Proca’s Abelian Sector: Approximate Equivalence. Thought of the Day 62.0


The underdetermination between the quantized Maxwell theory and the lower-mass quantized Proca theories is permanent (at least unless a photon mass is detected, in which case Proca wins). It does not immediately follow that our best science leaves the photon mass unspecified apart from empirical bounds, however. Electromagnetism can be unified with an SU(2) Yang-Mills field describing the weak nuclear force into the electroweak theory. The resulting electroweak unification of course is not simply a logical conjunction of the electromagnetic and weak theories; the theories undergoing unification are modified in the process. Maxwell’s theory can participate in this unification; can Proca theories participate while preserving renormalizability and unitarity? Probably they can. Thus evidently the underdetermination between Maxwell and Proca persists even in electroweak theory, though this unresolved rivalry is not widely noticed. There is some non-uniqueness in the photon mass term, partly due to the rotation by the weak mixing angle between the original fields in the SU(2) × U(1) group and the mass eigenstates after spontaneous symmetry breaking. Thus the physical photon is not simply the field corresponding to the original U(1) group, contrary to naive expectations. There are also various empirically negligible but perhaps conceptually important effects that can arise in such theories. Among these are charge dequantization – the charges of charged particles are no longer integral multiples of a smallest charge – and perhaps charge non-conservation. Crucial to the possibility of including a Proca-type mass term (as opposed to merely getting mass by spontaneous symmetry breaking) is the non-semi-simple nature of the gauge group SU(2) × U(1): this group has a subgroup U(1) that is Abelian and that commutes with the whole of the larger group. Were the electroweak theory to be embedded in some larger semi-simple group such as SU(5), then no Proca mass term could be included.

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