/home/www/ftp/data/quant-ph/dir_0103041/0103041.dvi


No place for particles in relativistic quantum
theories?

Hans Halvorson and Rob Clifton
Department of Philosophy, University of Pittsburgh

hphst1@pitt.edu, rclifton@pitt.edu

Abstract

Several recent arguments purport to show that there can be no relativistic,
quantum-mechanical theory of localizable particles and, thus, that relativity
and quantum mechanics can be reconciled only in the context of quantum
field theory. We point out some loopholes in the existing arguments, and we
provide two no-go theorems to close these loopholes. However, even with
these loopholes closed, it does not yet follow that relativity plus quantum me-
chanics exclusively requires a field ontology, since relativistic quantum field
theory itself might permit an ontology of localizable particles supervenient
on the fundamental fields. Thus, we provide another no-go theorem to rule
out this possibility. Finally, we allay potential worries about this conclusion
by arguing that relativistic quantum field theory can nevertheless explain the
possibility of “particle detections”, as well as the pragmatic utility of “parti-
cle talk.”

1 Introduction

It is a widespread belief, at least within the physics community, that there is no par-
ticle mechanics that is simultaneously relativistic and quantum-theoretic; and, thus,
that the only relativistic quantum theory is a field theory. This belief has received
much support in recent years in the form of rigorous “no-go theorems” by Mala-
ment (1996) and Hegerfeldt (1998a, 1998b). 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 super-
luminal spreading of the wavefunction must occur. Similarly, Malament shows the
inconsistency of a few intuitive desiderata for a relativistic, quantum-mechanical
theory of (localizable) particles. Thus, it appears that there is a fundamental con-
flict between the demands of relativistic causality and the requirements of a theory
of localizable particles.

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What is the philosophical lesson of this apparent conflict between relativistic
causality and localizability? One the one hand, if we believe that the assumptions
of Malament’s theorem must hold for any theory that is descriptive of our world,
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 the Malament’s assumptions must be false. Mala-
ment 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 Butter-
field 1999), or that we cannot judge the truth of the assumptions until we resolve
the interpretive issues of elementary quantum mechanics (cf. Barrett 2000).

Although we do not think that these arguments against Malament’s assump-
tions succeed, there are other reasons to doubt that Malament’s theorem is sufficient
to support a sound argument against the possibility of a relativistic quantum me-
chanics of localizable particles. First, Malament’s theorem depends on a specific
assumption about the structure of Minkowski spacetime—a “no preferred reference
frame” assumption—that could be seen as having less than full empirical warrant.
Second, 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, together, suffice to close these loopholes in the
argument against relativistic quantum mechanics. First, we present a strengthened
no-go theorem that subsumes the results of Malament and Hegerfeldt, and which
does not depend on the “no preferred frame” assumption (Theorem 1). Second,
we derive a generalized version of Malament’s 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 result show—or, are in-
tended to show—that a field ontology, rather than a particle ontology, is appro-
priate for relativistic quantum theories. While these results show that there are
no position observables that satisfy certain relativistic constraints, quantum field
theories—both relativistic and non-relativistic—already reject the notion of po-
sition observables in favor of “localized” field observables. Thus, no-go results
against relativistic position operators have nothing to say about the possibility that
relativistic quantum field theory 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 generic quantum theory to permit a particle interpretation, and we then show

2



that this condition fails in any relativistic theory (Theorem 3).
Since our world is presumably both relativistic and quantum-theoretic, these

results show that there are no localizable particles. However, in Section 7 we shall
argue that relativistic quantum field theory itself warrants an approximate use of
“particle talk” that is sufficient to save the phenomena.

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 1971) by showing that the assumption of “no superluminal
wavepacket spreading” can be replaced by the weaker assumption of “microcausal-
ity,” 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 space-
time M is an affine space, with a foliation S into spatial hyperplanes. (For ease,
we can think of an affine space as a vector space, so long as we do not assign any
physical significance to the origin.) This will permit us to consider a wide range
of relativistic (e.g., Minkowski) as well as non-relativistic (e.g., Galilean) space-
times. The pure states of our quantum-mechanical system are given by rays in
some Hilbert space H. We assume that there is a mapping ∆ 7→ E∆ of bounded
subsets of hyperplanes in M into projections on H. We think of E∆ as represent-
ing the proposition that the particle is localized in ∆; or, from a more operational
point of view, E∆ represents the proposition that a position measurement is certain
to find the particle within ∆. We also assume that there is a strongly continuous
representation a 7→ U(a) of the translation group of M in the unitary operators on
H. Here strong continuity means that for any unit vector ψ ∈ H, 〈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 condi-
tions hold, we say that the triple (H, ∆ 7→ E∆, a 7→ U(a)) is a localization system
over M.

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

Localizability: If ∆ and ∆′ are disjoint subsets of a single hyperplane, then
E∆E∆′ = 0.

Translation covariance: For any ∆ and for any translation a of M,
U(a)E∆U(a)∗ = E∆+a.

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Energy bounded below: For any timelike translation a of M, the generator H(a)
of the one-parameter group {U(ta) : t ∈ R} has a 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 partic-
ular, if we displace the particle by a spatial translation a, then the original wave-
function ψ will transform to some wavefunction ψa. Since the statistics for the
displaced detection experiment should be identical to the original statistics, we
have 〈ψ,E∆ψ〉 = 〈ψa,E∆+aψa〉. By Wigner’s theorem, however, the symme-
try is implemented by some unitary operator U(a). Thus, U(a)ψ = ψa, and
U(a)E∆U(a)∗ = E∆+a. In the case of time translations, the covariance con-
dition 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 free-falling observer, the particle has a lowest possi-
ble 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 Mala-
ment’s theorem. The special theory of relativity entails that there is a finite up-
per bound on the speed at which (detectable) physical disturbances can propagate
through space. Thus, if ∆ and ∆′ 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 ∆ to travel to ∆′. 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∆ψ〉 = 1 then 〈ψ,E∆′+taψ〉 = 0 whenever 0 ≤ t < �. This is equivalent to
the following assumption.

Strong causality: If ∆ and ∆′ are disjoint subsets of a single hyperplane, and if
the distance between ∆ and ∆′ is nonzero, then for any timelike translation
a, there is an � > 0 such that E∆E∆′+ta = 0 whenever 0 ≤ t < �.

(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”).

Microcausality: If ∆ and ∆′ are disjoint subsets of a single hyperplane, and if the
distance between ∆ and ∆′ is nonzero, then for any timelike translation a,
there is an � > 0 such that [E∆,E∆′+ta] = 0 whenever 0 ≤ t < �.

4



If E∆ can be measured within ∆, microcausality is equivalent to the assumption
that a measurement within ∆ cannot influence the statistics of measurements per-
formed in regions that are spacelike to ∆ (see Malament 1996, 5). Conversely, a
failure of microcausality would entail the possibility of act-outcome correlations at
spacelike separation. Note that both strong and weak causality 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.

Theorem (Malament). Let (H, ∆ 7→ E∆, a 7→ U(a)) be a localization system
over Minkowski spacetime that satisfies:

1. Localizability

2. Translation covariance

3. Energy bounded below

4. Microcausality

Then E∆ = 0 for all ∆.

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.

Several authors have claimed that Malament’s theorem is not sufficient to rule
out a relativistic quantum mechanics of localizable particles. In particular, these au-
thors argue that it is not reasonable to expect the conditions of Malament’s theorem
to hold for any relativistic, quantum-mechanical theory of particles. For example,
Dickson (1997) argues that a ‘quantum’ theory does not need a position opera-
tor (equivalently, a system of localizing projections) in order to treat position as
a physical quantity; Barrett (2000) argues that time-translation covariance is sus-
pect; and Fleming and Butterfield (1999) argue that the microcausality assumption
is not warranted by special relativity. We now show, however, that none of these
arguments is decisive against the assumptions of Malament’s theorem.

Dickson (1997, 214) cites the Bohmian interpretation of the Dirac equation as
a counterexample to the claim that any ‘quantum’ theory must represent position
by an operator. In order to see what Dickson might mean by this, recall that the
Dirac equation admits both positive and negative energy solutions. If H denotes
the Hilbert space of all (both positive and negative energy) solutions, then we may
define the ‘standard position operator’ Q by setting Qψ(x) = x · ψ(x) (Thaller
1992, 7). If, however, we restrict to the Hilbert space Hpos ⊂ H of positive en-
ergy solutions, then the probability density given by the Dirac wavefunction does

5



not correspond to a self-adjoint position operator (Thaller 1992, 32). According to
Holland (1993, 502), this lack of a position operator on Hpos precludes a Bohmian
interpretation of ψ(x) as a probability amplitude for finding the particle in an ele-
mentary volume d3x around x.

Since the Bohmian interpretation of the Dirac equation uses all states (both
positive and negative energy), and the corresponding position observable Q, it is
not clear what Dickson means by saying that the Bohmian interpretation of the
Dirac equation dispenses with a position observable. Moreover, since the energy
is not bounded below in H, this would not in any case give us a counterexample
to Malament’s theorem. However, Dickson could have developed his argument by
appealing to the positive energy subspace Hpos. In this case, we can talk about
positions despite the fact that we do not have a position observable in the usual
sense. In particular, we shall show in Section 5 that, for talk about positions, it
suffices to have a family of “unsharp” localization observables. (And, yet, we shall
show that relativistic quantum theories do not permit even this attenuated notion of
localization.)

Barrett (2000) argues that the significance of Malament’s theorem cannot be
assessed until we have solved the measurement problem:

If we might have to violate the apparently weak and obvious assump-
tions that go into proving Malament’s theorem in order to get a sat-
isfactory solution to the measurement problem, then all bets are off
concerning the applicability of the theorem to the detectible entities
that inhabit our world. (Barrett 2000, 16)

In particular, a solution to the measurement problem may require that we abandon
unitary dynamics. But if we abandon unitary dynamics, then the translation covari-
ance condition does not hold, and we need not accept the conclusion that there is
no relativistic quantum mechanics of (localizable) particles.

Unfortunately, it is not clear that we could avoid the upshot of Malament’s the-
orem by moving to a collapse theory. Existing (non-relativistic) collapse theories
take the empirical predictions of quantum theory seriously. That is, the “statistical
algorithm” of quantum mechanics is assumed to be at least approximately correct;
and collapse is introduced only to ensure that we obtain determinate properties
at the end of a measurement. However, in the present case, Malament’s theorem
shows that the statistical algorithm of any quantum theory predicts that if there are
local particle detections, then act-outcome correlations are possible at spacelike
separation. Thus, if a collapse theory is to stay close to these predictions, it too
would face a conflict between localizability and relativistic causality.

Perhaps, then, Barrett is suggesting that the price of accomodating localizable
particles might be a complete abandonment of unitary dynamics, even at the level

6



of a single particle. In other words, we may be forced to adopt a collapse theory
without having any underlying (unitary) quantum theory. But even if this is correct,
it wouldn’t count against Malament’s theorem, which was intended to show that
there is no relativistic quantum theory of localizable particles. Furthermore, noting
that Malament’s theorem requires unitary dynamics is one thing; it would be quite
another thing to provide a model in which there are localizable particles—at the
price of non-unitary dynamics—but which is also capable of reproducing the well-
confirmed quantum interference effects at the micro-level. Until we have such a
model, pinning our hopes for localizable particles on a failure of unitary dynamics
is little more than wishful thinking.

Like Barrett, Fleming (Fleming and Butterfield 1999, 158ff) disagrees with
the reasonableness of Malament’s assumptions. Unlike Barrett, however, Fleming
provides a concrete model in which there are localizable particles (viz., using the
Newton-Wigner position operator as a localizing observable) and in which Mala-
ment’s microcausality assumption fails. Nonetheless, Fleming argues that this fail-
ure of microcausality is perfectly consistent with relativistic causality.

According to Fleming, the property “localized in ∆” (represented by E∆) need
not be detectable within ∆. As a result, [E∆,E∆′] 6= 0 does not entail that it is
possible to send a signal from ∆ to ∆′. However, by claiming that local beables
need not be local observables, Fleming undercuts the primary utility of the notion
of localization, which is to indicate those physical quantities that are operationally
accessible in a given region of spacetime. Indeed, it is not clear what motivation
there could be—aside from indicating what is locally measurable—for assigning
observables to spatial regions. If E∆ is not measurable in ∆, then why should we
say that “E∆ is localized in ∆”? Why not say instead that “E∆ is localized in ∆′”
(where ∆′ 6= ∆)? Does either statement have any empirical consequences and,
if so, how do their empirical consequences differ? Until these questions are an-
swered, we maintain that local beables are always local observables; and a failure
of microcausality would entail the possibility of act-outcome correlations at space-
like separation. Therefore, the microcausality assumption is an essential feature of
any relativistic quantum theory with “localized” observables. (For a more detailed
argument along these lines, see Halvorson 2001, Section 6.)

Thus, the arguments against the four (explicit) assumptions of Malament’s the-
orem are unsuccessful; these assumptions are perfectly reasonable, and we should
expect them to hold for any relativistic, quantum-mechanical theory. However,
there is another difficulty with the argument against any relativistic quantum me-
chanics of (localizable) particles: Malament’s theorem makes tacit use of specific
features of Minkowski spacetime which—some might claim—have less than per-
fect empirical support. First, the following example shows that Malament’s theo-
rem fails if there is a preferred reference frame.

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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 ∈ ∆}, with t ∈ R, and ∆
a bounded open subset of R3, we assign the spectral projection E∆ of the position
operator for a particle in three dimensions. Let H(a) = 0 so that U(ta) = eit0 = I
for all t ∈ R. Since the energy in every state is zero, the energy condition is trivially
satisfied.

Note, however, that if the background spacetime is not regarded as having a
distinguished timelike direction, then this example violates the energy condition.
Indeed, the generator of an arbitrary timelike translation has the form

H(b) = b · P = b00 + b1P1 + b2P2 + b3P3 = b1P1 + b2P2 + b3P3, (1)

where b = (b0,b1,b2,b3) ∈ R4 is a timelike vector, and Pi are the three orthog-
onal components of the total momentum. But since each Pi has spectrum R, the
spectrum of H(b) is not bounded from below when b 6= a. 2

Malament’s theorem does not require the full structure of Minkowski spacetime
(e.g., the Lorentz group). Rather, it suffices to assume that the affine space M
satisfies the following condition.

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 all post-Galilean space-
times, there are some who claim that the existence of a (undetectable) preferred
reference frame is perfectly consistent with the empirical evidence on which rela-
tivistic theories are based (cf. Bell 1987, Chap. 9). What is more, the existence of
a preferred frame is an absolutely essential feature of a number of “realistic” inter-
pretations of quantum theory (cf. Maudlin 1994, Chap. 7). Thus, this tacit assump-
tion of Malament’s theorem has the potential to be a major source of contention
for those wishing to maintain that there can be a relativistic quantum mechanics of
localizable particles.

There is a further worry about the generality of Malament’s theorem: It is not
clear whether the result can be expected to hold for arbitrary relativistic space-
times, or whether it is an artifact of peculiar features of Minkowski spacetime
(e.g., that space is infinite). To see this, suppose that M is an arbitrary globally
hyperbolic manifold. (That is, M is a manifold that permits at least one foliation
S into spacelike hypersurfaces). Although M will not typically have a translation
group, we suppose 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 7→ U(g) in the

8



unitary operators on H; and, the translation covariance condition now says that
Eg(∆) = U(g)E∆U(g)

∗ for all g ∈ G.
The following example shows that Malament’s theorem fails even for the very

simple case where M is a two-dimensional cylinder.

Example 2. Let M = R ⊕ S1, where S1 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 S1 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 ∆ of Σ, let E∆ = I if µ(∆) ≥ 2/3, and let E∆ = 0 if
µ(∆) < 2/3. Then localizability holds, since for any pair (∆, ∆′) of disjoint open
subsets of Σ, either µ(∆) < 2/3 or µ(∆′) < 2/3. 2

Nonetheless, Examples 1 and 2 hardly serve as physically interesting coun-
terexamples to a strengthened version of 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. 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 strengthened version
of Malament’s theorem that applies to generic spacetimes.

3 Hegerfeldt’s Theorem

Hegerfeldt’s (1998a, 1998b) recent results on localization apply to arbitrary (glob-
ally hyperbolic) spacetimes, and they do not make us of the “no absolute velocity”
condition. Thus, we will suppose henceforth that M is a globally hyperbolic space-
time, and we will fix a foliation S of M, as well as a unique isomorphism between
any two hypersurfaces in this foliation. If Σ ∈ S, we will write Σ +t for the hyper-
surface that results from “moving Σ forward in time by t units”; and if ∆ is a subset
of Σ, we will use ∆+t to denote the corresponding subset of Σ+t. We assume that
there is a representation t 7→ Ut of the time-translation group R in the unitary op-
erators on H, and we will say that the localization system (H, ∆ 7→ E∆, t 7→ Ut)
satisfies time-translation covariance just in case UtE∆U−t = E∆+t for all ∆ and
all t ∈ R.

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

9



Lemma 1 (Hegerfeldt). Suppose that Ut = eitH, where H is a self-adjoint op-
erator with spectrum bounded from below. Let A be a positive operator (e.g., a
projection operator). Then for any state ψ, either

〈Utψ,AUtψ〉 6= 0 , for almost all t ∈ R,

or
〈Utψ,AUtψ〉 = 0 , for all t ∈ R .

Hegerfeldt claims that this lemma has the following consequence for localiza-
tion:

If there exist particle states which are strictly localized in some finite
region at t = 0 and later move towards infinity, then finite propagation
speed cannot hold for localization of particles. (Hegerfeldt 1998a,
243)

Hegerfeldt’s argument for this conclusion is as follows:

Now, if the particle or system is strictly localized in ∆ at t = 0 it
is, a fortiori, also strictly localized in any larger region ∆′ containing
∆. If the boundaries of ∆′ and ∆ have a finite distance and if finite
propagation speed holds then the probability to find the system in ∆′

must also be 1 for sufficiently small times, e.g. 0 ≤ t < �. But then
[Lemma 1], with A ≡ I−E∆′, states that the system stays in ∆′ for all
times. Now, we can make ∆′ smaller and let it approach ∆. Thus we
conclude that if a particle or system is at time t = 0 strictly localized
in a region ∆, then finite propagation speed implies that it stays in
∆ for all times and therefore prohibits motion to infinity. (Hegerfeldt
1998a, 242–243; notation adapted, but italics in original)

Let us attempt now to put this argument into a more precise form.
First, Hegerfeldt claims that the following is a consequence of “finite propaga-

tion speed”: If ∆ ⊆ ∆′, and if the boundaries of ∆ and ∆′ have a finite distance,
then a state initially localized in ∆ will continue to be localized in ∆′ for some
finite amount of time. We can capture this precisely by means of the following
condition.

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

10



(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” ∆, then it is localized in ∆. We can capture this assumption
in the following condition.

Monotonicity: If {∆n : n ∈ N} is a downward nested family of subsets of Σ such
that

⋂
n ∆n = ∆, then

∧
n E∆n = E∆.

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

Theorem (Hegerfeldt). Suppose that the localization system (H, ∆ 7→ E∆, t 7→
Ut) satisfies:

1. Monotonicity

2. Time-translation covariance

3. Energy bounded below

4. No instantaneous wavepacket spreading

Then UtE∆U−t = E∆ for all ∆ ⊂ Σ and all t ∈ R.
(For the proof of this theorem, see Appendix A.)

Thus, conditions 1–4 can be satisfied only if the particle has trivial dynamics. If
M is an affine space, and if we add “no absolute velocity” as a fifth condition in this
theorem, then we get the stronger conclusion that E∆ = 0 for all bounded ∆ (see
Lemma 2, appendix). Thus, there is an obvious similarity between Hegerfeldt’s
and Malament’s theorems. 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
H = L2(R), and let H = P 2/2m for some m > 0. Let ∆ 7→ EQ∆ denote
the spectral measure for Q. Fix some bounded subset ∆0 of R, and let E∆ =
E
Q
∆ ⊗ E

Q
∆0

(a projection operator on H ⊗ H) for all Borel subsets ∆ of R. Thus,
∆ 7→ E∆ is a (non-normalized) projection-valued measure. Let Ut = I⊗eitH, and
let E∆+t = UtE∆U−t for all t ∈ R. It is clear that monotonicity, time-translation

11



covariance, and energy bounded below hold. To see that strong causality holds, let
∆ and ∆′ be disjoint subsets of a single hyperplane Σ. Then,

E∆UtE∆′U−t = E
Q
∆E

Q
∆′ ⊗ E

Q
∆0
E
Q
∆0+t

= 0 ⊗ EQ∆E
Q
∆0+t

= 0 , (2)

for all t ∈ R. On the other hand, UtE∆U−t 6= E∆ for any nonempty ∆ and for
any t 6= 0. Thus, it follows from Hegerfeldt’s theorem that NIWS fails. 2

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 A Strengthened Hegerfeldt-Malament Theorem

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 physi-
cally interesting counterexample to a strengthened version of Hegerfeldt’s theorem.
In particular, if Σ is a fixed spatial hypersurface, and if {∆n : n ∈ N} is a covering
of Σ by bounded sets (i.e.,

⋃
n ∆n = Σ), then

∨
n E∆n = I ⊗E∆0 6= I ⊗I. Thus,

it is not certain that the particle will be detected somewhere or other in space. In
fact, if {∆n : n ∈ N} is a covering of Σ and {Πn : n ∈ N} is a covering of Σ + t,
then ∨

n∈N
E∆n = I ⊗ E∆0 6= I ⊗ E∆0+t =

∨
n∈N

EΠn. (3)

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
∨
n E∆n = I whenever {∆n :

n ∈ N} is a covering of Σ. This would be the case, for example, if the mapping
∆ 7→ E∆ (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.

Probability conservation: If {∆n : n ∈ N} is a covering of Σ, and {Πn : n ∈ N}
is a covering of Σ + t, then

∨
n E∆n =

∨
n EΠn .

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 {∆n : n ∈ N} and {Πn : n ∈ N} consist
of pairwise disjoint sets, then the localizability condition entails that

∨
n E∆n =∑

n E∆n and
∨
n EΠn =

∑
n EΠn . In this case, probability conservation is equiv-

alent to ∑
n∈N

Probψ(E∆n ) =
∑
n∈N

Probψ(EΠn ) , (4)

12



for any state ψ. Note, finally, that probability conservation is neutral with respect
to relativistic and non-relativistic models.1

Theorem 1 (Strengthened Hegerfeldt-Malament Theorem). Suppose that the lo-
calization system (H, ∆ 7→ E∆, t 7→ Ut) satisfies:

1. Localizability

2. Probability conservation

3. Time-translation covariance

4. Energy bounded below

5. Microcausality

Then UtE∆U−t = E∆ for all ∆ and all t ∈ R.
(For the proof of this theorem, see Appendix A.)

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∆ = 0 for all ∆ (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 each condition in Theorem 1 is indispens-
able; that is, no four of the conditions suffices to entail the conclusion. (Example 1
shows that conditions 1–5 can be simultaneously satisfied.) Suppose for simplicity
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 L2(R), and
let H = P 2/2m. Let Σ be a spatial hypersurface in M, and suppose that a coordi-
natization of Σ has been fixed, so that there is a natural association between each
bounded open subset ∆ of Σ and a corresponding spectral projection E∆ of Q.

(1+2+3+4) (a) Consider the standard localization system for a single non-relativistic
particle. That is, let Σ be a fixed spatial hyperplane, and let ∆ 7→ E∆ (with
domain the Borel subsets of Σ) be the spectral measure for Q. For Σ + t,
set E∆+t = UtE∆U−t, where Ut = eitH . (b) The Newton-Wigner approach
to relativistic QM uses the standard localization system for a non-relativistic

1Probability conservation would fail if a particle could escape to infinity in a finite amount of time
(cf. Earman 1986, 33). However, a particle can escape to infinity only if there is an infinite potential
well, and this would violate the energy condition. Thus, given the energy condition, probability
conservation should also hold for non-relativistic particle theories.

13



particle, only replacing the non-relativistic Hamiltonian P 2/2m with the rel-
ativistic Hamiltonian (P 2 + m2I)1/2, whose spectrum is also bounded from
below.

(1+2+3+5) (a) For a mathematically simple (but physically uninteresting) exam-
ple, take the first example above and replace the Hamiltonian P 2/2m with
P . In this case, microcausality trivially holds, since UtE∆U−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.

(1+2+4+5) Consider the the standard localization system for a non-relativistic par-
ticle, but set E∆+t = E∆ 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.

(1+3+4+5) (a) Let ∆0 be some bounded open subset of Σ, and let E∆0 be the
corresponding spectral projection of Q. When ∆ 6= ∆0, let E∆ = 0. Let
Ut = eitH, and let E∆+t = UtE∆U−t for all ∆. This example is physically
uninteresting, since the particle cannot be localized in any region besides
∆0, including proper supersets of ∆0. (b) See Example 3.

(2+3+4+5) Let ∆0 be some bounded open subset of Σ, and let E∆0 be the cor-
responding spectral projection of Q. When ∆ 6= ∆0, let E∆ = I. Let
Ut = eitH, and let E∆+t = UtE∆U−t for all ∆. Thus, the particle is always
localized in every region other than ∆0, and is sometimes localized in ∆0 as
well.

5 Are there Unsharply Localizable Particles?

We have argued that attempts to undermine the four explicit assumptions of Mala-
ment’s theorem are unsuccessful. We have also now shown that the tacit assump-
tion of “no absolute velocity” is not necessary to derive Malament’s conclusion.
And, yet, there is one more loophole in the argument against a relativistic quantum
mechanics of localizable particles. In particular, the basic assumption of a family
{E∆} of localizing projections is unnecessary; it is possible to have a quantum-
mechanical particle theory in the absence of localizing projections. What is more,
one might object to the use of localizing projections on the grounds that they repre-
sent 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

14



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” localizabil-
ity.

To see how we can define “particle talk” without having projection operators,
consider the relativistic theory of a single spin-1/2 electron (where we now restrict
to the subspace Hpos of positive energy solutions of the Dirac equation). In order
to treat the ‘x’ of the Dirac wavefunction as an observable, we need only 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 ∆ of Σ, we set

Probψ(x ∈ ∆) =
∫

∆
|ψ(x)|2dx . (5)

Now let ∆ 7→ E∆ be the spectral measure for the standard position operator on
the Hilbert space H (which includes both positive and negative energy solutions).
That is, E∆ multiplies a wavefunction by the characteristic function of ∆. Let F
denote the orthogonal projection of H onto Hpos. Then,∫

∆
|ψ(x)|2dx = 〈ψ,E∆ψ〉 = 〈ψ,FE∆ψ〉, (6)

for any ψ ∈ Hpos. Thus, we can apply the standard recipe to the operator FE∆
(defined on Hpos) to compute the probability that the particle will be found within
∆. However, FE∆ does not define a projection operator on Hpos. (In fact, it can
be shown that FE∆ 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 ∆, we
need only assume that there is an operator A∆ such that 〈ψ,A∆ψ〉 ∈ [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 (H, ∆ 7→ A∆, a 7→ U(a))
is an unsharp localization system over M just in case ∆ 7→ A∆ is a mapping
from subsets of hyperplanes in M to effects on H, and a 7→ U(a) is a continuous
representation of the translation group of M in unitary operators on H. (We assume
for the present that M is again 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

15



of E and vice versa (cf. Malament 1996, 5). This fact, along with the assumption
that E∆ is measurable in ∆, motivates the microcausality assumption. For the case
of an association of arbitrary effects with spatial regions, Busch (1999, Proposition
2) has shown that [A∆,A∆′] = 0 just in case for any state, the statistics for a mea-
surement of A∆ are not affected by a non-selective measurement of A∆′ 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 ∆ and ∆′ are disjoint subsets of a single hyperplane, then 〈ψ,E∆ψ〉 = 1
entails that 〈ψ,E∆′ψ〉 = 0. It is not difficult to see that this last condition is
equivalent to the assumption that E∆ + E∆′ ≤ I. That is,

〈ψ, (E∆ + E∆′)ψ〉 ≤ 〈ψ,Iψ〉 , (7)
for any state ψ. Now, it is an accidental feature of projection operators (as opposed
to arbitrary effects) that E∆ + E∆′ ≤ I is equivalent to E∆E∆′ = 0. Thus, the
apropriate generalization of localizability to unsharp localization systems is the
following condition.

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

That is, the probability for finding the particle in ∆, plus the probability for finding
the particle in some disjoint region ∆′, 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 ∆ plus the probability of finding a particle in ∆′ equals the
probability of finding a particle in ∆ ∪ ∆′. If this is true for all states ψ, we have:
Additivity: If ∆ and ∆′ are disjoint subsets of a single hyperplane, then

A∆ + A∆′ = A∆∪∆′.

With just these mild constraints, Busch (1999) was able to derive the following
no-go result.

Theorem (Busch). Suppose that the unsharp localization system (H, ∆ 7→ A∆, a 7→
U(a)) satisfies localizability, translation covariance, energy bounded below, mi-
crocausality, and no absolute velocity. Then, for all ∆, A∆ has no eigenvector
with eigenvalue 1.

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

16



Theorem 2. Suppose that the unsharp localization system (H, ∆ 7→ A∆, a 7→
U(a)) satisfies:

1. Additivity

2. Translation covariance

3. Energy bounded below

4. Microcausality

5. No absolute velocity

Then A∆ = 0 for all ∆.

(For the proof of this theorem, see Appendix B.)
Theorem 2 shows that invoking the notion of unsharp localization does nothing

to resolve the tension between relativistic causality and localizability. For exam-
ple, we can now show that the (positive energy) Dirac theory—in which there are
localizable particles—violates relativistic causality. Indeed, it is clear that the con-
clusion of Theorem 2 fails.2 On the other hand, additivity, translation covariance,
energy bounded below, and no absolute velocity hold. Thus, microcausality fails,
and the (positive energy) Dirac theory permits superluminal signalling.

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 7→ U(g) be a positive
energy representation of G in the unitary operators on a Hilbert space H. For
any Σ ∈ S, let µ denote the normalized rotation-invariant measure on Σ, and let
A∆ = µ(∆)I. Then, conditions 1–5 of Theorem 2 are satisfied, but the conclusion
of the theorem is false. 2

The previous counterexample can be excluded if we require there to be a fixed
positive constant δ such that, for each ∆, there is a state ψ with 〈ψ,A∆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 Appendix B.)
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., UtA∆U−t =
A∆ for all ∆. We conjecture that every counterexample to a generalized version of
Theorem 2 will have trivial dynamics.

2For any unit vector ψ ∈ Hpos, there is a bounded set ∆ such that
∫
∆
|ψ|2dx 6= 0. Thus,

A∆ 6= 0.

17



Theorem 2 strongly supports the conclusion that there is no relativistic quan-
tum mechanics of a single (localizable) particle; and that the only consistent com-
bination of special relativity and quantum mechanics is in the context of quantum
field theory. However, neither Theorem 1 nor Theorem 2 says anything about the
ontology of relativistic quantum field theory itself; they leave open the possibility
that relativistic quantum field theory might permit an ontology of localizable par-
ticles. 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∆ to represent the proposition that
a measurement is certain to find that all n particles lie within ∆, in which case
localizability should hold. Note, however, that when we alter the interpretation of
the localization operators {E∆}, 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 po-
sition operators that obey appropriate relativistic constraints. But it does not im-
mediately follow from this that we lack any notion of localization in relativistic
quantum theories. Indeed,

...a position operator is inconsistent with relativity. This compels us to
find another way of modeling localization of events. In field theory, we
model localization by making the observables dependent on position
in spacetime. (Ticiatti 1999, 11)

However, it is not a peculiar feature of relativistic quantum field theory that it lacks
a position operator: Any quantum field theory (either relativistic or non-relativistic)
will model localization by making the observables dependent on position in space-
time. Moreover, in the case of non-relativistic QFT, these “localized” observables

18



suffice to provide us with a concept of localizable particles. In particular, for each
spatial region ∆, there is a “number operator” N∆ whose eigenvalues give the
number of particles within the region ∆. 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 H is any operator with eigenvalues con-
tained in {0, 1, 2, . . .}. In order to describe the number of particles locally, we
require an association ∆ 7→ N∆ of subsets of spatial hyperplanes in M to number
operators on H, where N∆ represents the number of particles in the spatial region
∆. If a 7→ U(a) is a unitary representation of the translation group, we say that
the triple (H, ∆ 7→ N∆, a 7→ U(a)) is a system of local number operators over
M. Note that a localization system (H, ∆ 7→ E∆, a 7→ U(a)) is a special case of a
system of local number operators where the eigenvalues of each N∆ 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:3

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

Number conservation: If {∆n : n ∈ N} is a disjoint covering of Σ, then the
sum

∑
n N∆n converges to a densely defined, self-adjoint operator N on

H (independent of the chosen covering), and U(a)NU(a)∗ = N for any
timelike translation a of M.

Additivity asserts that, when ∆ and ∆′ are disjoint, the expectation value (in any
state ψ) for the number of particles in ∆ ∪ ∆′ is the sum of the expectations for
the number of particles in ∆ and the number of particles in ∆′. In the pure case, it
asserts that the number of particles in ∆ ∪ ∆′ is the sum of the number of particles

3Due to the unboundedness of number operators, we would need to take some care in giving
technically correct versions of the following conditions. In particular, the additivity condition should
technically include the clause that N∆ and N∆′ have a common dense domain, and the operator
N∆∪∆′ should be thought of as the self-adjoint closure of N∆ + N∆′ . In the number conservation
condition, the sum N =

∑
n
N∆n can be made rigorous by exploiting the correspondence between

self-adjoint operators and “quadratic forms” on H. In particular, we can think of N as deriving from
the upper bound of quadratic forms corresponding to finite sums.

19



in ∆ and the number of particles in ∆′. The “number conservation” condition tells
us that there is a well-defined total number of particles (at a given time), and that
the total number of particles does not change 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. Red-
head 1995). We will now derive the same conclusion from strictly weaker assump-
tions. In particular, we show that microcausality is the only specifically relativistic
assumption needed for this result. The relativistic spectrum condition—which re-
quires 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.4

Theorem 3. Suppose that the system (H, ∆ 7→ N∆, a 7→ U(a)) of local number
operators satisfies:

1. Additivity

2. Translation covariance

3. Energy bounded below

4. Number conservation

5. Microcausality

6. No absolute velocity

Then N∆ = 0 for all ∆.

(For the proof of the theorem, see Appendix C.)
Thus, in every state, there are no particles in any local region. This serves

as a reductio ad absurdum for any notion of localizable particles in a relativistic
quantum theory.

Unfortunately, Theorem 3 is not the strongest result we could hope for, since
“number conservation” can only be expected to hold 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 theo-
rem; cf. Streater and Wightman, 2000, 163) their dynamics are not unitarily imple-
mentable on a fixed Hilbert space. On the other hand it would be wrong to think of

4Microcausality is not only sufficient, but also necessary for the proof that there are no local
number operators. The Reeh-Schlieder theorem entails the cyclicity of the vacuum state. But the
cyclicity of the vacuum state alone does not entail that there are no local number operators; we must
also assume microcausality (cf. Halvorson 2001, Requardt 1982).

20



this as indicating a limitation on the generality of our conclusion: Haag’s theorem
also entails that interacting models of RQFT have no number operators—either
global or local.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, insofar as we can expect any rel-
ativistic quantum theory theory to satisfy a few basic conditions, these theories
do not admit (localizable) particles into their ontology. We also considered and
rejected several arguments which attempt to show that one (or more) of these con-
ditions can be jettisoned without doing violence to the theory of relativity or to
quantum mechanics. Thus, we have yet to find a good reason to reject one of the
premises on which our argument against localizable particles is based. However,
Segal (1964) and Barrett (2000) claim that we have independent grounds for re-
jecting the conclusion; that is, we have good reasons for believing that there are
localizable particles.

The argument for localizable particles appears to be very simple: Our experi-
ence 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 ob-
ject 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, relativistic quantum field theory itself shows how the “illusion” of local-
izable 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 rela-
tivistic quantum field theory, we shall now pursue the investigation in the frame-
work of algebraic quantum field theory.6 Here, one assumes that there is a cor-
respondence O 7→ R(O) between bounded open subsets of M and subalgebras
of observables on some Hilbert space H. Observables in R(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 ∈ R(O) and B ∈ R(O′). Furthermore,

5If a total number operator exists in a representation of the canonical commutation relations, then
that representation is quasiequivalent to a free-field (Fock) representation (Chaiken 1968). However,
Haag’s theorem entails that in relativistic theories, representations with nontrivial interactions are not
quasiequivalent to a free-field representation.

6For general information on algebraic quantum field theory, see (Haag 1992) and (Buchholz
2000). For specific information on particle detectors and “almost local” observables, see Chapter 6
of (Haag 1992) and Section 4 of (Buchholz 2000).

21



we assume that there is a continuous representation a 7→ U(a) of the translation
group of M in unitary operators on H, and that there is a unique “vacuum” state
Ω ∈ H 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, we again see that (strictly speaking) 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.)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 mea-
surement procedure we have performed. Thus, practically speaking, we can at best
determine a neighborhood of observables corresponding to a concrete measure-
ment procedure.

In the case of present interest, what we actually measure is always a local
observable—i.e., an element of R(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′ ∈ R(O), then a measurement of C′ will be practically indistinguish-
able from a measurement of C. Thus, if we let

Rδ(O) = {C : ∃C′ ∈ R(O) such that ‖C − C′‖ < δ}, (8)
denote the family of observables “almost localized” in O, then ‘FAPP’ (i.e., ‘for
all practical purposes’) we can locally measure any observable from Rδ(O). That
is, measurement of an element from Rδ(O) can be simulated to a high degree of
accuracy by local measurement of an element from R(O). However, for any local
region O, and for any δ > 0, Rδ(O) does contain (nontrivial) effects that anni-
hilate the vacuum.8 Thus, particle detections can always be simulated by purely
local measurements; and the appearance of (fairly-well) localized objects can be

7Recall that ‖C − C′‖ is defined as the supremum of ‖(C − C′)ψ‖ as ψ runs through the
unit vectors in H. It follows, then, from the Cauchy-Schwarz inequality that |〈ψ, (C − C′)ψ〉| ≤
‖C − C′‖ for any unit vector ψ.

8Suppose that A ∈ R(O), and let A(x) = U(x)AU(x)∗. If f is a test function on M whose
Fourier transform is supported in the complement of the forward light cone, then L =

∫
f(x)A(x)dx

is almost localized in O and 〈Ω,LΩ〉 = 0 (cf. Buchholz 2000, 7).

22



explained without the supposition 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 (in the strict sense) is not simply contrary to our manifest experience, but
would undermine the very possiblity of objective empirical science. For example,
Segal claims that,

...it is an elementary fact, without which experimentation of the usual
sort would not be possible, that particles are indeed localized in space
at a given time. (Segal 1965, 145; our italics)

Furthermore, “particles would not be observable without their localization in space
at a particular time” (1964, 139). In other words, experimentation involves obser-
vations 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, it is not (strictly speaking)
true that we observe particles. Rather, there are ‘observation events’, and these ob-
servation events are consistent (to a good degree of accuracy) with the supposition
that they are brought about by (localizable) particles.

Like Segal, Barrett (2000) claims that we will have trouble explaining how em-
pirical 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 lo-
cation 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 2000, 3).

However, it’s not clear what the difficulty here is supposed to be. On the
one hand, we have seen that relativistic quantum field theory does predict that
the appearances will be FAPP consistent with the supposition that there are local-
ized 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 ∈ Rδ(O) and C′ ∈ Rδ(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, it
also arises in the context of non-relativistic quantum field theory, where there is
no problem with describing localizable particles. Finally, Barrett might object that

23



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 measure-
ment 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 the attempt to reconcile quantum mechanics with relativity the-
ory...one is driven to a field theory; all talk about “particles” has to be
understood, at least in principle, as talk about the properties of, and
interactions among, quantized fields. (Malament 1996, 1)

We have argued that the first claim is correct—quantum mechanics and relativity
can be reconciled only in the context of quantum field theory. In order, however, to
close a couple of loopholes in Malament’s argument for this conclusion, we pro-
vided two further results (Theorems 1 and 2) which show that the conclusion con-
tinues to hold for generic spacetimes, as well as for “unsharp” localization observ-
ables. We then went on to show that relativistic quantum field theory also does not
permit an ontology of localizable particles; and so, strictly speaking, our talk about
localizable particles is a fiction. Nonetheless, relativistic quantum field theory 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, relativistic quantum field theory 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.

Acknowledgments: We would like to thank Jeff Barrett and David Malament
for helpful correspondence.

A Appendix

Theorem (Hegerfeldt). Suppose that the localization system (H, ∆ 7→ E∆, t 7→
Ut) satisfies monotonicity, time-translation covariance, energy bounded below, and
NIWS. Then UtE∆U−t = E∆ for all ∆ ⊂ Σ and all t ∈ R.

24



Proof. The formal proof corresponds directly to Hegerfeldt’s informal proof. Thus,
let ∆ be a subset of some spatial hypersurface Σ. If E∆ = 0 then obviously
UtE∆U−t = E∆ for all t ∈ R. So, suppose that E∆ 6= 0, and let ψ be a unit
vector such that E∆ψ = ψ. Since Σ is a manifold, and since ∆ 6= Σ, there is
a family {∆n : n ∈ N} of subsets of Σ such that, for each n ∈ N, the distance
between the boundaries of ∆n and ∆ is nonzero, and such that

⋂
n ∆n = ∆. Fix

n ∈ N. By NIWS and time-translation covariance, there is an �n > 0 such that
E∆nUtψ = Utψ whenever 0 ≤ t < �n. That is, 〈Utψ,E∆nUtψ〉 = 1 whenever
0 ≤ t < �n. Since energy is bounded from below, we may apply Lemma 1 with
A = I − E∆n to conclude that 〈Utψ,E∆nUtψ〉 = 1 for all t ∈ R. That is,
E∆nUtψ = Utψ for all t ∈ R. Since this holds for all n ∈ N, and since (by
monotonicity) E∆ =

∧
n E∆n , it follows that E∆Utψ = Utψ for all t ∈ R. Thus,

UtE∆U−t = E∆ for all t ∈ R.
Lemma 2. Suppose that the localization system (H, ∆ 7→ E∆, a 7→ U(a)) satis-
fies localizability, time-translation covariance, and no absolute velocity. Let ∆ be
a bounded spatial set. If U(a)E∆U(a)∗ = E∆ for all timelike translations a of
M, then E∆ = 0.

Proof. By no absolute velocity, there is a pair (a, b) of timelike translations such
that ∆ + (a − b) is in Σ and is disjoint from ∆. By time-translation covariance,
we have,

E∆+(a−b) = U(a)U(b)
∗E∆U(b)U(a)∗ = E∆. (9)

Thus, localizability entails that E∆ is orthogonal to itself, and so E∆ = 0.

Lemma 3. Let {∆n : n = 0, 1, 2, . . .} be a covering of Σ, and let E =
∨∞
n=0 E∆n .

If probability conservation and time-translation covariance hold, then UtEU−t =
E for all t ∈ R.
Proof. Since {∆n + t : n ∈ N} is a covering of Σ + t, probability conservation
entails that

∨
n E∆n+t = E. Thus,

UtEU−t = Ut
[ ∞∨
n=0

E∆n

]
U−t =

∞∨
n=0

[
UtE∆nU−t

]
(10)

=
∞∨
n=0

E∆n+t = E, (11)

where the third equality follows from time-translation covariance.

In order to prove the next result, we will need to invoke the following lemma
from Borchers (1967).

25



Lemma (Borchers). Let Ut = eitH, where H is a self-adjoint operator with spec-
trum bounded from below. Let E and F be projection operators such that EF = 0.
If there is an � > 0 such that

[E,UtFU−t] = 0 , 0 ≤ t < �,

then EUtFU−t = 0 for all t ∈ R.
Lemma 4. Let Ut = eitH, where H is a self-adjoint operator with spectrum
bounded from below. Let {En : n = 0, 1, 2, . . .} be a family of projection op-
erators such that E0En = 0 for all n ≥ 1, and let E =

∨∞
n=0 En. If UtEU−t = E

for all t ∈ R, and if for each n ≥ 1 there is an �n > 0 such that

[E0,UtEnU−t] = 0, 0 ≤ t < �n, (12)

then UtE0U−t = E0 for all t ∈ R.
Proof. If E0 = 0 then the conclusion obviously holds. Suppose then that E0 6= 0,
and let ψ be a unit vector in the range of E0. Fix n ≥ 1. Using (12) and Borchers’
lemma, it follows that E0UtEnU−t = 0 for all t ∈ R. Then,

‖EnU−tψ‖2 = 〈U−tψ,EnU−tψ〉 = 〈ψ,UtEnU−tψ〉 (13)
= 〈E0ψ,UtEnU−tψ〉 = 〈ψ,E0UtEnU−tψ〉 = 0 , (14)

for all t ∈ R. Thus, EnU−tψ = 0 for all n ≥ 1, and consequently, [
∨
n≥1 En]U−tψ =

0. Since E0 = E − [
∨
n≥1 En], and since (by assumption) EU−t = U−tE, it fol-

lows that
E0U−tψ = EU−tψ = U−tEψ = U−tψ , (15)

for all t ∈ R.

Theorem 1. Suppose that the localization system (H, ∆ 7→ E∆, t 7→ Ut) sat-
isfies localizability, probability conservation, time-translation covariance, energy
bounded below, and microcausality. Then UtE∆U−t = E∆ for all ∆ and all t ∈ R.
Proof. Let ∆ be an open subset of Σ. If ∆ = Σ then probability conservation and
time-translation covariance entail that E∆ = E∆+t = UtE∆U−t for all t ∈ R. If
∆ 6= Σ then, since Σ is a manifold, there is a covering {∆n : n ∈ N} of Σ\∆
such that the distance between ∆n and ∆ is nonzero for all n. Let E0 = E∆, and
let En = E∆n for n ≥ 1. Then 1 entails that E0En = 0 when n ≥ 1. If we
let E =

∨∞
n=0 En then probability conservation entails that UtEU−t = E for all

26



t ∈ R (see Lemma 3). By time-translation covariance and microcausality, for each
n ≥ 1 there is an �n > 0 such that

[E0,UtEnU−t] = 0, 0 ≤ t < �n. (16)
Since the energy is bounded from below, Lemma 4 entails that UtE0U−t = E0 for
all t ∈ R. That is, UtE∆U−t = E∆ for all t ∈ R.

B Appendix

Theorem 2. Suppose that the unsharp localization system (H, ∆ 7→ A∆, a 7→
U(a)) satisfies additivity, translation covariance, energy bounded below, micro-
causality, and no absolute velocity. Then A∆ = 0 for all ∆.

Proof. We prove by induction that ‖A∆‖ ≤ (2/3)m, for each m ∈ N, and for each
bounded ∆. For this, let F∆ denote the spectral measure for A∆.

(Base case: m = 1) Let E∆ = F∆(2/3, 1). We verify that (H, ∆ 7→ E∆, a 7→
U(a)) satisfies the conditions of Malament’s theorem. Clearly, no absolute ve-
locity and energy bounded below hold. Moreover, since unitary transformations
preserve spectral decompositions, translation covariance holds; and since spectral
projections of compatible operators are also compatible, microcausality holds. To
see that localizability holds, let ∆ and ∆′ be disjoint bounded subsets of a single
hyperplane. Then microcausality entails that [A∆,A∆′] = 0, and therefore E∆E∆′
is a projection operator. Suppose for reductio ad absurdum that ψ is a unit vector in
the range of E∆E∆′. By additivity, A∆∪∆′ = A∆ + A∆′ , and we therefore obtain
the contradiction:

1 ≥ 〈ψ,A∆∪∆′ψ〉 = 〈ψ,A∆ψ〉 + 〈ψ,A∆′ψ〉 ≥ 2/3 + 2/3 . (17)
Thus, E∆E∆′ = 0, and Malament’s theorem entails that E∆ = 0 for all ∆. There-
fore, A∆ = A∆F∆(0, 2/3) has spectrum lying in [0, 2/3], and ‖A∆‖ ≤ 2/3 for all
bounded ∆.

(Inductive step) Suppose that ‖A∆‖ ≤ (2/3)m−1 for all bounded ∆. Let
E∆ = F∆((2/3)m, (2/3)m−1). In order to see that Malament’s theorem applies
to (H, ∆ 7→ E∆, a 7→ U(a)), we need only check that localizability holds. For
this, suppose that ∆ and ∆′ are disjoint subsets of a single hyperplane. By micro-
causality, [A∆,A∆′] = 0, and therefore E∆E∆′ is a projection operator. Suppose
for reductio ad absurdum that ψ is a unit vector in the range of E∆E∆′ . Since
∆ ∪ ∆′ is bounded, the induction hypothesis entails that ‖A∆∪∆′‖ ≤ (2/3)m−1.
By additivity, A∆∪∆′ = A∆ + A∆′, and therefore we obtain the contradiction:

(2/3)m−1 ≥ 〈ψ,A∆∪∆′ψ〉 = 〈ψ,A∆ψ〉+〈ψ,A∆′ψ〉 ≥ (2/3)m+(2/3)m . (18)

27



Thus, E∆E∆′ = 0, and Malament’s theorem entails that E∆ = 0 for all ∆. There-
fore, ‖A∆‖ ≤ (2/3)m for all bounded ∆.

C Appendix

Theorem 3. Suppose that the system (H, ∆ 7→ N∆, a 7→ U(a)) of local number
operators satisfies additivity, translation covariance, energy bounded below, num-
ber conservation, microcausality, and no absolute velocity. Then, N∆ = 0 for all
bounded ∆.

Proof. Let N be the unique total number operator obtained from taking the sum∑
n N∆n where {∆n : n ∈ N} is a disjoint covering of Σ. Note that for any

∆ ⊆ Σ, we can choose a covering containing ∆, and hence, N = N∆ + A,
where A is a positive operator. By microcausality, [N∆,A] = 0, and therefore
[N∆,N] = [N∆,N∆ + A] = 0. Furthermore, for any vector ψ in the domain of
N, 〈ψ,N∆ψ〉 ≤ 〈ψ,Nψ〉.

Let E be the spectral measure for N, and let En = E(0,n). Then, NEn is a
bounded operator with norm at most n. Since [En,N∆] = 0, it follows that

〈ψ,N∆Enψ〉 = 〈Enψ,N∆Enψ〉 ≤ 〈Enψ,NEnψ〉 ≤ n, (19)

for any unit vector ψ. Thus, ‖N∆En‖ ≤ n. Since
⋃∞
n=1 En(H) is dense in H, and

since En(H) is in the domain of N∆ (for all n), it follows that if N∆En = 0, for
all n, then N∆ = 0. We now concentrate on proving the antecedent.

For each ∆, let A∆ = (1/n)N∆En. We show that the structure (H, ∆ 7→
A∆, a 7→ U(a)) satisfies the conditions of Theorem 2. Clearly, energy bounded
below and no absolute velocity hold. It is also straightforward to verify that addi-
tivity and microcausality hold. To check translation covariance, we compute:

U(a)A∆U(a)
∗ = U(a)N∆EnU(a)∗ = U(a)N∆U(a)∗U(a)EnU(a)∗ (20)

= U(a)N∆U(a)
∗En = N∆+aEn = A∆+a. (21)

The third equality follows from number conservation, and the fourth equality fol-
lows from translation covariance. Thus, N∆En = A∆ = 0 for all ∆. Since this
holds for all n ∈ N, N∆ = 0 for all ∆.

28



REFERENCES

Barrett, Jeffrey A. (2000), “On the nature of measurement records in relativistic
quantum field theory”, manuscript.

Bell, John S. (1987), Speakable and Unspeakable in Quantum Mechanics. New
York: Cambridge University Press.

Borchers, H.-J. (1967), “A remark on a theorem of B. Misra”, Communications
in Mathematical Physics 4: 315–323.

Buchholz, Detlev (2000), “Algebraic quantum field theory: A status report”,
math-ph/0011044.

Busch, Paul (1999), “Unsharp localization and causality in relativistic quantum
theory”, Journal of Physics A 32: 6535–6546.

Chaiken, Jan M. (1968), “Number operators for representations of the canon-
ical commutation relations”, Communications in Mathematical Physics 8: 164–
184.

Dickson, W. Michael (1998), Quantum Chance and Nonlocality. New York:
Cambridge University Press.

Earman, John (1986), A Primer on Determinism. Boston: D. Reidel.
Fleming, Gordon, and Jeremy Butterfield (1999), “Strange positions”, in J.

Butterfield and C. Pagonis (eds.), From Physics to Philosophy. NY: Cambridge
University Press, 108–165.

Haag, Rudolf (1992), Local Quantum Physics. New York: Springer.
Halvorson, Hans (2001), “Reeh-Schlieder defeats Newton-Wigner: On alterna-

tive localization schemes in relativistic quantum field theory”, Philosophy of Sci-
ence, forthcoming.

Hegerfeldt, Gerhard C. (1998a), “Causality, particle localization and positivity
of the energy”, in A. Böhm, et al. (eds.), Irreversibility and Causality. New York:
Springer, 238–245.

Hegerfeldt, Gerhard C. (1998b), “Instantaneous spreading and Einstein causal-
ity in quantum theory”, Annalen der Physik 7: 716–725.

Holland, Peter R. (1993), The Quantum Theory of Motion. New York: Cam-
bridge University Press.

Malament, David (1996), “In defense of dogma: Why there cannot be a rel-
ativistic quantum mechanics of (localizable) particles”, in Rob Clifton (ed.), Per-
spectives on Quantum Reality. Dordrecht: Kluwer, 1–10.

Maudlin, Tim (1994), Quantum Non-Locality and Relativity. Cambridge: Black-
well.

Redhead, Michael (1995), “The vacuum in relativistic quantum field theory”,
in David Hull, Micky Forbes, and Richard M. Burian (eds.), PSA 1994, v. 2. East
Lansing, MI: Philosophy of Science Association, 77–87.

29



Requardt, Manfred (1982), “Spectrum condition, analyticity, Reeh-Schlieder
and cluster properties in non-relativistic Galilei-invariant quantum theory”, Journal
of Physics A 15: 3715–3723.

Schlieder, S. (1971), “Zum kausalen Verhalten eines relativistischen quanten-
mechanischen System”, in S.P. Dürr (ed.), Quanten und Felder. Braunschweig:
Vieweg, 145–160.

Segal, Irving E. (1964), “Quantum fields and analysis in the solution mani-
folds of differential equations”, in William T. Martin and Irving E. Segal, (eds.),
Proceedings of a Conference on the Theory and Applications of Analysis in Func-
tion Space. Cambridge: MIT Press, 129–153.

Streater, Raymond F. and Arthur S. Wightman (2000), PCT, Spin and Statistics,
and All That. Princeton: Princeton University Press.

Thaller, Bernd (1992), The Dirac Equation. New York: Springer.
Ticiatti, Robin (1999), Quantum Field Theory for Mathematicians. New York:

Cambridge University Press.

30