Axiomatic Quantum Field Theory Research Articles

On mapping approaches in axiomatic quantum field theory

We here state a collection of results in axiomatic quantum field theory obtained under the general philosophy of ‘‘mapping approaches.’’ It is hoped that these results will stimulate further investigations along this direction, especially in connection with the problem of the existence and the construction of nontrivial four-dimensional quantum field theories.

Maxwell’s equations in axiomatic quantum field theory. II. Covariant and noncovariant gauges

The outlines of a general formalism for the description of the Maxwell field in the framework of axiomatic quantum field theory were given in a preceding paper. It was based on the use of locally convex topological spaces connected in a natural way with the distribution properties of the n-point functions of both the Aμ(x) and Fμν(x) fields. In this paper this approach is developed further. We discuss in particular aspects of the reconstruction theorem, gauges and their equivalence, and symmetries and gauge transformations. Finally a systematic and unified derivation is given of the various Lorentz covariant and noncovariant free field gauges together with the properties of the associated Aμ(x) operator field theories.

Instability of flat spacetime in semiclassical gravity

The semiclassical theory of gravity is considered in which an asymptotically flat background metric is coupled to quantized matter. We show that, in general, there are modes with spacelike wave vectors for small metric fluctuations around flat spacetime. Besides the usual axioms of quantum field theory in flat spacetime, the proof rests on the existence of a hard trace anomaly in the energy-momentum tensor due to matter self-couplings. Two possible interpretations of the result are discussed.

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.

Abstract concentratable and reducible linear functionals and generalized functions

In the class of linear mappings of the abstract (i.e., concretized by nothing) Poincaré invariant LIS (linear involutive space) L a f of test functions f: X∈ R 4 n → f( X)∈ C 1 into a LIS L and in the class of linear continuous mappings of the abstract Poincaré invariant locally convex topological LIS T a f into a locally convex topological LIS we axiomatize the notions of concentration of mappings in the closed sets δ⊆ R 4 n and reductions of mappings to the Borel sets δ⊆ R 4 a . The axiomatization is necessary both for constructing abstract axiomatic quantum field theory and for consistent introduction of the above notions in concrete classes of generalized functions whose test function spaces contain no nontrivial finite function.

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.

High-energy status of derivative analyticity relations

Asymptotic correlations between the phase delta (E) and the first derivative of the modulus vertical-barF (E) vertical-bar are proved to follow from the general principles of axiomatic quantum field theory (F (E) is the crossing-even or crossing-odd forward scattering amplitude), provided that certain additional assumptions concerning the properties of F (E) at infinite energy are satisfied. Analogous correlations hold, e.g., between ReF (E) and ImF (E). The asymptotic relations obtained resemble the derivative analyticity relations of Bronzan et al.; the main difference is that we introduce only the first derivative, deal with a considerably wider class of functions, and consider exclusively the high-energy asymptotic limit. Some of our relations represent a generalization of Gribov's and Midgal's result, extending its validity from the frame of Regge theory to a general consequence of axiomatic field theory. Problems related to applications are discussed.

Axiomatic restrictions on scattering amplitudes of hadrons

Problems of expsrimental verification of the quantum field theoretical principles by experimental test of any one of physical corollaries of the axiomatic quantum field theory are shortly discussed. A selective review of available axiomatic restrictions on scattering amplitudes of hadrons is given.

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.

Book Review: Introduction to axiomatic quantum field theory

Book Review: Introduction to axiomatic quantum field theory

Book reviews

Abstract Gauge Theories of Weak Interactions. By J. C. Taylor. (Cambridge University Press, 1976.) [Pp. xv+167.] £10.00. Scope: Treatise/Textbook. Level: Specialist/post-graduate. Introduction to Axiomatic Quantum Field Theory. By N. N. Bogolubov, A. A. Logunov and I. T. Todorov. (W. A. Benjamin, 1975.) [Pp. xxvi+707.] $32.50. Scope: Treatise/Textbook. Level: Specialist/post-graduate. Science in Focus. John Lenihan. (Blackie, 1975.) [Pp.64.] £2.25. Level: Undergraduate; School. Techniques of Applied Quantum Mechanics. J. P. Killingbeck. (Butterworths, 1975.) [Pp. xii + 224.] £9.00. Scope: Survey. Level: Undergraduate. Introduction to Tribology. By J. Halling. (Wykeham Publications Ltd., 1976.) [Pp. ix + 157.] £3.00. Scope: Textbook. Level: Undergraduate; school. Atlas of Optical Transforms. By G. Harburn, C. A. Taylor and T. R. Welberry. (G. Bell & Son, 1975.) [Pp. v + 33 + 31 plates.] £6.00 hardback, £3.25 paper. Level: Specialist/postgraduate. Laser Speckie and Related Phenomena. Edited by J. C. Dainty. (Springer-Verlag, 1975.) [Pp. xii + 286.] DM 94.80, $38.90. Level: For specialists only. Classical Field Theory. D. E. Soper. (Wiley Interscience, 1976.) [Pp. xi + 259.] £11.50, $23.00. Scope: Textbook. Level: Postgraduate; undergraduate. Fundamentals of Waves, Optics and Modern Physics (2nd edition). H. D. Young. (McGraw-Hill, 1976.) [Pp. xvii + 549.] £10.15. Scope: Textbook. Level: Undergraduate. Asimov's Guide to Science. Volume 1, The Physical Sciences, Volume 2, The Biological Sciences. By Isaac Asimov. (Pelican Books, 1975.) [Pp. Vol. 1, x+554; Vol. 2, vi+470.] £1.50, £1.25. Scope: General. Instabilities and Confinement in Toroidal Plasmas. Varenna Summer School, 1971. (Commission of the European Communities, Luxembourg, 1974.) [Pp. 472.] B.Fr. 1000. Scope: Conference Proceedings; Survey. Level: Specialist/post-graduate.

Theory of fourier hyperfunctions and its applications to quantum field theory

A new type of Fourier hyperfunctions is introduced. The axiomatic quantum field theory in terms of Fourier hyperfunctions is shown to include Wightman's formulation of tempered fields and its generalizations. The complete equivalence is established between the axioms for Wightman Fourier hyperfunctions and those for Green's functions by eliminating from the latter the linear growth condition of Osterwalder and Schrader.

Multiparticle eigenvectors for relativistic position operators

In the first part of the paper, the so-called generalized position operators proposed by Fleming are investigated. Compact expressions for these operators are given (a so-called transversal angular momentum operator is introduced for this purpose), and the properties of their algebraic structure are discussed. The second part is devoted to multiparticle eigenvectors for two of the three kinds of the position operators. The obtained solutions permit one to separate property the global motion of a physical system from its internal motions. The obtained internal variables are relativistic counterparts of relative velocities. The considerations are carried out in the general framework of quantum field theory and are based on the reconstruction theorem of the axiomatic quantum field theory. This also allows us to use the results for interacting systems. The discussion of Lorentz covariance of the localized states is carried out.

Solvable Models in Algebraic Statistical Mechanics

The methods of axiomatic quantum field theory have found application to quantum statistical mechanics, where it clarifies the conceptual and mathematical foundations. This book is a fairly detailed account of the four most important models in which the rigorous methods can be fully worked out.

Poincaré analytic vectors in axiomatic quantum field theory

It is demonstrated that in a Wightman quantum field theory with a denumerable set of field operators over the test function space SR, transforming covariantly among themselves under a unitary representation U(Λ, a) of the covering group of the Poincaré group, the domain D0 of vectors generated by the polynomial ring over the field operators applied on the vacuum state contains a dense invariant set of analytic vectors for U(Λ, a).

Adiabatic theorem in axiomatic quantum field theory

It is shown that Møller matricesS ± and scattering matrixS in axiomatic field theory can be expressed through their adiabatic analogs. In particular, it is proved under certain conditions that $$S_ - = \mathop {s\lim }\limits_{\alpha \to 0} S_\alpha (0,\infty )W_\alpha $$ whereW α is a trivial phase factor [i.e. a unitary operator of the form exp i / α ∝r(k)a + (k)a(k)dk]. Corresponding results in Hamiltonian approach are discussed.

Is the Maxwell Field Local?

Is the classical Maxwell field truly local? This question is raised by several observations, among them the Aharonov-Bohm effect; but the question cannot be answered without a systematic definition and characterization of local subalgebras of observables. This paper reformulates classical field theory in analogy to axiomatic quantum field theory and introduces a precise statement for local independence. (Synonyms for local independence are Einstein causality and principle of maximum signal velocity.) The formal answer of the analysis is: The free Maxwell field does not have local independence. This conclusion is critically discussed.

Local and Covariant Quantization of Linearized Einstein's Equations

An analysis is given of all the possible quantizations of the linearized Einstein equations in terms of a weakly local and/or covariant potential hμν(x). The discussion is done without making the apparently arbitrary choices which characterize the standard formulations, and special attention is paid in proving those general features which follow from basic principles and are therefore common to all local and/or covariant formulations. It is shown that the requirement of locality and/or covariance alone implies that the Einstein equations cannot hold as mean values on a dense set of states, and therefore unphysical states must be introduced in an essential way. Moreover, the requirement that the Einstein equations hold as mean values on the physical states forces the existence of states of negative norm in order to define hμν as a local and/or covariant operator. Thus the characteristic features of Gupta's formulation are shown to be shared by any local and/or covariant theory. The arbitrary choices which occur in the representation of the field operator hμν, in the definition of the metric operator and in the choice of the subsidiary condition which identifies the physical states, are shown to lead to only a one-parameter family of theories. They can be classified according to the subsidiary condition (∂μhμν+q∂νhμμ)+Ψ=0,each q ≠ −¼ identifying a possible theory. This arbitrariness, which makes the literature on the subject rather confusing, is resolved by proving that all the theories are (isometrically) equivalent. Such formulations are discussed in the framework of axiomatic quantum field theory, with particular emphasis on their group theoretical contents. Finally, an extensive treatment of Gupta's formulation is given along the lines discussed by Wightman and Gärding for quantum electrodynamics.

Bounds on the Asymptotic Behavior of an Elastic Scattering Amplitude near the Forward Direction

Upper and lower bounds on the asymptotic behavior of Im F(s, t)=A(s, v) and its higher derivatives on v for large s with v fixed to small non-positive values are derived using the results from axiomatic quantum field theory. Here, F(s, t) is a certain elastic scattering amplitude, sand t are the usual Mandelstam variables, v=t(ln(s/s 0))2 and s0 is an unknown scaling factor. The rough forms of the results are : i) C1 < A(s, v) < C2 for v1<v<O, S~co 84oo

Theorems on High Energy Collisions of Elementary Particles

This review gives a survey of the following topics: (i) results on analyticity and asymptotic growth that follow from axiomatic quantum field theory, (ii) bounds and inequalities involving cross sections at high energies, (iii) theorems and relations between cross sections requiring special assumptions, and (iv) a discussion of the Pomeranchuk theorem and of experimental and theoretical results that would hold if it were violated.