$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

Abstract

We perform the momentum-space quantization of a spin-less particle moving on the SU (2) group manifold, that is, the three-dimensional sphere S 3 , by using a non-canonical method entirely based on symmetry grounds. To achieve this task, non-standard (contact) symmetries are required as already shown in a previous article where the configuration-space quantization was given. The Hilbert space in the momentum space representation turns out to be made of a subset of (oscillatory) solutions of the Helmholtz equation in four dimensions. The most relevant result is the fact that both the scalar product and the generalized Fourier transform between configuration and momentum spaces deviate notably from the naively expected expressions, the former exhibiting now a non-trivial kernel, under a double integral, traced back to the non-trivial topology of the phase space, even though the momentum space as such is flat. In addition, momentum space itself appears directly as the carrier space of an irreducible representation of the symmetry group, and the Fourier transform as the unitary equivalence between two unitary irreducible representations.

Structure of gauge theories

“…The standard classical approach to a particle moving on a Riemann manifold with metric g i j (x) is established by the Lagrangian (see [38,39] and references therein):…”

Invariant Scalar Product and Associated Structures for Tachyonic Klein–Gordon Equation and Helmholtz Equation

“…A more rigorous and complete classification of the solutions of H (which extends trivially to KGt) can be found in [10] (Appendix B). We remark here that the initial value problem for ( 12) is well-posed only for the oscillatory initial conditions: the solutions' behavior changes smoothly for small perturbations when real exponential and linear solutions (the limiting case) are excluded.…”

“…Let us show that, for KGt, the scalar product (2) is crucial to achieve unitarity. The corresponding H version is obtained by changing | p | > κ by | p | < κ (see [10]).…”

“…The physical significance of the Helmholtz equation is quite diverse, ranging from wave optics [8,9] to a momentum space representation for a particle moving freely on a sphere [10].…”

Coherent states for equally spaced, homogeneous waveguide arrays

“…The issue of coherent states of the Eucliean E(2) group has been a challenge for a long time due to the lack of a resolution of the identity, with many proposals to circumvent it (see [30,31] for group theoretical approaches). In [19] it was proven that a resolution of the identity can be introduced by means of a nonlocal scalar product originally introduced in [32] and further developed in [33].…”