Axiomatic Quantum Field Theory Research Articles

A derivation of upper bounds for forward and fixed-momentum-transfer scattering amplitudes at high energies

The purpose of this short paper is to give, using continuous variables, a simple derivation of high-energy upper bounds for forward and nearly forward (fixed momentum transfer) scattering amplitudes which satisfy the analyticity properties of axiomatic quantum field theory. As usual, we consider the case of neutral scalar particles of equal mass. Our starting point is the Oran,s representation of the scattering amplitude (~)

Field-Theoretic Formulation of Quantum Statistical Mechanics

The notions of strong convergence of state vectors, introduced by Haag in his formalism of axiomatic quantum field theory, are extended to the case of vectors with an infinite number of particles but finite densities. Some general properties of nonequilibrium distribution functions are derived without the use of power series expansions or any other simplifying assumption. An integral representation is obtained for the distribution functions which makes it possible to discuss their behavior for small and large energies and to obtain some information about the singularities of these functions when continued analytically.

Difficulties with the definition of charge in axiomatic quantum field theory

Two basic approaches to define the charge are considered, and the difficulties arising from them in quantum field theory are studied. A mathematically compact treatment is given to the proposals which have been made to overcome them. The relationship between definition of charge and vanishing of the divergence of the current is investigated, and some related theorems are derived.

Restriction on the integrals of local currents

We prove a generalized version of two theorems proposed by S. Coleman. The proof is based on the usual axioms of quantum field theory.

On the Definition of Scattering in Relativistic Quantum Field Theory

A relativistic interacting theory for neutral bosons and in which there can be no stable bound states is studied. The theory is obtained by embedding two-particle relativistic quantum mechanics in quantum field theory by means of a procedure recently given by Sudarshan. This results in a theory in which there is scattering in the two-particle channel only. An « interacting » field operator ψx is defined for the theory. It is then found that under the usual formulation of axiomatic quantum field theory ψx must be regarded as a « local free-field operator ». However, there is inherent in ψx scattering information. A new asymptotic condition for field operators is then proposed, whereby this scattering information may be extracted. A heuristic discussion of the physical significance of the asymptotic condition is also given.

Multiplicative Symmetries in Axiomatic Quantum Field Theory

The group of all multiplicative symmetries defined for a finite number of interacting fields is studied in detail. Various theorems are proved connecting the existence of multiplicative symmetries with properties of the Wightman distributions of the fields.

Nuclear potential in axiomatic quantum field theory

It is shown here that a transition from axiomatic quantum field theory to Schrödinger equation can be accomplished provided the extrapolation of the scattering matrix off the energy shell is performed in a certain way. However, if one wants an equation which is valid for all physical energies, the appropriate equation is what is called here the “Dirac equation” from which the Schrödinger equation is obtained as an approximation. The potential appearing in the “Dirac equation” is expressed in closed form in terms of field-theory matrix elements.

Representation of Particle States in Quantum Field Theory

The usual axioms of quantum field theory are modified to allow a uniform treatment of stable and unstable particles without making explicit use of asymptotic conditions. A definition is proposed for the physical state of a single, neutral, scalar (or pseudoscalar) boson. The consistency of this definition requires the corresponding one-particle amplitude to satisfy an integral equation whose solutions depend on the mass spectrum and the preparation mechanism of the particle. The unstable particle decay law is obtained from the one-particle amplitude and at very long times appears likely to depend on the details of the preparation. For stable particles, the formulation given in this paper is shown to coincide in an asymptotic sense with the well-known Lehmann, Symanzik, and Zimmermann formulation. The generalizations to many-particle states and to particles with spin \textonehalf{} are indicated briefly.

Nuclear potential in quantum field theory

It is proposed to use the Low equation to define the potential between two nucleons. The work is entirely within the framework of axiomatic quantum field theory.

On the "Edge of the Wedge" Theorem

In the mathematical justification of the formal calculations of axiomatic quantum field theory and the theory of dispersion relations, a strategic role is played by a theorem on analytic functions of several complex variables which has been given the euphonious name of the edge of the wedge theorem. The statement of the theorem seems to be due originally to N. Bogoliubov (cf. 3, Mathematical Appendix, pp. 654-673) but no complete proof which is fully satisfactory from the mathematical point of view has yet appeared in the literature.

On the axioms of quantum field theory

Ein Satz von Axiomen fur die Quantenfeldtheorie wird diskutiert, der in einigen Punkten von dem Lehmann-Symanzik-Zimmermannschen abweicht, jedoch ebenfalls Lorentzinvarianz, Asymptotenbedingung und Kausalitat enthalt. Ausgangspunkt ist die nichtrelativistische kanonische Feldtheorie, welche in bestimmter Weise auf den relativistischen Fall ubertragen wird. Wegen des Haagschen Theorems mus dabei der Begriff des lokalen, kausalen Feldes verallgemeinert werden.