Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State
aa r X i v : . [ m a t h - ph ] F e b Yet More Ado About Nothing:The Remarkable Relativistic Vacuum State ∗ Stephen J. Summers
Department of MathematicsUniversity of FloridaGainesville FL 32611, USAFebruary 19, 2009
Abstract
An overview is given of what mathematical physics can currently say about the vacuumstate for relativistic quantum field theories on Minkowski space. Along with a review ofclassical results such as the Reeh–Schlieder Theorem and its immediate and controversialconsequences, more recent results are discussed. These include the nature of vacuumcorrelations and the degree of entanglement of the vacuum, as well as the striking factthat the modular objects determined by the vacuum state and algebras of observableslocalized in certain regions of Minkowski space encode a remarkable range of physicalinformation, from the dynamics and scattering behavior of the theory to the externalsymmetries and even the space–time itself. In addition, an intrinsic characterization ofthe vacuum state provided by modular objects is discussed.
For millenia, the concept of nothingness, in many forms and guises, has occupiedreflective minds, who have adopted an extraordinary range of stances towards thenotion — from holding that it is the Godhead itself, to rejecting it vehementlyas a foul blasphemy. Even among more scientifically inclined thinkers there hasbeen a similar range of views [49]. We have no intention here to sketch this vastrichness of thought about nothingness. Instead, we shall more modestly attempt ∗ This is an expanded version of an invited talk given at the Symposium ”Deep Beauty:Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World”,held at Princeton University on October 3–4, 2007.
1o explain what mathematical physics has to say about nothingness in its modernscientific guise: the relativistic vacuum state.What is the vacuum in modern science? Roughly speaking, it is that whichis left over after all which can possibly be removed has been removed, where“possibly” refers not to “technically possible” nor to “logically possible”, but to“physically possible” — that which is possible in light of (the current understand-ing of) the laws of physics. The vacuum is therefore an idealization which is onlyapproximately realized in the laboratory and in nature. But it is a most usefulidealization and a surprisingly rich concept.We shall discuss the vacuum solely in the context of the relativistic quantumtheory of systems in four spacetime dimensional Minkowski space, although weshall briefly indicate how similar states for quantum systems in other space–timescan be defined and studied. In a relativistic theory of systems in Minkowski space,the vacuum should appear to be the same at every position, and in every direction,for all inertial observers. In other words, it should be invariant under the Poincar´egroup, the group of isometries of Minkowski space. And since one can remove nofurther mass/energy from the vacuum, it should be the lowest possible (global)energy state. In a relativistic theory, when one removes all mass/energy, the totalenergy of the resultant state is 0.These desiderata of a vacuum are intuitively appealing, but it remains to givemathematical content to these intuitions. Once this is done, it will be seen thatthis state with “nothing in it” manifests remarkable properties, most of whichhave been discovered only in the past twenty years, and many of which are not intuitively appealing at first exposure. On the contrary, some properties of thevacuum state have proven to be decidedly controversial.In order to formulate in a mathematically rigorous manner the notion of a vac-uum state and to understand its properties, it is necessary to choose a mathemat-ical framework which is sufficiently general to subsume large classes of models, ispowerful enough to facilitate the proof of nontrivial assertions of physical interest,and yet is conceptually simple enough to have a direct, if idealized, interpreta-tion in terms of operationally meaningful physical quantities. Such a framework isprovided by algebraic quantum theory [3, 12, 13, 40, 52], also called local quantumphysics, which is based on operator algebra theory, itself initially developed by J.von Neumann for the express purpose of providing quantum theory with a rigorousand flexible foundation [74, 75]. This framework is briefly described in the nextsection, where a rigorous definition of a vacuum state in Minkowski space is given.In Section 3 the earliest recognized consequences of such a definition are dis-cussed, including such initially nonintuitive results as the Reeh–Schlieder Theorem.Rigorous results indicating that the vacuum is a highly entangled state are pre-sented in Section 4. Indeed, by many measures it is a maximally entangled state.Though some of these results have been proven quite recently, readers who arefamiliar with the heuristic picture of the relativistic vacuum as a seething broth ofvirtual particle–antiparticle pairs causing wide-ranging vacuum correlations may2ot be entirely surprised by their content. But there are concepts available in alge-braic quantum field theory (AQFT) which have no known counterpart in heuristicquantum field theory, such as the mathematical objects which arise in the modulartheory of M. Tomita and M. Takesaki [105], which is applicable in the setting ofAQFT. As explained in Sections 5 and 6, the modular objects associated with thevacuum state encode a truly astonishing amount of physical information and alsoserve to provide an intrinsic characterization of the vacuum state which admits ageneralization to quantum fields on arbitrary space–times. In addition, it is shownin Section 7 how these objects may be used to derive the space–time itself, therebyproviding, at least in principle, a means to derive from the observables and theirpreparation (the state) a space–time in which the former can be interpreted asbeing localized and evolving without any a priori input on the nature or evenexistence of a space–time. We make some concluding remarks in Section 8.
The operationally fundamental objects in a laboratory are the preparation ap-parata — devices which prepare in a repeatable manner the individual quantumsystems which are to be examined — and the measuring apparata — devices whichare applied to the prepared systems and which measure the “value” of some ob-servable property of the system. The physical notion of a “state” can be viewed asa certain equivalence class of such preparation devices, and the physical notion ofan “observable” (or “effect”) can be viewed as a certain equivalence class of suchmeasuring (or registration) devices [3, 70]. In principle, therefore, these quantitiesare operationally determined.In algebraic quantum theory, such observables are represented by self–adjointelements of certain algebras of operators, either W ∗ - or C ∗ -algebras. In this paperwe shall restrict our attention primarily to concretely represented W ∗ -algebras,which are commonly called von Neumann algebras in honor of the person whoinitiated their study [75]. The reader unfamiliar with these notions may simplythink of algebras M of bounded operators on some (separable) Hilbert space H (or see [59, 60, 106–108] for a thorough background). We shall denote by B ( H )the algebra of all bounded operators on H . Physical states are represented bymathematical states φ , i.e. linear, continuous maps φ : M → C from the algebraof observables to the complex number system which take the value 1 on the identitymap I on H and are positive in the sense that φ ( A ∗ A ) ≥ A ∈ M . Animportant subclass of states consists of normal states ; these are states such that φ ( A ) = Tr( ρA ), A ∈ M , for some density matrix ρ acting on H , i.e. a boundedoperator on H satisfying the conditions 0 ≤ ρ = ρ ∗ and Tr( ρ ) = 1. A special case Other sorts of algebras have also been seriously considered for various reasons; see e.g. [40, 88, 89]. Technicalities concerning topology will be systematically suppressed in this paper. We there-fore will not discuss the difference between C ∗ - and W ∗ -algebras.
3f such normal states is constituted by the vector states : if Φ ∈ H is a unit vectorand P Φ ∈ B ( H ) is the orthogonal projection onto the one dimensional subspace of H spanned by Φ, the corresponding vector state is given by φ ( A ) = h Φ , A Φ i = Tr( P Φ A ) , A ∈ M . Generally speaking, theoretical physicists tacitly restrict their attention to normalstates. In AQFT the spacetime localization of the observables is taken into account.Let R represent four dimensional Minkowski space and O denote an open subsetof R . Since any measurement is carried out in a finite spatial region and in afinite time, for every observable A there exist bounded regions O containing this“localization” of A . We say that the observable A is localized in any such region O and denote by R ( O ) the von Neumann algebra generated by all observableslocalized in O . Clearly, it follows that if O ⊂ O , then R ( O ) ⊂ R ( O ). Thisyields a net O 7→ R ( O ) of observable algebras associated with the experiment(s)in question. In turn, this net determines the smallest von Neumann algebra R on H containing all R ( O ). The preparation procedures in the experiment(s) thendetermine states φ on R , the global observable algebra.Given a state φ on R , one can construct [40, 59, 106] a Hilbert space H φ , adistinguished unit vector Ω φ ∈ H φ and a ( C ∗ -)homomorphism π φ : R → B ( H φ ),so that π φ ( R ) is a ( C ∗ -)algebra acting on the Hilbert space H φ , the set of vectors π φ ( R )Ω φ = { π φ ( A )Ω φ | A ∈ R} is dense in H φ and φ ( A ) = h Ω φ , π φ ( A )Ω φ i , A ∈ R . The triple ( H φ , Ω φ , π φ ) is uniquely determined up to unitary equivalence by theseproperties, and π φ is called the GNS representation of R determined by φ . Onlyif φ is a normal state is π φ ( R ) a von Neumann algebra and can ( H φ , Ω φ , π φ ) beidentified with (a subrepresentation of) ( H , Ω , R ) such that Ω φ ∈ H . Hence, astate determines a concrete, though idealized, representation of the experimentalsetting in a Hilbert space.In this setting, relativistic covariance is expressed through the presence of arepresentation P ↑ + ∋ λ α λ of the identity component P ↑ + of the Poincar´e group It is in the context of von Neumann algebras and normal states that classical probabilitytheory has a natural generalization to noncommutative probability theory; see e.g. [81]. It is clear from operational considerations that one could not expect to determine a minimallocalization region for a given observable experimentally. In [64] the possibility of determiningsuch a minimal localization region in the idealized context of AQFT is discussed at length.However, the existence of such a region is not necessary for any results in AQFT known to theauthor. If the state φ is not normal, then the state vectors in H φ are, in a certain mathematicallyrigorous sense, orthogonal to those in H . The vacuum state of a fully interacting model, asopposed to an interacting theory with various kinds of cutoffs introduced precisely so that it maybe realized in Fock space, is not normal with respect to Fock space, which is the representationspace for the corresponding free theory — see e.g. [8, 47]. For further perspective on this issue,see [100].
4y automorphisms α λ : R → R of R such that α λ ( R ( O )) = R ( λ O ) , for all O and λ , where α λ ( R ( O )) = { α λ ( A ) | A ∈ R ( O ) } and λ O = { λ ( x ) | x ∈ O} . One says that a state φ is Poincar´e invariant if φ ( α λ ( A )) = φ ( A ) for all A ∈ R and λ ∈ P ↑ + . In this case, there then exists a unitary represen-tation P ↑ + ∋ λ U φ ( λ ) acting on H φ , leaving Ω φ invariant, and implementing theaction of the Poincar´e group: U φ ( λ ) π φ ( A ) U φ ( λ ) − = π φ ( α λ ( A )) , for all A and λ . If the joint spectrum of the self–adjoint generators of the trans-lation subgroup U φ ( R ) is contained in the forward lightcone, then U φ ( P ↑ + ) is saidto satisfy the (relativistic) spectrum condition . This condition is a relativisticallyinvariant way of requiring that the total energy in the theory be nonnegative withrespect to every inertial frame of reference and that the quantum system is stablein the sense that it cannot decay to energies below that of the vacuum state.We can now present a standard rigorous definition of a vacuum state, whichincorporates all of the intuitive desiderata discussed above. Definition 2.1
A vacuum state is a Poincar´e invariant state φ on R such that U φ ( P ↑ + ) satisfies the spectrum condition. The corresponding GNS representation ( H φ , Ω φ , π φ ) is called a vacuum representation of the net of observable algebras. Note that after choosing an inertial frame of reference, the self–adjoint generator H of the time translation subgroup U φ ( t ), t ∈ R , carries the interpretation ofthe total energy operator and that, by definition, H Ω φ = 0, if φ is a vacuumstate. Moreover, the total momentum operator ~P and the total mass operator M ≡ p H − ~P ≥ M Ω φ = 0 = ~P Ω φ ).Such vacuum states, and hence such vacuum representations, actually exist. Inthe case of four dimensional Minkowski space, vacuum representations for quantumfield models with trivial S -matrix have been rigorously constructed by variousmeans (cf. e.g. [2,8,17,48,114]) and, more recently, the same has been accomplishedfor quantum field models with nontrivial scattering matrices [32, 33, 50]. For two,resp. three, dimensional Minkowski space, fully interacting quantum field modelsin vacuum representations have been constructed, cf. e.g. [8, 47, 48, 69]. Moreover,general conditions are known under which to a quantum field model without avacuum state can be (under certain conditions uniquely) associated a vacuumrepresentation which is physically equivalent and locally unitarily equivalent toit [18,20,38]. Hence, the mathematical existence of a vacuum state is often assuredeven in models which are not initially provided with one. Some authors just require of a vacuum state that it be invariant under the translation groupand satisfy the spectrum condition. For the purposes of this paper, it is convenient to adopt themore restrictive of the two standard definitions.
5t will be useful in the following to describe two special classes of spacetimeregions in Minkowski space. A double cone is a (nonempty) intersection of an openforward lightcone with an open backward lightcone. Such regions are bounded,and the set D of all double cones is left invariant by the natural action of P ↑ + uponit. An important class of unbounded regions is specified as follows. After choosingan inertial frame of reference, one defines the right wedge to be the set W R = { x = ( t, x , x , x ) ∈ R | x > | t |} and the set of wedges to be W = { λW R | λ ∈ P ↑ + } . The set of wedges is independent of the choice of referenceframe; only which wedge is designated the right wedge is frame-dependent. We now turn to some immediate consequences of the definition of a vacuum state.One of the most controversial was also one of the first to be noted. In order toavoid a too heavily laden notation, and since in this and the next section ourstarting point is a vacuum representation, we shall drop the subscript φ and thesymbol π φ ( i.e. we identify R ( O ) and π φ ( R ( O ))). A vacuum representation issaid to satisfy weak additivity if for each nonempty O the smallest von Neumannalgebra containing { U ( x ) R ( O ) U ( x ) − | x ∈ R } coincides with R . This is a weak technical assumption satisfied in most models;for example, it holds in any theory in which there is a Wightman field locallyassociated with the observable algebras (see, e.g. [6, 7, 37]).Let O be an open subset of R and let O ′ denote the interior of its causalcomplement, the set of all points in R which are spacelike separated from allpoints in O . A net O 7→ R ( O ) is said to be local (or to satisfy locality ) if whenever O ⊂ O ′ one has R ( O ) ⊂ R ( O ) ′ , where R ( O ) ′ , the commutant of R ( O ),represents the set of all bounded operators on H which commute with all elementsof R ( O ). Ordinarily, this property of locality is viewed as a manifestation ofEinstein causality, which posits that signals and causal influences cannot propagatefaster than the speed of light, and therefore spacelike separated quantum systemsmust be independent in some sense. As is the case with so many received notions,there is much more here than meets the eye initially; but this is not the place toaddress this matter (cf. [34, 96, 102] for certain aspects of this point). We wish toemphasize that locality will not be a standing assumption in this paper. If a netis not explicitly assumed to be local, then the property is not necessary for therespective result. And, in fact, locality will be derived in the settings discussed inSections 6 and 7. For vacuum representations of local nets in which weak additivity is satisfied,the Reeh–Schlieder Theorem holds (cf. [3, 8, 52, 58, 91]). For a very different derivation of locality, see [31]. heorem 3.1 Consider a vacuum representation of a local net fulfilling the con-dition of weak additivity. For every nonempty region O such that O ′ = ∅ , thevector Ω is cyclic and separating for R ( O ) , i.e. the set of vectors R ( O )Ω is densein H , resp. A ∈ R ( O ) and A Ω = 0 entail A = 0 . There are two distinct aspects to this theorem. First of all, the fact that thevacuum is separating for local observables means exactly that no nonzero localobservable can annihilate Ω. Hence, any event represented by a nonzero projection P ∈ R ( O ) must have nonzero expectation in the vacuum state: h Ω , P Ω i >
0. Inthe vacuum, any local event can occur! Moreover, 0 < C = C ∗ ∈ R ( O ) entailsthe existence of an element 0 = A ∈ R ( O ) such that C = A ∗ A ; thus h Ω , C Ω i = k A Ω k , which also yields h Ω , C Ω i > T ( x ) smeared with any test function with compactsupport cannot be a positive operator in a vacuum representation [41] (in fact,it is unbounded below), in contrast to the situation in classical physics, since itsvacuum expectation is zero. Furthermore, in light of the fact that the vacuumstate contains no real particles ( M Ω = 0), it follows that there can be no localizedparticle counters. Indeed, if C ∈ R is a particle counter for a particle described inthe model, then C = C ∗ > h Ω , C Ω i = 0. Therefore, C cannot be an elementof any algebra R ( O ) with O ′ nonempty. Hence the notion of particle in relativisticquantum field theory cannot be quite as simple as classical mechanics would haveit. It has even been argued that the notion is nonsensical in relativistic quantumfield theory, but this is not the place for further discussion of this point, either.(See, however, [19, 22, 45, 52, 55, 85].)Second, there is the cyclicity of the vacuum for all local algebras: every vectorstate in the vacuum representation can be arbitrarily well approximated usingvectors of the form A Ω, A ∈ R ( O ), no matter how small in extent O may be.Hence, the class of all states resulting from the action of arbitrary operationsupon the vacuum is effectively indistinguishable from the class of states resultingfrom operations performed in arbitrarily small spacetime regions upon the vacuum. Prima facie , such a state would seem to be different from the vacuum only in aregion which one can make as small as one desires. In our view, a reasonablephysical picture of this situation is indicated in this way: an experimenter inany given region O can, in principle, perform measurements designed to exploitnonlocal vacuum fluctuations (see the next section) in such a manner that anyprescribed state can be reproduced with any given accuracy. These consequencesof cyclicity also unleashed some controversy, some of which is well discussed in [54](see also [82]). We shall not elaborate upon these matters here, except to pointout the fact that the existing proposals to avoid Reeh–Schlieder by changing thenotion of localization (1) are necessarily restricted to free quantum field modelsand (2) introduce at least as many problems as they “solve”, see e.g. [54].We wish to emphasize that these (for some readers disturbing) properties areby no means unique to the vacuum — the Reeh–Schlieder Theorem is valid for Note that even if the net of observable algebras is not local, Ω is still cyclic for R ( O ). ny vector in the vacuum representation which is analytic for the energy [9]; inparticular, it holds for any vector with finite energy content. So its conclusionsand various consequences are true of all physically realizable vector states in thevacuum representation, since any preparation can only implement a finite exchangeof energy! We turn to what is rigorously known about the nature of vacuum correlations,preparing first some definitions to be used in this section. Given a pair ( M , N ) ofalgebras representing the observable algebras of two subsystems of a given quan-tum system, a state φ is said to be a product state across ( M , N ) if φ ( M N ) = φ ( M ) φ ( N ) for all M ∈ M , N ∈ N . In such states, the observables of the twosubsystems are not correlated and the subsystems manifest a certain kind of inde-pendence — see e.g. [96]. A normal state φ on M W N is separable if it is in thenorm closure of the convex hull of the normal product states across ( M , N ), i.e. itis a mixture of normal product states. Otherwise, φ is said to be entangled (across( M , N )). From the point of view of what is now called quantum informationtheory, the primary difference between classical and quantum theory is the exis-tence of entangled states in quantum theory. In fact, only if both subsystems arequantum, i.e. both algebras are noncommutative, do there exist entangled stateson the composite system [79]. Although not understood at that time in this man-ner, some of the founders of quantum theory realized as early as 1935 [39, 84] thatsuch entangled states were the source of the “paradoxical” behavior of quantumtheory (as viewed from the vantage point of classical physics). Today, entangledstates are regarded as a resource to be employed in order to carry out tasks whichcannot be done classically, i.e. only with separable states — cf. [57, 62, 112].Another direct consequence of the Reeh–Schlieder Theorem is that for all nonemptyspacelike separated O , O with nonempty causal complements, no matter how farspacelike separated they may be, there exist many projections P i ∈ R ( O i ) whichare positively correlated in the vacuum state, i.e. such that φ ( P P ) > φ ( P ) φ ( P ). Theorem 4.1
Consider a vacuum representation of a local net fulfilling the condi-tion of weak additivity, and let O , O be any nonempty spacelike separated regionswith nonempty causal complements. Let φ be any state induced by a vector analyticfor the energy (e.g. the vacuum state). Then for any projection P ∈ R ( O ) with = P = I there exists a projection P ∈ R ( O ) such that φ ( P P ) > φ ( P ) φ ( P ) . This is an immediate consequence of Theorem 3.1 and the following lemma, the also termed decomposable, classically correlated, or unentangled by various authors This terminology is becoming standard in quantum information theory, but there are stillphysicists who tacitly restrict their attention to vector states on mutually commuting algebrasof observables which are isomorphic to full matrix algebras, i.e. they consider only pure states,which are entangled if and only if they are not product states.
Lemma 4.2
Let M and N be von Neumann algebras on H with Ω ∈ H a unitvector cyclic for N and separating for M , and let ω be the corresponding stateinduced upon B ( H ) . Then for any projection P ∈ M with = P = I , there existsa projection Q ∈ N such that ω ( P Q ) > ω ( P ) ω ( Q ) . Proof.
Let P ∈ M be a projection with 0 = P = I . It suffices to establishthe existence of a projection Q ∈ N such that ω ( P Q ) = ω ( P ) ω ( Q ), since, ifnecessary, Q can be replaced by I − Q ∈ N to yield the assertion. So assumefor the sake of contradiction that ω ( P Q ) = ω ( P ) ω ( Q ), for all such Q . Then with b P = P − ω ( P ) · I ∈ M , one has ω ( b P Q ) = 0, for all projections Q ∈ N . By thespectral theorem, this entails ω ( b P N ) = 0, for all N ∈ N , i.e. h b P Ω , N Ω i = 0 , N ∈ N . Since Ω is cyclic for N , this yields b P Ω = 0, so that b P = 0, i.e. P = ω ( P ) · I . Since P = P , this entails k P Ω k = h P Ω , P Ω i = ω ( P ) ∈ { , } , i.e. either P Ω = 0 or P Ω = Ω. Since Ω is separating for M , this implies either P = 0 or P = I holds,a contradiction in either case. (cid:3) The fact that vacuum fluctuations enable such generic “superluminal correla-tions” has also generated controversy, since they seem to challenge received notionsof causality. This is another complex matter which we cannot go into here, but atleast some forms of causality have been proven in AQFT (for recent discussions,see e.g. [34, 80]) and therefore are completely compatible with such correlations.Of course, Theorem 4.1 entails that the vacuum is not a product state across( R ( O ) , R ( O )), but not yet that it is entangled across ( R ( O ) , R ( O )). Muchfiner analyses of the nature and degree of the entanglement of the vacuum statehave been carried out in the literature, and we shall explain some of these. Aquantitative measure of entanglement is provided by using Bell correlations . Thefollowing definition was made in [93].
Definition 4.3
Let M , N ⊂ B ( H ) be von Neumann algebras such that M ⊂ N ′ .The maximal Bell correlation of the pair ( M , N ) in the state φ is β ( φ, M , N ) ≡ sup 12 φ ( M ( N + N ) + M ( N − N )) , where the supremum is taken over all self-adjoint M i ∈ M , N j ∈ N with normless than or equal to 1. As explained in e.g. [94], the CHSH version of Bell’s inequalities can be formu-lated in algebraic quantum theory as β ( φ, M , N ) ≤ . (4.1)9f φ is separable across ( M , N ), then β ( φ, M , N ) = 1 [94]. Hence states whichviolate Bell’s inequalities are necessarily entangled, though the converse is not true(cf. [112] for a discussion and references). Whenever at least one of the systems isclassical, the bound (4.1) is satisfied in every state: Proposition 4.4 ( [94])
Let M , N ⊂ B ( H ) be mutually commuting von Neu-mann algebras. If either M or N is abelian, then β ( φ, M , N ) = 1 for all states φ on B ( H ) . If, on the other hand, both algebras are nonabelian, then there always exists astate in which the inequality (4.1) is (maximally) violated, as long as the Schliederproperty holds, i.e.
M N = 0 for M ∈ M and N ∈ N entail either M = 0 or N = 0 [68]. Because it is known [35, 94] that 1 ≤ β ( φ, M , N ) ≤ √
2, for all states φ on B ( H ), one says that if β ( φ, M , N ) = √
2, then the pair ( M , N ) maximallyviolates Bell’s inequalities in the state φ .In [95] it is shown under quite general physical assumptions that in a vacuumrepresentation of a local net one has β ( φ, R ( W ) , R ( W ′ )) = √
2, for every wedge W and every normal state φ . In particular, Bell’s inequalities are maximally violatedin the vacuum state. In addition, under somewhat more restrictive but still generalassumptions which include free quantum field theories and other physically relevantmodels, it is shown in [95] that β ( φ, R ( O ) , R ( O )) = √
2, for any two spacelikeseparated double cones whose closures intersect ( i.e. tangent double cones) and all normal states φ . Hence, such pairs of observable algebras also maximally violateBell’s inequalities in the vacuum.Commonly, physicists say that theories violating Bell’s inequalities are “nonlo-cal”; yet, here are fully local models maximally violating Bell’s inequalities. Thislinguistic confusion is probably so profoundly established by usage that it cannotbe repaired, but the reader should be aware of the distinct meanings of these twouses of “local”. The former refers to nonlocalities in certain correlations (in certainstates), while the latter refers to the commensurability of observables localized inspacelike separated spacetime regions. So the former is a property of states, whilethe latter is a property of observable algebras. The results discussed above estab-lish the generic compatibility of the former sort of “nonlocality” with the latterkind of “locality”. The wary reader should always ascertain which sense of “local”is being employed by a given author.In the now quite extensive quantum information theory literature, there arevarious attempts to quantify the degree of entanglement of a given state (cf. e.g. [57,62]), but these agree that maximal violation of inequality (4.1) entails max-imal entanglement. Thus, the vacuum state is maximally entangled and therebydescribes a maximally non-classical situation.The localization regions for the observable algebras which have been provento manifest maximal violation of Bell’s inequality in the vacuum (indeed, in ev-ery state) are spacelike separated but tangent. If the double cones have nonzero10pacelike separation, any violation of Bell’s inequality in the vacuum cannot bemaximal: Proposition 4.5 ( [93, 94, 97])
Let
O 7→ R ( O ) be a local net in an irreduciblevacuum representation with a lowest mass m > . Then for any pair ( O , O ) ofspacelike separated regions one has β ( φ, R ( O ) , R ( O )) ≤ √ − √
27 + 4 √ − e − md ( O , O ) ) (optimal for smaller d ( O , O ) ) and β ( φ, R ( O ) , R ( O )) ≤ e − md ( O , O ) (optimal for larger d ( O , O ) ), where φ is a vacuum state and d ( O , O ) is themaximal timelike distance O can be translated before it is no longer spacelikeseparated from O . Hence, if d ( O , O ) is much larger than a few Compton wavelengths of thelightest particle in the theory, then any violation of Bell’s inequality in the vacuumwould be too small to be observed. As explained in [94], if there are masslessparticles in the theory, then the best decay in the vacuum Bell correlation onecan expect is proportional to d ( O , O ) − . Although the decay in the masslesscase is much weaker, experimental apparata have nonzero lower bounds on theparticle energies they can effectively measure. Such nonzero sensitivity limitswould serve as an effective lowest mass, leading to an exponential decay onceagain [94]. Nonetheless, attempts have been made to obtain lower bounds on theBell correlation β ( φ, R ( O ) , R ( O )) as a function of d ( O , O ). As the publishedresults have only treated some very special models and very special observables,we shall refrain from discussing these here (but cf. [83] and references given there).Nonetheless, using properties of β ( φ, M , N ) established by the author and R.F.Werner [97], H. Halvorson and R. Clifton have proven the following result, whichentails that in a vacuum representation in which weak additivity and localityhold, the vacuum state (and any state induced by a vector analytic for the energy)is entangled across ( R ( O ) , R ( O )) for arbitrary nonempty spacelike separatedregions O , O . Theorem 4.6 ( [53])
Let M and N be nonabelian von Neumann algebras actingon H such that M ⊂ N ′ . If Ω ∈ H is cyclic for M and ω is the state on B ( H ) induced by Ω , then ω is entangled across ( M , N ) . The proof does not provide a lower bound on β ( φ, M , N ). For further discussionand references concerning the violation of Bell’s inequalities in algebraic quantumtheory, see [53, 82, 97, 99]. 11hough model independent lower bounds on β ( φ, R ( O ) , R ( O )) are not yetavailable, R. Verch and Werner [111] have obtained model independent resultson the nature of the entanglement of the vacuum state across nontangent pairs( R ( O ) , R ( O )) in terms of some further notions currently employed in quantuminformation theory, which go beyond Theorem 4.6. They proposed the followingdefinition [111]. Definition 4.7
Let M and N be von Neumann algebras acting upon a Hilbertspace H . A state φ on B ( H ) has the ppt property if for any choice of finitelymany M , . . . , M k ∈ M and N , . . . , N k ∈ N , one has X α,β φ ( M β M ∗ α N ∗ α N β ) ≥ . They show that this generalizes the notion of states with positive partial trans-pose familiar from quantum information theory [76], a notion restricted to finitedimensional Hilbert spaces prior to [111]. They also show that if a state is ppt,then it satisfies Bell’s inequalities, and they prove that any separable state is ppt.Indeed, in general the class of ppt states properly contains the class of separablestates.Another notion from quantum information theory is that of distillability (ofentanglement). Roughly speaking, this refers to being able to operate upon a givenstate in certain (local) ways to increase its entanglement across two subsystems.Separable states are not distillable; they are not entangled, and operating uponthem in the allowable manner will not result in an entangled state. We refer thereader to [111] for a discussion of the general case and restrict ourselves here to adiscussion of the following special case.
Definition 4.8 ( [111])
Let M and N be von Neumann algebras acting upona Hilbert space H . A state φ on B ( H ) is if there exist completelypositive maps T : B ( C ) → M and S : B ( C ) → N such that the functional ω ( X ⊗ Y ) ≡ φ ( T ( X ) S ( Y )) , X ⊗ Y ∈ B ( C ) ⊗ B ( C ) is not ppt. Verch and Werner show that 1-distillable states are distillable and not ppt.They also prove the following theorem.
Proposition 4.9 ( [111])
Let
O 7→ R ( O ) be a local net in a vacuum representa-tion satisfying weak additivity. Then if O and O are strictly spacelike separateddouble cones, the vacuum state is 1-distillable across the pair ( R ( O ) , R ( O )) . Hence, the vacuum is distillable and not ppt across ( R ( O ) , R ( O )) no matterhow large d ( O , O ) is. We remark that, once again, this theorem is valid also forstates induced by vectors in the vacuum representation which are analytic for theenergy [111]. For a discussion of some further aspects of the entanglement of thevacuum in AQFT, we refer the reader to [36].12 Geometric Modular Action
We emphasize that nearly all of the remarkable properties of the vacuum statediscussed to this point are shared by all vector states which are analytic for theenergy. In the remainder of this paper we shall be dealing with properties uniqueto the vacuum.A crucial breakthrough in the theory of operator algebras was the Tomita–Takesaki theory [105] (see also [60, 107]), which is proving itself to be equallypowerful and productive for the purposes of mathematical quantum theory. Oneof the settings subsumed by this theory is a von Neumann algebra M with acyclic and separating vector Ω ∈ H . The data ( M , Ω) then uniquely determinean antiunitary involution J ∈ B ( H ) and a strongly continuous group of unitaries∆ it , t ∈ R , such that J Ω = Ω = ∆ it Ω, J M J = M ′ and ∆ it M ∆ − it = M , for all t ∈ R , along with further significant properties. From the Reeh–Schlieder Theorem(Theorem 3.1), this theory is applicable to the pair ( R ( O ) , Ω), under the indicatedconditions. Since, as explained above, the algebras and states are operationallydetermined (in principle), the corresponding modular objects J O , ∆ it O are, as well.In pathbreaking work [6, 7], J.J. Bisognano and E.H. Wichmann showed thatfor a net of von Neumann algebras O 7→ R ( O ) locally associated with a finite–component quantum field satisfying the Wightman axioms [8, 58, 91] (and there-fore in a vacuum representation), the modular objects J W , ∆ itW determined bythe wedge algebras R ( W ), W ∈ W , and the vacuum vector Ω have a geometricinterpretation : ∆ itW = U ( λ W (2 πt )) , (5.2)for all t ∈ R and W ∈ W , where { λ W (2 πt ) | t ∈ R } ⊂ P ↑ + is the one-parametersubgroup of boosts leaving W invariant. Explicitly for W = W R , λ W R ( t ) = cosh t sinh t t cosh t . The relation (5.2) has come to be referred to as modular covariance . Moreover,for scalar Boson fields , one has J W R = Θ U π , (5.3)where Θ is the PCT-operator associated to the Wightman field and U π implementsthe rotation through the angle π about the 1-axis, with similar results for generalwedge W ∈ W . Hence, one has J W R R ( O ) J W R = R ( θ R O ) , (5.4) commonly called the modular conjugation or modular involution associated with ( M , Ω) ∆ is a certain, typically unbounded, positive operator called the modular operator associatedwith ( M , Ω) See also [37] for later advances in this particular setting. See [7] for arbitrary finite-component Wightman fields. O , where θ R ∈ P + is the reflection through the edge { (0 , , x , x ) | x , x ∈ R } of the wedge W R . This implies in turn that for all W ∈ W one has J W {R ( f W ) | f W ∈ W} J W = {R ( f W ) | f W ∈ W} . (5.5)Thus the adjoint action of the modular involutions J W , W ∈ W , leaves the set {R ( W ) | W ∈ W} of observable algebras associated with wedges invariant, i.e. wedge algebras are transformed to wedge algebras by this adjoint action.Although in the special case of the massless free scalar field [56] (and, moregenerally, for conformally invariant quantum field theories [14]) also the modularobjects corresponding to ( R ( O ) , Ω) for
O ∈ D have geometric meaning, someexplicit computations in the free massive field have indicated that this is not truein general. Moreover, as we shall see in the next section, only the vacuum vector Ωyields modular objects having any geometric content. This fact yields an intrinsiccharacterization of the vacuum state.But before we explore this noteworthy state of affairs, let us examine some of themore striking consequences of the above relations. For simplicity, we shall restrictthese remarks to the case of nets of algebras locally associated with a scalar Bosefield. Since every element e λ ∈ L + \L ↑ + of the complement of the identity component L ↑ + of the Lorentz group in the proper Lorentz group L + can be factored uniquelyinto a product e λ = θ R λ , with λ ∈ L ↑ + , it follows that by defining U ( e λ ) = J W R U ( λ )one obtains an (anti-)unitary representation of the proper Poincar´e group P + whichacts covariantly upon the original net of observables. Moreover, denoting by J the group generated by { J W | W ∈ W} and J + as the subgroup of J consisting ofproducts of even numbers of the generating involutions { J W | W ∈ W} , one has J = U ( P + ) and J + = U ( P ↑ + ) . (5.6)Hence, the modular involutions { J W | W ∈ W} encode the isometries of theunderlying space–time as well as a representation of the isometry group whichacts covariantly upon the observables. So in particular, U ( R ) ⊂ J + . Recallingthat the subgroup of translations U ( R ) determines the dynamics of the quantumfield, one sees that the modular involutions also encode the dynamics of the model!The dynamics need not be posited, but instead can be derived from the observablesand preparations of the quantum system, at least in principle, using the modularinvolutions.If the quantum field model is such that a scattering theory can be definedfor it and satisfies asymptotic completeness [3, 8, 58], then the original fields andthe asymptotic fields act on the same Hilbert space and have the same vacuum.Letting R (0) ( W ), W ∈ W , denote the observable algebras associated with thefree asymptotic field and J (0) W represent the modular involution corresponding to( R (0) ( W ) , Ω), one has, as was pointed out by B. Schroer [86], S = J W R J (0) W R , S is the scattering matrix for the original field model. Hence, the modularinvolutions associated with the wedge algebras and the vacuum state also encodeall information about the results of scattering processes in the given model! In addition, because of the connection between Tomita–Takesaki modular the-ory and KMS–states [13], modular covariance entails that when the vacuum stateis restricted to R ( W ) for any wedge W , then with respect to the automorphismgroup on R ( W ) generated by the boosts U ( λ W ( t )), it is an equilibrium stateat temperature 1 / π (in suitable units). Hence, any uniformly accelerated ob-servers find when testing the vacuum that it has a nonzero temperature [90]. Thisstriking fact is called the Unruh effect [110]. Moreover, because KMS–states arepassive [77], the vacuum satisfies the second law of thermodynamics with respectto boosts — an additional stability property.Modular covariance and/or the geometric action of the modular conjugations(5.4) have also been derived under other sets of assumptions (in addition to thosediscussed in the next section) which do not refer to the Wightman axioms, i.e. purely algebraic settings in which no appeal to Wightman fields is made [15,65,72,109] (see [11] for a review). Thus, these properties and their many consequenceshold quite generally. It is also of interest that some of these settings providealgebraic versions of the PCT Theorem and the Spin–Statistics connection [15,51, 63, 67, 72], but we shall not enter upon this topic here. We now turn to thoseconditions which provide an intrinsic characterization of the vacuum state. Though the definition of a vacuum state given in Definition 2.1 is standard, it isnot quite satisfactory, since it is not (operationally) intrinsic . It has been seen inSection 2 that the elements of quantum theory which are closest to its operationalfoundations are states and observables. However, in the definition of the vacuumstate one finds such notions as the spectrum condition and automorphic (andunitary) representations of the Poincar´e group, all of which are not expressed solelyin terms of these states and observables. This may not disturb some readers, solet us step back and locate the notion of Minkowski space vacuum state in a largercontext.One of the primary roles of the vacuum state in quantum field theory hasbeen to serve as a physically distinguished reference state with respect to whichother physical states can be defined and referred. Let us recall as an exampleof this that perturbation theory is performed with respect to the vacuum state, i.e. computations performed for general states of interest in quantum field theoryare carried out by suitably perturbing the vacuum. This role has proven to beso central that when theorists tried to formulate quantum field theory in space– Note that the same is not true about the modular unitaries, since both the original fieldand the asymptotic field are covariant under the same representation of P ↑ + . , they tried to find analogous states in thesenew settings, thereby running into some serious conceptual and mathematicalproblems. This is not the place to explain the range and scope of these difficulties,but one noteworthy problem is indicated by the question: what could replace thelarge isometry group (the Poincar´e group) of Minkowski space in the definition of“vacuum state”, in light of the fact that the isometry group of a generic space–timeis trivial? A further point is that in the definition of “vacuum state” the spectrumcondition serves as a stability condition; what could replace it even in such highlysymmetric space–times as de Sitter space, where the isometry group, though large,does not contain any translations?After much effort, a number of interesting selection criteria have been isolatedand studied; see, e.g. [10, 16, 24, 25, 29, 30, 43, 61, 66, 71, 78, 92]. Of these, all but oneeither select an entire folium of states — i.e. a representation, instead of a state— or are explicitly limited to a particular subclass of spacetimes (or both). Herewe shall discuss the selection criterion provided by the Condition of GeometricModular Action (CGMA), which in the special case of Minkowski space selectsthe vacuum state (as opposed to selecting the entire vacuum representation) butwhich can be formulated for general space–times.As we now no longer have a vacuum state/representation given, we return tothe notation of Section 2 and the initial data of a net O 7→ R ( O ) of observablealgebras and a state φ on R . The question we are now examining is: underwhich conditions, stated solely in terms of mathematical quantities completelydetermined by these initial data, is φ a vacuum state? Surprisingly, the core ofthe answer to this question is the relation (5.5). It will be convenient to introducethe notation R φ ( O ) ≡ π φ ( R ( O )) ′′ = ( π φ ( R ( O )) ′ ) ′ . We consider a special case ofthe condition first discussed in [21] and subsequently further generalized in [25]. Definition 6.1
A state φ on a net O 7→ R ( O ) satisfies the Condition of Geomet-ric Modular Action if the vector Ω φ is cyclic and separating for R φ ( W ) , W ∈ W ,and if the modular conjugation J W corresponding to ( R φ ( W ) , Ω φ ) satisfies J W {R φ ( f W ) | f W ∈ W} J W ⊂ {R φ ( f W ) | f W ∈ W} . (6.7) for all W ∈ W . Note that there is no prima facie reason why (5.5) should imply (5.4). Indeed,why should the action (5.5) even be implemented by point transformations on R ,much less by Poincar´e transformations? And since all Poincar´e transformationsmap wedges to wedges, why should (5.4) be the only solution, even if one did findoneself in the latter, fortunate situation?The following theorem was proven in [25,29]. The interested reader may consult[29] for the definition of the weak technical property referred to in hypothesis (c) After all, the space–time in which we find ourselves is not Minkowski space.
16f the following theorem— a property which involves only the net W
7→ R φ ( W )itself. Theorem 6.2 ( [25, 29])
Let φ be a state on a net O 7→ R ( O ) which satisfiesthe following constraints:(a) The map W ∋ W
7→ R φ ( W ) ∈ {R φ ( W ) | W ∈ W} is an order-preservingbijection.(b) If W ∩ W = ∅ , then Ω φ is cyclic and separating for R φ ( W ) ∩ R φ ( W ) .Conversely, if Ω φ is cyclic and separating for R φ ( W ) ∩R φ ( W ) , then W ∩ W = ∅ ,where the bar denotes closure.(c) The net W
7→ R φ ( W ) is locally generated.(d) The adjoint action of the modular conjugations J W , W ∈ W , acts transi-tively upon the set {R φ ( W ) | W ∈ W} , i.e. there exists a wedge W ∈ W suchthat { J W R φ ( W ) J W | W ∈ W} = {R φ ( W ) | W ∈ W} . Then there exists a continuous (anti-)unitary representation U of P + whichleaves Ω φ invariant and acts covariantly upon the net: U ( λ ) R φ ( O ) U ( λ ) − = R φ ( λ O ) , for all O and λ ∈ P + . Moreover, J = U ( P + ) , J + = U ( P ↑ + ) and J W R R ( O ) J W R = R ( θ R O ) , for all O . Furthermore, the wedge duality condition holds: R φ ( W ′ ) = R φ ( W ) ′ , for all W ∈ W , which entails that the net W
7→ R φ ( W ) is local. Hence, from the state and net are derived the isometry group of the space–time; a unitary representation of the isometry group formed from the modularinvolutions, leaving the state invariant and acting covariantly upon the net; thespecific geometric action of the modular involutions found in a special case byBisognano and Wichmann; the locality of the net; and even the dynamics etc . ofthe theory (see Section 5).The conceptually crucial observation is that all conditions in the hypothesis ofthis theorem are expressed solely in terms of the initial net and state, or algebraicquantities completely determined by them. Condition (a) entails that the adjointaction of the modular involutions J W upon the net induces an inclusion preservingbijection on the set W . Condition (b) assures that this bijection can be imple-mented by point transformations (indeed Poincar´e transformations) [25], and (c) In fact, hypothesis (c) may be dispensed with if the Modular Stability Condition (see below)is satisfied [29]. U ( P + ) is continuous [29]. Condition (d) strength-ens the Condition of Geometric Modular Action. Without this strengthening, theadjoint action of the J W can still be shown to be implemented by Poincar´e trans-formations [25], but the group J can then be isomorphic to a proper subgroup of P + [44].Although such a state φ is clearly a physically distinguished state, the spec-trum condition and modular covariance need not be fulfilled [25]. As an intrinsic stability condition, the Modular Stability Condition has been proposed. Definition 6.3 ( [25])
For any W ∈ W , the elements ∆ itW , t ∈ R , of the modulargroup corresponding to ( R φ ( W ) , Ω φ ) are contained in J . Note that in this condition no reference is made to the space–time, its isometrygroup, or any representation of the isometry group. This condition can be posed formodels on any space–time [25]. Together with the CGMA, this modular stabilitycondition then yields both the spectrum condition and modular covariance (5.2).
Theorem 6.4 ( [25, 29])
If, in addition to the hypothesis of Theorem 6.2, theModular Stability Condition is satisfied, then after choosing suitable coordinateson R , the spectrum condition is satisfied by U ( P + ) and modular covariance holds.The associated representation ( H φ , π φ , Ω φ ) is therefore a vacuum representation. Of course, this is not, strictly speaking, a characterization of arbitrary vacuumstates; this theorem provides an intrinsic characterization of those vacuum stateswhich manifest further desirable properties, properties which are also manifestedin the models in the special circumstances considered by Bisognano and Wich-mann. But since these latter circumstances are precisely those expected to arisein standard quantum field theory, the vacuum states characterized in Theorems6.2 and 6.4 are probably the vacuum states of most direct physical interest.
Although the hypothesis of Theorem 6.4 makes no explicit or implicit referenceto an underlying space–time, Theorem 6.2 does so implicitly through use of theset of wedges W . However, the results of the preceding section did suggest thepossibility that, without any a priori reference to a space–time, the space–time Note that the continuity of the representation of the translation group follows without con-dition (c) [23]. In fact, only a four dimensional real manifold with a coordinatization is required in order toformulate and prove the theorems in Section 6, but it is nonetheless clear that the introductionof wedges as defined tacitly appeals to Minkowski space. {A i } i ∈ I of unitalC ∗ –algebras indexed by “laboratories” i ∈ I . A i is interpreted as the algebra gen-erated by all observables measurable in the laboratory i . Since it makes sense tospeak of one laboratory as being contained in another, the set I of laboratories isprovided with a natural partial order ≤ . It is then immediate that if i ≤ j then A i ⊂ A j . Hence, the map I ∋ i
7→ A i ∈ {A i } i ∈ I is order preserving. We shallassume that this map is a bijection, since otherwise there would be some redun-dancy in the description of the system. If ( I, ≤ ) is a directed set, then {A i } i ∈ I isa net and the inductive limit A of {A i } i ∈ I exists and may be used as a referencealgebra. But even if {A i } i ∈ I is not a net, it is possible [46] to naturally embed A i , i ∈ I , into a C ∗ –algebra A so that the inclusion relations are preserved. It will notbe necessary to distinguish between these cases in the results, and we shall referto states φ upon A as being states upon the net {A i } i ∈ I .Given such a state φ , we proceed to the corresponding GNS representation anddefine R i = π φ ( A i ) ′′ , i ∈ I . We shall assume that the implementing vector Ω φ iscyclic and separating for all R i , i ∈ I , and denote by J i , ∆ i , the correspondingmodular objects. Again, let J denote the group generated by the involutions J i , i ∈ I . Note that J Ω φ = Ω φ , for all J ∈ J . In this abstract context, theCGMA is the requirement that the adjoint action of each J i upon the elements of {R i } i ∈ I leaves the set {R i } i ∈ I invariant [25]. Among other matters, the CGMAhere entails that the set { J i } i ∈ I is an invariant generating set for the group J , and such a structure is the starting point for the investigations of the branch ofgeometry known as absolute geometry, see e.g. [1, 4, 5]. From such a group anda suitable set of axioms to be satisfied by the generators of that group, absolutegeometers derive various “metric” spaces such as Minkowski spaces and Euclideanspaces upon which the abstract group J now acts as the isometry group of themetric space. Different sets of axioms on the group yield different metric spaces.This affords us with the possibility of deriving a space–time from the group J , so The index set can be naturally refined by further encoding the time (with respect to somereference clock in the laboratory) during which the measurement is carried out without changingthe validity of the following assertions. In other words, the smallest group containing { J i } i ∈ I is J and J { J i } i ∈ I J − ⊂ { J i } i ∈ I forall J ∈ J . φ, {A i } i ∈ I ) would determine the space–time in whichthe quantum systems could naturally be considered to be evolving. We emphasizethat different groups J would verify different sets of algebraic relations and wouldthus lead to different space–times.For the convenience of the reader, we summarize our standing assumptions,which refer solely to objects which are completely determined by the data ( φ, {A i } i ∈ I ). Standing Assumptions
For the net {A i } i ∈ I of nonabelian C ∗ -algebras and thestate φ on A we assume(i) i
7→ R i is an order-preserving bijection;(ii) Ω φ is cyclic and separating for each algebra R i , i ∈ I ;(iii) the adjoint action of each J i leaves the set {R i } i ∈ I invariant.Already these assumptions restrict significantly the class of admissible groups J [25]. In general, it may be necessary to pass to a suitable subcollection of {R i } i ∈ I in order for the Standing Assumptions to be satisfied [25] (if, indeed, they aresatisfied at all) — see [101] for a brief discussion of this point.We must introduce some notation in order to concisely formulate the alge-braic requirements upon J which lead to the construction of three dimensionalMinkowski space. We use lower case Latin letters to denote arbitrary modularinvolutions J i , i ∈ I , upper case Latin letters to denote involutions in J of theform ab , and lower case Greek letters for arbitrary elements of J . By ξ | η we shallmean “ ξη is an involution”, and α, β | ξ, η is shorthand for “ α | ξ , β | ξ , α | η , and β | η ”. Theorem 7.1 ( [101])
Assume in the above setting that the following relationshold in J :1. For every P, Q there exists a g with P, Q | g .2. If P, Q | g, h , then P = Q or g = h .3. If a, b, c | P , then abc ∈ { J i : i ∈ I } .4. If a, b, c | g , then abc ∈ { J i : i ∈ I } .5. There exist g, h, j such that g | h but j | g, h, gh are all false.6. For each P and g with P | g false, there exist exactly two distinct elements h , h such that h , h | P is true and g, h i | R, c are false for all
R, c , i = 1 , .Then there exists a model (based on J ) of three dimensional Minkowski spacein which each J i , i ∈ I , is identified as a spacelike line (and every spacelike lineis such an element) and on which each J i , i ∈ I , acts adjointly as the reflectionabout the spacelike line to which it corresponds. J is isomorphic to P +22 andforms in a canonical manner a strongly continuous (anti)unitary representation U the proper Poincar´e group for three dimensional Minkowski space f P + . Moreover, there exists a bijection χ : I → W such that after defining R ( χ ( i )) = R i , the resultant net {R ( χ ( i )) } of wedge algebras on Minkowski spaceis covariant under the action of the representation U ( P + ) . Furthermore, one has R ( χ ( i )) ′ = R ( χ ( i ) ′ ) for all i ∈ I . Thus, if the map χ : I → W is order–preserving,then the net {R ( χ ( i )) } is local.If, further, ∆ itj ∈ J for all j ∈ I , t ∈ R , then modular covariance is satisfiedand the state ω is a vacuum state on the net {R ( χ ( i )) } . We emphasize that assumptions 1–6 are purely algebraic in nature and involveonly the group J , which is completely determined by the initial data ( φ, {A i } i ∈ I ).Although we do not propose the verification of such conditions as a practicalprocedure to determine space–time, it is, in our view, a noteworthy conceptualpoint that such a derivation is possible in principle. It is also noteworthy thatthe derived structure is so rigid and provides such a complete basis for physicalinterpretation. Indeed, from the observables and state can be derived a space–time, an identification of the localizations of the observables in that space–timeand a continuous unitary representation of the isometry group of the space–timesuch that the resultant, re–interpreted net is covariant under the action of theisometry group and the re–interpreted state is a vacuum state. It is perhaps worthmentioning that the modular symmetry group J of a theory on four dimensionalMinkowski space as discussed in Section 6 does not verify assumptions 1–6 above.Moreover, models on three dimensional Minkowski space satisfying the CGMA doverify assumptions 1–6.In [103, 104, 113] sets of algebraic conditions on J have also been found so thatthe space derived is three dimensional de Sitter space, respectively four dimensionalMinkowski space. We anticipate that similar results can be proven for other highlysymmetric space–times such as anti–de Sitter space and the Einstein universe, butnot for general space–times. It is a striking fact that, in the senses indicated above, the modular involutionsassociated with the vacuum state (and only the vacuum state) encode the followingphysically significant matters. • the space–time in which the quantum systems may be viewed as evolving • the isometry group of the space–time • a strongly continuous unitary representation of this isometry group which actscovariantly upon the net of observable algebras and leaves the state invariant • the locality, i.e. the Einstein causality, of the quantum systems the set of wedges in three dimensional Minkowski space the dynamics of the quantum systems • the scattering behavior of the quantum systems • the spin–statistics connection in the quantum systems • the stability of the quantum systems • the thermodynamic behavior of the quantum systemsIt has also become clear through examples — quantum field theories on deSitter space [10, 25, 44], anti-de Sitter space [24, 30], a class of positively curvedRobertson–Walker space–times [26,27], as well as others [92,98] — that the encod-ing of crucial physical information by modular objects and the subsequent utilityof this approach are not limited to Minkowski space theories.It is necessary to distinguish between the, in some sense, maximal results ofSection 7 and those of Section 6. The former cannot be expected to be reproduciblein most space–times, since the isometry groups are not large enough to determinethe space–time, and the arguments in Section 7 rely tacitly upon the possibility ofinterpreting the modular group J as (a suitably large subgroup of) the isometrygroup of some space–time. However, most of the results of Section 6, and hencemost of the list above, can be expected to be attainable in more general space–times, without regard to the size of the isometry group of the space–time. Ashas been verified in a class of models in a family of Robertson–Walker space–times [26], the CGMA and the encoding of crucial physical information by modularinvolutions associated with certain observable algebras and select states can holdeven when the modular symmetry group J is strictly larger than the isometrygroup of the space–time (in fact, in these examples a significant portion of J isnot associated with any kind of pointlike transformations upon the space–time).In other words, it is quite possible that the fact that the modular symmetry groupgives no more than (a subgroup of) the isometry group of the space–time in thepresence of the CGMA for theories on Minkowski or de Sitter space is an accidentdue to the fact that these space–times are maximally symmetric. Moreover, it ispossible that using the CGMA and Modular Stability Condition to select statesof physical interest yields a modular symmetry group J containing, along withthe standard symmetries expected from classical theory, new and purely quantumsymmetries encoding unexpected physical information (further evidence for thisspeculation which goes beyond [26] can be adduced in [42]).Finally, we mention that modular objects associated with privileged algebrasof observables and states (usually the vacuum) are also proving to be useful in theconstruction of quantum field models in two, three and four dimensional Minkowskispace, which cannot be constructed by previously known techniques of constructivequantum field theory [17, 28, 32, 33, 50, 69, 73, 87]. But such matters go well beyondthe scope of this paper. 22 eferences [1] J. Ahrens, Begr¨undung der absoluten Geometrie des Raumes aus dem Spiegelungsbegriff, Math. Zeitschr., , 154–185 (1959).[2] H. Araki, Von Neumann algebras of local observables for free scalar field, J. Math. Phys., , 1–13 (1964).[3] H. Araki, Mathematical Theory of Quantum Fields , (Oxford University Press, Oxford)1999.[4] F. Bachmann,
Aufbau der Geometrie aus dem Spiegelungsbegriff , second edition (Springer-Verlag, Berlin, New York) 1973.[5] F. Bachmann, A. Baur, W. Pejas and H. Wolff, Absolute geometry, in:
Fundamentals ofMathematics , Vol. 2, edited by H. Behnke, F. Bachmann, K. Fladt and H. Kunle (MITPress, Cambridge, Mass.) 1986.[6] J.J. Bisognano and E.H. Wichmann, On the duality condition for a hermitian scalar field,
J. Math. Phys., , 985–1007 (1975).[7] J.J. Bisognano and E.H. Wichmann, On the duality condition for quantum fields, J. Math.Phys., , 303–321 (1976).[8] N.N. Bogolubov, A.A. Logunov and I.T. Todorov, Introduction to Axiomatic QuantumField Theory , (W.A. Benjamin, Reading, Mass.) 1975 (translation of Russian original,published in 1969).[9] H.-J. Borchers, On the converse of the Reeh–Schlieder theorem,
Commun. Math. Phys., , 269–273 (1968).[10] H.-J. Borchers and D. Buchholz, Global properties of vacuum states in de Sitter space, Ann. Inst. Henri Poincar´e, , 23–40 (1999).[11] H.-J. Borchers, On revolutionizing quantum field theory with Tomita’s modular theory, J. Math. Phys., , 3604–3673 (2000).[12] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics I ,(Springer Verlag, Berlin, Heidelberg, New York) 1979.[13] O. Bratteli and D.W. Robinson,
Operator Algebras and Quantum Statistical MechanicsII , (Springer Verlag, Berlin, Heidelberg, New York) 1981.[14] R. Brunetti, D. Guido and R. Longo, Modular structure and duality in conformal quantumfield theory,
Commun. Math. Phys., , 201–219 (1993).[15] R. Brunetti, D. Guido and R. Longo, Group cohomology, modular theory and space–timesymmetries,
Rev. Math. Phys., , 57–71 (1995).[16] R. Brunetti, K. Fredenhagen and M. K¨ohler, The microlocal spectrum condition andWick polynomials of free fields on curved spacetimes, Commun. Math. Phys., , 633–652 (1996).[17] R. Brunetti, D. Guido and R. Longo, Modular localization and Wigner particles,
Rev.Math. Phys., , 759–785 (2002).[18] D. Buchholz and K. Fredenhagen, Locality and the structure of particle states, Commun.Math. Phys., , 1–54 (1982).[19] D. Buchholz, M. Porrmann and U. Stein, Dirac versus Wigner: Towards a universalparticle concept in local quantum field theory, Phys. Lett.,
B 267 , 377-381 (1991).[20] D. Buchholz and R. Wanzenberg, The realm of the vacuum,
Commun. Math. Phys., ,577–589 (1992).[21] D. Buchholz and S.J. Summers, An algebraic characterization of vacuum states inMinkowski space,
Commun. Math. Phys. , 449–458 (1993).
22] D. Buchholz, On the manifestations of particles, in:
Mathematical Physics Towards the21st Century , edited by R. Sen and A. Gersten (Ben Gurion University Press, Beer-Sheva)1994.[23] D. Buchholz, M. Florig and S.J. Summers, An algebraic characterization of vacuum statesin Minkowski space, II: Continuity aspects,
Lett. Math. Phys., , 337–350 (1999).[24] D. Buchholz, M. Florig and S.J. Summers, The second law of thermodynamics, TCPand Einstein causality in anti-de Sitter space–time, Class. Quantum Grav., , L31–L37(2000).[25] D. Buchholz, O. Dreyer, M. Florig and S.J. Summers, Geometric modular action andspacetime symmetry groups, Rev. Math. Phys., , 475–560 (2000).[26] D. Buchholz, J. Mund and S.J. Summers, Transplantation of local nets and geometricmodular action on Robertson–Walker space–times, in: Mathematical Physics in Mathe-matics and Physics (Siena) , edited by R. Longo,
Fields Institute Communications, ,65–81 (2001).[27] D. Buchholz, J. Mund and S.J. Summers, Covariant and quasi-covariant quantum dy-namics in Robertson–Walker space–times, Class. Quant. Grav., , 6417–6434 (2002).[28] D. Buchholz and G. Lechner, Modular nuclearity and localization, Ann. Henri Poincar´e, , 1065–1080 (2004).[29] D. Buchholz and S.J. Summers, An algebraic characterization of vacuum states inMinkowski space, III: Reflection maps, Commun. Math. Phys., , 625–641 (2004).[30] D. Buchholz and S.J. Summers, Stable quantum systems in Anti-de Sitter space: Causal-ity, independence and spectral properties,
J. Math. Phys., , 4810–4831 (2004).[31] D. Buchholz and S.J. Summers, Quantum statistics and locality, Phys. Lett. A,
J. Phys. A, , 2147–2163 (2007).[33] D. Buchholz and S.J. Summers, Warped convolutions: A novel tool in the constructionof quantum field theories, , in: Quantum Field Theory and Beyond , edited by E. Seilerand K. Sibold (World Scientific, Singapore), pp. 107–121, 2008.[34] J. Butterfield, Stochastic Einstein locality revisited,
Brit. J. Phil. Sci., , 805–867 (2007).[35] B.S. Cirel’son, Quantum generalization of Bell’s inequality, Lett. Math. Phys., , 93–100(1980).[36] R. Clifton and H. Halvorson, Entanglement and open systems in algebraic quantum fieldtheory, Stud. Hist. Phil. Mod. Phys., , 1–31 (2001).[37] W. Driessler, S.J. Summers and E.H. Wichmann, On the connection between quantumfields and von Neumann algebras of local operators, Commun. Math. Phys., , 49–84(1986).[38] W. Dybalski, A sharpened nuclearity condition and the uniqueness of the vacuum in QFT,
Commun. Math. Phys., , (2008).[39] A. Einstein, B. Podolsky and N. Rosen, Can quantum mechanical description of physicalreality be considered complete?,
Phys. Rev., , 777–780 (1935).[40] G.G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory , (JohnWiley & Sons, New York) 1972.[41] H. Epstein, V. Glaser and A. Jaffe, Nonpositivity of the energy density in quantized fieldtheories,
Nuovo Cim., , 1016–1022 (1965).
42] L. Fassarella and B. Schroer, The modular origin of chiral diffeomorphisms and theirfuzzy analogs in higher–dimensional quantum field theories,
Phys. Lett. B, , 415–425(2002).[43] C.J. Fewster and R. Verch, Stability of quantum systems at three scales: Passivity, quan-tum weak energy inequalities and the microlocal spectrum condition,
Commun. Math.Phys., , 329–375 (2003).[44] M. Florig,
Geometric Modular Action , Ph.D. Dissertation, University of Florida, 1999.[45] D. Fraser, The fate of “particles” in quantum field theories with interactions,
Stud. Hist.Phil. Mod. Phys., , 841–859 (2008).[46] K. Fredenhagen, Global observables in local quantum physics, in: Quantum and Non-Commutative Analysis , (Kluwer Academic Publishers, Amsterdam) 1993.[47] J. Glimm and A. Jaffe, Boson quantum field theory models, in:
Mathematics of Contem-porary Physics , edited by R.F. Streater, (Academic Press, London) 1972.[48] J. Glimm and A. Jaffe,
Quantum Physics , (Springer Verlag, Berlin) 1981.[49] E. Grant,
Much Ado About Nothing: Theories of Space and Vacuum from the MiddleAges to the Scientific Revolution , (Cambridge University Press, Cambridge) 1981.[50] H. Grosse and G. Lechner, Wedge–local quantum fields and noncommutative Minkowskispace,
JHEP, , 012 (2007).[51] D. Guido and R. Longo, An algebraic spin and statistics theorem, I, Commun. Math.Phys., , 517–533 (1995).[52] R. Haag,
Local Quantum Physics , (Springer-Verlag, Berlin) 1992.[53] H. Halvorson and R. Clifton, Generic Bell correlation between arbitrary local algebras inquantum field theory,
J. Math. Phys., , 1711–1717 (2000).[54] H. Halvorson, Reeh-Schlieder defeats Newton-Wigner: On alternative localizationschemes in relativistic quantum field theory, Philos. Sci., , 111–133 (2001).[55] H. Halvorson and R. Clifton, No place for particles in relativistic quantum theory?, Phil.Sci., , 1–28 (2002).[56] P.D. Hislop and R. Longo, Modular structure of the local algebras associated with thefree massless scalar field theory, Commun. Math. Phys., , 71–85 (1982).[57] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement,preprint quant-ph/0702225.[58] R. Jost, General Theory of Quantized Fields , (American Mathematical Society, Provi-dence, RI) 1965.[59] R.V. Kadison and J.R. Ringrose,
Fundamentals of the Theory of Operator Algebras , Vol-ume I, (Academic Press, Orlando) 1983.[60] R.V. Kadison and J.R. Ringrose,
Fundamentals of the Theory of Operator Algebras , Vol-ume II, (Academic Press, Orlando) 1986.[61] B.S. Kay and R.M. Wald, Theorems on the uniqueness and thermal properties of station-ary, nonsingular, quasi-free states on spacetimes with a bifurcate Killing horizon,
Phys.Rep., , 49–136 (1991).[62] M. Keyl, Fundamentals of quantum information theory,
Phys. Rep., , 431–548 (2002).[63] B. Kuckert, A new approach to spin & statistics,
Lett. Math. Phys., , 319-331 (1995).[64] B. Kuckert, Localization regions of local observables, Commun. Math. Phys., , 197–216 (2000); erratum,
Commun. Math. Phys., , 589–590 (2002).[65] B. Kuckert, Two uniqueness results on the Unruh effect and on PCT-symmetry,
Commun.Math. Phys., , 77–100 (2001).
66] B. Kuckert, Covariant thermodynamics of quantum systems: Passivity, semipassivity andthe Unruh effect,
Ann. Phys., , 216–229 (2002).[67] B. Kuckert and R. Lorenzen, Spin, statistics and reflections, II, Lorentz invariance,
Com-mun. Math. Phys., , 809–831 (2007).[68] L.J. Landau, On the violation of Bell’s inequality in quantum theory,
Phys. Lett. A, ,54–56 (1987).[69] G. Lechner, Construction of quantum field theories with factorizing S-matrices,
Commun.Math. Phys., , 821–860 (2008).[70] G. Ludwig,
Foundations of Quantum Mechanics , Volume I (Springer Verlag, Berlin, Hei-delberg and New York) 1983.[71] C. L¨uders and J.E. Roberts, Local quasiequivalence and adiabatic vacuum states,
Com-mun. Math. Phys., , 29–63 (1990).[72] J. Mund, The Bisognano–Wichmann theorem for massive theories,
Ann. Henri Poincar´e, , 907–926 (2001).[73] J. Mund, B. Schroer and J. Yngvason, String–localized quantum fields and modularlocalization, Commun. Math. Phys., , 621–672 (2006).[74] J. von Neumann,
Mathematische Grundlagen der Quantenmechanik (Springer Verlag,Berlin) 1932; English translation:
Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton) 1955.[75] J. von Neumann,
Collected Works , Volume I, edited by A.H. Taub, (Pergamon Press,New York and Oxford) 1962.[76] A. Peres, Separability criterion for density matrices,
Phys. Rev. Lett., , 1413–1415(1996).[77] W. Pusz and S.L. Woronowicz, Passive states and KMS states for general quantum sys-tems, Commun. Math. Phys., , 273–290 (1978).[78] M.J. Radzikowski, Micro-local approach to the Hadamard condition in quantum fieldtheory on curved space–time, Commun. Math. Phys., , 529–553 (1996).[79] G.A. Raggio, A remark on Bell’s inequality and decomposable normal states,
Lett. Math.Phys., , 27–29 (1988).[80] M. R´edei and S.J. Summers, Local primitive causality and the common cause principlein quantum field theory, Found. Phys., , 335–355 (2002).[81] M. R´edei and S.J. Summers, Quantum probability theory, Stud. Hist. Phil. Mod. Phys., , 390–417 (2007).[82] M. Redhead, More ado about nothing, Found. Phys., , 123–137 (1995).[83] B. Reznik, A. Retzker and J. Silman, Violating Bell’s inequality in vacuum, Phys. Rev.A, , 042104 (2005).[84] E. Schr¨odinger, Die gegenw¨artige Situation in der Quantenmechanik, Naturwis-senschaften, , 807–812, 812–828, 844–849 (1935).[85] B. Schroer, Infrateilchen in der Quantenfeldtheorie, Fortschr. Phys., , 1–31 (1963).[86] B. Schroer, Wigner representation theory of the Poincare group, localization, statisticsand the S-matrix, Nucl. Phys. B, , 519–546 (1997).[87] B. Schroer, Modular localization and the bootstrap–formfactor program,
Nucl. Phys. B, , 547–568 (1997).[88] J. Schwinger, The algebra of microscopic measurements,
Proc. Natl. Acad. Sci., USA, ,1542–1553 (1959).
89] I.E. Segal, Postulates for general quantum mechanics,
Ann. Math., , 930–948 (1947).[90] G.L. Sewell, Quantum fields on manifolds: PCT and gravitationally induced thermalstates, Ann. Phys., , 201–224 (1982).[91] R.F. Streater and A.S. Wightman,
PCT, Spin and Statistics, and All That , (Ben-jamin/Cummings Publ. Co., Reading, Mass.) 1964.[92] R. Strich, Passive states for essential observers,
J. Math. Phys., , 022301 (2008).[93] S.J. Summers and R. Werner, The vacuum violates Bell’s inequalities, Phys. Lett., , 257–259 (1985).[94] S.J. Summers and R. Werner, Bell’s inequalities and quantum field theory, I. Generalsetting,
J. Math. Phys., , 2440–2447 (1987).[95] S.J. Summers and R. Werner, Maximal violation of Bell’s inequalities for algebras ofobservables in tangent spacetime regions, Ann. Inst. Henri Poincar´e, , 215–243 (1988).[96] S.J. Summers, On the independence of local algebras in quantum field theory, Rev. Math.Phys., , 201–247 (1990).[97] S.J. Summers and R.F. Werner, On Bell’s inequalities and algebraic invariants, Lett.Math. Phys., , 321–334 (1995).[98] S.J. Summers and R. Verch, Modular inclusion, the Hawking temperature and quantumfield theory in curved space–time, Lett. Math. Phys., , 145–158 (1996).[99] S.J. Summers, Bell’s inequalities and algebraic structure, in: Operator Algebras and Quan-tum Field Theory , edited by S. Doplicher, R. Longo, J.E. Roberts, and L. Zsido (Interna-tional Press, distributed by AMS, Providence, RI), pp. 633–646, 1997.[100] S.J. Summers, On the Stone–von Neumann uniqueness theorem and its ramifications, in:
John von Neumann and the Foundations of Quantum Physics , edited by M. R´edei andM. Stoelzner, (Kluwer Academic Publishers, Dordrecht), pp. 135–152, 2001.[101] S.J. Summers and R.K. White, On deriving space–time from quantum observables andstates,
Commun. Math. Phys., , 203–220 (2003).[102] S.J. Summers, Subsystems and independence in relativistic microscopic physics, to appearin:
Stud. Hist. Phil. Mod. Phys., , (2009).[103] S.J. Summers and R.K. White, On deriving space–time from quantum observables andstates, II: Three dimensional de Sitter space, manuscript in preparation.[104] S.J. Summers and R.K. White, On deriving space–time from quantum observables andstates, III: Four dimensional Minkowski space, manuscript in preparation.[105] M. Takesaki,
Tomita’s Theory of Modular Hilbert Algebras and Its Applications , (SpringerVerlag, Berlin, Heidelberg and New York) 1970.[106] M. Takesaki,
Theory of Operator Algebras , Volume I, (Springer Verlag, Berlin, Heidelbergand New York) 1979.[107] M. Takesaki,
Theory of Operator Algebras , Volume II, (Springer Verlag, Berlin, Heidelbergand New York) 2003.[108] M. Takesaki,
Theory of Operator Algebras , Volume III, (Springer Verlag, Berlin, Heidel-berg and New York) 2003.[109] S. Trebels, ¨Uber die geometrische Wirkung modularer Automorphismen , Ph.D. Disserta-tion, University of G¨ottingen, 1997.[110] W.G. Unruh, Notes on black–hole evaporation,
Phys. Rev. D, , 870–892 (1976).[111] R. Verch and R.F. Werner, Distillability and positivity of partial transposes in generalquantum field systems, Rev. Math. Phys., , 545–576 (2005). Quant. Inform. Comput., , no. 3, 1–25 (2001).[113] R. White, An Algebraic Characterization of Minkowski Space , Ph.D. Dissertation, Uni-versity of Florida, 2001.[114] A.S. Wightman and L. G˚arding, Fields as operator–valued distributions in relativisticquantum field theory,
Ark. f. Fys., , 129–184 (1964)., 129–184 (1964).