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

Abstract

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

What Are Observables in Hamiltonian Einstein–Maxwell Theory?

“…Recently the author showed that for massive electromagnetism, the requirement that equivalent theories have equivalent observables (in other words, that gauge-fixing/un-fixing doesn't change the observable content) is inconsistent with the separate first-class constraint view but fits perfectly with the gauge generator G [47,48]. Massive electromagnetism approaches massless (Maxwell) as m → 0, whether classically or in quantum field theory [5,24,44]. It is a commonplace in quantum field theory that the de Broglie-Proca formulation without gauge freedom is useful for showing unitarity, whereas the Stueckelberg-Utiyama formulation with gauge freedom is useful for showing renormalizability [43, pp.…”

Space-time constructivism vs. modal provincialism: Or, how special relativistic theories needn't show Minkowski chronogeometry

“…Such issues are relevant to whether it is epistemically possible for us now that such a theory is true of the actual world. If one is interested only in whether theories with similar features are metaphysically possible, and epistemically possible as of the 1910s rather than 2016, then one can use a massive scalar theory instead [Pitts, 2011b, Pitts, 2011a, Pitts, 2016b, doing to Nordström's theory what Hugo von Seeliger and Carl Neumann in the 1890s had done and Einstein in 1917 [Einstein, 1923] would do to Newton's theory. Hence the general philosophical idea of Freund, Maheshwari and Schonberg manifestly could be realized for scalar gravity theories, whether or not it could be for tensor theories such as they considered.…”

“…More definitively troublesome is the fact, that Norton's (3) is false even within what is presumably a sector of Special Relativity, namely, universally coupled massive scalar gravity, which is just Poincaré-invariant. These theories, though they have roots in the 1890s in the work of Seeliger and Neumann, and would have gravity satisfy the rather familiar Klein-Gordon equation in the lowest approximation, have hardly ever been studied thoroughly, especially for philosophical lessons, until recently [Pitts, 2011a, Pitts, 2011b, Pitts, 2016b. One cannot assume that physicists have already done all the work that philosophy requires.…”

“…[Einstein and Grossmann, 1996] In the scalar case, only the trace of stress-energy serves as such a source. It is not obvious by inspection, but is nonetheless demonstrable, that as a result the matter field equations have matter coupled to an effective geometry that is curved (though conformally flat) because the gravitational potential merges with the original volume element [Kraichnan, 1955, Freund and Nambu, 1968, Deser and Halpern, 1970, Pitts, 2011a, Pitts, 2011b, Pitts, 2016b. Thus the field equations follow from a Lagrangian density of this form (with u being matter, g giving effective volumes via the effective positive volume element √ −g,η µν being the unimodular Weyl-flat tensor density giving the conformal geometry, and η being the determinant of the flat metric):…”

Massive Nordström scalar (density) gravities from universal coupling

“…Thus there is a problem of underdetermination between the massless theory and its massive variants for sufficiently small masses [28]. The analog of this instance of underdetermination was already clearly understood by Seeliger in the 1890s.…”