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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
Methods of teaching algebra in the high school with reference to selected topics
Thesis (M.A.)--Boston Universit
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