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In Matthew Schwartz's Quantum Field Theory and the Standard Model, he says (section 8.6, page 132) that it is possible to avoid introducing the redundant gauge degrees of freedom in QED by quantizing $F^{\mu\nu}$ instead of $A^\mu$. He points out several difficulties with this approach, including that it requires using a non-local lagrangian, which makes it to show that the theory is local and causal.

I'd never heard of this approach before, and I'd like to know more about, but I'm having trouble finding literature on it. What is this formulation called? What would the lagrangian look like?

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  • $\begingroup$ I don't know specifically about quantizing $F_{\mu\nu}$ directly, but you can choose quantize QED in Coulomb gauge which has no unphysical degrees of freedom, but the Lagrangian is not manifestly local or Lorentz invariant. You can read about this in Sections 6.2.1 and 6.4 of Tong's notes. $\endgroup$
    – Andrew
    Mar 16, 2023 at 18:14

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Schwartz has likely manifestly gauge-invariant formulations of QED in mind, such as e.g. Ref. 1. Its eq. (5) displays a non-local Lagrangian.

References:

  1. I. Goldberg, Gauge-Invariant Quantum Electrodynamics II, Phys. Rev. 139 (1965) B1665.
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