What Is Quantum Field Theory About?
The previous page left particles as things that can be counted but perhaps not individuated. Quantum field theory is worse for them. A series of results, none interpretive and all technical, makes it doubtful that "particle" picks out anything fundamental at all — and the natural replacement, the field, turns out not to supply well-behaved bearers either.
The technical material is developed in the book's quantum field theory section, notably the Fock space inventory and particles as excitations. What follows is the ontological moral those pages do not draw.
Four results against particles
Malament's theorem. There is no relativistic quantum theory of localisable particles satisfying a small set of modest conditions — that particles cannot be found in two disjoint regions at once, that localisation is translation-covariant, and that no particle can be detected outside the light cone of its earlier location. Any theory meeting these conditions has zero probability of finding a particle anywhere. The moral is that "particle" and "relativistic quantum theory" do not fit together.
Related: there is no good position operator. The Newton–Wigner operator, the standard candidate, is frame-dependent — states localised in one inertial frame are not localised in another. Localisability, the most object-like feature a particle could have, is not frame-invariant.
The Unruh effect. An observer accelerating through the Minkowski vacuum detects a thermal bath of particles. Particle number is therefore not an observer-independent quantity: what is empty space to one observer is populated to another. A notion of existence that varies with the observer's acceleration is not the notion of a fundamental entity.
Reeh–Schlieder. The vacuum is a cyclic and separating vector for local algebras: acting on it with operations confined to any bounded region can produce states approximating anything at all, anywhere. Consequently the vacuum is entangled across every pair of regions — "empty" space is not empty of correlations, and no region is causally or informationally isolated.
Haag's theorem. The interaction picture does not exist. The free-field and interacting-field representations are unitarily inequivalent, so the Fock-space notion of a particle — defined by the free-field number operator — does not survive interaction. Since everything in the world interacts, the particle concept applies exactly where nothing happens.
Taken together: particles are approximate, contextual, observer-relative, and available only asymptotically. That is precisely the profile of a derivative entity — which puts the ordinary objects problem at the alleged fundamental level, rather than only at the level of tables.
Is the field better?
The natural response is that the fundamental entities are fields and the particles are their excitations, which is how the physics is usually described informally.
This helps less than it appears. Quantum field operators are operator-valued distributions, not operator-valued functions: is not well defined at a point, and only smeared quantities over test functions are legitimate observables. So there is no such thing as the value of the field at a point, and the picture of a field as a property distributed across spacetime points — the classical picture that makes field ontology intuitive — does not survive quantisation.
Further difficulties: gauge fields carry surplus structure (below); the field configuration is not what the state assigns definite values to, so the interpretive problems of quantum mechanics reappear unchanged; and in the algebraic approach the physical content is carried by the net of local algebras rather than by any field values.
The result is a genuine vacancy. Neither particles nor fields deliver the well-behaved bearers that a substance–attribute ontology requires — which is the strongest contemporary motivation for ontic structural realism, the position that relations are all there is. It should be kept distinct from the argument for the same conclusion from theory change; this one says there are no objects, that one says only that we could not know them.
Gauge and surplus structure
The general lesson of the section in miniature.
In a gauge theory, distinct mathematical configurations represent the same physical situation: the electromagnetic potential and are physically identical, differing by a gauge transformation. The formalism has more structure than the world does, and the physical content is what survives quotienting by the symmetry. Redhead's term for the excess is surplus structure.
This is why reading ontology naively off a formalism always over-counts. It is also why the hole argument is best understood as a gauge argument: diffeomorphism invariance in general relativity means that which point plays which role is surplus, and the substantivalist who insists that the permuted models represent different situations is treating gauge freedom as physical difference — exactly what the haecceity option did in the particle case.
The methodological principle: the ontology is what is invariant. Where that principle is followed, several disputes turn out to concern surplus structure rather than the world. Where it cannot be followed — because it is unclear which transformations are symmetries — the ontology is correspondingly unclear.
Where this sits
This page bears on the inventory and category questions at the level where they were supposed to be easiest. One expects controversy about numbers and holes and clarity about particles; the situation is the reverse.
It also constrains earlier pages in a specific way. Mereological nihilism needs simples, and QFT does not supply them. The substance tradition needs bearers, and QFT does not supply them either. What it does supply is structure — symmetry groups, algebras, invariants — which is why the position that takes structure as fundamental has better physics credentials than its metaphysical strangeness suggests.