1. Introduction

This essay aims to provide a partial but accessible introduction to the interpretation of quantum field theory (QFT), focusing on the simplest case of linear field theories.Footnote ¹ It is organized as follows. Section 2 considers the project of interpreting physical theories, as well as constraints to which interpretation answers. Section 3 sketches QFT in a way that highlights how those constraints might apply to its interpretation. Sections 4–8 explain how several questions, which are recognizably species of Section 2's interpretative genus, arise in connection with QFT. The questions I catalog are not exhaustive. Nor are the answers I place in the mouths of two fictional and extremist interpreters of QFT, the Hilbert Space Conservative and the Algebraic Imperialist.Footnote ² These characters are expository devices, intended to make interpretative projects vivid, by starkly illustrating their pursuit. From the clash of these extremists emerge several moderating suggestions, which depend on such overtly philosophical questions as the nature of physical possibility. My primary aim here is neither to develop nor to defend these suggestions, but to convey a sense of the interpretative landscape on which they appear. Although my exposition is often only rough, I try to direct those desirous of more precision or detail to references which might satisfy them.Footnote ³

2. Interpreting Physical Theories

This section presents one take on the project of interpreting physical theories. While I do not claim that the construal of interpretative projects offered here is the only, or even the best, construal of such projects, I do suggest that many archetypal interpretative questions are interpretative questions in this section's sense, and that many prominent and probing criticisms of interpretations call those interpretations to task for violating constraints this section announces.

One business of philosophers of physics is the interpretation of physical theories. The interpreter asks: “Under what conditions is this theory true? What does it say the world is like?” (Van Fraassen Reference Van Fraassen1991, 242). A standard sort of answer (see Van Fraassen Reference Van Fraassen1989, ch. 9 and references therein) proceeds in two stages. At the first, structure-specifying, stage, the interpreter characterizes the structures by which the theory would represent physical reality.Footnote ⁴ At the second, semantic, stage, the interpreter characterizes the physical worlds which model the theory so structured. Thus an interpretation of a theory identifies the physical worlds possible according to the theory.Footnote ⁵

For that bulk of central physical theories admitting a Hamiltonian formulation, the first-stage specification of theory structure can be decomposed into three elements. The first is an identification of the theory's state space. In the mercifully simple case of the classical theory of a particle of mass m moving in one dimension, the state space is comprised of ordered pairs (q, p) of position and momentum values. The second element of structure specification is an identification of the set of physical magnitudes or observables the theory recognizes. In the same simple case, observables are functions from elements of the state space to the reals; e.g., the physical magnitude kinetic energy E is given by the function. f_E(q, p) = p ²/2m. The third element of structure specification is the identification of the theory's dynamics, its account of how the states of systems change with time. In the simple case, dynamics takes the form of trajectories through state space determined by Hamilton's equations once a Hamiltonian function H(q, p) is provided.

With these elements in hand, interpretation can proceed to its second, semantic stage of characterizing the physical models of the theory. A semantics can take the form of an account of which propositions attributing determinate values to observable magnitudes recognized by the theory are true of a system represented by a state of the theory. A natural semantics for the simple classical theory is: “Observable O has value x in state (q, p)” is true just in case f_o(q, p) = x. With a fix on a theory's state space, observables, and dynamics, the interpreter can use a semantics for the theory to characterize the worlds of which the theory is true.

Examples of familiar interpretative questions posed about each of these four elements are not hard to come by.Footnote ⁶ A question about a physical theory's state space: does absolute position parameterize the state space of Newtonian theory? A question about the observables appropriate to a physical theory: must the observables of general relativity be generally covariant, and if so, in what sense? A question about the dynamics of a physical theory: does the wave function of a quantum system collapse upon measurement or doesn't it? A question about that same theory's semantics: do all propositions attributing determinate values to observables pertaining to quantum systems have truth values, or just some of them? If the latter, which ones?

Interpretative questions of the sorts just canvassed can interact. The venerable debate between substantivalists and relationalists about space(time) can be profitably cast as the issue of how best to harmonize answers to the three structure-specifying questions. In the quantum mechanical measurement problem, structure-specifying questions interact with semantic questions: if one adopts the no-collapse answer to the question about dynamics—that is, if one takes quantum dynamics to be unitary dynamics—while adopting a semantics that limits the observables with determinate values to those whose values the theory predicts with certainty, one is committed to the unpalatable consequence that most measurements leave laboratory apparatus in states we cannot take to represent measurement outcomes.

The measurement problem demonstrates that interpretations of physical theories can be subject to stringent intrinsic criteria of adequacy. By intrinsic criteria, I mean criteria we can articulate without making reference to other physical theories, criteria that (so to speak) arise between us, the theory we're interpreting, and the world to which the theory would apply. Intrinsic criteria include logical coherence, consistency with empirical data, and adequacy to experimental practice. They may even include simplicity and mathematical elegance. A vast expanse of the literature on the interpretation of quantum mechanics engages the intrinsic concern of whether it is possible to give an interpretation of quantum mechanics consistent with the empirical predictions of the minimally interpreted quantum statistical algorithm. Another expanse addresses the measurement problem just sketched, which constrains interpretation to secure the fundamental presupposition of laboratory practice, that experiments have outcomes.

Interpretation needn't be a hermetic activity. We may also subject interpretations to extrinsic criteria of adequacy, criteria that make reference to theories beside the theory interpreted, criteria that (so to speak) arise between us, the theory we're interpreting, and other theories to which we're committed—perhaps even, depending on the level of our commitment to these theories, their interpretations. We may, for instance, ask: how well does the theory as interpreted fit with environing theories? Or: How promising a resource is the theory as interpreted for the construction of novel theories? Examinations of whether Bohmian interpretations of quantum mechanics violate the letter or the spirit of the special theory of relativity express the first sort of extrinsic concern. Working physicists—for instance, quantum gravity theorists taking a stake in the interpretation of spacetime theories—often express the second sort of extrinsic concern.

With these preliminary remarks in mind, I want to survey some interpretative questions that arise in connection with QFT. Cushing, for one, might seem to query the point of such an exercise: “My contention is that at the foundational level there is essentially nothing new in quantum field theory that is not already present in non-relativistic quantum mechanics” (Cushing Reference Cushing, Brown and Harre1987, 25).Footnote ⁷ My contention is that there is sufficient novelty, both in the interpretation of the theory proper and in the fit of its interpretations with the interpretations of environing physical theories, that the quantum field is fertile ground for the philosopher of physics.

3. Quantum Field Theory

This section characterizes QFT in two ways. First, extrinsically (and so with a view toward extrinsic constraints on its interpretation), in terms of how it relates to other theories, developed and imagined. Second, (semi-) intrinsically, in terms of the features constitutive of a quantization of a classical field theory. The reference to the classical field theory is what makes this characterization only semi-intrinsic.

3.2. (Semi-) Intrinsic Characterization

Traditionally, to quantize a classical theory is to cast the theory in Hamiltonian form and then to promote its canonical observables q_k, p_k to symmetric operators $\hat{q}_{k}$,$\hat{p}_{k}$ obeying commutation relations corresponding to the Poisson brackets of the classical theory. The familiar Heisenberg form of the canonical commutation relations (CCRs), constituting a quantization of our mercifully simply classical theory (q, p), is

The standard realization of these CCRs is the Schrödinger Representation, on the Hilbert space L ² (ℝ) of square-integrable complex-valued functions ψ(x) of the reals, with

The Heisenberg form (3.1) of the CCRs is, however, mathematically infelicitous (see Summers Reference Summers, Redei and Stölzner2001). The unboundedness of the position and momentum operators is to blame: they are not jointly defined on all of L ² (ℝ), so that (3.1) fails to be an operator equation for that Hilbert space. While this can perhaps be addressed by painstaking attention to domains of definition for canonical operators, there is a more convenient fix. It is to exponentiate $\hat{q}$ and $\hat{p}$ to obtain unitary operators, bounded and everywhere defined, and to re-express (3.1) in terms of those unitaries. In particular, where $\hat{U}\left(a\right)=\mathrm{exp}\,\left(\frac{-i2\pi a\hat{q}}{h}\right)$ and $\hat{V}\left(a\right)=\mathrm{exp}\,\left(\frac{-i2\pi a\hat{q}}{h}\right)$, if (3.1) holds, then for all a, b ∊ ℝ,

(3.2)

Call (3.2) the Weyl form of the CCRs, or Weyl relations for short. Unitary operators satisfying the Weyl relations express the quantization of the one-dimensional classical theory. According to Stone's theorem, provided the families $\left\{\hat{U}\left(a\right)\right\}$ and $\left\{\hat{V}\left(b\right)\right\}$ are (weakly) continuous with respect to the real parameters a and b, the canonical operators $\hat{q}$ and $\hat{p}$ can be recovered as the infinitesimal generators of those families—the Heisenberg CCRs (3.1) are essentially the infinitesimal form of the Weyl relations (3.2).Footnote ⁸

The quantization of a classical field theory to obtain a QFT proceeds along analogous lines. Textbook approaches (e.g., Kaku Reference Kaku1993, Ch. 3.2) begin with a classical field equation, for instance, the Klein-Gordon equation on a spacetime (M, g_ab) (n.b. $x=\left(\vec{x},t\right)$ is a point in the four manifold M):

This arises from a Hamiltonian H that is a function of the configuration variable ϕ (x) and a canonically conjugate momentum density; $\pi \left(x\right)\equiv \frac{\partial L}{\partial \varphi \left(x\right)}$ where L is the theory's Lagrangian density. To quantize Klein-Gordon theory is to replace ϕ and π with Hilbert space operators $\hat{\varphi }$ and $\hat{\pi }$ obeying the canonical commutation relations (CCRs) answering to the Poisson brackets of the classical field theory:

Once again, to avoid technical difficulties involving the domains of definition of the unbounded operators appearing in these CCRs, they are typically recast in exponential form as Weyl relations.Footnote ⁹

Thus to quantize a classical field theory is to find a Hilbert space on which act operators satisfying the Weyl relations arising from that theory. Where is a Hilbert space and { Ŵ (y)} are operators on satisfying the Weyl relations, I will call (, Ŵ (y))) a representation of the Weyl relations for that theory. We can, without prejudice to interpretive questions we encounter down the road, use these Weyl relations to characterize what features of a system must be held constant when asking what states are possible for that system. These features are, minimally, that the Weyl relations involved in the quantization of the system hold of the set of observables pertaining to the system. So a state over a set of observables satisfying canonical anti-commutation relations characteristic of the free fermion field is not a state possible, in this sense, for the free boson field, which is a system characterized by appeal to canonical commutation relations.

In what follows, I will have occasion to refer to some features of a textbook heuristic for quantizing the Klein-Gordon field on Minkowski spacetime to obtain the free boson field. This frequency-splitting heuristic proceeds, very roughly, as follows (Kaku, Reference Kaku1993; see also Wald Reference Wald1994, Ch. 4.3). By a plane wave expansion, solutions to the classical Klein-Gordon equation can be Fourier-decomposed into uncoupled normal modes with angular frequency ω_k, and so the classical field can be modeled as an infinite collection of independent oscillators. The heuristic quantization builds the single particle Hilbert space for the QFT from classical solutions which oscillate with positive frequency with respect to time. Exploiting the analogy with the simple harmonic oscillator, the textbook approach recasts the CCRs in terms of “creation” and “annihilation” operators â $\stackrel{\dagger }{k}$ and â_k for positive frequency field modes ω_k:

(3.4)

Formal expressions for operators $\hat{\varphi }\left(x\right)$ and $\hat{\pi }\left(x\right)$ satisfying the CCRs can be constructed from these. The operators in (3.3) act on a symmetric Fock space (see Wald Reference Wald1994, A.2) which is a direct sum of n-fold symmetrized tensor products of the single particle Hilbert space constructed from positive frequency solutions.

This Fock space features a state |0〉 such that â_k|0〉 = 0 for all k. |0〉 is the lowest energy eigenstate of the quantum Hamiltonian for the free boson field (also handled by analogy with the simple harmonic oscillator). The state $( \mbox{\^a} \stackrel{\dagger }{k}) ^{n}\vert 0\rangle $ is an eigenstate of the Hamiltonian with the same energy a system of n particles each with energy hω_k would have—provided the momenta and rest masses of these particles are given by standard relativistic expressions. Thus the quantization tempts a particle interpretation: the vacuum state |0〉 is the no particle state; the state $( \mbox{\^a} \stackrel{\dagger }{k}) \vert 0\rangle $ obtained by acting on the vacuum once with the creation operator $\mbox{\^a} \stackrel{\dagger }{k}$ describes one particle of energy hω_k; the state $\mbox{\^a} \stackrel{\dagger }{k}\mbox{\^a} \stackrel{\dagger }{k}$|0〉 obtained by acting on the vacuum twice with the creation operator $\mbox{\^a} \stackrel{\dagger }{k}$ describes two particles, each of energy hω_k; the state obtained from this state by acting on it with an annihilation operator â_k describes one particle of energy hω_k again, and so on. Every state in the representation can be approximated to arbitrary precision by acting on |0〉 with linear combinations of creation and annihilation operators. That is to say, |0〉 is cyclic with respect to the operators {â_k, $\mbox{\^a} \stackrel{\dagger }{k}$}.

Of particular significance is the operator $\hat{N}_{k}=\hat{a}^{\dagger }_{k}\hat{a}_{k}$ − the number operator for particles of type k. Its expectation value in a state gives the mean number of particles of type k one would find in that state. Hence, $\hat{N}=\sum_{j}\hat{a}^{\dagger }_{j}\hat{a}_{j}$ is the total number operator. That for all k, 〈0| $\hat{N}_{k}$|0〉 = 〈0| $\hat{N}$ |0〉 = 0 reinforces |0〉's claim to be the vacuum or no-particle state.Footnote ¹⁰ The set of states available in a quantization with a Fock space structure can be understood in terms of excitations of an underlying ground state. Thus Fock space representations aid and abet the understanding of QFT as particle physics.

4. Unitary Inequivalence And The First Question Of Interpretation

The topic of this section is the availability of unitarily inequivalent representations of the Weyl relations constituting the quantization of a classical field theory, an availability which drives the interpretative questions I consider in this essay. To describe what unitary inequivalence amounts to, I must review some features of the Hilbert space approach to quantum mechanics. On the Hilbert space approach, the space of states available for a quantum system is a space of normed vectors in (more generally, density operators on) a Hilbert space ; quantum observables are bounded, self-adjoint operators on ; once the Hamiltonian operator Ĥ is provided, Schrödinger's equation determines dynamical trajectories through state space. Thus we may associate a Hilbert space and a collection of operators with a quantum theory on the Hilbert space approach.

Now a definition. A Hilbert space , and a collection of operators {Ô_i} is unitarily equivalent to (′, {$\hat{O}^{\prime }_{i}$}) if and only if there exists a one-to-one, linear, norm-preserving transformation (“unitary map”) U: → ′ such that U ⁻¹$\hat{O}^{\prime }_{i}$U = Ô_i for all values of i. The Born Rule invests a quantum theory with empirical content by assigning an observable Ô in a state | ψ 〉 the expectation value 〈 ψ | O| ψ 〉. Unitary equivalence is a sufficient condition for every state in a theory (, { Ô_i}) to have a counterpart in a theory (′, {$\hat{O}^{\prime }_{i}$}), in this sense: the expectation values assigned observables Ô_i by a state | ψ 〉 in the first theory are duplicated by those assigned $\hat{O}^{\prime }_{i}$ by the counterpart state U| ψ 〉 in the second. Because unitary maps are invertible, the counterpart relation is symmetric. In the case that two theories' observables are the sets of bounded, self-adjoint operators on their respective Hilbert spaces, the unitary equivalence of theories is not only sufficient but also necessary for every state in the first theory to have a counterpart in the second, and vice versa.Footnote ¹¹

The physical content of a state consists at least in its assignment of expectation values to the observables the theory recognizes. Thus the physical content of a theory consists at least in the set of such assignments it countenances. Hence two theories are physically equivalent only if their sets of expectation value assignments are suitably isomorphic. For quantum theories on the Hilbert space approach, according to the results of the previous paragraph, the suitable isomorphism in question is that furnished by unitary equivalence. Ergo it is prima facie reasonable to suppose that, on the Hilbert space approach, in order for two quantum theories to be physically equivalent, they must be unitarily equivalent. And so it is prima facie reasonable to regard unitarily inequivalent representations of the Weyl relations expressing the quantization of some classical theory as distinct physical theories.

I do not say that unitary equivalence is sufficient for physical equivalence on the Hilbert space approach, because I take physical equivalence to be a relation that obtains between interpreted theories, and I can readily imagine unitarily equivalent quantum theories which agree in the state sets they recognize but which under different interpretations assign different determinate properties to systems in their domain. Take a Bohmified quantum theory which makes position always determinate and one which makes momentum always determinate, for instance. Though unitarily equivalent, these interpreted theories aren't obviously physically equivalent. Hence my scruples about the sufficiency of unitary equivalence for physical equivalence. As far as I can tell, they don't affect the questions developed in this essay, which (put roughly) concern whether unitary equivalence is necessary for physical equivalence.

If unitary equivalence is necessary for physical equivalence, and if to quantize a classical theory is to find a representation of the appropriate Weyl relations, quantization yields a unique physical theory only if all representations of the Weyl relations are unitarily equivalent. Provided that the state space of the classical theory is finite-dimensional, this unitary equivalence is secured by the Stone-Von Neumann theorem, which states, in essence, that all representations of the Weyl relations expressing the quantization of a finite dimensional classical theory are unitarily equivalent to the Schrödinger representation.Footnote ¹² But QFTs are quantizations of—well, field theories, theories with infinitely many degrees of freedom. The Stone-von Neumann theorem does not cover their quantization. Indeed, the Weyl relations encapsulating the quantization of classical Klein-Gordon theory admit continuously many inequivalent representations. Given our working criterion of physical equivalence, this means that there are continuously many physically inequivalent quantum theories of the free boson field. This dramatic non-uniqueness prompts worries about the foundations of QFT, e.g., is there any such thing as the free boson field to begin with?!

Supporting evidence that unitarily inequivalent quantizations are physically inequivalent takes the following form. Suppose that (, Ŵ (y)) and (′, Ŵ ′ (y))—henceforth, the unprimed and primed representations, respectively—are unitarily inequivalent quantizations of the Klein-Gordon field, each with Fock space structure. Then the vacuum state |0〉 of the unprimed representation cannot be expressed as a state in the primed representation, and mutatis mutandis. What's more, the unprimed total number operator $\hat{N}$ cannot appear as an operator in the primed representation, and mutatis mutandis. The primed and unprimed theorists cannot recognize one another's fundamental particle notions. Indeed, using one theorist's notions to plumb the particle content of the quantum field precludes using the other's. Any pair of states which one can distinguish by appeal to the probabilities they ascribe to propositions concerning umprimed particle contents are indistinguishable by appeal to the probabilities they ascribe to propositions concerning primed particle content. And mutatis mutandis.Footnote ¹³ Provided that we're committed to articulating the physical content of a QFT in terms of particle notions—or even more cautiously, in terms of quanta of excitation of an underlying ground state—the differences between the physical contents of the primed and unprimed representations seem irreconcilable.

The questions of interpretation sketched here revolve around the availability of unitarily inequivalent representations of the Weyl relations for QFT. One question of interpretation this availability prompts is: what is the state space of a QFT? Is it the set of unit vectors (more generally, density matrices) residing in a single (unitary equivalence class of) representation(s) of the Weyl relations? Is every unit vector in every representation of the Weyl relations a possible state for the theory? Or does the state space of the theory lie somewhere between the stingy and the magnanimous options? Concerning the state space proper to QFT, these are interpretative questions in section 2's sense. Insofar as the Stone-von Neumann theorem banishes the specter of unitarily inequivalent representations from the quantization of classical theories whose configuration spaces are ℝⁿ, they are interpretative questions that arise along with the attempt to quantize classical field theories.

5. Extremist Answers To The First Question

Here I develop two extremes in the continuum of possible answers to this first interpretative question. In what follows, as I raise additional interpretative questions born from QFT, I'll try to keep track of what these extremists have vested in their answers. These extremists are fictional devices, introduced to illustrate how foundational investigation of QFT might go, and how in its course interpretative questions intertwine. Against the backdrop of this rough illustration, the details of particular positions can be developed, and those positions accurately attributed to particular thinkers. But that task is subsequent to my present one, which is primarily to construct the backdrop.

According to my first extremist, whom I'll label the Hilbert Space Conservative (or just Conservative, for short), quantum states are what quantum states always were: normed elements of a Hilbert space. And the physical equivalence of quantum theories is what physical equivalence always was: unitary equivalence (or something for which unitary equivalence is necessary). The Conservative therefore maintains only one unitary equivalence class of representations of the Weyl relations to be physical, to bear the physical content of the theory. Enjoining us to pursue QFT in the ambit of this unitary equivalence class, she spurns representations of the Weyl relations outside of it as spurious, because they are physically inequivalent to the representations bearing the physical content of the theory.

Notice that the Hilbert spaces bearing the content of quantum theory, according to the Conservative, can be quite complicated. They can, like the Fock spaces of particle physics, be the direct sum of n-fold symmetrized tensor products of a single particle Hilbert space, or they can, like the Hilbert space for a single particle with both spin 1/2 and position degrees of freedom, be the Hilbert space obtained by taking the tensor product of a two-dimensional Hilbert space and an infinite-dimensional Hilbert space. The Conservative is not committed to the absurdity that the possible states of a system must all be representable as density matrices on some simple, non-composite Hilbert space. She is rather committed to the position that, however complicated the Hilbert space structure bearing a representation of the Weyl relations characteristic of a system, all states possible for the system can be represented on that Hilbert space structure.

Now, an immediate problem for the Conservative is the problem of privilege: by what criterion does she select the unitary equivalence class of physical representations from the infinitely many representations available? And, supposing she can articulate such a criterion, how is she to justify it?

Not much is made in QFT textbooks of the availability of unitarily inequivalent representations. That textbook QFT is almost invariably conducted in Minkowski spacetime explains this. Minkowski spacetime is highly symmetric and well structured. It is first of all globally hyperbolic. That is to say, it has a Cauchy surface, a spacelike hypersurface every inextendible causal curve crosses exactly once. A globally hyperbolic spacetime can be foliated by a one-parameter family of Cauchy surfaces, each of which has the same topology. Thus such a (four-dimensional) spacetime is topologically Σ × ℝ, where Σ gives the topology of a three manifold. Minkowski spacetime is moreover stationary. That is to say, it is globally hyperbolic and it hosts a one-parameter family of isometries whose orbits are timelike. Thus a stationary spacetime admits a foliation by Cauchy surfaces such that the timelike transformations between these surfaces are symmetry transformations of the metric. Inertial time translations from one Cauchy slice to another in Minkowski spacetime are symmetries of the Minkowski metric (see Wald Reference Wald1984, Ch. 8, for more on causal structure).

These symmetries furnish criteria for privilege. Recall the heuristic “frequency-splitting” quantization mentioned at the end of Section 3. This quantization builds the single particle Hilbert space of the QFT from classical solutions to the Klein-Gordon equation which are “positive frequency” with respect to time. A natural query to raise in this relativistic setting is: time according to whom? And the resoundingly satisfying answer available in Minkowski spacetime is: time according to some family of inertial observers, that is, time associated with some family of symmetries of the metric. It doesn't matter which; following the heuristic for different families of inertial observers yields unitarily equivalent quantizations. The frequency splitting prescription is unambiguous in Minkowski spacetimes because propagating a classical solution from one Cauchy surface to another, isometrically related to the first, does not alter its status as positive frequency.

To say that a heuristic for quantization may be unambiguously applied does not privilege the result of its application over other quantizations available. But here again the isometries of Minkowski spacetime serve the standard construction well. For in the presence of spacetime symmetries, a reasonable demand to place on a quantization with a Fock space structure is that its vacuum state be invariant under the full isometry group of the spacetime. Axiomatic approaches to QFT issue this demand in the form of an axiom. Its rough and ready justification is that the result of subjecting a spacetime with nothing in it to any of its symmetry transformations should be a spacetime with nothing in it. Suppose it is further demanded that this vacuum state be annihilated by all “negative frequency” operators (i.e., annihilation operators a_k associated with positive frequency modes ω _k).Footnote ¹⁴ The rough and ready justification of this demand is that the vacuum state be fully empty in the sense that it is not possible to obtain another state by removing something from the vacuum. The quantization meeting both demands is unique up to unitary equivalence. It is, in fact, the standard quantization obtained by the frequency-splitting heuristic. Thus the Hilbert space Conservative can solve her problem of privilege in Minkowski spacetime by appeal to that spacetime's symmetries: the standard quantization is the only quantization satisfying prima facie reasonable symmetry-presupposing and symmetry-driven demands.

Of course, the strategy of invoking spacetime symmetries to solve the problem of privilege fails when we consider the pursuit of QFT in more general spacetime settings. Recalling that QFT on curved spacetime might be one signpost on the road to quantum gravity, let us suppose that the Conservative is well-mannered enough to want her interpretation of QFT to extend to generic spacetime settings. Curved spacetimes won't in general share the symmetries of Minkowski spacetime—generic curved spacetimes will lack symmetries altogether. Interpreting QFT on curved spacetime, the Conservative can't appeal to spacetime symmetries to implement the frequency-splitting heuristic or to resolve her problem of privilege. The consideration of curved spacetimes shortcircuits the Conservative's privileging strategy, and forces those interested in the foundations of QFT to take the availability of unitarily inequivalent representations seriously. Absent the consideration of curved spacetime settings, the interpretative questions I survey here are not gripping. Considering curved spacetime settings presents the Conservative with the problem of privilege. Leaving her with it for now, I turn next to my other extremist, the Algebraic Imperialist, or Imperialist, for short.

What spurs Algebraic Imperialism is the suggestive fact that each concrete Hilbert space representation of the Weyl relations gives rise to an abstract algebra, the Weyl algebra, which is representation-independent. To obtain this algebra, start with a concrete representation of the Weyl relations; construct finite, complex, linear combinations of the unitary operators bearing this representation to obtain a subalgebra of , the set of bounded linear operators on and complete this subalgebra in the Hilbert space operator norm. The result is a C* algebra,Footnote ¹⁵ the Weyl algebra. The key result for Imperialists is that Weyl algebras constructed from (possibly unitarily inequivalent) representations of the Weyl relations (, Ŵ (y)) and (′, Ŵ ′(y)) are equivalent up to *isomorphism.

Abstract algebras can be realized in concrete Hilbert spaces. A map π: , from elements of an abstract algebra to the set of bounded linear operators on a Hilbert space , furnishes a Hilbert space representation of the algebra.Footnote ¹⁶ Hilbert space representations of Weyl algebras for QFT have structure in excess of the algebraic structure. For instance, the map π: has as its image only a proper subset of . That is, there are bounded, self-adjoint operators on concrete Hilbert space representations of Weyl algebras without direct correlates in those algebras. Concrete representations also make available operator topologies—for instance, the weak topology, which defines convergence for sequences of operators in terms of the Hilbert space inner product, or the strong topology, which defines convergence in terms of Hilbert space norm—which can't be defined in terms of the abstract algebra, whose natural topology is furnished by the C* norm ‖·‖. According to the Algebraic Imperialists, all this extra structure is extraneous-the Weyl algebra,Footnote ¹⁷ and not any of its concrete but disparate Hilbert space realizations, encapsulates the physical content of QFT: “[T]he important thing here is that the observables form some algebra, and not the representation Hilbert space on which they act” (Segal Reference Segal and Lurcat1967, 128)).

Now, the algebraic approach to QFT identifies physical observables with self-adjoint elements of the Weyl algebra (i.e., elements A such that A* = A). To assign these observables expectation values, the algebraic approach dispenses with the intermediaries of Hilbert space representations and Hilbert space states. Algebraic states assign observables expectation values directly. A state ω on an abstract algebra is a linear functional ω : that is normed (ω (I) = 1)) and positive (ω (A* A) ≥ 0 for all A ∊ ). The expectation value of an observable A ∊ in state ω is simply ω (A). And the Algebraic Imperialist's answer to the first question of interpretation is that the state space of QFT is simply the space of algebraic states on the abstract Weyl algebra.

We can draw connections between the Conservative's and the Imperialist's notions of state. A state | ψ 〉 in a Hilbert space carrying a representation π : of an algebra naturally gives rise to the algebraic state ω (A) = 〈 ψ | π (A) | ψ 〉 for all A ∊ . Somewhat less obviously, an algebraic state ω over an algebra can be recast as state in a Hilbert space bearing a faithful representation of that algebra (a representation π of is faithful iff π (A) = 0 ⇒ A = 0). Gel'fand, Naimark, and Segal showed how to construct this GNS representation (see Bratelli and Robinson Reference Bratelli and Robinson1987, § 3.2):

1. Theorem 1 (GNS). Given a C*-algebra and a state ω: over that algebra, there exists a Hilbert space _ω, a faithful representation π_ω (_ω), and a cyclic vector |Ψ_ω〉 ∊ _ω such that ω (A) = 〈Ψ_ω|π_ω(A)|Ψ_ω〉 for all A ∊ . Moreover the triple (_ω, π_ω, |Ψ_ω〉) is unique up to unitary equivalence.

Thus we can translate every algebraic state to a Hilbert space state, and every Hilbert space state to an algebraic state. But it does not follow that the Conservative and the Imperialist notions of state are equivalent. Consider abstract algebraic states ω and ω ′ (for simplicity's sake assumed throughout to be pure statesFootnote ¹⁸) which give rise via the GNS construction to unitarily inequivalent concrete representations (, π) and (′, π ′). (Two representations (₁, π ₁) and (₂, π ₂) of an abstract algebra are unitarily equivalent just in case there is a unitary map U: ₁ ⇒ ₂ such that Uπ ₁(A)U ⁻¹ = π ₂(A) for all A ∊ .) Both ω and ω ′ are, according to the Imperialist, possible states of the quantum field.

But not according to the Conservative. The set of states representable as density matrices on the GNS construction ( _ω, π_ω, | Ψ_ω>) for ω is called the folium of ω . Giving rise to unitarily inequivalent GNS constructions, ω and ω ′ reside in different folia: one can't be represented as a density matrix on the GNS representation of the other. Now, the Conservative believes that all physically possible states lie in a single folium-this is just another way of saying that the state space of the theory is the space of density matrices on a Hilbert space. Because ω and ω ′ reside in different folia, the Conservative can't consider them both physically possible.

Let us pause to remark how stringent, and how bivalent, the Conservative's notion of physical possibility is.Footnote ¹⁹ Alerted to the availability of unitarily inequivalent representations of the Weyl relations, a more tolerant advocate of the Hilbert space approach might say: quantum-theoretic states reside in a single folium. But it's an empirical question which folium. She might even borrow algebraic resources to characterize the proper folium: where ω happens to be the algebraic state of the universe, it's ω 's folium, that is, it's the set of states expressible as density matrices in the GNS construction arising from ω .Footnote ²⁰ The tolerant Conservative regards these states as, so to speak, more possible than those residing in folia (e.g., ω ′'s) disjoint from ω 's. But she thinks it excessive to relegate states outside ω 's folium to outright physical impossibility. After all, had the state of the universe been ω ′ rather than ω, a folium of these outlying states would have comprised the quantum-theoretic state space. Intuitively, there are nearby, physically possible worlds the correct quantum theory of which is set in ω ′'s folium.

Considerations from the thermodynamic limit of quantum statistical mechanics suggest that this gentler way with physical possibility is also saner. Space prohibits developing these considerations here, but this is the general idea: equilibrium states at different temperatures can correspond to unitarily inequivalent representations. Suppose that the state of the universe ω is an equilibrium state. Refusing to admit that physical possibility comes in degrees, the extreme Conservative would declare states outside ω 's folium to be physically impossible. But in this setting, that amounts to declaring non-actual temperatures physically impossible, which seems modally draconian. Recognizing degrees of physical possibility, a gentler Conservative might begin to reconcile Algebraic and Hilbert space senses of the state space of QFT.

An extreme Conservative, on the other hand, riled by the physical significance with which the Imperialist would invest unitarily inequivalent representations, might challenge that Imperialist with a question of principle. What notion of physical equivalence supports and is supported by the Imperialist's identification of the state space of QFT? As a matter of historical fact, advocates of algebraic approaches did seem to regard their recognition of unitarily inequivalent representations as requiring some sort of apology, an apology some couched in terms of the notion of weak equivalence. The idea is that two representations are weakly equivalent exactly when no finite set of expectation values specified to finite accuracy can locate a state in one, rather than the other, representation. More precisely, (, π) and (′, π ′) are weakly equivalent if and only if, given any Hilbert space state on the first representation and the algebraic state ω it yields, and given any finite set of elements of the abstract algebra of observables and any finite set of non-zero margins of experimental error, there exists a Hilbert space state on the second representation yielding an algebraic state ω ′ that reproduces within those margins of errors ω 's assignment of expectation values to those observables.

Now the apologizing Imperialist “define[s] two representations to be physically equivalent if and only if they are weakly equivalent” (Robinson Reference Robinson, Chretien and Deser1966, 488). A rough and ready verificationism suggests a justification of the equation of physical and weak equivalence. Run at the level of algebraic states, the justification goes like this. Any actual measurement performed on a system in the algebraic state ω will fix the expectation values of some finite collection of observables A_i, i = 1, 2, …, n within experimental error margins ∊_i > 0. Accordingly, such a measurement cannot distinguish ω from any other state ω ′ satisfying | ω (A_i) − ω ′(A_i)| < ∊ _i for all i. Such states are, for all practical purposes, physically equivalent. And so, representations are for all practical purposes physically equivalent if they are weakly equivalent. What's more, all interesting concrete representations of the Weyl algebra are weakly equivalent (see Wald Reference Wald1994, Ch. 4.5):

1. Theorem 2 (Fell). Let (₁, π ₁) and (₂, π ₂) be faithful but possibly unitarily inequivalent representations of the C*-algebra . Let A ₁, …A_n ∊ , and let ∊ ₁, …,∊_n > 0. Suppose ω ₁ is an algebraic state corresponding to a Hilbert space state on ₁. Then there exists a state ω ₂ corresponding to a Hilbert space state on ₂ such that for all i = 1, …, n, it is the case that |ω ₁(A_i) − ω ₂(A_i)| < ∊_i.

Weyl algebras are simple, and any non-zero representation of a simple algebra is faithful. Thus Fell's theorem encompasses all non-zero representations of Weyl algebras. Defining physical equivalence as weak equivalence, the apologetic Imperialist can conclude that, “All faithful representations of [the Weyl algebra] are physically equivalent” (Robinson Reference Robinson, Chretien and Deser1966, 488)—and chastise the Conservative for elevating surplus structure to physical significance:

Weak equivalence underwrites a satisfactory response to the Conservative's challenge only if weak equivalence is an adequate explication of physical equivalence. We just now imagined a verificationist justification of the explication. But such a justification fails, even by operationalist lights. For, as Summers (Reference Summers, Redei and Stölzner2001) has observed, even verificationists ought to take states to be predictive instrumentalities. They would therefore expect the equivalence of states to extend to their predictions concerning future measurements. But Fell's theorem offers us no assurance that a state on for all practical purposes indistinguishable from a state on ′ with respect to some set { A_i} of observables will continue to mimic the first state's predictions with respect to an expanded or altered set of observables. The practical indistinguishability of states that secures the weak equivalence of the representations bearing them is only a backward-looking or static sort of indistinguishability. Before recognizing a state in one theory as a suitable counterpart to a state in another, an operationalist should demand not only backward-looking but forward-looking indistinguishability: the indistinguishability of putative counterpart states conceived as predictive instrumentalities. Thus the appeal to Fell's theorem should not convince even the operationalist that weak equivalence is an acceptable notion of physical equivalence.

Apologies for algebraic liberality invoking weak equivalence fall flat. I believe that, at this stage at least, the best way for an Imperialist to articulate his criterion of physical equivalence is the bold way. Just as his notion of algebraic state deposes the Hilbert space notion of state, so too a criterion of physical equivalence suited to the algebraic approach should depose the Hilbert space criterion of unitary equivalence. Placing the physical content of a quantum theory in an algebra of observables, the bold Imperialist would maintain that two quantum theories are physically equivalent only if their associated algebras are *-isomorphic. The bold Imperialist would not flinch in the face of the observation that he thereby elevates a continuum of states giving rise to unitarily inequivalent GNS representations to physical significance. For, he would maintain, he has no reason to take unitary equivalence seriously as a notion of physical equivalence to begin with.

As Clifton and Halvorson (Reference Clifton and Halvorson2001) have aptly observed, we encounter here a hurdle liable to be faced by attempts to use accounts of physical equivalence, gleaned from the general philosophy of science literature, to adjudicate the claims of the Imperialist and the Conservative. Take Reichenbach's (Reference Reichenbach1938) notion, that physical theories are equivalent if and only if they are confirmed equally by every possible empirical datum. Conservative and Imperialist alike can agree with Reichenbach-provided they disagree with each other about which empirical data are possible. The Conservative will characterize possible data in terms of observables associated with bounded operators on concrete Hilbert spaces; the Imperialist will characterize the same data in terms of elements of abstract Weyl algebras.

The boldness of championing *-isomorphism as a criterion, adapted to the algebraic approach, of physical equivalence, does not preclude a conciliatory gesture on the part of the Imperialist. Joining the gentle Conservative in recognizing degrees of physical possibility, he could concede that states in the folium of the actual state of the universe ω —states representable as density matrices on a single Hilbert space—are more physically significant than states in outlying folia. When it comes to answering the first interpretative question, invoking degrees of possibility may serve to reconcile algebraic and Hilbert space ways of thinking.

6. The Second Interpretative Question: Observables

I've sketched an interpretative question concerning QFT's state space. Closely related questions concern the observables the theory honors. Given that the Imperialist reckons as observables only self-adjoint elements of the abstract Weyl algebra, and given that the image of these elements under a representation π in a concrete Hilbert space forms only a proper subset of self-adjoint operators on that Hilbert space—and so only a proper subset of the observables we're accustomed, on the Hilbert space approach, to recognize—one might anticipate that algebraic liberality about states comes at the apparent cost of stinginess about observables. This section rehearses several versions of the charge that the Algebraic Imperialist recognizes fewer observables than he should.

Applied to a particle moving in one dimension, unmitigated Imperialism can deny even the canonical position and momentum magnitudes observable status. Recall (cf. equations (3.1) and (3.2)) that position $\hat{q}$ and momentum $\hat{p}$ are recoverable from the families of unitaries $\left\{\hat{U}\left(a\right)\right\}$ and $\left\{\hat{V}\left(b\right)\right\}$ satisfying the Weyl relations only if those families are (weakly) continuous. Recall also that the Weyl algebra is based on the integral, not the infinitesimal, form of the CCRs—speaking loosely, the unitaries, but not their infinitesimal generators, are in the algebra. The fact underlying the first stinginess charge that there are algebraic states—called irregular, or singular states—in whose GNS representations the families of operators satisfying the Weyl relations aren't weakly continuous. This foils the recovery of position or momentum as the infinitesimal generators of those families of unitary operators. Put bluntly, there are algebraic states from whose GNS representations position or momentum observables are missing! These are precisely the states which make possible what the Hilbert space approach could not accomplish: the assignment of precise, punctal, and dispersion-free values of position or momentum to quantum systems (see Clifton and Halvorson Reference Clifton and Halvorson2001).Footnote ²¹ Still, the cost of withholding observable status from position or momentum and the inconvenience of having to deal with non-separable Hilbert spaces that represent singular states, may strike some as unattractive. A slightly moderated Imperialism follows Segal (Reference Segal and Lurcat1967, 127) in admitting only regular states as physically reasonable. The regularity requirement applies to the QFT context as well-there too continuity requirements may be imposed on representations of the Weyl algebra.

Let π : be a representation of the Weyl algebra in a concrete Hilbert space. Some Hilbert space operators are in the image of the representation; others, like position and momentum (provided the representation is regular), are recoverable from these. But any concrete Hilbert space representation harbors a host of operators without straightforward correlates, direct or indirect, in the algebra. These parochial operators, wage-earning observables on the Hilbert space approach, are otiose accretions according to the Imperialist. Underlying further charges of stinginess against the Imperialist are the facts that number operators and projection operators are, in general, parochial operators. They appear only as (what the Imperialist reckons) surplus structure in concrete representations of the Weyl algebra, and have no analogs in the algebra itself. But number operators play a prominent roles in the in-practice understanding of QFT as particle physics, and projection operators are central to many foundational treatments of quantum theory. Surely, these stinginess charges run, number and projection operators should be counted as observables; surely the algebraic approach is deficient for failing to so count them.

One sort of Imperialist defense would be to demonstrate the harmlessness of the deficiency. Such a demonstration might take the form of a recovery project: we use number operators to interpret scattering experiments and projection operators to calculate probabilities. If algebraic resources alone are sufficient to this pair of purposes, the absence from those resources of literal, rather than functional, counterparts to number operators and projection operators shouldn't vex us.Footnote ²² This recovery project is obviously a daunting interpretative task. Impartial judges will count this as grounds for the development, not the dismissal, of algebraic approaches—and see how, or if, that development progresses.

Another sort of defense against these stinginess charges moderates extreme Imperialism. The moderating move is to attribute physical significance to observables not represented by self-adjoint elements of the abstract algebra. Where ω is the actual state of the universe, moderated Imperialism elevates to observable status self-adjoint operators on the Hilbert space of the GNS construction associated with ω . These will include a total number operator and a full set of projection operators, although, of course, which total number operator and which set of projection operators is not specifiable in advance of the state of the universe. Suggesting that which physical magnitudes there are (in addition to those represented by self-adjoint elements of the abstract algebra) depends on what the state of the universe is, Imperialism so moderated, is no doubt controversial. But the core idea of Imperialism so moderated—that some observables are automatically physically relevant, others contingently so—is a natural counterpart to the gentle Conservative's maneuver of reckoning some possibilities more physical than others.

A final stinginess charge against the Imperialist is grounded, not in expectations tutored by the Hilbert space approach, but in extrinsic constraints on the interpretation of QFT. Consider the stress-energy tensor, which the Einstein field equations relate to spacetime curvature. The Imperialist's algebra will not in general contain an element corresponding to the stress-energy tensor (see Wald Reference Wald1994, Ch. 4.6). But to erect semi-classical quantum gravity as a signpost on the road to quantum gravity, we must be able to assign the stress-energy tensor an expectation value. Now, we can add the stress-energy tensor to QFT's catalog of observables by following a mathematical prescription for assigning it an expectation value in a given algebraic state. But the prescription succeeds only for some algebraic states, viz., those possessed of a leading singularity structure of Hadamard form (see Wald Reference Wald1994, ch. 4.6). Algebraic states in this way sustaining the stress-energy tensor as an observable are known as Hadamard states. If one's interpretative predilections are to insist that the stress energy tensor is an observable, then one might maintain (as do Kay and Wald (Reference Kay and Wald1991)) that for an algebraic state to be physical, it must be Hadamard. Call this the Hadamard condition.

Imposing this condition would have repercussions for the debate between the Hilbert Space Conservative and the Algebraic Imperialist. One strikes generally against the Imperialist: the case of the stress-energy tensor establishes that there are physically significant magnitudes without counterpart in the abstract algebra. Another strikes against the apologetic Imperialist's explication of physical equivalence as weak equivalence. Any two algebraic states give rise to weakly equivalent representations, but not all algebraic states are Hadamard. If states must be Hadamard to be physical, two representations can be weakly equivalent even if only one of them is physically possible. But a physically possible representation can't be physically equivalent to a physically impossible one! Thus weak equivalence is not physical equivalence. Similar considerations would depose the Imperialist's bold notion of physical equivalence as *-isomorphism.

The Hadamard condition actually reinforces Hilbert Space Conservatism. In spacetimes admitting a compact Cauchy surface—i.e. closed universes—all Hadamard vacuum states define unitarily equivalent representations. In such spacetimes, the Conservative can solve her problem of privilege simply by imposing the Hadamard condition. Recognizing only those representations of the Weyl relations whose vacua are Hadamard, she recognizes only a unitary equivalence class of representations. “There does not appear to be any necessity even to consider states lying outside this single Hilbert space” (Wald Reference Wald1994, 97).

In spacetimes with noncompact Cauchy surfaces—open universes—things don't work out quite so nicely for the Conservative. There it is not the case that all Hadamard vacuum states give rise to unitarily equivalent representations. Still, a result due to Verch extends the Conservative a flimsier sort of equivalence between (possibly unitarily inequivalent) representations arising from Hadamard states ω and ω ′. These states are locally indistinguishable, in the sense that (roughly) for any finite region of spacetime, and any Hilbert space state in the GNS representation (, π,) arising from ω , there's a state in the GNS representation (′, π ′) arising from ω ′ reproducing exactly the first state's assignment of expectation values to all the observables pertaining to that region of spacetime. “Consequently, the states in unitarily inequivalent quantum field theory constructions arising from different Hadamard vacuum states can be distinguished only by making measurements of observables over unbounded regions of spacetime” (Wald Reference Wald1994, 97). If she privileges a unitary equivalence class of representations arising from Hadamard states in an open universe, the Conservative can argue, the only otherwise (by the lights of the Hadamard condition) physically reasonable states she excludes from consideration are those indistinguishable from states she recognizes, even with respect to infinitely many observables measured infinitely accurately, so long as each of those observables pertains to a bounded spacetime region.

But now perhaps the Hilbert Space Conservative has said too much. She has suggested that it doesn't matter which unitary equivalence class of Hadamard vacua she privileges, because one choice is locally indistinguishable from any other. Thereby granting legitimacy to an operationalism according to which only locally distinguishable differences are genuine differences, she undermines principled resistance to operationalist equations of weak and physical equivalence. Still, she can oppose the equation on the grounds, cultivated above, that it falls short of requirements even operationalists would impose on equivalence.

Anyway, her reinforcement of Hilbert Space Conservatism is only as strong as the Hadamard condition to which it appeals. And the following might count as reasons to reject the Hadamard condition as a criterion of physical reasonableness. First, not all (by a variety of criteria) “physically interesting” states are Hadamard states. For example, neither the Unruh nor the Boulware vacua of extended Schwarzschild spacetime are Hadamard. Imposing the Hadamard condition disqualifies these states as physically significant. Second, there are spacetimes which harbor no Hadamard states—for example, Kerr Spacetime (see Kay and Wald Reference Kay and Wald1991). Imposing the Hadamard condition would declare such spacetimes unphysical! Finally, as the next section explains, there might be a sense of “dynamical evolution” adapted to QFT for which it is not the case that the set of Hadamard states is closed under dynamical evolution. Assuming that it would be shocking for a “physical” state of affairs to be carried to an “unphysical” state of affairs by the dynamics of a physical theory, such breakdown of dynamical closure would be a strike against the Hadamard condition as a criterion of physical reasonableness.

7. A Third Interpretative Question: Dynamics?

I've canvassed interpretative questions that arise in connection both with the state space and with the observable set of QFT. I turn next to an interpretative question concerning the theory's dynamics. I revert to my extremists to introduce the problem.

8. Conclusion

By shameless use of the rhetorical device of those fictional extremists, the Algebraic Imperialist and the Hilbert Space Conservative, I've tried to illustrate some of the questions of interpretation which arise in connection with QFT. They include questions about how to characterize the state space of the theory, questions about which physical magnitudes belong in the set of observables the theory wields, and questions about how to implement the dynamics of the theory. I've also tried to sketch, albeit impressionistically, moderating responses to these questions. One moderating proposal takes the Algebraist's state space to pick out physical possibilities in a broad sense, and takes the elements of his algebra to correspond to a core constituency of observables. It then uses the GNS construction for the actual state of the universe ω to identify a more narrow space of physical possibility (states in the folium of ω) and to supplement the core constituency of observables with specially appointed ones (self adjoint operators on the GNS construction for ω). Thus algebraic notions identify possibilities in the broad sense and the core set of observables; Hilbert space notions characterize possibility in the narrow sense and the supplemented set of observables; and part of what we've learned from QFT is that narrowing possibility and supplementing the core set of observables are state-dependent matters. Although possibly dubious intuitions about dynamical closure could undo this reconciliation, other moderating suggestions—for instance, to narrow the full space of algebraic possibility by imposing the Hadamard condition—are in the offing. We can be interpreters of QFT without being extremists.

Confronting the questions presented here, we confront interpretative issues not prominent in philosopher's discussion of ordinary quantum mechanics. We also confront extrinsic constraints which (by hypothesis of QFT's situation in the network of physical theories being different from the situation of ordinary quantum mechanics) we might miss were we to confine our attention to the interpretation of non-relativistic quantum mechanics. Philosophers of physics more subtle than my extremists should turn—they are turning—their attention to such questions.