NOTE-162: French — Unitary inequivalence and structural realism

Contribution

The paper takes Howard’s objection seriously. Ladyman’s founding case for OSR relied on Stone–von Neumann equivalence, and in QFT that equivalence fails “of necessity”, not by accident. French then maps the realist’s options. What he adds to the literature is an argument that the two physically significant uses of inequivalent representations, SSB and superselection/statistics, both favour a structuralist reading once structure is taken to include the relations between representations and to be modally extended.

Key insight

Howard’s objection assumes that inequivalent representations are competitors, so that choosing one is non-structural and keeping all is incoherent. French argues that they are not competitors in the relevant sense. What does the physical work in SSB (the symmetry connecting the degenerate vacua) and in quantum statistics (the category of DHR representations) is precisely structure that “cut[s] across different representations” (Earman). A structuralist who lets structure span possibilities, like the blades of a Swiss Army knife, can take that work at face value.

Assumptions

• OSR’s notion of structure consists of laws plus symmetry principles, with natural kinds given group-theoretically: fermion/boson by the permutation group, mass and spin by irreducible representations of the Poincaré group (Introduction).
• Howard’s premise: unitary equivalence is necessary for physical equivalence (Clifton–Halvorson). French later rejects it, following Baker.
• Physics inputs taken on authority: the Stone–von Neumann theorem and its failure for infinitely many degrees of freedom; Haag’s theorem; the GNS construction; Earman (2004) on SSB and non-unitarily-implementable symmetries; the DHR selection criterion, which excludes long-range forces such as electromagnetism; and Halvorson & Müger (2006) on the category of DHR representations, which is symmetric in ≥4 spacetime dimensions and braided in 2 or 3.
• Realism as fallible and provisional. Both conventional QFT and AQFT are expected to be superseded, by something like loop quantum gravity or string theory.
• Approximate truth via partial structures (da Costa & French 2003), offered as the framework for inter-theory relations.

Key results

• Option 1 (conventional QFT): with Wallace, short-distance inequivalence is a mathematical artefact of dropping the cut-off, and long-distance inequivalence encodes information that is inaccessible from a finite region. So the realist can say the world’s structure is approximately that of cut-off QFT (Option 1; Conclusion). The cost is Fraser’s objection that the cut-off variant does not have “even approximately the same content” as AQFT, and that SSB needs inequivalent representations.
• Against Fraser’s preference for AQFT: AQFT lacks realistic interacting models, so it too is a programme treated as a staging post. Its “wholly true” advantage fails because it omits gravity (“Response” section).
• SSB (Case 1): broken symmetry is a relation between a group and a subgroup, for example O(3)→O(2) for a buckled ping-pong ball, or SU(3)×SU(2)×U(1)→SU(3)×U(1) at about 100 GeV, so structuralism can accommodate it. The whole ensemble of degenerate solutions keeps the symmetry (“symmetry spreading”, Stewart & Golubitsky). This yields a dilemma for the critic (“Back to Ruetsche”). If SSB is invoked to explain asymmetric phenomena, the inequivalence serves a symmetry-based ontology. If the representations cannot be dismissed, what matters is cross-representation structure. Either way the structuralist approves.
• Statistics (Case 2): under DHR, charge superselection sectors correspond one-to-one to irreducible representations of the global gauge group, in Minkowski dimension or higher. An Abelian gauge group gives Bose/Fermi statistics. A non-Abelian one gives parastatistics, or ordinary statistics plus hidden variables (Han–Nambu colour). So statistics “arise naturally from within the (AQFT) formulation”, and inequivalent representations “help her cause”.
• Surplus structure and plenitude: the non-actual (para-) sectors should be kept, since they were heuristically useful in the history of QCD. The principle of plenitude need not be accepted. The tension is dissolved by making structure modal.
• Intertranslatability: Baker’s proposal, that the existence of a symmetry mapping between representations suffices for them to be intertranslatable, would let the structural realist treat the SSB-related inequivalent representations as intertranslatable. This “remove[s] the assumption behind Howard’s challenge”.
• Conclusion: “Far from presenting a problem for structural realism, then, inequivalent representations indicate how that position can be further developed and strengthened.”

Claims

id	claim	strength	support
C1	Structural realists can adopt conventional cut-off QFT and dismiss inequivalent representations, following Wallace.	moderate	argument relying on Wallace (2006)
C2	Fraser’s “compelling reason” to prefer AQFT fails, since AQFT is equally provisional.	moderate	informal argument, “Response” section
C3	SSB in QFT supports rather than threatens OSR, because the symmetry is preserved across the ensemble of vacuum representations.	moderate–weak	dilemma argument relying on Earman (2004) and Brading & Castellani (2008); the justification of SSB itself is left open
C4	The DHR analysis derives quantum statistics and charge kinds from the algebra, removing permutation invariance as an independent posit.	strong (for the physics) / moderate (for the OSR moral)	the technical result is Halvorson & Müger (2006) and Haag (1992); the ontological reading is French’s
C5	Baker’s symmetry-mapping criterion removes Howard’s assumption that unitary inequivalence entails physical inequivalence.	weak–moderate	relies on Baker’s preprint; Baker leaves open whether the condition is also necessary
C6	Accommodating inequivalent representations requires a “profoundly modal”, trans-world conception of structure.	weak	proposal; French calls it “more profound (and perhaps problematic)” and does not develop it
C7	The two options can be combined: modal algebraic structure plus actual Standard Model structure.	weak	sketched in the Conclusion; “obviously requires further elaboration”

Concepts

• Unitarily inequivalent representations: Hilbert-space representations of the canonical commutation relations (or of an algebra) that no unitary map relates. They are guaranteed to be absent for finitely many degrees of freedom (Stone–von Neumann) and generically present for infinitely many.
• Algebraic vs Hilbert space chauvinism (Ruetsche): the view that all physical content is in the abstract net of algebras, versus the view that it is in the bounded operators of one chosen representation.
• Swiss Army Knife (Ruetsche): content is specified in tiers. The universal representation holds every observable “folded up”, and contingent features pick out the relevant representations (blades) for a situation.
• Trans-world structuralism: French’s name for structure that spans physically possible worlds, with actuality emerging patchwise.
• Symmetry spreading (Stewart & Golubitsky): SSB read as the symmetry being carried by the whole set of degenerate asymmetric solutions.
• DHR criterion: expectation values approach vacuum values far from the origin. It selects the charge superselection sectors of particle physics.
• Representational realism (Halvorson & Müger): the content of QFT is the net, the dynamics, and the structure of relations between representations.

Connections

This paper is the physics-side test of the OSR programme surveyed in the SEP entry (LIT-045, Ladyman’s), whose founding example is the one Howard’s objection targets. Its Option 1 leans on Wallace’s defence of conventional QFT. Wallace’s later math-first statement of structural realism (LIT-151, read as x5) gives a general version of the attitude French borrows: coarse-grained, mathematically defined equivalence, with ontology derivative on the mathematics. Its modal, dispositional turn (“causal empowerment”, counterfactuals via laws and symmetries) touches Vetter (LIT-109). Its group-theoretic account of kinds connects to Wigner on symmetry and conservation laws (LIT-180). The introduction’s quasi-set option for non-individual quanta is the line developed in LIT-107. Chakravartty (LIT-196, read as x6) argues that mass and charge are intrinsic, identity-fixing properties. French’s framework treats such properties as fixed by irreducible group representations, and that is the point at which the two papers meet head-on.

Bearing on metaphysics: the paper presupposes OSR, on which structure (laws and symmetries) is the fundamental ontological category and objects are derivative or eliminated. It defends a modal thesis: fundamental structure spans possible worlds, and the actual is a coarse “blade”. The tag is justified.

Bearing on the record

Any reading or THEORY in this record that treats OSR as refuted by QFT should answer this paper, and any that endorses OSR should note that its QFT defence rests on the unfinished modal proposal and on Baker’s unpublished criterion. There is nothing here for ML practice or for the Anthology of the SOTA.

Limitations

• It is a 2010 draft with visible loose ends: the “(see vF?)” placeholder, citation-date slips, and endnotes detached in this extraction. The published text may differ.
• Every technical claim is imported: Wallace, Fraser, Earman, Ruetsche, Baker (a preprint at the time), and Halvorson & Müger. The paper’s own contribution is interpretive and is argued informally.
• The modal “trans-world” conception of structure is proposed but not developed. French explicitly declines to tie it to possible worlds or to causation.
• SSB’s realist credentials are left hanging on physics. “If the Higgs isn’t found, all bets are off” — the Higgs was announced in July 2012, after this draft, although Earman’s and Morrison’s doubts about SSB are not only about the Higgs’s existence.
• The combined “third option” is flagged by French himself as needing elaboration, and he records Fraser’s objection that the two kinds of representation differ in kind.

Open questions

• Can “trans-world” modal structure be stated precisely, perhaps category-theoretically, as the endnotes suggest via Landry, without collapsing into the algebraic chauvinism French rejects?
• Is Baker’s symmetry-mapping condition necessary as well as sufficient for physical equivalence? If it is only sufficient, representations related by no symmetry, such as some IR-inequivalent ones, stay unaddressed.
• Does the DHR derivation of statistics extend to theories with long-range forces (QED), which the DHR criterion excludes?

Corrections to the seeded skim

• The dossier writes “‘trans-world structuralism’, combined with Ruetsche’s ‘representational realism’”. “Representational Realism” is Halvorson and Müger’s term (Halvorson & Müger 2006, p. 119), not Ruetsche’s. French adopts it in the Case 2 discussion as a form of structural realism. Ruetsche’s contribution is the Swiss Army Knife and its tiered content specification.
• The dossier leaves out the argument that actually “removes the assumption behind Howard’s challenge” (“Back to Ruetsche”). Ruetsche’s unitary-equivalence-as-physical-equivalence rests on Clifton and Halvorson’s translation scheme. Baker argues that the scheme sits badly with relationalism and sophisticated substantivalism about spacetime symmetries, and proposes that a symmetry mapping between representations is sufficient for intertranslatability. French adopts this on the structuralist’s behalf.
• The dossier leaves out the Case 2 claim that inequivalent representations help the structural realist. On the DHR analysis, Bose/Fermi (and para-) statistics and the charge-sector kind structure fall out of the algebra, with no need to impose permutation invariance as an extra symmetry, so “the structure of the world has one less fundamental feature”. This is the paper’s strongest pro-OSR claim.
• The dossier leaves out the “Response: AQFT, Inequivalence and Underdetermination” section. There French argues that Fraser’s “compelling reason” to prefer AQFT (QFT must be made consistent) is weakened, because AQFT too is a programme with no realistic interacting models, and so is also a staging post. He also argues that partial truth applies to AQFT as much as to conventional QFT.
• The dossier says the realist chooses between two options. The Conclusion states a third: the algebra is modal, trans-world structure, and for actual structure the realist adopts the Standard Model and conventional cut-off QFT, investing in SU(3)×SU(2)×U(1). French credits Fraser with the objection that the conventional/AQFT representations differ in kind from the SSB representations, and says the option “obviously requires further elaboration”.
• The draft carries artefacts that the published version may have fixed. There is a live placeholder, “(see vF?)”, among the justifications of Curie’s principle. Ruetsche is cited in the text as 2003 but listed as “Ruetsche 2002” in the references, and the Swiss Army Knife pages are given as “pp. 1339-1441”.
• metaphysics tag: justified. The paper’s positive proposal is metaphysical: a “profoundly modal” conception of structure, “trans-world structuralism”, on which the fundamental structure spans the physically possible worlds and actuality “emerge[s] … in a patchwork manner”. It also takes a stance on the metaphysics of symmetry breaking and natural kinds. philosophy-of-science rightly leads, since the frame is scientific realism and the interpretation of QFT. quantum-foundations is justified.