1. Introduction

It is a widespread belief, at least within the physics community, that there is no relativistic quantum theory of (localizable) particles; and, thus, that the only relativistic quantum theory is a theory of fields. This belief has received much support in recent years in the form of rigorous no-go theorems by Malament (Reference Malament and Clifton1996) and Hegerfeldt (Reference Hegerfeldt and Böhm1998a, Reference Hegerfeldt1998b). In particular, Hegerfeldt shows that in a generic quantum theory (relativistic or non-relativistic), if there are states with localized particles, and if there is a lower bound on the system's energy, then superluminal spreading of the wavefunction must occur. Similarly, Malament shows the inconsistency of a few intuitive desiderata for a relativistic quantum mechanics of (localizable) particles. Thus, it appears that quantum theory engenders a fundamental conflict between relativistic causality and localizability.

What is the philosophical lesson of this conflict between relativistic causality and localizability? One the one hand, if we believe that the assumptions of Malament's theorem must hold for any empirically adequate theory, then it follows that our world cannot be correctly described by a particle theory. On the other hand, if we believe that our world can be correctly described by a particle theory, then one (or more) of Malament's assumptions must be false. Malament clearly endorses the first response; that is, he argues that his theorem entails that there is no relativistic quantum mechanics of localizable particles (insofar as any relativistic theory precludes act-outcome correlations at spacelike separation). Others, however, have argued that the assumptions of Malament's theorem need not hold for any relativistic, quantum-mechanical theory (cf. Fleming and Butterfield Reference Fleming, Butterfield, Butterfield and Pagonis1999), or that we cannot judge the truth of the assumptions until we resolve the interpretive issues of elementary quantum mechanics (cf. Barrett Reference Barrett2001).

We do not think that these objections to the soundness of Malament's argument are cogent. However, there are other tacit assumptions of Malament's theorem that some might be tempted to question. For example, Malament's theorem depends on the assumption that there is no preferred inertial reference frame, which some believe to have very little empirical support (cf. Cushing Reference Cushing and Clifton1996). Furthermore, Malament's theorem establishes only that there is no relativistic quantum mechanics in which particles can be completely localized in spatial regions with sharp boundaries; it leaves open the possibility that there might be a relativistic quantum mechanics of “unsharply” localized particles.

In this paper, we present two new no-go theorems which show that these tacit assumptions of Malament's theorem are not needed to sustain an argument against localizable particles. First, we derive a no-go theorem against localizable particles that does not assume the equivalence of all inertial frames (Theorem 1). Second, we derive a no-go theorem that shows that there is no relativistic quantum mechanics of unsharply localized particles (Theorem 2).

However, it would be a mistake to think that these results show—or, are intended to show—that a field ontology, rather than a particle ontology, is appropriate for relativistic quantum theories. While these results show that there is no position observable that satisfies relativistic constraints, quantum field theories—both relativistic and non-relativistic—already reject the notion of a position observable in favor of localized field observables. Thus, our first two results have nothing to say about the possibility that relativistic quantum field theory (RQFT) might permit a “particle interpretation,” in which localized particles are supervenient on the underlying localized field observables. To exclude this latter possibility, we formulate (in Section 6) a necessary condition for a quantum field theory to permit a particle interpretation, and we then show that this condition fails in any relativistic theory (Theorem 3).

Presumably, any empirically adequate theory must be able to reproduce the predictions of special relativity and of quantum mechanics. Therefore, our no-go results show that the existence of localizable particles is, strictly speaking, ruled out by the empirical data. However, in Section 7 we defuse this counterintuitive consequence by showing that RQFT itself explains how the illusion of localizable particles can arise, and how “particle talk”—although strictly fictional—can still be useful.

2. Malament's Theorem

Malament's theorem shows the inconsistency of a few intuitive desiderata for a relativistic quantum mechanics of (localizable) particles. It strengthens previous results (e.g., Schlieder Reference Schlieder and Dürr1971) by showing that the assumption of “no superluminal wavepacket spreading” can be replaced by the weaker assumption of “microcausality,” and by making it clear that Lorentz invariance is not needed to derive a conflict between relativistic causality and localizability.

In order to present Malament's result, we assume that our background spacetime M is an affine space, with a foliation into spatial hyperplanes. This will permit us to consider a wide range of relativistic (e.g., Minkowski) as well as non-relativistic (e.g., Galilean) spacetimes. The pure states of our quantum-mechanical system are given by rays in some Hilbert space . We assume that there is a mapping Δ ↦ E _Δ of bounded subsets of hyperplanes in M into projections on . We think of E _Δ as representing the proposition that the particle is localized in Δ; or, from a more operational point of view, E _Δ represents the proposition that a position measurement is certain to find the particle within Δ. We also assume that there is a strongly continuous representation a ↦ U(a) of the translation group of M in the unitary operators on . Here strong continuity means that for any unit vector ψ ∊ , 〈ψ, U(a)ψ〉 → 1 as a → 0; and it is equivalent (via Stone's theorem) to the assumption that there are energy and momentum observables for the particle. If all of the preceding conditions hold, we say that the triple (, Δ↦ E _Δ, a ↦ U(a)) is a localization system over M.

The following conditions should hold for any localization system—either relativistic or non-relativistic—that describes a single particle.

1. Localizability: If Δ and Δ′ are disjoint subsets of a single hyperplane, then E _ΔE _Δ′ = 0.

2. Translation covariance: For any Δ and for any translation a of M, U(a)E _ΔU(a)* = E _Δ+a.

3. Energy bounded below: For any timelike translation a of M, the generator H(a) of the one-parameter group {U(t a) : t ∊ ℝ} has spectrum bounded from below.

We recall briefly the motivation for each of these conditions. “Localizability” says that the particle cannot be detected in two disjoint spatial sets at a given time. “Translation covariance” gives us a connection between the symmetries of the spacetime M and the symmetries of the quantum-mechanical system. In particular, if we displace the particle by a spatial translation a, then the original wavefunction ψ will transform to some wavefunction ψ_a. Since the statistics for a displaced detection experiment should be identical to the original statistics, we have 〈ψ, E _Δψ〉 = 〈ψ_a, E _Δ+aψ_a〉. By Wigner's theorem, however, the symmetry is implemented by some unitary operator U(a). Thus, U(a)ψ = ψ_a, and U(a)E _ΔU(a)* = E _Δ+a. In the case of time translations, the covariance condition entails that the particle has unitary dynamics. (This might seem to beg the question against a collapse interpretation of quantum mechanics; we dispell this worry at the end of this section.) Finally, the “energy bounded below” condition asserts that, relative to any inertial observer, the particle has a lowest possible energy state. If it were to fail, we could extract an arbitrarily large amount of energy from the particle as it drops down through lower and lower states of energy.

We now turn to the “specifically relativistic” assumptions needed for Malament's theorem. The special theory of relativity entails that there is a finite upper bound on the speed at which (detectable) physical disturbances can propagate through space. Thus, if Δ and Δ′ are distant regions of space, then there is a positive lower bound on the amount of time it should take for a particle localized in Δ to travel to Δ′. We can formulate this requirement precisely by saying that for any timelike translation a, there is an ∊ > 0 such that, for every state ψ, if 〈ψ, E _Δψ〉 = 1 then 〈ψ, E _{Δ′+t a}ψ〉 = 0 whenever 0 ≤ t < ε. This is equivalent to the following assumption.

1. Strong causality: If Δ and Δ′ are disjoint subsets of a single hyperplane, and if the distance between Δ and Δ′ is nonzero, then for any timelike translation a, there is an ∊ > 0 such that E _ΔE _{Δ′+t a} = 0 whenever 0 ≤ t < ε.

(Note that strong causality entails localizability.) Although strong causality is a reasonable condition for relativistic theories, Malament's theorem requires only the following weaker assumption (which he himself calls “locality”).

1. Microcausality: If Δ and Δ′ are disjoint subsets of a single hyperplane, and if the distance between Δ and Δ′ is nonzero, then for any timelike translation a, there is an ∊ > 0 such that [E _Δ, E _{Δ′+t a}] = 0 whenever 0 ≤ t < ε.

If E _Δ can be measured within Δ, microcausality is equivalent to the assumption that a particle detection measurement within Δ cannot influence the statistics of particle detection measurements performed in regions that are spacelike to Δ (see Malament Reference Malament and Clifton1996, 5). Thus, a failure of microcausality would entail the possibility of act-outcome correlations at spacelike separation. Note that both strong causality and microcausality make sense for non-relativistic spacetimes (as well as for relativistic spacetimes); though, of course, we should not expect either causality condition to hold in the non-relativistic case.

1. Theorem (Malament). Let (, Δ ↦ E _Δ, a ↦ U(a)) be a localization system over Minkowski spacetime that satisfies:

   1. 1. Localizability

   2. 2. Translation covariance

   3. 3. Energy bounded below

   4. 4. Microcausality

2. Then E _Δ = 0 for all Δ.

Thus, in every state, there is no chance that the particle will be detected in any local region of space. As Malament claims, this serves as a reductio ad absurdum of any relativistic quantum mechanics of a single (localizable) particle.

2.2. Tacit Assumptions of Malament's Theorem

The objections to the four assumptions of Malament's theorem are unconvincing. By any reasonable understanding of special relativity and of quantum theory, these assumptions should hold for any theory that is capable of reproducing the predictions of both theories. Nonetheless, we anticipate that further objections could be directed against the more or less tacit assumptions of Malament's theorem.

As we noted earlier, Malament's theorem does not make use of the full structure of Minkowski spacetime (e.g., Lorentz invariance). However, the following example shows that the theorem fails if there is a preferred inertial reference frame.

Example 1. Let M = ℝ¹ ⊕ ℝ³ be full Newtonian spacetime with a distinguished timelike direction a. To any set of the form {(t, x): x ∊ Δ}, with t ∊ ℝ, and Δ a bounded open subset of ℝ³, we assign the spectral projection E _Δ of the position operator for a particle in three dimensions. Thus, the conclusion of Malament's theorem is false, while both the microcausality and localizability conditions hold. Let P ₀ = 0, and for i = 1, 2, 3, let P _i = −i(d/dx _i). For any four-vector b, let U(b) = exp{i(b·P)}, where

Thus, translation covariance holds, and since the energy is identically zero, the energy condition trivially holds. (Note, however, that if M is not regarded as having a distinguished timelike direction, then this example violates the energy condition.) □

A brief inspection of Malament's proof shows that the following assumption on the affine space M is sufficient for his theorem to go through.

1. No absolute velocity: Let a be a spacelike translation of M. Then there is a pair (b, c) of timelike translations of M such that a = b − c.

Despite the fact that “no absolute velocity” is a feature of both Galilean and Minkowski spacetimes, there are some who claim that the existence of a (undetectable) preferred reference frame is perfectly consistent with all current empirical evidence (cf. Cushing Reference Cushing and Clifton1996). What is more, the existence of a preferred frame is an absolutely essential feature of a number of “realistic” interpretations of quantum theory (cf. Maudlin Reference Maudlin1994, Chap. 7). Thus, this tacit assumption of Malament's theorem could be a source of contention for those wishing to maintain the possibility of a relativistic quantum mechanics of localizable particles.

Second, some might wonder whether Malament's result is an artifact of special relativity, and whether a notion of localizable particles might be restored in the context of general relativity. Indeed, it is not difficult to see that Malament's result does not automatically generalize to arbitrary relativistic spacetimes.

To see this, suppose that M is an arbitrary globally hyperbolic manifold. (That is, M is a manifold that permits at least one foliation into spacelike hypersurfaces). Although M will not typically have a translation group, we assume that M has a transitive Lie group G of diffeomorphisms. (Just as a manifold is locally isomorphic to ℝⁿ, a Lie group is locally isomorphic to a group of translations.) We require that G has a representation g ↦ U(g) in the unitary operators on ; and, the translation covariance condition now says that E _g(Δ) = U(g)E _ΔU(g)* for all g ∊ G. The following example then shows that Malament's theorem fails even for the very simple case where M is a two-dimensional cylinder.

Example 2. Let M = ℝ ⊕ S ¹, where S ¹ is the one-dimensional unit circle, and let G denote the Lie group of timelike translations and rotations of M. It is not difficult to construct a unitary representation of G that satisfies the energy bounded below condition. (We can use the Hilbert space of square-integrable functions from S ¹ into ℂ, and the procedure for constructing the unitary representation is directly analogous to the case of a single particle moving on a line.) Fix a spacelike hypersurface Σ, and let μ denote the normalized rotation-invariant measure on Σ. For each open subset Δ of Σ, let E _Δ = I if μ(Δ) ≥ 2/3, and let E _Δ = 0 if μ(Δ) < 2/3. Then localizability holds, since for any pair (Δ, Δ′) of disjoint open subsets of Σ, either μ(Δ) < 2/3 or μ(Δ′) < 2/3. □

Obviously, Examples 1 and 2 are not physically interesting counterexamples to Malament's theorem. In particular, in Example 1 the energy is identically zero, and therefore the probability for finding the particle in a given region of space remains constant over time. Similarly, in Example 2 the particle is localized in every region of space with volume greater than 2/3, and the particle is never localized in a region of space with volume less than 2/3. In the following two sections, then, we will formulate explicit conditions to rule out such pathologies, and we will use these conditions to derive a no-go theorem that applies to generic spacetimes.

3. Hegerfeldt's Theorem

Hegerfeldt's (Reference Hegerfeldt and Böhm1998a, Reference Hegerfeldt1998b) recent results on localization apply to arbitrary (globally hyperbolic) spacetimes, and they do not make use of the “no absolute velocity” condition. Thus, we will suppose henceforth that M is a globally hyperbolic spacetime, and we will fix a foliation of M, as well as a unique isomorphism between any two hypersurfaces in this foliation. If Σ ∊ , we will write Σ + t for the hypersurface that results from “moving Σ forward in time by t units”; and if Δ is a subset of Σ, we will use Δ + t to denote the corresponding subset of Σ + t. We assume that there is a representation t ↦ U _t of the time-translation group ℝ in the unitary operators on ; and we say that the localization system (, Δ ↦ E _Δ, t ↦ U _t) satisfies time-translation covariance just in case U _tE _ΔU _−t = E _Δ+t for all Δ and all t ∊ ℝ.

Hegerfeldt's result is based on the following root lemma.

1. Lemma 1 (Hegerfeldt). Suppose that U _t= e ^itH, where H is a self-adjoint operator with spectrum bounded from below. Let A be a positive operator (e.g., a projection operator). Then for any unit vector ψ, either

   
   or
   

Hegerfeldt claims that this lemma has the following consequence for localization:

Hegerfeldt's argument for this conclusion is as follows:

Let us attempt now to formalize this argument.

First, Hegerfeldt claims that the following is a consequence of “finite propagation speed”: If Δ ⊆ Δ′, and if the boundaries of Δ and Δ′ have a finite distance, then a state initially localized in Δ will continue to be localized in Δ′ for some finite amount of time. We can capture this precisely by means of the following condition.

1. No instantaneous wavepacket spreading (NIWS): If Δ ⊆ Δ′, and the boundaries of Δ and Δ′ have a finite distance, then there is an ∊ > 0 such that E _Δ ≤ E _Δ′+t whenever 0 ≤ t < ε.

(Note that NIWS plus localizability entails strong causality.) In the argument, Hegerfeldt also assumes that if a particle is localized in every one of a family of sets that “approaches” Δ, then it is localized in Δ. We can capture this assumption in the following condition.

1. Monotonicity: If {Δ_n : n ∊ ℕ} is a downward nested family of subsets of Σ such that ∩_n Δ_n = Δ, then ∧_n E _Δn = E _Δ.

Using this assumption, Hegerfeldt argues that if NIWS holds, and if a particle is initially localized in some finite region Δ, then it will remain in Δ for all subsequent times. In other words, if E _Δψ = ψ, then E _ΔU _tψ = U _tψ for all t ≥ 0. We can now translate this into the following formal no-go theorem.

1. Theorem (Hegerfeldt). Suppose that the localization system (, Δ ↦ E _Δ, t ↦ U _t) satisfies:

   1. 1. Monotonicity

   2. 2. Time-translation covariance

   3. 3. Energy bounded below

   4. 4. No instantaneous wavepacket spreading

2. Then U _tE _ΔU _−t = E _Δ for all Δ ⊂ Σ and all t ∊ ℝ.

(The proof of this and all subsequent theorems can be found in the appendix.)

Thus, conditions 1–4 can be satisfied only if the particle has trivial dynamics. The following Lemma then shows how to derive Malament's conclusion from Hegerfeldt's theorem.

1. Lemma 2. Let M be an affine space. Suppose that the localization system (,Δ ↦ E _Δ, a ↦ U(a)) satisfies localizability, time-translation covariance, and no absolute velocity. For any bounded spatial set Δ, if U (a) E _ΔU(a)* = E _Δ for all timelike translations a of M, then E _Δ = 0.

Thus, if we add “no absolute velocity” to the assumptions of Hegerfeldt's theorem, then it follows that E _Δ = 0 for all bounded Δ. However, NIWS is a stronger causality assumption than microcausality. In fact, while NIWS plus localizability entails strong causality (and hence microcausality), the following example shows that NIWS is not entailed by the conjunction of strong causality, monotonicity, time-translation covariance, and energy bounded below.

Example 3. Let Q, P denote the standard position and momentum operators on = L ₂ (ℝ), and let H = P ²/2m for some m > 0. Let $\Delta \mapsto E^{Q}_{\Delta }$ denote the spectral measure for Q. Fix some bounded subset Δ₀ of ℝ, and let $E_{\Delta }=E^{Q}_{\Delta }\otimes E^{Q}_{\Delta _{0}}$ (a projection operator on ⊗ ) for all Borel subsets Δ of ℝ. Thus, Δ ↦ E _Δ is a (non-normalized) projection-valued measure. Let U _t = I ⊗ e ^itH, and let E _Δ+t = U _tE _ΔU _−t for all t ∊ ℝ. It is clear that monotonicity, time-translation covariance, and energy bounded below hold. To see that strong causality holds, let Δ and Δ′ be disjoint subsets of a single hyperplane Σ. Then,

for all t ∊ ℝ. On the other hand, U _tE _Δ U _−t ≠ E _Δ for any nonempty Δ and for any t ≠ 0. Thus, it follows from Hegerfeldt's theorem that NIWS fails. □

Thus, we could not recapture the full strength of Malament's theorem simply by adding “no absolute velocity” to the conditions of Hegerfeldt's theorem.

4. Doing without “No Absolute Velocity”

Example 3 shows that Hegerfeldt's theorem fails if NIWS is replaced by strong causality (or by microcausality). On the other hand, Example 3 is hardly a physically interesting counterexample to a strengthened version of Hegerfeldt's theorem. In particular, if Σ is a fixed spatial hypersurface, and if {Δ_n: n ∊ ℕ} is a covering of Σ by bounded sets (i.e., ∪_nΔ_n = Σ), then ∨_n E _Δn = I ⊗ E _Δ0 ≠ I ⊗ I. Thus, it is not certain that the particle will be detected somewhere or other in space. In fact, if {Δ_n: n ∊ ℕ} is a covering of Σ and {Π_n : n ∊ ℕ} is a covering of Σ + t, then

Thus, the total probability for finding the particle somewhere or other in space can change over time.

It would be completely reasonable to require that $\vee _{n}E_{\Delta _{n}}=I$ whenever {Δ_n : n ∊ ℕ} is a covering of Σ. This would be the case, for example, if the mapping Δ ↦ E _Δ (restricted to subsets of Σ) were the spectral measure of some position operator. However, we propose that—at the very least—any physically interesting model should satisfy the following weaker condition.

1. Probability conservation: If {Δ_n : n ∊ ℕ} is a covering of Σ, and {Π_n: n ∊ ℕ} is a covering of Σ + t, then $\vee _{n}E_{\Delta _{n}}=\vee _{n}E_{\Pi _{n}}$.

Probability conservation guarantees that there is a well-defined total probability for finding the particle somewhere or other in space, and this probability remains constant over time. In particular, if both {Δ_n : n ∊ ℕ} and {Π_n: n ∊ ℕ} consist of pairwise disjoint sets, then the localizability condition entails that $\vee _{n}E_{\Delta _{n}}=\sum_{n}E_{\Delta _{n}}$ and $\vee _{n}E_{\Pi _{n}}=\Sigma _{n}E_{\Pi _{n}}$. In this case, probability conservation is equivalent to

for any state ψ. Note, finally, that it is reasonable to require probability conservation for both relativistic and non-relativistic models.Footnote 1 With this in mind, we can now formulate a no-go result that generalizes aspects of both Malament's and Hegerfeldt's theorems.

1. Theorem 1. Suppose that the localization system (, Δ ↦ E _Δ, t ↦ U _t) satisfies:

   1. 1. Localizability

   2. 2. Probability conservation

   3. 3. Time-translation covariance

   4. 4. Energy bounded below

   5. 5. Microcausality

   Then U _tE _ΔU _−t = E _Δ for all Δ and all t ∊ ℝ.

If M is an affine space, and if we add “no absolute velocity” as a sixth condition in this theorem, then it follows that E _Δ = 0 for all Δ (see Lemma 2). Thus, modulo the probability conservation condition, Theorem 1 recaptures the full strength of Malament's theorem. Moreover, we can now trace the difficulties with localization to microcausality alone: There are localizable particles only if it is possible to have act-outcome correlations at spacelike separation.

We now give examples to show that all five assumptions of Theorem 1 are essential for the result. (Example 1 shows that these assumptions can be simultaneously satisfied.) For simplicity, suppose that M is two-dimensional. (All examples work in the four-dimensional case as well.) Let Q, P be the standard position and momentum operators on L ₂(ℝ), and let H = P ²/2m. Let Σ be a spatial hypersurface in M, and suppose that a coordinatization of Σ has been fixed, so that there is a natural association between each bounded open subset Δ of Σ and a corresponding spectral projection E _Δ of Q.

1. (1 + 2 + 3 + 4) (a) Consider the standard localization system for a single non-relativistic particle. That is, let Δ ↦ E _Δ (with domain the Borel subsets of Σ) be the spectral measure for Q. For Σ + t, set E _Δ+t = U _tE _ΔU _−t, where U _t = e ^itH. (b) The Newton-Wigner approach to relativistic QM uses the standard localization system for a non-relativistic particle, only replacing the non-relativistic Hamiltonian P ²/2m with the relativistic Hamiltonian (P ² + m ²I)^1/2, whose spectrum is also bounded from below.

2. (1 + 2 + 3 + 5) (a) For a mathematically simple (but physically uninteresting) example, take the first example above and replace the Hamiltonian P ²/2m with P. In this case, microcausality trivially holds, since U _tE _ΔU _−t is just a shifted spectral projection of Q. (b) For a physically interesting example, consider the relativistic quantum theory of a single spin-1/2 electron (see Section 2). Due to the negative energy solutions of the Dirac equation, the spectrum of the Hamiltonian is not bounded from below.

3. (1 + 2 + 4 + 5) Consider the the standard localization system for a non-relativistic particle, but set E _Δ+t = E _Δ for all t ∊ ℝ. Thus, we escape the conclusion of trivial dynamics, but only by disconnecting the (nontrivial) unitary dynamics from the (trivial) association of projections with spatial regions.

4. (1 + 3 + 4 + 5) (a) Let Δ₀ be some bounded open subset of Σ, and let $E_{\Delta _{0}}$ be the corresponding spectral projection of Q. When Δ ≠ Δ₀, let E _Δ = 0. Let U _t = e ^itH, and let E _Δ+t = U _tE _ΔU _−t for all Δ. This example is physically uninteresting, since the particle cannot be localized in any region besides Δ₀, including proper supersets of Δ₀. (b) See Example 3.

5. (2 + 3 + 4 + 5) Let Δ₀ be some bounded open subset of Σ, and let $E_{\Delta _{0}}$ be the corresponding spectral projection of Q. When Δ ≠ Δ₀, let E _Δ = I. Let U _t = e ^itH, and let E _Δ+t = U _tE _ΔU _−t for all Δ. Thus, the particle is always localized in every region other than Δ₀, and is sometimes localized in Δ₀ as well.

5. Are There Unsharply Localizable Particles?

We have argued that attempts to undermine the four explicit assumptions of Malament's theorem are unsuccessful. We have also now shown that the “no absolute velocity” condition is not necessary to rule out localizable particles. However, there is one further question that might arise concerning the soundness of Malament's argument. In particular, some might argue that it is possible to have a quantum-mechanical particle theory in the absence of a family {E _Δ} of localizing projections. What is more, one might argue that localizing projections represent an unphysical idealization—viz., that a “particle” can be completely contained in a finite region of space with a sharp boundary, when in fact it would require an infinite amount of energy to prepare a particle in such a state. Thus, there remains a possibility that relativistic causality can be reconciled with “unsharp” localizability.

To see how we can define “particle talk” without having projection operators, consider again the relativistic theory of a single spin-1/2 electron (where we now restrict to the subspace _pos of positive energy solutions of the Dirac equation). In order to treat the argument ‘x’ of the Dirac wavefunction as an observable, it would be sufficient to define a probability amplitude and density for the particle to be found at x; and these can be obtained from the Dirac wavefunction itself. That is, for a subset Δ of Σ, we set

Now let Δ ↦ E _Δ be the spectral measure for the standard position operator on the Hilbert space (which includes both positive and negative energy solutions). That is, E _Δ multiplies a wavefunction by the characteristic function of Δ. Let F denote the orthogonal projection of onto _pos. Then,

for any ψ ∊ _pos. Thus, we can apply the standard recipe to the operator FE _Δ (defined on _pos) to compute the probability that the particle will be found within Δ. However, FE _Δ is not a projection operator. (In fact, it can be shown that FE _Δ does not have any eigenvectors with eigenvalue 1.) Thus, we do not need a family of projection operators in order to define probabilities for localization.

Now, in general, to define the probability that a particle will be found in Δ, we need only assume that there is an operator A _Δ such that 〈ψ, A _Δψ〉 ∊ [0, 1] for any unit vector ψ. Such operators are called effects, and include the projection operators as a proper subclass. Thus, we say that the triple (, Δ ↦ A _Δ, a ↦ U(a)) is an unsharp localization system over M just in case Δ ↦ A _Δ is a mapping from subsets of hyperplanes in M to effects on , and a ↦ U(a) is a continuous representation of the translation group of M in unitary operators on . (We assume again for the present that M is an affine space.)

Most of the conditions from the previous sections can be applied, with minor changes, to unsharp localization systems. In particular, since the energy bounded below condition refers only to the unitary representation, it can be carried over intact; and translation covariance also generalizes straightforwardly. However, we will need to take more care with microcausality and with localizability.

If E and F are projection operators, [E, F] = 0 just in case for any state, the statistics of a measurement of F are not affected by a non-selective measurement of E and vice versa (cf. Malament Reference Malament and Clifton1996, 5). This fact, along with the assumption that E _Δ is measurable in Δ, motivates the microcausality assumption. For the case of an association of arbitrary effects with spatial regions, Busch (Reference Busch1999, Prop. 2) has shown that [A _Δ, A _Δ′] = 0 just in case for any state, the statistics for a measurement of A _Δ are not affected by a non-selective measurement of A _Δ′ and vice versa. Thus, we may carry over the microcausality assumption intact, again seen as enforcing a prohibition against act-outcome correlations at spacelike separation.

The localizability condition is motivated by the idea that a particle cannot be simultaneously localized (with certainty) in two disjoint regions of space. In other words, if Δ and Δ′ are disjoint subsets of a single hyperplane, then 〈ψ, E _Δψ〉 = 1 entails that 〈ψ, E _Δ′ψ〉 = 0. It is not difficult to see that this last condition is equivalent to the assumption that E _Δ + E _Δ′ ≤ I. That is,

for any unit vector ψ. Now, it is an accidental feature of projection operators (as opposed to arbitrary effects) that E _Δ + E _Δ′ ≤ I is equivalent to E _ΔE _Δ′ = 0. Thus, the appropriate generalization of localizability to unsharp localization systems is the following condition.

1. Localizability: If Δ and Δ′ are disjoint subsets of a single hyperplane, then A _Δ + A _Δ′ ≤ I.

That is, the probability for finding the particle in Δ, plus the probability for finding the particle in some disjoint region Δ′, never totals more than 1. It would, in fact, be reasonable to require a slightly stronger condition, viz., the probability of finding a particle in Δ plus the probability of finding a particle in Δ′ equals the probability of finding a particle in Δ ∪ Δ′. If this is true for all states ψ, we have:

1. Additivity: If Δ and Δ′ are disjoint subsets of a single hyperplane, then A _Δ + A _Δ′ = A _Δ∪Δ′.

With just these mild constraints, Busch (Reference Busch1999) was able to derive the following no-go result.

1. Theorem (Busch). Suppose that the unsharp localization system (, Δ ↦ A _Δ, a ↦ U(a)) satisfies localizability, translation covariance, energy bounded below, microcausality, and no absolute velocity. Then, for any Δ, A _Δ has no eigenvector with eigenvalue 1.

Thus, it is not possible for a particle to be localized with certainty in any bounded region Δ. Busch's theorem, however, leaves it open whether there are (nontrivial) “strongly unsharp” relativistic localization systems. The following result shows that there are not.

1. Theorem 2. Suppose that the unsharp localization system (, Δ ↦ A _Δ, a ↦ U(a)) satisfies:

   1. 1. Additivity

   2. 2. Translation covariance

   3. 3. Energy bounded below

   4. 4. Microcausality

   5. 5. No absolute velocity

2. Then A _Δ = 0 for all Δ.

Theorem 2 shows that invoking the notion of unsharp localization does nothing to resolve the tension between relativistic causality and localizability. For example, we can now conclude that the positive energy Dirac theory violates microcausality.Footnote 2

Unfortunately, Theorem 2 does not generalize to arbitrary globally hyperbolic spacetimes, as the following example shows.

Example 4. Let M be the cylinder spacetime from Example 2. Let G denote the group of timelike translations and rotations of M, and let g ↦ U(g) be a positive energy representation of G in the unitary operators on a Hilbert space . For any Σ ∊ let μ denote the normalized rotation-invariant measure on Σ, and let A _Δ = μ(Δ)I. Then, conditions 1–5 of Theorem 2 are satisfied, but the conclusion of the theorem is false. □

The previous counterexample can be excluded if we require there to be a fixed positive constant δ such that, for each Δ, there is a state ψ with 〈ψ, A _Δ ψ〉 ≥ δ. In fact, with this condition added, Theorem 2 holds for any globally hyperbolic spacetime. (The proof is an easy modification of the proof we give in the Appendix.) However, it is not clear what physical motivation there could be for requiring this further condition. Note also that Example 4 has trivial dynamics; i.e., U _tA _ΔU _−t = A _Δ for all Δ. We conjecture that every counterexample to a generalized version of Theorem 2 will have trivial dynamics.

Theorem 2 strongly supports the conclusion that there is no relativistic quantum mechanics of a single (localizable) particle; and therefore that special relativity and quantum mechanics can be reconciled only in the context of a quantum field theory. However, neither Theorem 1 nor Theorem 2 says anything about the ontology of relativistic quantum field theories; they leave it fully open that such theories might permit an ontology of localizable particles. To eliminate this latter possibility, we will now proceed to present a more general result which shows that there are no localizable particles in any relativistic quantum theory.

6. Are There Localizable Particles in RQFT?

The localizability assumption is motivated by the idea that a “particle” cannot be detected in two disjoint spatial regions at once. However, in the case of a many-particle system, it is certainly possible for there to be particles in disjoint spatial regions. Thus, the localizability condition does not apply to many-particle systems; and Theorems 1 and 2 cannot be used to rule out a relativistic quantum mechanics of n > 1 localizable particles.

Still, one might argue that we could use E _Δ to represent the proposition that a measurement is certain to find that all n particles lie within Δ, in which case localizability should hold. Note, however, that when we alter the interpretation of the localization operators {E _Δ}, we must alter our interpretation of the conclusion. In particular, the conclusion now shows only that it is not possible for all n particles to be localized in a bounded region of space. This leaves open the possibility that there are localizable particles, but that they are governed by some sort of “exclusion principle” that prohibits them all from clustering in a bounded spacetime region.

Furthermore, Theorems 1 and 2 only show that it is impossible to define position operators that obey appropriate relativistic constraints. But it does not immediately follow from this that we lack any notion of localization in relativistic quantum theories. Indeed,

However, it is not a peculiar feature of RQFT that it lacks a position operator: All quantum field theories (both relativistic and non-relativistic) model localization by making the observables dependent on position in spacetime. Moreover, in the case of non-relativistic QFT, these “localized” observables suffice to provide us with a concept of localizable particles. In particular, for each spatial region Δ, there is a “number operator” N _Δ whose eigenvalues give the number of particles within the region Δ. Thus, we have no difficultly in talking about the particle content in a given region of space, despite the absence of any position operator.

Abstractly, a number operator N on is any operator with eigenvalues {0, 1, 2,…}. In order to describe the number of particles locally, we require an association Δ ↦ N _Δ of subsets of spatial hyperplanes in M to number operators on , where N _Δ represents the number of particles in the spatial region Δ. If a ↦ U(a) is a unitary representation of the translation group, we say that the triple (, Δ ↦ N _Δ, a ↦ U(a)) is a system of local number operators over M.

Note that a localization system (, Δ ↦ E _Δ, a ↦ U(a)) is a special case of a system of local number operators where the eigenvalues of each N _Δ are restricted to {0, 1}. Furthermore, if we loosen our assumption that number operators have a discrete spectrum, and instead require only that they have spectrum contained in [0, ∞), then we can also include unsharp localization systems within the general category of systems of local number operators. Thus, a system of local number operators is the minimal requirement for a concept of localizable particles in any quantum theory.

In addition to the natural analogues of the energy bounded below condition, translation covariance, and microcausality, we will be interested in the following two requirements on a system of local number operators:Footnote 3

1. Additivity: If Δ and Δ′ are disjoint subsets of a single hyperplane, then N _Δ + N _Δ′ = N _Δ∪Δ′.

2. Number conservation: If {Δ_n : n ∊ ℕ} is a disjoint covering of Σ, then the sum Σ_n N _Δn converges to a densely defined, self-adjoint operator N on (independent of the chosen covering), and U (a) NU (a)* = N for any timelike translation a of M.

Additivity asserts that, when Δ and Δ′ are disjoint, the expectation value (in any state) for the number of particles in Δ ∪ Δ′ is the sum of the expectations for the number of particles in Δ and the number of particles in Δ′. In the pure case, it asserts that the number of particles in Δ ∪ Δ′ is the sum of the number of particles in Δ and the number of particles in Δ′. The “number conservation” condition tells us that there is a well-defined global number operator, and that its expectation value remains constant over time. This condition holds for any non-interacting model of QFT.

It is a well-known consequence of the Reeh-Schlieder theorem that relativistic quantum field theories do not admit systems of local number operators (cf. Redhead Reference Redhead, Hull, Forbes and Burian1995). We will now derive the same conclusion from strictly weaker assumptions. In particular, microcausality is the only specifically relativistic assumption needed for this result. The relativistic spectrum condition—which requires that the spectrum of the four-momentum lie in the forward light cone, and which is used in the proof of the Reeh-Schlieder theorem—plays no role in our proof.Footnote 4

1. Theorem 3. Suppose that the system (, Δ ↦ N _Δ, a ↦ U (a)) of local number operators satisfies:

   1. 1. Additivity

   2. 2. Translation covariance

   3. 3. Energy bounded below

   4. 4. Number conservation

   5. 5. Microcausality

   6. 6. No absolute velocity

   7. Then N _Δ = 0 for all Δ.

Thus, in every state, there are no particles in any local region. This serves as a reductio ad absurdum for a notion of localizable particles in any relativistic quantum theory.

Unfortunately, Theorem 3 is not the strongest result we could hope for, since “number conservation” holds only in the (trivial) case of non-interacting fields. However, we would need a more general approach in order to deal with interacting relativistic quantum fields, because (due to Haag's theorem; cf. Streater and Wightman Reference Streater and Wightman2000, 163) their dynamics are not unitarily implementable on a fixed Hilbert space. On the other hand, this hardly indicates a limitation on the generality of our conclusion, since Haag's theorem also entails that interacting models of RQFT have no number operators—not even a global number operator.Footnote 5 Still, it would be interesting to recover this conclusion (perhaps working in a more general algebraic setting) without using the full strength of Haag's assumptions.

7. Particle Talk Without Particle Ontology

The results of the previous sections show that relativistic quantum theories do not admit (localizable) particles into their ontology. We also considered and rejected several objections to our characterization of relativistic quantum theories. Thus, we have yet to find a good reason to reject one of the premises of our argument against localizable particles. However, according to Segal (Reference Segal, Martin and Segal1964) and Barrett (Reference Barrett2001), there are independent grounds for believing that there are localizable particles—and therefore for rejecting one of the premises of the no-go results.

The argument for localizable particles appears to be very simple: Our experience shows us that objects (particles) occupy finite regions of space. But the reply to this argument is just as simple: These experiences are illusory! Although no object is strictly localized in a bounded region of space, an object can be well-enough localized to give the appearance to us (finite observers) that it is strictly localized. In fact, RQFT itself shows how the “illusion” of localizable particles can arise, and how talk about localizable particles can be a useful fiction.

In order to assess the possibility of “approximately localized” objects in RQFT, we shall now pursue the investigation in the framework of algebraic quantum field theory.Footnote 6 Here, one assumes that there is a correspondence O ↦ (O) between bounded open subsets of M and subalgebras of observables on some Hilbert space . Observables in (O) are considered to be “localized” (i.e., measurable) in O. Thus, if O and O′ are spacelike separated, we require that [A, B] = 0 for any A ∊ (O) and B ∊ (O′). Furthermore, we assume that there is a continuous representation a ↦ U(a) of the translation group of M in unitary operators on , and that there is a unique “vacuum” state Ω ∊ such that U(a)Ω = Ω for all a. This latter condition entails that the vacuum appears the same to all observers, and that it is the unique state of lowest energy.

In this context, a particle detector can be represented by an effect C such that 〈Ω, CΩ〉 = 0. That is, C should register no particles in the vacuum state. However, the Reeh-Schlieder theorem entails that no positive local observable can have zero expectation value in the vacuum state. Thus, it is impossible to detect particles by means of local measurements; instead, we will have to think of particle detections as “approximately local” measurements.

If we think of an observable as representing a measurement procedure (or, more precisely, an equivalence class of measurement procedures), then the norm distance ‖C − C′‖ between two observables gives a quantitative measure of the physical similarity between the corresponding procedures. (In particular, if ‖C − C′‖ < δ, then the expectation values of C and C′ never differ by more than δ.)Footnote 7 Moreover, in the case of real-world measurements, the existence of measurement errors and environmental noise make it impossible for us to determine precisely which measurement procedure we have performed. Thus, practically speaking, we can at best determine a neighborhood of observables corresponding to a concrete measurement procedure.

In the case of present interest, what we actually measure is always a local observable—i.e., an element of (O), where O is bounded. However, given a fixed error bound δ, if an observable C is within norm distance δ from some local observable C′ ∊ (O), then a measurement of C′ will be practically indistinguishable from a measurement of C. Thus, if we let

denote the family of observables “almost localized” in O, then ‘FAPP’ (i.e., ‘for all practical purposes’) we can locally measure any observable from _δ(O). That is, measurement of an element from _δ(O) can be simulated to a high degree of accuracy by local measurement of an element from (O). However, for any local region O, and for any δ > 0, _δ(O) does contain (nontrivial) effects that annihilate the vacuum.Footnote 8 Thus, particle detections can always be simulated by purely local measurements; and we can explain the appearance of macroscopically localized objects without assuming that there are localizable particles in the strict sense.

However, it may not be easy to pacify Segal and Barrett with a FAPP solution to the problem of localization. Both appear to think that the absence of localizable particles is not simply contrary to our manifest experience, but would undermine the very possibility of objective empirical science. For example, Segal claims that,

Furthermore, “particles would not be observable without their localization in space at a particular time” (1964, 139). In other words, experimentation involves observations of particles, and these observations can occur only if particles are localized in space. Unfortunately, Segal does not give any argument for these claims. It seems to us, however, that the moral we should draw from the no-go theorems is that Segal's account of observation is false. In particular, we do not observe particles; rather, there are ‘observation events’, and these observation events are consistent (to a good degree of accuracy) with the supposition that they are brought about by localizable particles.

Like Segal, Barrett (Reference Barrett2001) claims that we will have trouble explaining how empirical science can work if there are no localizable particles. In particular, Barrett claims that empirical science requires that we be able to keep an account of our measurement results so that we can compare these results with the predictions of our theories. Furthermore, we identify measurement records by means of their location in space. Thus, if there were no localized objects, then there would be no identifiable measurement records, and “. . . it would be difficult to account for the possibility of empirical science at all” (Barrett Reference Barrett2001, 3).

However, it's not clear what the difficulty here is supposed to be. On the one hand, we have seen that RQFT does predict that the appearances are consistent with the supposition that there are localized objects. So, for example, we could distinguish two record tokens at a given time if there were two disjoint regions O and O′ and particle detector observables C ∊ _δ(O) and C′ ∊ _δ(O′) (approximated by observables strictly localized in O and O′ respectively) such that 〈ψ, Cψ〉 ≈ 1 and 〈ψ, C′ψ〉 ≈ 1. Now, it may be that Barrett is also worried about how, given a field ontology, we could assign any sort of trans-temporal identity to our record tokens. But this problem, however important philosophically, is distinct from the problem of localization. Indeed, the problem of trans-temporal identification of particles also arises in the context of non-relativistic QFT, where there is no problem of localization. Finally, Barrett might object that once we supply a quantum-theoretical model of a particle detector itself, then the superposition principle will prevent the field and detector from getting into a state where there is a fact of the matter as to whether a particle has been detected in the region O. But this is simply a restatement of the standard quantum measurement problem that infects all quantum theories—and we have made no pretense of solving that here.

8. Conclusion

Malament claims that his theorem justifies the belief that,

In order to buttress Malament's argument for this claim, we provided two further results (Theorems 1 and 2) which show that the conclusion continues to hold for generic spacetimes, as well as for unsharp localization observables. We then went on to show that RQFT does not permit an ontology of localizable particles; and so, strictly speaking, our talk about localizable particles is a fiction. Nonetheless, RQFT does permit talk about particles—albeit, if we understand this talk as really being about the properties of, and interactions among, quantized fields. Indeed, modulo the standard quantum measurement problem, RQFT has no trouble explaining the appearance of macroscopically well-localized objects, and shows that our talk of particles, though a façon de parler, has a legitimate role to play in empirically testing the theory.