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Constraints and spontaneous symmetry breaking in quantum field theory

1994
114 pages
Supervisor:
  • N.P. Landsman
Thesis: PhD
  • Cambridge U.
(1994)
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Abstract:
The first part of this thesis applies the Rieffel induction procedure, recently advocated by Landsman for the quantization of systems with constraints, to certain linear quantum field theories. After a preparatory chapter in which Rieffel induction is used to implement constraints on the Heisenberg algebra [qμ, P1.1] = -i9μ1.1, Landsman's proposal is applied to free quantum electrodynamics (QED). Starting from the Fock representation of the unconstrained field algebra, a new representation of the field algebra on the Rieffel-induced Hilbert space Hphys is constructed, which carries a trivial action of the gauge group. This leads to a new type of gauge fixing, lying conceptually between the Coulomb gauge and' the Gupta-Bleuler gauges. The characteristic features of this formulation of free QED arc presented in detail (Hamiltonian, propagator, n-point correlation functions, (semi)-positivity of the metric, implementation of the Poincarc group, action of gauge transformations, etc.). Also, some steps are undertaken to apply the method to functional representations and for a 3-dimensional vector potential with Ao = 0. Subsequently, the Rieffel induction procedure is applied to a simple model showing spontaneous symmetry breaking, viz. the Sti.ickelberg-Kibble model. Here, a physical state space Hphys is constructed which carries a massive representation of the Poincare group. Its longitudinal one-particle component arises from a particular Bogoliubov-transformation of the five (unphysical) degrees of freedom one has started with. The second, smaller part of this thesis contains two more chapters on spontaneous symmetry breaking. In the first one, formal properties of the effective potential of a scalar field theory are investigated. One finds that the effective potential is exactly one-fold differentiable at the value of the vacuum expectation value, and this turns out to be crucial for the validity of the perturbative loop expansion. In a second chapter, the algebraic characterization of the vacuum expectation value of a scalar field as an element in the center of the weak closure of certain representations of the field algebra is used in an attempt to simplify a particular type of gauge-invariant interaction terms.