Functions In Quantum Field Theory Research Articles

A generalization of the classical moment problem on *-algebras with applications to relativistic quantum theory. I.

A (non-commutative) generalization of the classical moment problem is formulated on arbitrary *-algebras with units. This is used to produce aC*-algebra associated with the space of test functions for quantum fields. ThisC*-algebra plays a role in theories of bounded localized observables in Hilbert space which is similar to that of the space of test functions in quantum field theories (namely it is represented in Hilbert space). The case of local quantum fields which satisfy a slight generalization of the growth condition is investigated.

Legendre transforms of the generating functionals in quantum field theory

Legendre transforms of the generating functionals in quantum field theory

On entire functionals in quantum field theory

The paper contains an investigation of convergence and asymptotic behaviour of the generating functionals on mass shell in the Quantum Field Theory of one real, scalar, selfinteracting field. Also the relations between the Hilbert space fo entire functionals and the various spaces of exponentially bounded entire functionals are treated in some detail. There result various estimates for the functional series and double series representing vectors, operators and their functional derivatives.

Analytic Functionals in Quantum Field Theory and the Regularization of Ultraviolet Divergences

The commutator and propagator distributions associated with free-particle fields are represented as boundary values of analytic kernels in complex space-time. A distribution product of such quantities is then defined as a boundary value of the product of the analytic kernels of its various factors, with the proviso that the product kernel be understood in the sense of Hadamard's ``finite part'' in those cases when it would otherwise be nonintegrable. For the case of quantum electrodynamics, the momentum-space representation of the products of causal propagators are derived. It is demonstrated by explicit calculation that the second-order scattering matrix elements thus obtained are identical to those usually obtained by means of Pauli-Villars regularization and renormalization.

Analytic Functionals in Quantum Field Theory

A class of Lorentz invariant generalized functions can be defined as analytic functionals, i.e., as continuous linear functionals which are contour integrals over a suitable space of test functions. These generalized functions include in particular the invariant functions of quantum field theory, but also include ``propagators with higher order poles.'' The analysis shows exactly which of these are well defined, especially in the important special case of zero mass. Various applications to quantum field theory are indicated.

Use of the Generalized Jost Function in Quantum Field Theory

The structure of the S-matrix is investigated in terms of the so-called generalized Jost function. Starting from the partial wave dispersion relation, the analytic properties of this function can be derived in the single-channel scattering, and it is shown that there exists a close connection between the analyticity of this function and the generalized Levinson relation. In the case where the one-particle singularity in scattering amplitudes is caused by the elementary particle, the corresponding Jost function has a pole, while in the case where this singularity is caused by the composite bound) state, the Jost function does not have the pole. Singularities corresponding to the Castillejo- Dalitz-Dyson ambiguity are completely separated. Thus, by making the conjecture that the analyticity in the low-momentum region still remains in the multichannel case, the Levinson theorem can be extended to the multichannel case and to the S- matrix theory where the Hamiltonian is not used explicitly. A new representation of the S-matrix, and how the structure of the Smatrix changes according to each case mentioned above, is clarified.

Asymptotic Properties of the Wave Function for a Bound Nonrelativistic Three-Body System

The asymptotic properties of a Schrödinger wave function which represents the bound ground state of a system of three interacting particles are examined. It is assumed that the interaction can be described by a static potential which is the sum of three two-body potentials and one three-body potential, where the potentials have the property that if any one of the particles is separated from the other two by a distance which tends to infinity, then the part of the potential energy which depends on the position of that particle tends to zero. The problem is treated nonrelativistically. Decreasing exponential bounds on the ground-state three-body wave function are established in configuration space. It is shown that these bounds depend only on the masses of the three particles, on the ground-state energy of the three-body system, and on the lower bounds on the spectra of the Hamiltonians for the three two-body systems arising when one of the three particles is removed and the remaining two interact through the corresponding two-body potential. It is furthermore shown that the three-body wave function in momentum space admits of analytic continuation into a tube-region in complex momentum space, which tube-region is the product of real momentum space and a convex bounded region in imaginary momentum space. The tube-region is explicitly determined. The implications of the results for the theory of the vertex function in quantum field theory are discussed. The relevancy of the results obtained in this paper to the variational computation of energy levels and wave functions of three-body systems is also briefly discussed. In the course of the derivation some generalizations are pointed out; in particular, the final results remain valid for a class of more general interactions, which cannot be described by static potentials. Likewise, the results remain valid for a class of spin-dependent interactions.

On distributions and spaces of sequences, II. On the multiplications of improper functions in quantum field theory

On distributions and spaces of sequences, II. On the multiplications of improper functions in quantum field theory

Wightman Functions, Retarded Functions and Their Analytic Continuations

Recent studies on the analyticity of the general n-point Wightman functions and the retarded functions in quantum field theory are reviewed. According to Wightman, the conventional axiom of quantum field theory is transcribed into an equivalent set of properties of the Wightman functions. It is then shown that the Wightman functions are boundary values of analytic functions in complex coordinate spaces and the linear properties of the Wightman functions are transcribed into an equivalent requirement on domains of analyticity of these analytic functions. For the study of retarded functions, the contributions of vacuum intermediate states are systematically subtracted from the Wightman functions. The truncated Wightman functions thus obtained have the same linear properties as those of the Wightman functions except for a smaller support in the momentum space. A generali­ zation of 0-function (the step function of Heaviside) to the characteristic function of a convex polyhedral cone in an n-dimensional vector space is defined and is used to define the generalized retarded functions. The properties of the latter which are equivalent to the linear properties of the Wightman functions are stated. It is then shown that the generalized retarded functions are boundary values of analytic functions in complex momentum spaces and a part of the properties of the former are transcribed into an equivalent requirement on domains of analyticity of the latter. The whole discussion is carried out regardless of the choice between local commutativity and anticommutativity. The problem of obtaining the envelope of holomorphie remains unsolved.

On the Many-Particle Structure of Green's Functions in Quantum Field Theory

The structure of the expectation values of retarded multiple commutators (r functions) is analyzed in terms of the number of particles in the decomposition of absorptive parts. As to the one-particle structure, it is found that an r function is a sum of a finite number of terms, each of them except one (that one being called one-particle irreducible) being in momentum space a product of one-particle irreducible factors, linked by one-particle propagation functions. As to the two-particle structure, it is found that a one-particle irreducible function is the solution of an inhomogeneous Bethe-Salpeter equation, whose kernel and inhomogeneous term both are two-particle irreducible functions. This structure, which could be generalized to higher particle numbers, closely resembles perturbation theory but is here derived from locality and the asymptotic condition alone, by converting the nonlinear system of integral equations for r functions stepwise into one in which neither one- or two-particle reducible functions, nor one- or two-particle intermediate states appear. The implication of such structure analysis for an interpretation of perturbation theory, improvements of present methods to derive analytic properties of scattering amplitudes, and a formalism with unstable particles are discussed, and the strength of singularities of various functions investigated.

High Energy Limit of Vertex Function in Quantum Field Theory

High Energy Limit of Vertex Function in Quantum Field Theory

High-Energy Behavior in Quantum Field Theory

An attack is made on the problem of determining the behavior at high energies and momenta of the Green's functions of quantum field theory, using new mathematical methods from the theory of real variables. We define a class ${A}_{n}$ of functions of $n$ real variables, whose behavior may be specified in a certain manner by means of certain asymptotic coefficients. The Feynman integrands of perturbation theory (with energies taken imaginary) belong to such classes. We then prove that if certain conditions on the coefficients are satisfied then an integral over $k$ of the variables converges, and belongs to the class ${A}_{n\ensuremath{-}k}$ with new coefficients simply related to the old ones. When applied to perturbation theory this theorem validates the renormalization procedure of Dyson and Salam, proving that the renormalized integrals actually do always converge, and provides a simple rule for calculating the behavior of any Green's function to any order of perturbation theory.

Formal Theory of Scattering in the Quantum Field Theory

The advantages of the exponential form of the transformation function are exploited. It is shown how to define the representation in which all effects of the self-field of physical particles on themselves are eliminated while the remainder is presented in the form of an effective velocity-dependent potential among them. One can speak of the transformation function which transforms eigenfunctions of the free-particle Hamiltonian into eigenfunctions of the total Hamiltonian: the paradox, which has been pointed out by Van Hove and further studied by Haag with regard to the existence of such a transformation function in quantum field theory, can be circumvented by considering the exponent as its representative. This exponent has a Lorentz-invariant representation. The exponent of the $S$ matrix is related in a very simple manner to the exponent of the transformation function over a finite interval of time. The asymptotic condition as used in the quantum field theory is also analyzed from our viewpoint.

One particle singularities of green functions in quantum field theory

The time ordered Green’s functions are expanded with respect to the singularities arising from intermediate one-particle states.

Many-Body Problem in Quantum Field Theory

A formalism for the covariant description of scattering processes involving composite particles is derived from first principles. Its basic ingredients are the Green's functions of quantum field theory combined with a physically sound formulation of the adiabatic hypothesis. Explicit expressions are presented for various examples of nucleon-nucleus and boson-nucleus collisions. It is shown that the impulse approximation can be formulated for such problems. A general method of obtaining electromagnetic moments of nuclei, based on the scattering formalism, is also derived. The normalization condition for covariant amplitudes is discussed and its application to bound-state problems reviewed. In particular, a method of carrying out perturbation theory for the discrete spectrum is suggested.

Structure of Green's Functions in Quantum Field Theory. II

The Green's functions are vacuum expectation values of time ordered products of field operators in the Heisenberg representation. They give us vital information about the nature of the interacting particles and quanta represented by these field operators. It is shown that the Fourier transforms of the Green's functions (for up to four operators) are expressible as parametric integrals involving invariant energy denominators and real, scalar weight functions which are termed the spectral functions. Relativity, causality, and some other fundamental assumptions of field theory are required to derive the result.The spectral functions have a simple physical interpretation, and completely specify the structure of the Green's functions. The equations of motion which hold between Green's functions of different order can be translated into the corresponding relations between the spectral functions. Renormalization can be carried out explicitly, bringing the equations for the spectral functions into a manifestly renormalized form. The causality condition also serves as a means of obtaining renormalized quantities without recourse to the usual subtraction procedure. The observed masses and coupling constants occur in the equations for the spectral functions as external parameters fixing the boundary condition. No unobservable bare masses and couplings, nor the (infinite) renormalization constants ever appear in the equations. These quantities, however, are shown to be expressible in terms of the spectral functions.The Green's functions involving more than four field operators are not considered in this paper.

Structure of Green's Functions in Quantum Field Theory

The Green's functions are vacuum expectation values of time ordered products of field operators in the Heisenberg representation. They give us vital information about the nature of the interacting particles and quanta represented by these field operators. It is shown that the Fourier transforms of the Green's functions (for up to four operators) are expressible as parametric integrals involving invariant energy denominators and real, scalar weight functions which are termed the spectral functions. Relativity, causality, and some other fundamental assumptions of field theory are required to derive the result.The spectral functions have a simple physical interpretation, and completely specify the structure of the Green's functions. The equations of motion which hold between Green's functions of different order can be translated into the corresponding relations between the spectral functions. Renormalization can be carried out explicitly, bringing the equations for the spectral functions into a manifestly renormalized form. The causality condition also serves as a means of obtaining renormalized quantities without recourse to the usual subtraction procedure. The observed masses and coupling constants occur in the equations for the spectral functions as external parameters fixing the boundary condition. No unobservable bare masses and couplings, nor the (infinite) renormalization constants ever appear in the equations. These quantities, however, are shown to be expressible in terms of the spectral functions.The Green's functions involving more than four field operators are not considered in this paper.