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In field theory, the Stueckelberg action (named after Ernst Stueckelberg[1]) describes a massive spin-1 field as an R (the real numbers are the Lie algebra of U(1)) Yang–Mills theory coupled to a real scalar field . This scalar field takes on values in a real 1D affine representation of R with as the coupling strength.
This is a special case of the Higgs mechanism, where, in effect, λ and thus the mass of the Higgs scalar excitation has been taken to infinity, so the Higgs has decoupled and can be ignored, resulting in a nonlinear, affine representation of the field, instead of a linear representation — in contemporary terminology, a U(1) nonlinear σ-model.
Gauge-fixing , yields the Proca action.
This explains why, unlike the case for non-abelian vector fields, quantum electrodynamics with a massive photon is, in fact, renormalizable, even though it is not manifestly gauge invariant (after the Stückelberg scalar has been eliminated in the Proca action).
^Stückelberg, Ernst C.G. (1938). "Die Wechselwirkungskräfte in der Elektrodynamik und in der Feldtheorie der Kräfte". Helvetica Physica Acta (in German). 11: 225.
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