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Classical and quantum field theory provide not only realistic examples ofextant notions of empirical equivalence, but also new notions of empiricalequivalence, both modal and occurrent. A simple but modern gravitational casegoes back to the 1890s, but there has been apparently total neglect of thesimplest relativistic analog, with the result that an erroneous claim has takenroot that Special Relativity could not have accommodated gravity even if therewere no bending of light. The fairly recent acceptance of nonzero neutrinomasses shows that widely neglected possibilities for nonzero particle masseshave sometimes been vindicated. In the electromagnetic case, there is permanentunderdetermination at the classical and quantum levels between Maxwell's theoryand the one-parameter family of Proca's electromagnetisms with massive photons,which approximate Maxwell's theory in the limit of zero photon mass. WhileYang-Mills theories display similar approximate equivalence classically,quantization typically breaks this equivalence. A possible exception, includingunified electroweak theory, might permit a mass term for the photons but notthe Yang-Mills vector bosons. Underdetermination between massive and massless(Einstein) gravity even at the classical level is subject to contemporarycontroversy.

british journal for the philosophy of science

Permanent Underdetermination from Approximate Empirical Equivalence in  Field Theory: Massless and Massive Scalar Gravity, Neutrino, Electromagnetic,  Yang-Mills and Gravitational Theories