Physical states in quantum electrodynamics

Daniel Zwanziger

    Research output: Contribution to journalArticle

    Abstract

    Because of difficulties with the Gupta-Bleuler subsidiary condition in the charged sectors, an alternative scheme for identifying physical states in the indefinite-metric space I of quantum electrodynamics is proposed: Any vector ΦI is a physical state if it is positive on the observables, θΦ,θΦ0, Φ,Φ=1, for θ any element of the algebra of observables. Observables θ, in turn, are selected by the requirement that they commute with the generators of the restricted gauge transformations of the second kind, Aμ→Aμ+μλ, ψ→ψexp(ieλ), with λ(x)=c-number, 2λ=0. This is equivalent to the requirement [B(x),θ]=0, where B(x)=•A(x) in the Feynman gauge. It is proved that the substitute Gupta-Bleuler condition B(-)(x)Φ=b(-)(x)Φ provides a subspace I[b] of physical states, where b(-)(x) is the negative-frequency part of any real c-number solution of the wave equation 2b(x)=0 satisfying b(x)d3x=q, with q an eigenvalue of the charge operator. Different functions b(x) characterize different superselection sectors which are eigenspaces of generators G(λ) of the restricted gauge transformations of the second kind with eigenvalues G(λ)=λ(x)0b(x)d3x. In a given superselection sector Maxwell's equations take the form μFμν=Jν-νb, where -νb is interpreted as a classical external current which is induced by the quantum-mechanical current Jν. The proof relies on the axiom of asymptotic completeness I=Iin=IoutandIin and is specified by the ansatz of infrared coherence, namely, limω→0aμin(k)∼-(2π)-32ieipipi•k, where aμin(k) is the photon annihilation operator and pi is the momentum of an incoming particle of charge ei, and in → out. The spectral decomposition of the infrared-coherent space is effected. Its singularity in the neighborhood of the electron mass agrees with the singularity of the electron propagator in the Feynman gauge, which allows an on-shell normalization of the charged field ψ.

    Original languageEnglish (US)
    Pages (from-to)2570-2589
    Number of pages20
    JournalPhysical Review D - Particles, Fields, Gravitation and Cosmology
    Volume14
    Issue number10
    DOIs
    StatePublished - 1976

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    quantum electrodynamics
    sectors
    eigenvalues
    generators
    metric space
    real numbers
    operators
    requirements
    subsidiaries
    electron mass
    completeness
    Maxwell equation
    wave equations
    algebra
    substitutes
    momentum
    decomposition
    propagation
    photons
    electrons

    ASJC Scopus subject areas

    • Physics and Astronomy (miscellaneous)

    Cite this

    Physical states in quantum electrodynamics. / Zwanziger, Daniel.

    In: Physical Review D - Particles, Fields, Gravitation and Cosmology, Vol. 14, No. 10, 1976, p. 2570-2589.

    Research output: Contribution to journalArticle

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