Local Gauss Law and Local Gauge Symmetries in QFT

Local gauge symmetries reduce to the identity on the observables, as well as on the physical states (apart from reflexes of the local gauge group topology) and therefore their use in Quantum Field Theory (QFT) asks for a justification of their strategic role. They play an intermediate role in deriving the validity of Local Gauss Laws on the physical states (for the currents which generate the related global gauge group); conversely, we show that local gauge symmetries arise whenever a vacuum representation of a local field algebra \(\mathcal{F}\) is used for the description/construction of physical states satisfying Local Gauss Laws, just as global compact gauge groups arise for the description of localizable states labeled by superselected quantum numbers. The above relation suggests that the Gauss operator \(\bf{G}\), which by locality cannot vanish in \(\mathcal{F}\), provides an intrinsic characterization of the realizations of a gauge QFT in terms of a local field algebra \(\mathcal{F}\) and of the related local gauge symmetries generated by \(\bf{G}\).

1. 1.

   For the problems arising for a non-local dynamics, see e.g. 9, Appendix A.In the case of DHR states the existence of a local field algebra for their description is guaranteed by general principles; in the case of LGL states it may be motivated by the standard way of treating gauge field theory models, e.g. in the perturbative treatment of QED or more generally of the standard model. Some hint is provided, e.g. in QED, by giving a small mass \(\mu\) to the photon, so that the charged states become DHR states and DRH analysis applies with the existence of a local field algebra \(\mathcal{F}\) for their description. As shown by Blanchard and Seneor the vacuum correlations of \(\mathcal{F}\) have a limit for \(\mu\rightarrow 0\) (preserving locality) and the problem is reduced to the construction of the physical LGL states in terms of the vacuum representation of the local algebra \(\mathcal{F}\). Actually, in Symanzik’s treatment of the Proca theory, with the use of the Stuckelberg field \(B\), in analogy with QED, (Lectures on Lagrangian Field Theory, DESY report T-71/1), the fields \(\psi_{g}=\exp^{-ie[(-\Updelta)^{-1}\partial_{i}A^{i}]}\psi\), \(A_{g}^{\mu}=A^{\mu}-\partial^{\mu}[(-\Updelta)^{-1}\partial_{i}A^{i}]\), with \(\psi,\,A_{\mu}\) the Proca fields, commute with \(B\), which in the limit \(\mu\rightarrow 0\) generates the local gauge transformations, (with gauge parameters \({\varepsilon}(x)\) satisfying \(\square{\varepsilon}=0\)), and should yield the (Coulomb) physical charged states.

2. 2.

   For a more detailed discussion see5 Chapter 8, Section 2.1.

3. 3.

   For a general discussion of the occurrence of local gauge symmetries in QED, in a \(\mathrm{C}^{*}\)-algebra setting, see F. Ciolli, G. Rizzi, E. Vasselli, QED Representation for the Net of Causal Loops, Rev. Math. Phys., 27, 1550012 (2013); D. Buchholz, F. Ciolli, G. Rizzi, E. Vasselli, The universal algebra of the electromagnetic field. III. Static charges and emergence of gauge fields, arXiv: 2111.01538 [math-ph]. When the present note was in reparation a very important result was obtained by the same authors, for QED in the presence of external charges 12.

I am indebted to Francesco Serra for a stimulating discussion.

Authors and Affiliations

1. Dipartimento di Fisica “E. Fermi”, Università di Pisa, Pisa, Italy

   Franco Strocchi

Corresponding author

Correspondence to Franco Strocchi .

Editors and Affiliations

1. Unit of Pisa, National Research Council (CNR), Pisa, Italy

   Andrea Cintio

2. Inst for App. Math. and Hausdorff Centre, University of Bonn, Bonn, Nordrhein-Westfalen, Germany

   Alessandro Michelangeli

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