The Clash of Causality and Localizability

The Consequences of the Reeh-Schlieder Theorem

The Reeh-Schlieder theorem¹¹ (1961) is a central analyticity result from algebraic QFT (AQFT), the axiomatic reformulation of QFT which was introduced in chapter 3. From a physical point of view the Reeh-Schlieder theorem is based on vacuum correlations. Although the theorem stems from an analysis of the vacuum state, the 0-particle state, it can eas-ily be extended to other N-particle states with N 6= 0.¹² This already demonstrates the scope of its importance. In short the upshot of the Reeh-Schlieder theorem is that local measurements can never decide whether we observe an N-particle state. We begin with a technical statement of the theorem. With Ω being the vacuum state and R(O) the von Neumann algebra as introduced in 3 the following result can be derived on the basis of the axioms of AQFT (see appendix B):

Reeh-Schlieder Theorem: Forany bounded open region O in space-time the set {AΩ : A ∈ R(O)} is dense in Hilbert space H.

The definition ofdense as well as of other technical concepts can be found in the physics glossary. For a rough-and-ready explanation one can say that one set lies dense in another set if its elements are so finely distributed over the whole second set that for any given element in this second set and any given distance one can find an element in the first set which lies closer to this element in the second set than expressed by the given distance. The set {AΩ : A ∈ R(O)} is said to be generated from the vacuum state Ω by the von Neumann algebra R(O) of local observables associated with O because it is the set which you get when all the operators A in R(O) are applied to the vacuum state. Recall that an operator is a mathematical entity which “can be thought of as an animal that eats vectors and spits

11Although Reeh and Schlieder (1961) is the original source later accounts render an easier access. For references confer the titles which are cited in the following footnotes.

12See Haag (1996), p. 102.

CHAPTER 6. THE PARTICLE INTERPRETATION OF QFT 83 out other vectors”¹³ . With these explanations the content of the Reeh-Schlieder theorem can be expressed as follows: choosing suitable elements of R(O) and acting with them on Ω any vector in H can be approximated arbitrarily closely.

The statement of the Reeh-Schlieder theorem is different and much stronger than the well-known fact that the Hilbert spaceHcan be spanned by eigenstates of the number operator which can be build up from the vacuum state by application of suitable creation operators. It can be shown that local algebras R(O) - about which the Reeh-Schlieder theorem talks - never contain pure creation (or annihilation) operators. The strength of the Reeh-Schlieder theorem becomes clearer when one considers that O can be a small neighborhood of a point in space-time. What the Reeh-Schlieder theorem now asserts is that acting on the vacuum state Ω with elements ofR(O) we can approximate as closely as we like any state inH, in particular one that is very different from the vacuum in some space-like separated regionO⁰. The Reeh-Schlieder theorem is thus clearly exploiting long distance correlations of the vacuum.

Even though the Reeh-Schlieder theorem is an astonishing result it is not immediately obvious what the conceptual consequences actually are.

To this end we need the following

Corollary of the Reeh-Schlieder theorem: Ω is a separating vec-tor for R(O), i. e. two elements A1, A2 ∈ R(O) which yield the same result when acting on Ω must be one and the same operator, or in shortAΩ = 0⇒A = 0, whereA ≡(A1 −A2).

The use of this corollary yields an interesting interpretive result. Again, the assumptions are the standard axioms of AQFT. It will be shown that:

Local measurements can never decide whether we observe an N-particle state.¹⁴

13See p. 251 in Salmon et al. (1992).

14Cf. Redhead (1995a).

CHAPTER 6. THE PARTICLE INTERPRETATION OF QFT 84 Let us call this statement ‘Redhead’s claim’ because it was he who ex-pressed this conclusion more explicitly than anybody else, at least to my knowledge. The underlying mathematical result is the fact that a projec-tion operatorP_Ψ which corresponds to an N-particle state Ψ can never be an element of a local algebra R(O). Since the proof of ‘Redhead’s claim’

is very instructive and comparatively easy I will restate it here.

Proof

Given the Reeh-Schlieder theorem and its above-mentioned corollary the proof is a straightforward reductio ad absurdum. Consider an arbitrary N-particle state Ψ (with N 6= 0). Since Ψ is orthogonal to the 0-particle state Ω the corresponding projector P_Ψ satisfies P_ΨΩ = 0. Let us now ask whether Ψ is an element of a local algebra R(O) corresponding to a bounded region O. If this were the case then one could decide by a local measurement, restricted to the region O, whether we have an N-particle state Ψ or not, or, to be more cautious, whether we will find such a state or not when we perform such a measurement. Let us assume now as a trial that P_Ψ is an element of a local algebra R(O). In this case our corollary is applicable so that P_ΨΩ = 0 would imply P_Ψ = 0. This, however, contradicts our assumption that P_Ψ is the projection operator corresponding to the N-particle state Ψ, which cannot be0 unless Ψ itself is 0. We are forced, therefore, to drop our assumption thatP_Ψis an element of a local algebraR(O).

q. e. d.

The exact meaning and range of Redhead’s claim will become clearer when we compare it with the ensuing no-go theorem by David Malament.

The comparison itself will follow the exposition of the two results to be compared and makes up the core of this section. We will conclude the section with some critical remarks about the legitimacy of the respective interpretations.

CHAPTER 6. THE PARTICLE INTERPRETATION OF QFT 85 Malament’s No-go Theorem

Malament’s no-go theorem Malament (1996) is another consequence of an-alyticity which rests on a lemma of Borchers Borchers (1967). In short it says that a quantum theory of a fixed number of particles satisfying in particular the very weak locality condition of statistical independence of measurements in space-like related spatial sets predicts a zero probability for finding a particle in any spatial set.

Malament’s no-go theorem rests on four conditions. We assume the existence of projection operators P_∆ on Hilbert space H representing the proposition that a particle detector would respond if a position measure-ment were performed in the spatial set ∆. ∆ is taken to be a bounded open subset of a spacelike hyperplane in Minkowski space-time M. Furthermore we will assume that there is a strongly continuous, unitary representation U(a),a ∈ M (in H) of the translation subgroup of the Poincar´e group in M. Malament’s conditions now are the following:

(i) Translation Covariance Condition:

P_∆+a =U(a)P_∆U(−a) (6.1) for alla in M and all spatial sets ∆. ∆ +adenotes the set ∆ after a translation by a.

(ii) Energy Condition: The spectrum of the Hamiltonian operator H(a) is bounded below (see glossary) provided that H(a) satisfies U(ta) = e^−itH(a) for all unit vectors ain M which are future directed and timelike.

(iii) Localizability Condition: If ∆₁ and ∆₂ are disjoint spatial sets in the same hyperplane,

P_∆1P_∆2 =P_∆2P_∆1 =0. (6.2) (iv) Locality Condition: If ∆₁and ∆₂ are spatial sets (not necessarily

in the same hyperplane) that are space-like related,

P_∆1P_∆2 =P_∆2P_∆1. (6.3)

CHAPTER 6. THE PARTICLE INTERPRETATION OF QFT 86 What do these conditions mean? Condition (iii) is the essential in-gredient of the particle concept: A particle - in contrast to a field - cannot be found in two disjoint spatial sets at the same time. P_∆1 and P_∆2 must therefore be orthogonal. The condition is very weak since it does not set any finite limit to the travelling speed of a particle. Condition (iv) is the relativistic part of Malament’s assumptions: Measurements in ∆₁ must be statistically independent from measurements in the space-like related ∆₂. P_∆1 and P_∆2 must commute therefore. This condition again is very weak since it does not require for P_∆1 and P_∆2 to be orthogonal which would rule out that the particle can travel at superluminal speed.

How does Malament’s no-go theorem work? Using a lemma of Borchers and each of the four conditions above Malament derives that

P_∆= 0 for any spatial set ∆. (6.4) This means that the probability for finding a particle anywhere in space is 0 no matter how large ∆ is. Since that is an unacceptable conclusion Mala-ment’s proof has the weight of a no-go theorem provided that we acccept his four conditions as natural assumptions for a particle interpretation.

What exactly does this say about the possibility of a particle interpretation? A relativistic quantum theory of a fixed number of par-ticles, satisfying in particular the localizability and the locality condition, has to assume a world devoid of particles (more precisely: a world in which particles can never be detected) in order not to contradict itself. Mala-ment’s no-go theorem thus shows that there is no middle ground between QM and QFT, i. e. no theory with a fixed number of particles (like in QM) and which is relativistic (like QFT) without running into the localizability problem of the no-go theorem. One is forced towards QFT (without a fixed number of particles). A particle interpretation of QFT is not directly ruled out though!

CHAPTER 6. THE PARTICLE INTERPRETATION OF QFT 87

6.3.2 Locating the Origin of Non-Localizability