### 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 language | English (US) |
---|---|

Pages (from-to) | 2570-2589 |

Number of pages | 20 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 14 |

Issue number | 10 |

DOIs | |

State | Published - 1976 |

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### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*Physical Review D - Particles, Fields, Gravitation and Cosmology*,

*14*(10), 2570-2589. https://doi.org/10.1103/PhysRevD.14.2570

**Physical states in quantum electrodynamics.** / Zwanziger, Daniel.

Research output: Contribution to journal › Article

*Physical Review D - Particles, Fields, Gravitation and Cosmology*, vol. 14, no. 10, pp. 2570-2589. https://doi.org/10.1103/PhysRevD.14.2570

}

TY - JOUR

T1 - Physical states in quantum electrodynamics

AU - Zwanziger, Daniel

PY - 1976

Y1 - 1976

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevD.14.2570

DO - 10.1103/PhysRevD.14.2570

M3 - Article

VL - 14

SP - 2570

EP - 2589

JO - Physical review D: Particles and fields

JF - Physical review D: Particles and fields

SN - 1550-7998

IS - 10

ER -