A C*-algebra formulation of the quantization of the electromagnetic field
A presentation of the Fermi, Gupta--Bleuler, and radiation gauge methods for quantizing the free electromagnetic field is given in the Weyl algebra formalism for quantum field theory first introduced by Segal. The abstract Weyl algebra of the vector potential is defined using the formalism of Manuceau. Then the Fermi and Gupta--Bleuler methods are given as schemes for constructing representations of the algebra. The algebra of the physical photons is shown to be a factor algebra of a certain subalgebra of the original algebra of the vector potential. In this formalism, the application of the supplementary condition in the Fermi method, and the supplementary condition and indefinite metric in the Gupta--Bleuler method, can be interpreted as the means by which a representation of this factor algebra is obtained. The Weyl algebra of the physical photons is the Weyl algebra associated with the radiation gauge method. It is also shown that in the Fock representation of the Weyl algebra given by the Fermi method, automorphisms of the algebra corresponding to Lorentz transformations cannot always be implemented by unitary transformations. This leads us to construct a new representation of the Weyl algebra which provides a covariant representation for the vector potential.
- Research Organization:
- Department of Mathematical Physics, University of Adelaide, South Australia 5001
- OSTI ID:
- 7225926
- Journal Information:
- J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 18:4; ISSN JMAPA
- Country of Publication:
- United States
- Language:
- English
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657007 -- Electricity & Magnetism-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
BANACH SPACE
ELECTROMAGNETIC FIELDS
FIELD ALGEBRA
FIELD THEORIES
GAUGE INVARIANCE
HILBERT SPACE
INVARIANCE PRINCIPLES
LORENTZ TRANSFORMATIONS
MATHEMATICAL SPACE
QUANTUM FIELD THEORY
SPACE
TRANSFORMATIONS
WEYL UNIFIED THEORY