# A Local Gauge-Invariant Formulation of Quantum Electrodynamics

## Abstract

In this article I want to discuss a local and manifestly gauge-invariant formulation of quantum electrodynamics which has been developed by R. Menikoff and myself.^{(1,2)} The manifest gauge invariance is achieved by using the electromagnetic field strengths, rather than potentials, to describe the electromagnetic field, and by using local currents, rather than canonical fields, to describe the matter. In order to represent this theory in Hilbert space, we study the continuous unitary representations of the group obtained from the exponentiated currents and electromagnetic field strengths. These operators can be represented on a Hilbert space having positive norm, so that the necessity for an indefinite metric does not arise here, and the equations of motion hold as operator equations in this Hilbert space.

## Keywords

Hilbert Space Local Current Quantum Electrodynamic Current Algebra Nonrelativistic Quantum## Preview

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## References and Notes

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- 7.A brief review of the work on local currents in nonrelativistic quantum theory through 1974 can be found in D. H. Sharp, What we have learned about representing local nonrelativistic current algebras, in
*Local Currents and Their Applications*,Eds. D. H. Sharp and A. S. Wightman, North-Holland, Amsterdam (1974), pp. 85–98. Further developments are described in Reference (8).Google Scholar - 8.
- 9.G. A. Goldin and D. H. Sharp, Lie algebras of local currents and their representations in
*Group Representations in Mathematics and Physics*,*Battelle Seattle 1969 Rencontres*, Ed. V. Bargmann, Springer-Verlag, New York (1970), pp. 300–311.Google Scholar - 10.G. A. Goldin, J. Grodnik, R. T. Powers, and D.
**H**. Sharp,*J. Math. Phys*.**15**, 88 (1974).MathSciNetADSCrossRefGoogle Scholar - 11.I. Gelfand and N. Vilenkin,
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*Commun. Math. Phys*.**35**, 25 (1974).MathSciNetADSCrossRefGoogle Scholar - 16.G. A. Goldin, R. Menikoff, and D. H. Sharp, Particle statistics from induced representations of a local current group, accepted for publication in
*J. Math. Phys*.Google Scholar