Wightman Theory Research Articles

Reeh-Schlieder Theorem for Ultrahyperfunctional Wightman Theory

It will be shown that the Reeh-Schlieder property holds for states of quantum fields for ultrahyperfunctional Wightman theory. As by product, it is shown that the Reeh-Schlieder property also holds for states of quantum fields on a non-commutative Minkowski space in the setting ultrahyperfunctional.

Noncommutative deformations of Wightman quantum field theories

Quantum field theories on noncommutative Minkowski space are studied in a model-independent setting by treating the noncommutativity as a deformation of quantum field theories on commutative space. Starting from an arbitrary Wightman theory, we consider special vacuum representations of its Weyl-Wigner deformed counterpart. In such representations, the effect of the noncommutativity on the basic structures of Wightman theory, in particular the covariance, locality and regularity properties of the fields, the structure of the Wightman functions, and the commutative limit, is analyzed. Despite the nonlocal structure introduced by the noncommutativity, the deformed quantum fields can still be localized in certain wedge-shaped regions, and may therefore be used to compute noncommutative corrections to two-particle S-matrix elements.

FROM EUCLIDEAN FIELD THEORY TO QUANTUM FIELD THEORY

In order to construct examples for interacting quantum field theory models, the methods of Euclidean field theory turned out to be powerful tools since they make use of the techniques of classical statistical mechanics. Starting from an appropriate set of Euclidean n-point functions (Schwinger distributions), a Wightman theory can be reconstructed by an application of the famous Osterwalder–Schrader reconstruction theorem. This procedure (Wick rotation), which relates classical statistical mechanics and quantum field theory, is, however, somewhat subtle. It relies on the analytic properties of the Euclidean n-point functions. We shall present here a C*-algebraic version of the Osterwalder–Schrader reconstruction theorem. We shall see that, via our reconstruction scheme, a Haag–Kastler net of bounded operators can directly be reconstructed. Our considerations also include objects, like Wilson loop variables, which are not point-like localized objects like distributions. This point of view may also be helpful for constructing gauge theories.

Generalized free field-like U(1)-Gauge theories within the wightman framework

Generalized free field-like U(1)-gauge theories are considered as an application of the Wightman theory to some special gauge field theories within the Borchers-Uhlmann approach to general QFT (without linear program). Along these lines, conditions upon the pseudo-Wightman functional are given such that the reconstructed indefinite-metric space D of state vectors decomposes into the direct sum of charge sectors labelled by the elements of the Pontrjagin dual group of a compact and Abelian gauge group G . All these gauge groups G are characterized in such a way that the positive definiteness of states of the vacuum sector holds. Furthermore, observable fields not carrying any charges as well as charge carrying fields are given. Finally, an example covering charged scalar free fields of mass m > 0 is discussed within that context.

Relativistic invariance and charge conjugation in quantum field theory

We prove that superselection sectors with finite statistics in the sense of Doplicher, Haag, and Roberts are automatically Poincare covariant under natural conditions (e.g. split property for space-like cones and duality for contractible causally complete regions). The same holds for topological charges, namely sectors localized in space-like cones, providing a converse to a theorem of Buchholz and Fredenhagen. We introduce the notion of weak conjugate sector that turns out to be equivalent to the DHR conjugate in finite statistics. The weak conjugate sector is given by an explicit formula that relates it to the PCT symmetry in a Wightman theory. Every Euclidean convariant sector (possibly with infinite statistics) has a weak conjugate sector and the converse is true under the above natural conditions. On the same basis, translation covariance is equivalent to the property that sectors are sheaf maps modulo inner automorphisms, for a certain sheaf structure given by the local algebras. The construction of the weak conjugate, sector also applies to the case of local algebras onS1 in conformal theories. Our main tools are the Bisognano-Wichmann description of the modular structure of the von Neumann algebras associated with wedge regions in the vacuum sector and the relation between Jones index theory for subfactors and the statistics of superselection sectors.

Discrete vacuum superselection rule in Wightman theory with essentially self-adjoint field operators

The main results of earlier work by the author, Sushko, and Khoruzhii describing the algebraic structure of quantum-field systems with (discrete) vacuum superselection rules are generalized to the large class of Wightman theories with essentially self-adjoint field operators (a very strong restriction was imposed on the theory, namely, that the polynomial Op algebra of the Wightman fields /rho/ belongs to the class II, i.e., /rho/ /sub s'/ =/rho/ /sub w'/). It is also shown that the field Op algebra of a Wightman theory with discrete vaccum superselection rule possesses a class II extension.

Euclidean invariant integral representations for Schwinger functionals

It is shown that if the Schwinger functions of a Wightman theory with a mass gap can be represented as moments of a complex measure, there is also such a representation by a measure that is invariant under the Euclidean group. A necessary and sufficient condition is also given for the exponential of a quadratic form on the test function space to be the characteristic functional of a complex measure.

An axiomatic framework for supersymmetries

A framework is proposed in which supersymmetries can be defined in the language of axiomatic quantum field theory. This framework contains a nuclear *-algebra into which the Borchers’ algebra of a general supermultiplet is embedded through a map which, in some sense, generalizes the concept of a superfield. The algebra of supersymmetry is represented on the constructed nuclear *-algebra as an algebra of graded derivations. Since a graded derivation cannot be integrated to a usual automorphism group (only to a formal group), it is assumed that conditions at the Lie algebraic level are strong enough to produce supersymmetric behavior. Thus it is conjectured that if a functional of the nuclear *-algebra is annihilated by the algebra of supersymmetry, and if this functional is related to a state of the Borchers’ algebra through the embedding map, this state through the GNS construction gives rise to a supersymmetric Wightman theory. This supersymmetric condition produces an infinite number of correlations among the n-point functions. The 2-point Wightman functions for a general supermultiplet are completely analyzed and it is found that their behavior is similar to the perturbative results. Finally it is proved that the free fields satisfy these supersymmetric conditions.

Consequences of weakening the positivity property of Wightman quantum field theories

A weaker form of the notion of a Wightman theory is introduced. The weakened form embodies all the requirements of a Wightman theory. However, the consequences of local commutativity are obtained without an assertion that local commutation can be stated in terms of the field operators. Models are presented that illustrate the usefulness of the weakened requirements.

Ultradistributions and quantum fields: Fourier-Laplace transforms and boundary values of analytic functions

The distribution theory is the mathematical framework of the axiomatic quantum field theory of A.S. Wightman. The axioms are satisfied in the case of the free field and in some non-trivial models studied in the constructive quantum field theory introduced by J. Glimm and A. Jaffe (two and three dimensions). No non-trivial example in four dimensions is known. The vacuum expectation values in the Wightman theory can have at most polynomial increase in momentum space. A. Jaffe has extended the axioms in order to allow non-polynomial increase in momentum space. In this paper we discuss the ultradistribution framework which is the most general framework for Jaffe fields (strictly localizable fields). The ultradistributions have been introduced by A. Beurling, G. Björck and independently by C. Roumieu. Ultradistribution theory is a natural generalization of the distribution theory. We study the Fourier-Laplace transform of ultradistributions, extenting results of A. Jaffe [8,9] in several directions. A Bochner-Schwartz theorem for ultradistributions is also shown to be valid. We expect ultradistribution theory to play a role in constructive quantum field theory.

The Algebraic Structure of the Thirring Model

AbstractWe summarize the representation theory of the group SU(1,1) as needed for the massless Thirring model. Representations of the current operator algebra are given taking account of conformal covariance. The conformal covariance transformation behaviour of the Thirring field is investigated. The Haag‐Araki‐Kastler observable algebra of the Thirring field is reconstructed from the Wightman theory of this model.

Maxwell’s equations in axiomatic quantum field theory. I. Field tensor and potentials

An approach to the investigation of the Maxwell field in the framework of axiomatic quantum field theory is presented which employs Borchers’ algebraic reformulation of Wightman theory in a modified form adapted to the special features of the electromagnetic field. This makes it possible to clarify the relation between tensor and potential field operators, the meaning and properties of different gauges, the sense in which field equations hold and the properties of state spaces with their special subspaces.

Euclidean Markov fields of higher integer spin. II. Massless case

The Euclidean spin-1 massless vector field and the Euclidean spin-2 massless tensor field in general convariant gauges are shown to be Markovian. However, the reflection property is not satisfied. This indicates a close connection between the requirement of indefinite metric Hilbert space in Wightman theory and the violation of the reflection property in the Euclidean theory of electromagnetic and 'gravitational' potentials.

Euclidean Markov fields of higher integer spin. I. Massive case

The general Lagrangian approach of Takahashi and Umezawa (1953) is used for the construction of free massive spin-1 and spin-2 Euclidean Markov fields. It is shown that these fields can be put in the framework of Markov field theory. An example of a Euclidean field which is Markovian but non-reflexive is also given. This field does not lead to a Wightman theory in the Minkowski region.

Gauge Problem in Quantum Field Theory. II. Difficulties of Combining Einstein Equations and Wightman Theory

The results of a previous paper on the gauge problem in quantum electrodynamics are generalized to the gravitational case. In particular, the difficulties connected with the quantization of the gravitational field are analyzed in the framework of axiomatic field theory. For convenience, the simple case of weak gravitational field in vacuo is discussed. Even in this simple case, inconsistencies arise if one wants to combine the Einstein equations and quantum field theory. Under the assumptions: (1) existence of the vacuum, invariant under the Poincar\'e group, (2) existence of a representation of the Poincar\'e group such that the fields have tensor transformation properties, and (3) analyticity of the two-point function in the forward tube, it is proved that the Einstein equations for the gravitational potential have no solution apart from the trivial one ${R}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}\ensuremath{\sigma}}=0$. The result is obtained without assuming either local commutativity or the spectral condition or a positive metric in the Hilbert space. Thus the difficulties which arise in the quantization of the gravitational field have very little to do with the Hilbert-Lorentz condition, indefinite metric, etc.; rather, they are strongly connected with the definition of the Riemann tensor in terms of the gravitational potential. As a corollary of the above result, the representations of the Poincar\'e group for massless spin-2 particles in quantum field theory are shown to be essentially different from the corresponding ones of the classical case.

Spontaneous breakdown of symmetry in axiomatic theory

We prove two results; the first is that it is possible to prove Goldstone’s theorem in a Wightman theory in which there is a conserved current with a meaning for a sharp time and which generates a broken symmetry. The second is that for such theories there exist infinitely many positive linear functionals on the algebra of quasi-local operators, that are Lorentz invariant and lead to a theory with positive energy. That is, we may regard the vacuum functional as non-unique. These two results are illustrated in the theory of the free massless neutral scalar field.