Heisenberg Operators in Quantum Electrodynamics. II

F. J. Dyson
Phys. Rev. 83, 608 – Published 1 August 1951
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Abstract

The equations of motion of quantum electrodynamics are set up in the interaction representation, using a formalism due to Gupta. A new representation, called the intermediate representation, is defined by constructing explicitly a unitary operator S(t), which transforms the state-vector of the interaction representation into the state-vector of the new representation. The intermediate representation is intermediate in behavior between the interaction and Heisenberg representations. In it the low frequency changes in the state of a system are represented by changes in the state-vector, as in the interaction representation, while the high frequency fluctuations are represented by the time-variation of the field operators, as in the Heisenberg representation.

The program of this and a forthcoming paper is to prove that the intermediate representation provides a complete and divergence-free formulation of quantum electrodynamics, with a divergence-free Schrödinger equation which describes accurately the behavior of any physical system. In this paper the mathematical technique is developed which will enable the divergences to be eliminated from the Schrödinger equation. For simplicity, the technique is explained by applying it first to the analysis of the electromagnetic potential operators. The greater part of the paper is occupied with a proof that any fourier component of an electromagnetic potential operator in the intermediate representation is divergence-free after renormalizations have been consistently carried out. The cancellation of the divergences by appropriate compensating terms arising from S(t) is an extremely intricate process, the success of which could not be foreseen without carrying through the calculations in detail.

In a final section, it is explained how Heisenberg operators are to be regarded as a special limiting case of intermediate representation operators. It follows from the preceding analysis, that averages over finite space-time regions of Heisenberg field operators in quantum electrodynamics are divergence-free after renormalization.

  • Received 29 March 1951

DOI:https://doi.org/10.1103/PhysRev.83.608

©1951 American Physical Society

Authors & Affiliations

F. J. Dyson

  • Department of Mathematical Physics, University of Birmingham, Birmingham, England

See Also

Heisenberg Operators in Quantum Electrodynamics. I

F. J. Dyson
Phys. Rev. 82, 428 (1951)

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Vol. 83, Iss. 3 — August 1951

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