Fields on the Poincaré group: Arbitrary spin description and relativistic wave equations.

*(English)*Zbl 1006.83001In field theory, particles with different spins are normally described by multicomponent spin-tensor fields on Minkowski space. However, for this purpose scalar functions too, depending on both Minkowski space coordinates and on continuous bosonic variables corresponding to spin degrees of freedom, can be used as fields. Such fields were introduced by V. L. Ginzburg and I. E. Tamm [Zh. Eksp. Teor. Fiz. 17, 227-237 (1947)], V. Bargmann and E. P. Wigner [Proc. Natl. Acad. Sci. USA 34, 211-223 (1948; Zbl 0030.42306)], H. Yukawa [Phys. Rev. (2) 77, 219-226 (1950; Zbl 0036.26705)] and D. V. Shirkov [Zh. Eksp. Teor. Fiz. 21, 748-760 (1951)] in connection with the problem of constructing relativistic wave equations (RWE). Fields of this type may be treated as fields on homogeneous spaces of the Poincaré group. D. Finkelstein [Phys. Rev. (2) 100, 924-931 (1955; Zbl 0065.44305)] gave a systematic development of this point of view along with a classification and explicit construction of homogeneous spaces of the Poincare group which contains Minkowski space. Further, in order to give a dynamical role to the spin, F. Lurcat [Phys. 1, 95 (1964)] suggested the construction of Quantum field theory on the Poincaré group. In this direction one can refer to the pioneering work of H. Bacry and A. Kihlberg [J. Math. Phys. 10, 2132-2141 (1969; Zbl 0191.27002)], A. Kihlberg [Ann. Inst. Henri Poincaré 13, 57-76 (1970)], C. P. Boyer and G. N. Fleming [J. Math. Phys. 15, 1007-1024 (1974)], H. Arodź [Acta Phys. Pol., B 7, 177-190 (1976)], M. Toller [Nuovo Cimento B (11) 44, 67-98 (1978); J. Math. Phys. 37, 2694-2730 (1996; Zbl 0862.22018)] and W. Drechsler [ J. Math. Phys. 38, 5531-5558 (1997; Zbl 0892.58089)].

In this paper, the approach is just given from the group theoretic point of view. The authors consider the scalar fields \(f(x,z)\), \(x\) being real coordinates on Minkowski space whereas \(z\), describing spin degrees of freedom, are complex coordinates on the Lorentz subgroup. The authors develop a general approach to describe particles with different spins in the framework of a theory of scalar fields on the Poincaré group and show that these fields are generating functions for the usual spin-tensor fields. Employing tools from harmonic analysis (i.e. with the help of various sets of commuting operators on the group), the authors classify the scalar fields and obtain a description of irreps. of the group. The authors then go on to formulate a general scheme of constructing RWE for particles with definite mass and spin in this language in any number of dimensions. Further, the authors introduced discrete transformations in the space of scalar functions and relate these transformations to automorphisms of the proper Poincaré group. In the sequel, the authors apply the above general scheme to elaborate the study of scalar fields on two, three, four-dimensional groups so as to construct RWE, analyze their solutions and explore the general features of these equations. In the course of this study, it turns out that there exist two different types of scalar functions, one related to a finite dimensional non-unitary representation and the other to an infinite dimensional unitary representation of the Lorentz subgroup.

Any way, the work, “an useful contribution”, is quite interesting as well as stimulating.

In this paper, the approach is just given from the group theoretic point of view. The authors consider the scalar fields \(f(x,z)\), \(x\) being real coordinates on Minkowski space whereas \(z\), describing spin degrees of freedom, are complex coordinates on the Lorentz subgroup. The authors develop a general approach to describe particles with different spins in the framework of a theory of scalar fields on the Poincaré group and show that these fields are generating functions for the usual spin-tensor fields. Employing tools from harmonic analysis (i.e. with the help of various sets of commuting operators on the group), the authors classify the scalar fields and obtain a description of irreps. of the group. The authors then go on to formulate a general scheme of constructing RWE for particles with definite mass and spin in this language in any number of dimensions. Further, the authors introduced discrete transformations in the space of scalar functions and relate these transformations to automorphisms of the proper Poincaré group. In the sequel, the authors apply the above general scheme to elaborate the study of scalar fields on two, three, four-dimensional groups so as to construct RWE, analyze their solutions and explore the general features of these equations. In the course of this study, it turns out that there exist two different types of scalar functions, one related to a finite dimensional non-unitary representation and the other to an infinite dimensional unitary representation of the Lorentz subgroup.

Any way, the work, “an useful contribution”, is quite interesting as well as stimulating.

Reviewer: Om Prakash Singh (Aligarh)