γ 6ανµβ = 0. The relativistic perturbation calculations of the processes including the S = 1 bosons will require the development technical methods analogous to those which have been elaborated for the fermion-fermion interaction namely reducing contracted products of the corresponding Γ matrices [ ]. Our aim with this paper is to find the formulae of the relativistic scalar products like that γ µα... γ βµ. The following relations can be deduced by straightforward calculations: 2 γ µα γ βµ = 3δ αβ i 2 γ 5αβ (5) γ µα γ 5 γ βµ = 3γ 5 δ αβ i 4 ɛ αβστ γ 5στ (6) γ µα γ στ γ βµ = 2γ στ δ αβ + γ αβ δ στ γ ασ δ τβ γ ατ δ σβ γ βσ δ ατ γ βτ δ ασ iɛ αβσµ γ 4τµ iɛ αβτµ γ 4σµ (7) γ µα γ 4στ γ βµ = 2γ 4στ δ αβ γ 4αβ δ στ + γ 4ασ δ τβ + γ 4ατ δ σβ + + γ 4βσ δ ατ + γ 4βτ δ ασ iɛ αβσµ γ τµ iɛ αβτµ γ σµ (8) γ µα γ 5στ γ βµ = 2γ 5στ δ αβ + 2γ 5ασ δ βτ + 2γ 5τβ δ ασ 2γ 5σβ δ ατ 2γ 5ατ δ σβ + 12i (δ ασ δ τβ δ ατ δ σβ ) + (9) + 12iɛ αστβ γ 5 γ µα γ 6στρφ γ βµ = 0. (10) The formulae for the S = 1 matrices which have been used above are presented in Appendix. 2 We have also used the Wolfram MATEMATICA programm to check them. 3

4 Appendix Here we present the set of algebraic relations for S = 1 spin matrices cf. [20 21]. We imply a summation on the repeated indices. and S k S i S k = S i (1) S k S i S j S k = 2δ ij S j S i (2) S k S i S j S l S k = S l S i S j + S j S l S i S j δ il (3) S k S i S j S l S m S k = δ ij δ lm + δ im δ jl S m S l S j S i (4) S ij S k = δ ij S k δ jks i δ iks j + i 2 (ɛ ikls jl + ɛ jkl S il ) (5) S ik S jl + S jl S ik = 2δ ik S jl + 2δ jl S ik + (ɛ ilm ɛ jkn ɛ ijm ɛ kln ) S mn (6) S l S ij S m = 2δ ij δ lm δ im δ jl δ jm δ il δ lm S ij + +δ im S l S j + δ jm S l S i + δ il S j S m + δ jl S i S m (7) S l S ij S m S m S ij S l = δ il (S j S m S m S j ) + δ jl (S i S m S m S i ) + +δ im (S l S j S j S l ) + δ jm (S l S i S i S l ) (8) or = iɛ ilm S j iɛ jlm S i 2δ ij (S m S l S l S m ) (9) S i S j S k + S j S k S i + S k S i S j = S i δ jk + S k δ ij + S j δ ik + + i 4 (ɛ ijls lk + ɛ kil S jl + ɛ jkl S il ). (10) This set supplies the known formulae for S = 1 spin matrices e. g. presented in [21]: S i S j S k + S k S j S i = δ ij S k + δ jk S i (11) S ik S j = i ] [δ ij S4k + δ jk S4i + ɛ jil Slk + ɛ jkl Sil (12) 2 Σ 2 i = S3 2 no summation (13) Σ i Σ j + Σ j Σ i = 2δ ij S 2 3 (14) Σ i Σ j Σ j Σ i = 2iɛ ijk Σ k (15) where Σ 1 S 2 1 S 2 2 Σ 2 S 12 = S 1 S 2 + S 2 S 1 Σ 3 S 3. (16) References [1] S. Weinberg Phys. Rev. 133 B1318 (1964). [2] L. H. Ryder Quantum Field Theory. (Cambridge Univ. Press 1985). [3] W. Greiner Relativistic Quantum Mechanics. (Springer-Verlag Berlin-Heidelberg 1990). 4

5 [4] V. V. Dvoeglazov and N. B. Skachkov Yad. Fiz [English translation: Sov. J. Nucl. Phys. 1065] (1988); JINR Communications R (1987); V. V. Dvoeglazov and S. V. Khudyakov Hadronic J (1998); V. V. Dvoeglazov Rev. Mex. Fis. Suppl (1994) hep-th/ [5] D. V. Ahluwalia M. B. Johnson and T. Goldman Phys. Lett. B (1993). [6] V. V. Dvoeglazov Int. J. Theor. Phys (1998). [7] V. V. Dvoeglazov Int. J. Mod. Phys. B (2006); J. Phys. CS (2007); Int. J. Mod. Phys. CS (2011). [8] I. M. Gelfand and M. L. Tsetlin ZhETF (1956); G. A. Sokolik ZhETF (1957). [9] E. P. Wigner in Group Theoretical Concepts and Methods in Elementary Particle Physics Lectures of the Istanbul Summer School of Theoretical Physics Ed. by F. Gürsey (Gordon and Breach New York-London-Paris 1964). [10] Z. K. Silagadze Yad. Fiz [English translation: Sov. J. Nucl. Phys. 392] (1992). [11] V. N. Bolotov et al. Phys. Lett. B (1990). [12] D. N. Castaño Dark Matter Constrains from High Energy Astrophysical Observations. Ph. D. Thesis (2012) [13] A. Barut I. Muzinich and D. N. Williams Phys. Rev (1963). [14] A. Sankaranarayanan and R. H. Good Nuovo Cimento (1965); D. Shay and R. H. Good Jr. Phys. Rev (1969). [15] R. H. Tucker and C. L. Hammer Phys. Rev. D (1971). [16] J. S. R. Chisholm Proc. Cambridge Phil. Soc (1952). [17] E. R. Caianiello and S. Fubini Nuovo Cimento (1952). [18] R. H. Good Jr. Rev. Mod. Phys (1955). [19] J. S. R. Chisholm Nuovo Cimento (1963); J. Kahane J. Math. Phys (1968); J. S. R. Chisholm Comp. Phys. Comm (1972). [20] D. A. Varshalovich et al. Quantum Theory of Angular Momentum. (World Scientific Singapore 1988). [21] D. L. Weaver Am. J. Phys (1978); idem. J. Math. Phys (1978). 5