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https://doi.org/10.1103/physrevd.14.2570
Copy DOIJournal: Physical Review D | Publication Date: Nov 15, 1976 |
Citations: 41 |
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 $\mathcal{I}$ of quantum electrodynamics is proposed: Any vector $\ensuremath{\Phi}\ensuremath{\in}\mathcal{I}$ is a physical state if it is positive on the observables, $〈\ensuremath{\theta}\ensuremath{\Phi}, \ensuremath{\theta}\ensuremath{\Phi}〉\ensuremath{\ge}0$, $〈\ensuremath{\Phi}, \ensuremath{\Phi}〉=1$, for $\ensuremath{\theta}$ any element of the algebra of observables. Observables $\ensuremath{\theta}$, in turn, are selected by the requirement that they commute with the generators of the restricted gauge transformations of the second kind, ${A}_{\ensuremath{\mu}}\ensuremath{\rightarrow}{A}_{\ensuremath{\mu}}+{\ensuremath{\partial}}_{\ensuremath{\mu}}\ensuremath{\lambda}$, $\ensuremath{\psi}\ensuremath{\rightarrow}\ensuremath{\psi}\mathrm{exp}(ie\ensuremath{\lambda})$, with $\ensuremath{\lambda}(x)=c\ensuremath{-}\mathrm{number}$, ${\ensuremath{\partial}}^{2}\ensuremath{\lambda}=0$. This is equivalent to the requirement $[B(x), \ensuremath{\theta}]=0$, where $B(x)=\ensuremath{\partial}\ifmmode\cdot\else\textperiodcentered\fi{}A(x)$ in the Feynman gauge. It is proved that the substitute Gupta-Bleuler condition ${B}^{(\ensuremath{-})}(x)\ensuremath{\Phi}={b}^{(\ensuremath{-})}(x)\ensuremath{\Phi}$ provides a subspace ${\mathcal{I}}_{[b]}$ of physical states, where ${b}^{(\ensuremath{-})}(x)$ is the negative-frequency part of any real $c$-number solution of the wave equation ${\ensuremath{\partial}}^{2}b(x)=0$ satisfying $\ensuremath{\int}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{b}(x){d}^{3}x=q$, with $q$ an eigenvalue of the charge operator. Different functions $b(x)$ characterize different superselection sectors which are eigenspaces of generators $G(\ensuremath{\lambda})$ of the restricted gauge transformations of the second kind with eigenvalues $G(\ensuremath{\lambda})=\ensuremath{\int}\ensuremath{\lambda}(x){\stackrel{\ensuremath{\leftrightarrow}}{\ensuremath{\partial}}}_{0}b(x){d}^{3}x$. In a given superselection sector Maxwell's equations take the form ${\ensuremath{\partial}}_{\ensuremath{\mu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}}={J}^{\ensuremath{\nu}}\ensuremath{-}{\ensuremath{\partial}}^{\ensuremath{\nu}}b$, where $\ensuremath{-}{\ensuremath{\partial}}^{\ensuremath{\nu}}b$ is interpreted as a classical external current which is induced by the quantum-mechanical current ${J}^{\ensuremath{\nu}}$. The proof relies on the axiom of asymptotic completeness $\mathcal{I}={\mathcal{I}}^{\mathrm{in}}={\mathcal{I}}^{\mathrm{out}} \mathrm{and} {\mathcal{I}}^{\mathrm{in}}$ and is specified by the ansatz of infrared coherence, namely, ${\mathrm{lim}}_{\ensuremath{\omega}\ensuremath{\rightarrow}0}{a}_{\ensuremath{\mu}}^{\mathrm{in}}(k)\ensuremath{\sim}\ensuremath{-}{(2\ensuremath{\pi})}^{\ensuremath{-}\frac{3}{2}}\frac{{\ensuremath{\Sigma}}_{i}{e}_{i}{p}_{i}}{{p}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}k}$, where ${a}_{\ensuremath{\mu}}^{\mathrm{in}}(k)$ is the photon annihilation operator and ${p}_{i}$ is the momentum of an incoming particle of charge ${e}_{i}$, and in \ensuremath{\rightarrow} 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 $\ensuremath{\psi}$.
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