Source
French (2010), LIT-170, read in full in NOTE-162 from the author’s 2010 PhilSci-Archive preprint. The published version (Studies in History and Philosophy of Modern Physics 43(2), 121–136, 2012) was not reached, and the preprint is unpaginated, so locations are section names.
What was actually shown
The objection. Ladyman’s founding case for ontic structural realism used the Stone–von Neumann equivalence of matrix and wave mechanics. For infinitely many degrees of freedom that equivalence fails “of necessity”. Howard argues that the realist is then left with non-isomorphic, competing structures: choosing one is not structural, and keeping all is incoherent.
Option 1, conventional QFT (“Option 1”). Following Wallace, a cut-off QFT treats short-distance inequivalence as an artefact of dropping the cut-off and long-distance inequivalence as information inaccessible from a finite region. Fraser’s reply, that cut-off QFT lacks “even approximately the same content” as AQFT, is answered by noting that AQFT has no realistic interacting models and is also a staging post (“Response” section).
Option 2, the Swiss Army Knife (after Ruetsche). Structure spans the representations, and contingent features pick the relevant one.
- Symmetry breaking (Case 1): broken symmetry is a group–subgroup relation (O(3) → O(2); SU(3)×SU(2)×U(1) → SU(3)×U(1)), and the ensemble of degenerate vacua keeps the full symmetry. If SSB explains asymmetric phenomena, the inequivalence serves a symmetry-based ontology; if the representations cannot be dismissed, what matters is cross-representation structure (“Back to Ruetsche”).
- Statistics (Case 2): under the DHR criterion, charge superselection sectors correspond to irreducible representations of the global gauge group. Abelian groups give Bose and Fermi statistics, non-Abelian ones parastatistics. Statistics then “arise naturally from within the (AQFT) formulation”, and permutation invariance need not be posited. The reading grades the physics strong and the moral for OSR moderate.
Howard’s premise. Ruetsche’s identification of unitary equivalence with physical equivalence rests on Clifton and Halvorson’s translation scheme. French adopts Baker’s proposal that a symmetry mapping between representations suffices for intertranslatability, which “remove[s] the assumption behind Howard’s challenge”.
What could have come out otherwise. The DHR analysis could have needed statistics imposed by hand, and SSB could have been explicable only in a single representation. Neither is the case in the results French imports. His own steps are arguments, and no step of his could have failed empirically.
What this does not say
- That OSR is vindicated by QFT. The conclusion that inequivalent representations “indicate how that position can be further developed and strengthened” is French’s reading of others’ results.
- What modal structure is. French calls it “more profound (and perhaps problematic)” and declines to tie it to possible worlds or causation.
- That every inequivalence is removed. Baker’s condition is sufficient, and Baker leaves open whether it is necessary. Representations related by no symmetry, such as some infrared-inequivalent ones, are not addressed.
- That the two options are combined. The Conclusion sketches a third option (modal algebraic structure plus actual Standard Model structure) and records Fraser’s objection that the representations involved differ in kind.
- That statistics are derived for every field theory. The DHR criterion excludes long-range forces such as electromagnetism.
- Anything about the published text. The 2012 version may revise this draft, which has visible loose ends (“(see vF?)”).
Connections
- THEORY-042 states what a superselection rule does operationally. French’s Case 2 relies on the DHR account of the same sectors.
- THEORY-017 holds that in a Hilbert-space model only unitarily invariant structure is intrinsic. French’s problem is the case where there is more than one Hilbert space and no unitary between them.
- THEORY-034 places structural realisms against Chakravartty’s dilemma (LIT-196). French’s kinds are fixed by irreducible group representations, which is where he and Chakravartty meet head on (NOTE-162). He is not placed there, and this account does not place him.
- De Haro and Butterfield (THEORY-690) argue that dualities give no support to structural realism. That concerns dualities, not inequivalent representations of one theory, so the two do not conflict.
- The SEP survey of structural realism is LIT-045; Wallace’s math-first version, which French borrows for Option 1, is LIT-151.