Geodesic
Unitary representations of the Wigner group $ISL(2,\mathbb C)$ and a two-spinor description of massive particles with an arbitrary spin
Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 3, pp. 331-361 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Based on Wigner unitary representations for the covering group $ISL(2,\mathbb C)$ of the Poincaré group, we obtain spin-tensor wave functions of free massive particles with an arbitrary spin that satisfy the Dirac–Pauli–Fierz equations. In the framework of a two-spinor formalism, we construct spin-polarization vectors and obtain conditions that fix the corresponding density matrices (the Berends–Fronsdal projection operators) determining the numerators in the propagators of the fields of such particles. Using these conditions, we find explicit expressions for the particle density matrices with integer (Berends–Fronsdal projection operators) and half-integer spin. We obtain a generalization of the Berens–Fronsdal projection operators to the case of an arbitrary number $D$ of space–time dimensions.
Keywords: Wigner unitary representation, Berends–Fronsdal projection operator
Mots-clés : Poincaré group, Dirac–Pauli–Fierz equation. 
@article{TMF_2018_195_3_a0,
     author = {A. P. Isaev and M. A. Podoinicin},
     title = {Unitary representations of {the~Wigner} group $ISL(2,\mathbb C)$ and a~two-spinor description of~massive particles with an~arbitrary spin},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {331--361},
     year = {2018},
     volume = {195},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_195_3_a0/}
}
TY  - JOUR
AU  - A. P. Isaev
AU  - M. A. Podoinicin
TI  - Unitary representations of the Wigner group $ISL(2,\mathbb C)$ and a two-spinor description of massive particles with an arbitrary spin
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2018
SP  - 331
EP  - 361
VL  - 195
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2018_195_3_a0/
LA  - ru
ID  - TMF_2018_195_3_a0
ER  - 
%0 Journal Article
%A A. P. Isaev
%A M. A. Podoinicin
%T Unitary representations of the Wigner group $ISL(2,\mathbb C)$ and a two-spinor description of massive particles with an arbitrary spin
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2018
%P 331-361
%V 195
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2018_195_3_a0/
%G ru
%F TMF_2018_195_3_a0
A. P. Isaev; M. A. Podoinicin. Unitary representations of the Wigner group $ISL(2,\mathbb C)$ and a two-spinor description of massive particles with an arbitrary spin. Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 3, pp. 331-361. http://geodesic.mathdoc.fr/item/TMF_2018_195_3_a0/

[1] E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group”, Ann. Math., 40:1 (1939), 149–204 ; V. Bargmann, E. P. Wigner, “Group theoretical discussion of relativistic wave equations”, Proc. Nat. Acad. Sci. USA, 34:5 (1948), 211–223 | DOI | MR | MR | Zbl

[2] P. A. M. Dirac, “Relativistic wave equations”, Proc. Roy. Soc. London. Ser. A, 155:886 (1936), 447–459 | DOI

[3] M. Fierz, “Über den Drehimpuls von Teilichen mit Ruhemasse null und beliebigem Spin”, Helvetica Phys. Acta, 13:12 (1940), 45–60 | MR

[4] M. Fierz, W. Pauli, “On relativistic wave equations for particles of arbitrary spin in an electromagnetic field”, Proc. Roy. Soc. London. Ser. A, 173:953 (1939), 211–232 | DOI | MR

[5] A. P. Isaev, V. A. Rubakov, Teoriya grupp i simmetrii. Konechnye gruppy. Gruppy i algebry Li, URSS, M., 2018

[6] S. Weinberg, “Feynman rules for any spin”, Phys. Rev., 133:5B (1964), B1318–B1332 | DOI | MR | Zbl

[7] Yu. V. Novozhilov, Vvedenie v teoriyu elementarnykh chastits, Nauka, M., 1972 | MR | MR

[8] J. A. de Azcárraga, A. Frydryszak, J. Lukierski, C. Miquel-Espanya, “Massive relativistic particle model with spin from free two-twistor dynamics and its quantization”, Phys. Rev. D, 73:10 (2006), 105011, 9 pp. | DOI | MR

[9] J. A. Azcárraga, S. Fedoruk, J. M. Izquierdo, J. Lukierski, “Two-twistor particle models and free massive higher spin fields”, JHEP, 4 (2015), 10, 39 pp. | DOI

[10] R. E. Behrends, C. Fronsdal, “Fermi decay for higher spin particles”, Phys. Rev., 106:2 (1957), 345–353 | DOI | MR

[11] N. N. Bogolyubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, Obschie printsipy kvantovoi teorii polya, Nauka, M., 1987 | MR | MR | Zbl

[12] I. L. Buchbinder, S. M. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity. Or a Walk Through Superspace, IOP, Bristol, 1995 | MR

[13] Yu. B. Rumer, A. I. Fet, Teoriya grupp i kvantovannye polya, Nauka, M., 1977 | MR

[14] R. Penrose, M. A. H. MacCallum, “Twistor theory: an approach to the quantization of fields and space-time”, Phys. Rep., 6:4 (1973), 241–315 | DOI | MR

[15] K. P. Tod, “Some symplectic forms arising in twistor theory”, Rep. Math. Phys., 11:3 (1977), 339–346 | DOI | MR

[16] Z. Perjés, “Twistor variables of relativistic mechanics”, Phys. Rev. D, 11:8 (1975), 2031–2041 | DOI | MR

[17] L. P. Hughston, Twistors and Particles, Lecture Notes in Physics, 97, Springer, Berlin, 1979 | MR

[18] A. Bette, “On a pointlike relativistic massive and spinning particle”, J. Math. Phys., 25:8 (1984), 2456–2460 ; “Directly interacting massless particles – a twistor approach”, J. Math. Phys., 37:4 (1996), 1724–1734 | DOI | MR | DOI | MR

[19] A. Bette, J. A. de Azcárraga, J. Lukierski, C. Miquel-Espanya, “Massive relativistic free fields with Lorentz spins and electric charges”, Phys. Lett. B, 595:1–4 (2004), 491–497 | DOI | MR

[20] S. Fedoruk, J. Lukierski, “Massive twistor particle with spin generated by Souriau–Wess–Zumino term and its quantization”, Phys. Lett. B, 733 (2014), 309–315 | DOI

[21] L. C. Biedenharn, H. W. Braden, P. Truini, H. van Dam, “Relativistic wavefunctions on spinor spaces”, J. Phys. A, 21:18 (1988), 3593–3610 | DOI | MR

[22] C. Fronsdal, “On the theory of higher spin fields”, Nuovo Cimento, 9, supp. 2 (1958), 416–443 | DOI | MR

[23] R. Penrouz, V. Rindler, Spinory i prostranstvo-vremya, v. 2, Spinornye i tvistornye metody v geometrii prostranstva-vremeni, Mir, M., 1988 | MR | MR

[24] R. Penrose, “Zero rest-mass fields including gravitation: asymptotic behaviour”, Proc. Roy. Soc. London. Ser. A, 284:1397 (1965), 159–203 | DOI | MR

[25] E. Witten, “Perturbative gauge theory as a string theory in twistor space”, Commun. Math. Phys., 252:1–3 (2004), 189–258 | DOI | MR

[26] H. Elvang, Y.-T. Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge Univ. Press, Cambridge, 2015 | DOI | MR