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Free quantum fields on the Poincaré group. (English) Zbl 0862.22018
Summary: A class of free quantum fields defined on the Poincaré group is described by means of their two-point vacuum expectation values. They are not equivalent to fields defined on the Minkowski space-time and they are “elementary” in the sense that they describe particles that transform according to irreducible unitary representations of the symmetry group, given by the product of the Poincaré group and of the group \(SL(2, \mathbb{C})\) considered as an integral symmetry group. Some of these fields describe particles with positive mass and arbitrary spin and particles with zero mass and arbitrary helicity or with an infinite helicity spectrum. In each case the allowed \(SL(2, \mathbb{C})\) internal quantum numbers are specified. The properties of local commutativity and the limit in which one recovers the usual field theories in Minkowski space-time are discussed. By means of a superposition of elementary fields, one obtains an example of a field that presents a broken symmetry with respect to the group \(Sp(4, \mathbb{R})\) that survives in the short-distance limit. Finally, the interaction with an accelerated external source is studied and it is shown that, in some theories, the average number of particles emitted per unit of proper time diverges when the acceleration exceeds a finite critical value.

22E70 Applications of Lie groups to the sciences; explicit representations
81T10 Model quantum field theories
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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