Publications

Publication List of Stephen J. Summers

1. Appendix to K.Osterwalder and R.Schrader, Axioms for Euclidean Green’s functions, II

S.J. Summers, Communications in Mathematical Physics, 42, 281-305 (1975).

2. A new proof of the asymptotic nature of perturbation theory in P(φ)2 models

S.J. Summers, Helvetica Physica Acta, 53, 1-30 (1980).

Abstract

A new method of proving the asymptotic nature of perturbation theory (for Schwinger and generalized Schwinger functions) in P(φ)2 quantum field models, which does not presume the convergence of a cluster expansion, is presented. The method recaptures all previous such results and, also, (in application to certain models) results not yet attainable by previous methods. An explicit proof is given for the 4)2 model in the two-phase region to illustrate the essential points. Application to all P(φ)2 models with mean field limits is discussed.

3. On the phase diagram of a P(φ)2 quantum field model

S.J. Summers, Annales de l’Institut Henri Poincaré, 34, 173-229 (1981).

Abstract

The phase diagram of a two-dimensional Bose quantum field model (with polynomial self-interaction of degree six) is rigorously verified, except in a neighborhood of the expected critical points, by the construction of distinct states satisfying the Osterwalder-Schrader axioms coexisting along the expected phase transition lines of the diagram. Perturbation theory in the respective states is proven to be asymptotic (without the use of a convergent cluster expansion), yielding asymptotic expansions to arbitrary order for the generalized Schwinger functions throughout the diagram. A strong estimate on the positions (in parameter space) of the double and triple points is given.

4. Normal product states for fermions and twisted duality for CCR- and CAR-type algebras with application to the Yukawa2 quantum field model

S.J. Summers, Communications in Mathematical Physics, 86, 111-141 (1982).

Abstract

We present sufficient conditions that imply duality for the algebras of local observables in all Abelian sectors of all locally normal, irreducible representations of a field algebra if twisted duality obtains in one of these representations. It is verified that the Yukawa model in two space-time dimensions satisfies these conditions, yielding the first proof of duality for the observable algebra in all coherent charge sectors in this model. This paper also constitutes the first verification of the assumptions of the axiomatic study of the structure of superselection sectors by Doplicher, Haag and Roberts in an interacting model with nontrivial sectors. The existence of normal product states for the free Fermi field algebra and, thus, the verification of the “funnel property” for the associated net of local algebras are demonstrated.

5. Nonexistence of quantum fields associated with two-dimensional spacelike planes

W. Driessler and S.J. Summers, Communications in Mathematical Physics, 89, 221-226 (1983).

Abstract

It is shown that in a relativistic quantum field theory satisfying Wightman’s axioms, there are no nontrivial field-like operators, or even bilinear forms, associated to a two- (or less) dimensional spacelike manifold in Minkowski space. This generalizes Wightman’s result that fields cannot be defined as operators at a point and stands in contrast to Borchers’ result that field operators can be associated with one-dimensional timelike manifolds.

6. On commutators and self-adjointness

W. Driessler and S.J. Summers, Letters in Mathematical Physics, 7, 319-326 (1983).

Abstract

For A a symmetric and H a self-adjoint (not necessarily semi-bounded) operator on a Hilbert space, we give conditions in terms of the boundedness of operators of the form (H + z)-p(ad H)n(A)(H + z)-q, z complex and n,p,q natural numbers, which imply essential self-adjointness of A on any core of some power of H. By specializing to the case of semibounded H and/or A, we arrive at the same conclusions under weaker assumptions. Our results generalize several previous ones of the same nature and are best-possible. Applications to quantum mechanics and quantum field theory are indicated.

7. A dense set of cyclic vectors for quantum field polynomial algebras

W. Driessler and S.J. Summers, Journal of Mathematical Physics, 24, 2809-2819 (1983).

Abstract

It is shown that in the Hilbert space of a quantum field theory with a nonzero mass gap there exists a dense set of vectors, each entire analytic for the energy-momentum operators and each cyclic for the global and local polynomial algebras. It is proven that for every vector Φ from this dense set there exists an element Q of the polynomial algebra which maps Φ onto the vacuum vector and which annihilates the vacuum. A similar, stronger result is proven for free field theories (including mass zero).

8. Central decomposition of Poincaré-invariant nets of local field algebras and absence of spontaneous breaking of the Lorentz group

W. Driessler and S.J. Summers, Annales de l’Institut Henri Poincaré, 43, 147-166 (1985).

Abstract

We study reducible, Poincaré-invariant representations of nets of local field algebras and prove a number of structure results, some of which are generalizations of previous work on nets of observable algebras by Araki and some of which are quite new. Using these we examine the central decomposition of such nets, study the spontaneous breaking of the Lorentz group symmetry under such decompositions into pure phases, and consider the significance of the modular automorphism groups of the wedge algebras in this context.

9. The vacuum violates Bell’s inequalities

S.J. Summers and R.F. Werner, Physics Letters, 110 A, 257-259 (1985).

Abstract

It is found that the vacuum state in any Bose or Fermi free quantum field theory violates Bell’s inequalities maximally, i.e. in principle, with suitable detectors, maximal violations of Bell’s inequalities may be obtained without setting up a source. We explain, however, why it would be difficult to measure such violations.

10. On the decomposition of relativistic quantum field theories into pure phases

W. Driessler and S.J. Summers, Helvetica Physica Acta, 59, 331-348 (1986).

Abstract

We give two new independent sufficient conditions which individually insure that the extremal decomposition of a Wightman state on the polynomial algebra (equivalently, the Borchers algebra) of a relativistic quantum field is actually a decomposition into pure phases, i.e. the clustering property is satisfied in each extremal state occurring in the decomposition. Moreover, the corresponding representation also decomposes into a direct integral of irreducible representations with unique vacua.

11. On the connection between quantum fields and von Neumann algebras of local operators

W. Driessler, S.J. Summers and E.H. Wichmann, Communications in Mathematical Physics, 105, 49-84 (1986).

Abstract

The relationship between a standard local quantum field and a net of local von Neumann algebras is discussed. Two natural possibilities for such an association are identified, and conditions for these to obtain are found. It is shown that the local net can naturally be so chosen that it satisfies the Special Condition of Duality. The notion of an intrinsically local field operator is introduced, and it is shown that such an operator defines a local net with which the field is locally associated. A regularity condition on the field is formulated, and it is shown that if this condition holds, then there exists a unique local net with which the field is locally associated if and only if the field algebra contains at least one intrinsically local operator. Conditions under which a field and other fields in its Borchers class are associated with the same local net are found, in terms of the regularity condition mentioned.

12. Bell’s inequalities and quantum field theory, I. General setting

S.J. Summers and R.F. Werner, Journal of Mathematical Physics, 28, 2440-2447 (1987).

Abstract

Bell’s inequalities are briefly presented in the context of order-unit spaces and then studied in some detail in the framework of C*-algebras. The discussion is then specialized to quantum field theory. Maximal Bell correlations β(φ,A(O1),A(O2)) for two subsystems localized in spacetime regions O1 and O1 and constituting a system prepared in the state φ are defined, along with the concept of maximal Bell violations. After the study of these ideas in general, properties of these correlations in vacuum states of arbitrary quantum field models are studied. For example, it is shown that in the vacuum state the maximal Bell correlations decay exponentially with the product of the lowest mass and the spacelike separation of O1 and O2. This paper is also preparation for the proof in Paper II that Bell’s inequalities are maximally violated in the vacuum state.

13. Bell’s inequalities and quantum field theory, II. Bell’s inequalities are maximally violated in the vacuum

S.J. Summers and R.F. Werner, Journal of Mathematical Physics, 28, 2448-2456 (1987).

Abstract

In the context of the study of Bell’s inequalities carried out in Paper I, it is proven that Bell’s inequalities are maximally violated in the vacuum state by suitable spacelike separated observables for both Bose and Fermi free quantum field theories.

14. Maximal violation of Bell’s inequalities is generic in quantum field theory

S.J. Summers and R.F. Werner, Communications in Mathematical Physics, 110, 247-259 (1987).

Abstract

Under weak technical assumptions on a net of local von Neumann algebras in a Hilbert space, which are fulfilled by any net associated to a quantum field satisfying the standard axioms, it is shown that for every vector state in the Hilbert space, there exist observables localized in complementary wedge-shaped regions in Minkowski space-time which maximally violate Bell’s inequalities in the given vector state. If, in addition, the algebras corresponding to wedge-shaped regions are injective (which is known to be true in many examples), then the maximal violation occurs in any state given by a density matrix on the Hilbert space.

15. Concerning the condition of additivity in quantum field theory

S.J. Summers and E.H. Wichmann, Annales de l’Institut Henri Poincaré, 47, 113-124 (1987).

Abstract

The condition of additivity for local von Neumann algebras is discussed within the framework of local quantum field theory. It is shown that this condition holds for algebras of observables associated with wedge-shaped regions in Minkowski space-time if the system of local algebras is associated with a local quantum field in a weak sense. And under somewhat stronger conditions, additivity is shown to hold for arbitrary regions for the algebras of a certain minimal net generated by the quantum field.

16. From algebras of local observables to quantum fields: Generalized H-bounds

S.J. Summers, Helvetica Physica Acta, 60, 1004-1023 (1987).

Abstract

Previous results on obtaining quantum fields as limits of sequences of bounded, local operators (local observables) are extended to generalized H-bounds and ultradistribution fields. A topology on the net of local observable algebras is specified such that each limit point of suitable sequences in the topology determines an ultradistribution (resp. tempered distribution) quantum field which is associated to the net in a certain strong sense and which satisfies an L1-continuous generalized H-bound. And it is shown that an ultradistribution (or tempered distribution) quantum field which satisfies an L1-continuous generalized H-bound is associated to a net of local observable algebras if and only if it is obtainable as such a limit.

17. Maximal violation of Bell’s inequalities for algebras of observables in tangent spacetime regions

S.J. Summers and R.F. Werner, Annales de l’Institut Henri Poincaré, 49, 215-243 (1988).

Abstract

We continue our study of Bell’s inequalities and quantum field theory. It is shown in considerably broader generality than in our previous work that algebras of local observables corresponding to complementary wedge regions maximally violate Bell’s inequality in all normal states. Pairs of commuting von Neumann algebras which maximally violate Bell’s inequalities in all normal states are characterized. Also, algebras of local observables corresponding to tangent double cones are shown to maximally violate Bell’s inequalities in all normal states in dilatation-invariant theories, in free quantum field models, and in a class of interacting models. Further, it is proven that such algebras are not split in any theory with an ultraviolet scaling limit.

18. Bell’s inequalities and quantum field theory

S.J. Summers, in: Quantum Probability and Applications, V, edited by L. Accardi and W. von Waldenfels, (Berlin, Springer-Verlag) Lecture Notes in Mathematics, # 1442, pp. 393-413, 1990.

Abstract

The present state of mathematically rigorous results about Bell’s inequalities in relativistic quantum field theory is reviewed. In addition, the nature of the statistical independence of algebras of observables associated to spacelike separated spacetime regions is discussed.

19. On the independence of local algebras in quantum field theory

S.J. Summers, Reviews in Mathematical Physics, 2, 201-247 (1990).

Abstract

A review is made of the multitude of different mathematical formalizations of the physical concept “two observables (or two systems) are independent” which have been proposed in quantum theories, particularly relativistic quantum field theory. The most basic mathematical formulation of independence in any quantum theory is what one may call kinematical independence: the two observables, resp. the observables of the two quantum systems, which are represented by elements of a C*-algebra, resp. two subalgebras of a C*-algebra, are required to commute. This is related to a mathematical formulation of the notion of the coexistence (or compatibility) of two observables. Another basic notion of independence, generally called statistical independence in the literature, is, roughly speaking, two quantum systems are said to be statistically independent if each can be prepared in any state, how ever the other system has been prepared. There are numerous mathematical formulations of this notion, and their interrelationships are explained. Statistical independence and kinematical independence are shown to be logically independent. Additional notions such as strict locality and their relationship to statistical independence are discussed. The mathematics of a more quantitative measure of statistical independence, Bell’s inequalities, is reviewed, and its relations with previously introduced notions are indicated. All of these notions are then viewed in application to relativistic quantum field theory.

20. An algebraic characterization of vacuum states in Minkowski space

D. Buchholz and S.J. Summers, Communications in Mathematical Physics, 155, 449-458 (1993).

Abstract

An algebraic characterization of vacuum states on nets of C*-algebras over Minkowski space is given and space-time translations are reconstructed with the help of the modular structures associated with such states. The result suggests that a “condition of geometrical modular action” might hold in quantum field theories on a wider class of spacetime manifolds. It also yields a derivation of the dynamics of quantum field theories from the operationally given data of an experiment – the preparation and the observables.

21. Beyond coherent states: Higher order representations

G. Reents and S.J. Summers, in: On Klauder’s Path: A Field Trip, edited by G.G. Emch, G.C. Hegerfeldt and L. Streit, (Singapore, World Scientific), pp. 179-188, 1994.

Abstract

A class of representations of the canonical commutation relations is presented which naturally subsumes coherent and symplectic (i.e. quasifree) representations. Necessary and sufficient conditions are presented for the unitary inequivalence of these representations with the Fock representation. Hence a new class of representations for bosonic systems with infinitely many degrees of freedom is opened up.

22. On Bell’s inequalities and algebraic invariants

S.J. Summers and R.F. Werner, Letters in Mathematical Physics, 33, 321-334 (1995).

Abstract

Some algebraic invariants associated with Bell’s inequalities are defined for inclusions of von Neumann algebras and studied in the context of general algebraic quantum theory. More special results are proven for quantum field theory, which establish that these invariants take infinitely many values. Sharp short-distance bounds on the Bell correlations are also demonstrated in the context of relativistic quantum field theory.

23. Quadratic representations of the canonical commutation relations

M. Proksch, G. Reents and S.J. Summers, Publications of the Research Institute of Mathematical Sciences, Kyoto University, 31, 755-804 (1995).

Abstract

This paper studies a class of representations (called quadratic) of the canonical commutation relations over symplectic spaces of arbitrary dimension, which naturally generalizes coherent and symplectic (i.e. quasifree) representations and which has previously been heuristically employed in the special case of finite degrees of freedom in the physics literature. An explicit characterization of canonical quadratic transformations in terms of a “standard form” is given, and it is shown that they can be exponentiated to give representations of the Weyl algebra. Necessary and sufficient conditions are presented for the unitary equivalence of these representations with the Fock representation. Possible applications to quantum optics and quantum field theory are briefly indicated.

24. Modular inclusion, the Hawking temperature and quantum field theory in curved space-time

S.J. Summers and R. Verch, Letters in Mathematical Physics, 37, 145-158 (1996).

Abstract

A recent result by Borchers connecting geometric modular action, modular inclusion and the spectrum condition, is applied in quantum field theory on spacetimes with a bifurcate Killing horizon (these are generalizations of black hole space-times, comprising the familiar black hole spacetime models). Within this framework we give sufficient, model-independent conditions ensuring that the temperature of thermal equilibrium quantum states is the Hawking temperature. And we verify that these conditions are satisfied by the net of local algebras associated to the free quantum field on any space-time with bifurcate Killing horizon.

25. Geometric modular action and transformation groups

S.J. Summers, Annales de l’Institut Henri Poincaré, 64, 409-432 (1996).

Abstract

We study a weak form of geometric modular action, which is naturally associated with transformation groups of partially ordered sets and which provides these groups with projective representations. Under suitable conditions it is shown that these groups are implemented by point transformations of topological spaces serving as models for space-times, leading to groups which may be interpreted as symmetry groups of the space-times. As concrete examples, it is shown that the Poincaré group and the de Sitter group can be derived from this condition of geometric modular action. Further consequences and examples are discussed.

26. On the statistical independence of algebras of observables

M. Florig and S.J. Summers, Journal of Mathematical Physics, 38, 1318-1328 (1997).

Abstract

We re-examine various notions of statistical independence presently in use in algebraic quantum theory, establishing alternative characterizations for such independence, some of which are also valid without assuming that the observable algebras mutually commute. In addition, in the context which holds in concrete applications to quantum theory, the equivalence of three major notions of statistical independence is proven.

27. Bell’s inequalities

S.J. Summers, commissioned article in: Encyclopaedia of Mathematics: Supplement Volume 1, edited by M. Hazewinkel (Dordrecht, Kluwer Academic Publishers), pp. 94-95, 1997.

Abstract

This is a brief introduction to Bell’s inequalities for the Encyclopaedia of Mathematics.

28. Bell’s inequalities and algebraic structure

S.J. Summers, in: Operator Algebras and Quantum Field Theory, edited by S. Doplicher, R. Longo, J.E. Roberts, and L. Zsido (International Press, distributed by AMS), pp. 633-646, 1997.

Abstract

We provide an overview of the connections between Bell’s inequalities and algebraic structure.

29. An algebraic characterization of vacuum states in Minkowski space, II: Continuity aspects

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    D. Buchholz, M. Florig and S.J. Summers, Letters in Mathematical Physics, 49, 337-350 (1999).

    Abstract

    An algebraic characterization of vacuum states in Minkowski space is given which relies on recently proposed conditions of geometric modular action and modular stability for algebras of observables associated with wedge-shaped regions. In contrast to previous work, continuity properties of these algebras are not assumed but derived from their inclusion structure. Moreover, a unique continuous unitary representation of spacetime translations is constructed from these data. Thus the dynamics of relativistic quantum systems in Minkowski space is encoded in the observables and state and requires no prior assumption about any action of the spacetime symmetry group upon these quantities.

    30. Further representations of the canonical commutation relations

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    M. Florig and S.J. Summers, Proceedings of the London Mathematical Society, 80, 451-490 (2000).

    Abstract

    We construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (i.e. the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize which such perturbations lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, resp. quasi-equivalent, with Fock, coherent or quasifree representations.

    31. Geometric modular action and spacetime symmetry groups

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    D. Buchholz, O. Dreyer, M. Florig and S.J. Summers, Reviews in Mathematical Physics, 12, 475-560 (2000).

    Abstract

    A condition of geometric modular action is proposed as a selection principle for physically interesting states on general space-times. This condition is naturally associated with transformation groups of partially ordered sets and provides these groups with projective representations. Under suitable additional conditions, these groups induce groups of point transformations on these space-times, which may be interpreted as symmetry groups. The consequences of this condition are studied in detail in application to two concrete space-times – four-dimensional Minkowski and three-dimensional de Sitter spaces – for which it is shown how this condition characterizes the states invariant under the respective isometry group. An intriguing new algebraic characterization of vacuum states is given. In addition, the logical relations between the condition proposed in this paper and the condition of modular covariance, widely used in the literature, are completely illuminated.

    32. The second law of thermodynamics, TCP, and Einstein causality in anti-de Sitter space-time

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    D. Buchholz, M. Florig and S.J. Summers, Classical and Quantum Gravity, 17, L31-L37 (2000).

    Abstract

    If the vacuum is passive for uniformly accelerated observers in anti-de Sitter space-time ( i.e. cannot be used by them to operate a perpetuum mobile ), they will (a) register a universal value of the Hawking-Unruh temperature, (b) discover a TCP symmetry, and (c) find that observables in complementary wedge-shaped regions are commensurable (local) in the vacuum state. These results are model independent and hold in any theory which is compatible with some weak notion of space-time localization.

    33. On the Stone-von Neumann uniqueness theorem and its ramifications

S.J. Summers, in: John von Neumann and the Foundations of Quantum Physics, edited by M. Rédei and M. Stölzner (Vienna Circle Yearbook series, Kluwer Academic Press), pp. 135-152, 2001. See the Vienna Circle’s web site for further information.

Abstract

A brief history of the Stone-von Neumann uniqueness theorem and its ramifications is provided. The influence of this theorem on the development of quantum theory, which was its initial source of motivation, is emphasized. In addition, its impact upon mathematics itself is suggested by considering certain subsequent developments in originally unanticipated directions.

34. Transplantation of local nets and geometric modular action on Robertson-Walker space-times

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    D. Buchholz, J. Mund and S.J. Summers, Fields Institute Communications, 30, 65-81 (2001).

    Abstract

    A novel method of transplanting algebras of observables from de Sitter space to a large class of Robertson-Walker space-times is exhibited. It allows one to establish the existence of an abundance of local nets on these spaces which comply with a recently proposed condition of geometric modular action. The corresponding modular symmetry groups appearing in these examples also satisfy a condition of modular stability, which has been suggested as a substitute for the requirement of positivity of the energy in Minkowski space. Moreover, they exemplify the conjecture that the modular symmetry groups are generically larger than the isometry and conformal groups of the underlying space-times.

    35. Local primitive causality and the common cause principle in quantum field theory

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    M. Rédei and S.J. Summers, Foundations of Physics, 32, 335-355 (2002).

    Abstract

    If {A(V)} is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V1 and V2 are spacelike separated spacetime regions, then the system (A(V1),A(V2),φ) is said to satisfy the Weak Reichenbach’s Common Cause Principle iff for every pair of projections A ε A(V1), B ε A(V2) correlated in the normal state φ there exists a projection C belonging to a von Neumann algebra associated with a spacetime region V contained in the union of the backward light cones of V1 and V2 and disjoint from both V1 and V2, a projection having the properties of a Reichenbachian common cause of the correlation between A and B. It is shown that if the net has the local primitive causality property then every local system (A(V1),A(V2),φ) with a locally normal and locally faithful state φ and open bounded V1 and V2 satisfies the Weak Reichenbach’s Common Cause Principle.

    36. Covariant and quasi-covariant quantum dynamics in Robertson-Walker space-times

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    D. Buchholz, J. Mund and S.J. Summers, Classical and Quantum Gravity, 19, 6417-6434 (2002).

    Abstract

    We propose a canonical description of the dynamics of quantum systems on a class of Robertson-Walker space-times. We show that the worldline of an observer in such space-times determines a unique orbit in the identity component SO0(4,1) of the local conformal group of the space-time and that this orbit determines a unique transport on the space-time. For a quantum system on the space-time modeled by a net of local algebras, the associated dynamics is expressed via a suitable family of “propagators”. In the best of situations, this dynamics is covariant, but more typically the dynamics will be “quasi-covariant” in a sense we make precise. We then show by using our technique of “transplanting” states and nets of local algebras from de Sitter space to Robertson-Walker space that there exist quantum systems on Robertson-Walker spaces with quasi-covariant dynamics. The transplanted state is locally passive, in an appropriate sense, with respect to this dynamics.

    37. On deriving space-time from quantum observables and states

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    S.J. Summers and R.K. White, Communications in Mathematical Physics, 237, 203-220 (2003). Note that the copyright is held by Springer-Verlag.

    Abstract

    We prove that, under suitable assumptions, operationally motivated quantum data completely determine a space-time in which the quantum systems can be interpreted as evolving. At the same time, the dynamics of the quantum system is also determined. To minimize technical complications, this is done in the example of three-dimensional Minkowski space.

    38. An algebraic characterization of vacuum states in Minkowski space, III: Reflection maps

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    D. Buchholz and S.J. Summers, Communications in Mathematical Physics, 246, 625-641 (2004). Note that the copyright is held by Springer-Verlag.

    Abstract

    Employing the algebraic framework of local quantum physics, vacuum states in Minkowski space are distinguished by a property of geometric modular action. This property allows one to construct from any locally generated net of observables and corresponding state a continuous unitary representation of the proper Poincaré group which acts covariantly on the net and leaves the state invariant. The present results and methods substantially improve upon previous work. In particular, the continuity properties of the representation are shown to be a consequence of the net structure, and surmised cohomological problems in the construction of the representation are resolved by demonstrating that, for the Poincaré group, continuous reflection maps are restrictions of continuous homomorphisms.

    39. Stable quantum systems in Anti-de Sitter space: Causality, independence and spectral properties

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    D. Buchholz and S.J. Summers, Journal of Mathematical Physics, 45, 4810-4831 (2004).

    Abstract

    If a state is passive for uniformly accelerated observers in n-dimensional (n \geq 2) anti-de Sitter space-time, i.e. cannot be used by them to operate a perpetuum mobile, they will (a) register a universal value of the Unruh temperature, (b) discover a PCT symmetry, and (c) find that observables in complementary wedge-shaped regions necessarily commute with each other in this state. The stability properties of such a passive state induce a “geodesic causal structure” on AdS and concommitant locality relations. It is shown that observables in these complementary wedge-shaped regions fulfill strong additional independence conditions. In two-dimensional AdS these even suffice to enable the derivation of a nontrivial, local, covariant net indexed by bounded spacetime regions. All these results are model-independent and hold in any theory which is compatible with a weak notion of space-time localization. Examples are provided of models satisfying the hypotheses of these theorems.

    40. Remarks on causality in relativistic quantum field theory

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    M. Rédei and S.J. Summers, International Journal of Theoretical Physics, 44, 1029-1039 (2005). Republished, this time with the correct (!!!) authors, in: International Journal of Theoretical Physics, 46, 2053-2062 (2007).

    Abstract

    It is shown that the correlations predicted by relativistic quantum field theory in locally normal states between projections in local von Neumann algebras A(V1),A(V2) associated with spacelike separated spacetime regions V1,V2 have a (Reichenbachian) common cause located in the union of the backward light cones of V1 and V2. Further comments on causality and independence in quantum field theory are made.

    41. Quantum statistics and locality

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    D. Buchholz and S.J. Summers, Physics Letters A, 337, 17-21 (2005).

    Abstract

    It is shown that two observers have mutually commuting observables if they are able to prepare in each subsector of their common state space some state exhibiting no mutual correlations. This result establishes a heretofore missing link between statistical and locality (commensurability) properties of observables in relativistic quantum physics. The analysis is based on a discussion of coincidence experiments and leads to a quantitative measure of deviation from locality. Hence, it may be applied in intrinsically nonlocal theories such as string theory and field theory on noncommutative spacetime.

    42. Geometric modular action and spontaneous symmetry breaking

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    D. Buchholz and S.J. Summers, Annales Henri Poincaré, 6, 607-624 (2005).

    Abstract

    We study spontaneous symmetry breaking for field algebras on Minkowski space in the presence of a condition of geometric modular action (CGMA) proposed earlier as a selection criterion for vacuum states on general space-times. We show that any internal symmetry group must commute with the representation of the Poincaré group (whose existence is assured by the CGMA) and each translation-invariant vector is also Poincaré invariant. The subspace of these vectors can be centrally decomposed into pure invariant states and the CGMA holds in the resulting sectors. As positivity of the energy is not assumed, similar results may be expected to hold for other space-times.

    43. Tomita-Takesaki modular theory

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    S.J. Summers, commissioned article in: Volume 5 of the Encyclopedia of Mathematical Physics , edited by J.-P. Françoise, G. Naber and T.S. Tsun, (Elsevier Press), pp. 251-257, 2006.

    Abstract

    We provide an brief overview of Tomita-Takesaki modular theory and some of its applications to mathematical physics.

    44. Scattering in relativistic quantum field theory: Fundamental concepts and tools

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    D. Buchholz and S.J. Summers, commissioned article in: Volume 4 of the Encyclopedia of Mathematical Physics , edited by J.-P. Françoise, G. Naber and T.S. Tsun, (Elsevier Press), pp. 456-465, 2006.

    Abstract

    We provide a brief overview of the basic tools and concepts of quantum field theoretical scattering theory.

    45. Quantum probability theory

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    M. Rédei and S.J. Summers, Studies in History and Philosophy of Modern Physics, 38, 390-417 (2007).

    Abstract

    The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.

    46. String- and brane-localized causal fields in a strongly nonlocal model

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    D. Buchholz and S.J. Summers, Journal of Physics A, 40, 2147-2163 (2007).

    Abstract

    We study a weakly local, but nonlocal model in spacetime dimension d \geq 2 and prove that it is maximally nonlocal in a certain specific quantitative sense. Nevertheless, depending on the number of dimensions d, it has string-localized or brane-localized operators which commute at spatial distances. In two spacetime dimensions, the model even comprises a covariant and local subnet of operators localized in bounded subsets of Minkowski space which has a nontrivial scattering matrix. The model thus exemplifies the algebraic construction of local operators from algebras associated with nonlocal fields.

    47. Warped convolutions: A novel tool in the construction of quantum field theories

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    D. Buchholz and S.J. Summers, in: Quantum Field Theory and Beyond , edited by E. Seiler and K. Sibold (World Scientific, Singapore), pp. 107-121, 2008.

    Abstract

    Recently, Grosse and Lechner introduced a novel deformation procedure for non-interacting quantum field theories, giving rise to interesting examples of wedge-localized quantum fields with a non-trivial scattering matrix. In the present article we outline an extension of this procedure to the general framework of quantum field theory by introducing the concept of warped convolutions: given a theory, this construction provides wedge-localized operators which commute at spacelike distances, transform covariantly under the underlying representation of the Poincaré group and admit a scattering theory. The corresponding scattering matrix is nontrivial but breaks the Lorentz symmetry, in spite of the covariance and wedge-locality properties of the deformed operators.

    48. Subsystems and independence in relativistic microscopic physics

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    S.J. Summers, Studies in History and Philosophy of Modern Physics, 40, 133-141 (2009).

    Abstract

    The analyzability of the universe into subsystems requires a concept of the “independence” of the subsystems, of which the relativistic quantum world supports many distinct notions which either coincide or are trivial in the classical setting. The multitude of such notions and the complex relations between them will only be adumbrated here. The emphasis of the discussion is placed upon the warrant for and the consequences of a particular notion of subsystem independence, which, it is proposed, should be viewed as primary and, it is argued, provides a reasonable framework within which to sensibly speak of relativistic quantum subsystems.

    49. When are quantum systems operationally independent?

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    M. Rédei and S.J. Summers, International Journal of Theoretical Physics, 49, 3250-3261 (2010).

    Abstract

    We propose some formulations of the notion of “operational independence” of two subsystems S1,S2 of a larger quantum system S and clarify their relation to other independence concepts in the literature. In addition, we indicate why the operational independence of quantum subsystems holds quite generally, both in nonrelativistic and relativistic quantum theory.

    50. Yet more ado about nothing: The remarkable relativistic vacuum state

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    S.J. Summers, in: Deep Beauty , edited by H. Halvorson (Cambridge University Press, Cambridge and New York), pp. 317–341, 2011.

    Abstract

    An overview is given of what mathematical physics can currently say about the vacuum state for relativistic quantum field theories on Minkowski space. Along with a review of classical results such as the Reeh-Schlieder Theorem and its immediate and controversial consequences, more recent results are discussed. These include the nature of vacuum correlations and the degree of entanglement of the vacuum, as well as the striking fact that the modular objects determined by the vacuum state and algebras of observables localized in certain regions of Minkowski space encode a remarkable range of physical information, from the dynamics and scattering behavior of the theory to the external symmetries and even the space-time itself. These modular objects also provide an intrinsic characterization of the vacuum state itself, a fact which is of particular relevance to the search for criteria to select physically significant reference states for quantum field theories on curved space-times.

    51. Warped convolutions, Rieffel deformations and the construction of quantum field theories

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    D. Buchholz, G. Lechner and S.J. Summers, Communications in Mathematical Physics, 304, 95-123 (2011).

    Abstract

    Warped convolutions of operators were recently introduced in the algebraic framework of quantum physics as a new constructive tool. It is shown here that these convolutions provide isometric representations of Rieffel’s strict deformations of C*-dynamical systems with automorphic actions of Rn, whenever the latter are presented in a covariant representation. Moreover, the device can be used for the deformation of relativistic quantum field theories by adjusting the convolutions to the geometry of Minkowski space. The resulting deformed theories still comply with pertinent physical principles and their Tomita-Takesaki modular data coincide with those of the undeformed theory; but they are in general inequivalent to the undeformed theory and exhibit different physical interpretations.

    52. A perspective on constructive quantum field theory

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    S.J. Summers, in: Fundamentals of Physics (Eolss Publishers, Oxford), 2012; this is a volume of UNESCO’s scientific encyclopedia, Encyclopedia of Life Support Systems , available online here and in hardcopy in the near future. An extended and occasionally updated version of this paper is available here .

    Abstract

    An overview of the accomplishments of constructive quantum field theory is provided.

    Other Publications

  • Commissioned Book Review of: An Axiomatic Basis for Quantum Mechanics, Vol. I and II, by G. Ludwig; Physics Today, 72-74 (August, 1988).
  • Commissioned Book Review of: The Algebraic Theory of Superselection Sectors: Introduction and Recent Results, edited by D. Kastler; Mathematical Reviews, 5162-5163 (September, 1993).
  • Commissioned Book Review of: Local Quantum Physics, by R. Haag; Mathematical Reviews, 2308 (April, 1994).
  • Commissioned Book Review of: Quantum Groups, Quantum Categories and Quantum Field Theory, by J. Fröhlich and T. Kerler; Mathematical Reviews, 3700-3701 (June, 1995).
  • Commissioned Book Review of: Operatoralgebraic Methods in Quantum Field Theory, by H. Baumgärtel; Mathematical Reviews, 3302-3303 (May, 1997).
  • Commissioned Book Review of: Mathematical Theory of Quantum Fields, by H. Araki; Mathematical Reviews (online) (2002).
  • 218 commissioned reviews of research papers in: Mathematical Reviews.
  • “On the Impossibility of a Poincare-Invariant Vacuum State with Unit Norm” Refuted, online preprint at arXiv.org: arXiv:0802.2935 .
  • A nontechnical (no formulae) overview of recent advances in the construction of quantum field models in AQFT is HERE . I am keeping this overview up to date with current developments.
  • Last updated on February 10, 2014