Particle spin from representations of the diffeomorphism group
It is shown that one can obtain particle spin directly from irreducible representations of the diffeomorphism group, without the need for anticommuting field operators. The semidirect product \(S\wedge K\) of Schwartz's space of functions on \(\mathbb R^ 3\) with the group \(K\) of diffeomorphisms of \(\mathbb R^ 3\) provides a model for quantum field theory. The representations of \(\mathrm{SL}(3,\mathbb R)\) are used to induce representations of the diffeomorphism group \(K\). The local currents are interpreted in these representations; the local measurement operators are then constructed by taking appropriate limits in a representation of \(S\wedge K\). The Hilbert space decomposes into invariant subspaces of fixed spin under the local rigid rotations. The generalization of the results to the case of \(N\) particles is also given. Finally, the physical interpretation of the procedure is discussed. It is pointed out that the description of particles with spin in which the local \(\mathrm{SU}(2)\) current group is adjoined to the group \(S\wedge K\) can be obtained from a unified point of view. The unitarily inequivalent representations of the same group \(S\wedge K\) can describe many different physical systems and, in particular, particles with integers or half-integer spin. The universality of the given model is possible owing to the infinitely many degrees of freedom embodied in the group of diffeomorphisms.
scientific article; zbMATH DE number 3852059
scientific article