XXXVI Workshop on Geometric Methods in Physics 2-8.07.2017
VI School on Geometry and Physics 26-30.06.2017

Gerald Goldin

Diffeomorphism group representations in relativistic quantum field theory

Unitary representations of diffeomorphism groups and their semidirect products are known to play a fundamental role in nonrelativistic (Galilean) quantum theory, describing the kinematics of a wide variety of physical systems. The infinitesimal generators of appropriate 1-parameter subgroups are the self-adjoint mass density and momentum flux density, which form a local current algebra. Furthermore, the irreducible, unitary diffeomorphism group representations fall naturally into hierarchies, whose intertwining operators create and annihilate configurations of the same kind. These intertwining operators have an interpretation as ``second-quantized'' fields. This talk focuses on the role played by the diffeomorphism group and its representations in relativistic quantum field theories. Here the mass and momentum densities are reference frame-dependent constructs, not local in spacetime and not Lorentz covariant. Nevertheless, they describe actual quantum measurements. Thus, we propose to start in a fixed reference frame, and with respect to that frame, to introduce the quantum kinematics described by the diffeomorphism group of ``space’’ at fixed time. Only after constructing intertwining fields do we introduce the spacetime symmetry group (the Poincaré group in this case), which provides the information needed to connect descriptions in the different frames of reference. Then we construct local relativistic fields out of the (noncovariant) intertwining fields. We work out this idea for the case of free relativistic scalar bosons, where we formally express the Hamiltonian explicitly in terms of the original density and current operators. The talk is based on joint work with D. H. Sharp (Los Alamos).

Event sponsored by:
Centre de recherches mathématiques          University
of Bialystok
University of Bialystok

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