Massless particles, electromagnetism, and Rieffel induction
Jun, 199449 pages
Published in:
- Rev.Math.Phys. 7 (1995) 923-958
e-Print:
- hep-th/9411174 [hep-th]
Report number:
- DESY-94-109
Citations per year
Abstract: (arXiv)
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.Note:
- LaTeX, 52 pages
- gauge field theory: U(1)
- axiomatic field theory
- Hamiltonian formalism
- phase space: Hilbert space
- operator: algebra
- algebra: representation
- mathematical methods
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