550 research outputs found
Groups, Jacobi functions, and rigged Hilbert spaces
This paper is a contribution to the study of the relations between special functions, Lie algebras
and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are
in correspondence with the representations of their algebra of symmetry, that induce discrete and
continuous bases coexisting on a rigged Hilbert space supporting the representation. Meaningful
operators are shown to be continuous on the spaces of test vectors and its dual. Here, the chosen
special functions, called “Algebraic Jacobi Functions” are related to the Jacobi polynomials and
the Lie algebra is su(2, 2). These functions with m and q fixed, also exhibit a su(1, 1)-symmetry.
Different discrete and continuous bases are introduced. An extension in the spirit of the associated
Legendre polynomials and the spherical harmonics is presented introducing the “Jacobi Harmonics”
that are a generalization of the spherical harmonics to the three-dimensional hypersphere S3
Massless particles, electromagnetism, and Rieffel induction
The connection between space-time covariant representations (obtained by
inducing from the Lorentz group) and irreducible unitary representations
(induced from Wigner's little group) of the Poincar\'{e} group is re-examined
in the massless case. In the situation relevant to physics, it is found that
these are related by Marsden-Weinstein reduction with respect to a gauge group.
An analogous phenomenon is observed for classical massless relativistic
particles. This symplectic reduction procedure can be (`second') quantized
using a generalization of the Rieffel induction technique in operator algebra
theory, which is carried through in detail for electro- magnetism. Starting
from the so-called Fermi representation of the field algebra generated by the
free abelian gauge field, we construct a new (`rigged') sesquilinear form on
the representation space, which is positive semi-definite, and given in terms
of a Gaussian weak distribution (promeasure) on the gauge group (taken to be a
Hilbert Lie group). This eventually constructs the algebra of observables of
quantum electro- magnetism (directly in its vacuum representation) as a
representation of the so-called algebra of weak observables induced by the
trivial representation of the gauge group.Comment: LaTeX, 52 page
- …