Unitary Representations of the Wigner Group ISL(2, ℂ) and A Two-Spinor Description of Massive Particles With An Arbitrary Spin

A. P. Isaev1, M. A. Podoinicin1
1Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Oblast, Russia, Dubna State University, Dubna, Moscow Oblast, Russia

Tóm tắt

Based on Wigner unitary representations for the covering group ISL(2,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 Behrends–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 (Behrends–Fronsdal projection operators) and half-integer spin. We obtain a generalization of the Behrends–Fronsdal projection operators to the case of an arbitrary number D of space–time dimensions.

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Tài liệu tham khảo

E. P. Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Ann. Math., 40, 149–204 (1939)

V. Bargmann and E. P. Wigner, “Group theoretical discussion of relativistic wave equations,” Proc. Nat. Acad. Sci. USA, 34, 211–223 (1948).

P. A. M. Dirac, “Relativistic wave equations,” Proc. Roy. Soc. London. Ser. A, 155, 447–459 (1936).

M. Fierz, “Über den Drehimpuls von Teilichen mit Ruhemasse null und beliebigem Spin,” Helvetica Phys. Acta, 13, No. 12, 45–60 (1940).

M. Fierz and W. Pauli, “On relativistic wave equations for particles of arbitrary spin in an electromagnetic field,” Proc. Roy. Soc. London. Ser. A, 173, 211–232 (1939).

I. P. Isaev and V. A. Rubakov, Theory of Groups and Symmetries: Finite Groups. Groups, and Lie Algebras [in Russian], URSS, Moscow (2018).

S. Weinberg, “Feynman rules for any spin,” Phys. Rev., 133, B1318–B1332 (1964); “Feynman rules for any spin: II. Massless particles,” Phys. Rev., 134, B882–B896 (1964).

Yu. V. Novozhilov, Introduction to Elementary Particle Theory [in Russian], Nauka, Moscow (1972); English transl. (Intl. Ser. Monogr. Nat. Phil., Vol. 78), Pergamon, Oxford (1975).

J. A. de Azcárraga, A. Frydryszak, J. Lukierski, and C. Miquel-Espanya, “Massive relativistic particle model with spin from free two-twistor dynamics and its quantization,” Phys. Rev. D, 73, 105011 (2006).

J. A. Azcárraga, S. Fedoruk, J. M. Izquierdo, and J. Lukierski, “Two-twistor particle models and free massive higher spin fields,” JHEP, 1504, 10 (2015).

R. E. Behrends and C. Fronsdal, “Fermi decay for higher spin particles,” Phys. Rev., 106, 345–353 (1957).

N. N. Bogolyubov, A. Logunov, A. I. Oksak, and I. T. Todorov, General Principles of Quantum Field Theory [in Russian], Nauka, Moscow (1987); English transl. (Math. Phys. Appl. Math., Vol. 10), Kluwer, Dordrecht (1990).

I. L. Buchbinder and S. M. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity: Or a Walk Through Superspace, IOP, Bristol (1995).

Ya. B. Rumer and A. I. Fet, Group Theory and Quantized Fields [in Russian], Nauka, Moscow (1977).

R. Penrose and M. A. H. MacCallum, “Twistor theory: An approach to the quantization of fields and space–time,” Phys. Rep., 6, 241–315 (1973).

K. P. Tod, “Some symplectic forms arising in twistor theory,” Rep. Math. Phys., 11, 339–346 (1977).

Z. Perjés, “Twistor variables of relativistic mechanics,” Phys. Rev. D, 11, 2031–2041 (1975).

L. P. Hughston, Twistors and Particles (Lect. Notes Phys., Vol. 97), Springer, Berlin (1979).

A. Bette, “On a pointlike relativistic massive and spinning particle,” J. Math. Phys., 25, 2456–2460 (1984)

A. Bette,“Directly interacting massless particles—a twistor approach,” J. Math. Phys., 37, 1724–1734 (1996).

A. Bette, J. A. de Azcárraga, J. Lukierski, and C. Miquel-Espanya, “Massive relativistic free fields with Lorentz spins and electric charges,” Phys. Lett. B, 595, 491–497 (2004).

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

L. C. Biedenharn, H. W. Braden, P. Truini, and H. van Dam, “Relativistic wavefunctions on spinor spaces,” J. Phys. A, 21, 3593–3610 (1988).

C. Fronsdal, “On the theory of higher spin fields,” Nuovo Cimento, 9, Supp. 2, 416–443 (1958).

R. Penrose and W. Rindler, Spinors and Space–Time, Vol. 2, Spinor and Twistor Methods in Space–Time Geometry, Cambridge Univ. Press, Cambridge (1986).

R. Penrose, “Zero rest-mass fields including gravitation: Asymptotic behaviour,” Proc. Roy. Soc. London. Ser. A, 284, 159–203 (1965).

E. Witten, “Perturbative gauge theory as a string theory in twistor space,” Commun. Math. Phys., 252, 189–258 (2004).

H. Elvang and Y. -T. Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge Univ. Press, Cambridge (2015).