New relationships between Feynman integrals

New types of relationships between Feynman integrals are presented. It is shown that Feynman integrals satisfy functional equations connecting integrals with different values of scalar invariants and masses. A method is proposed for obtaining such relations. The derivation of functional equations for one-loop propagator- and vertex-type integrals is given. It is shown that a propagator-type integral can be written as a sum of two integrals with modified scalar invariants and one propagator massless. The vertex-type integral can be written as a sum over vertex integrals with all but one propagator massless and one external momenta squared equal to zero. It is demonstrated that the functional equations can be used for the analytic continuation of Feynman integrals to different kinematic domains.     

On the orthonormalization of generalized free Fermi-field-functionals

Dynamics of quantum field theory can be formulated by functional equations. To develop a complete functional quantum theory the physical information has to be given by functional operations only. In order to calculate the global physical observables in the functional map a functional scalar product has to be defined. This has been done in preceding papers by Stumpf. It is shown that the free Fermi fields are orthonormalized with this scalar product.     

Magnetisierungsprozesse in Kollektiven von Einbereichsteilchen. II

Certain mathematical aspects of the static pion-nucleon theory are investigated. We start from the fact that the theory in its uniformized form (cuts transformed away) leads to a system of functional equations for the S-matrix. The nonlinear mapping involved in the functional equations is a second-order Cremona transformation. After a summary of the general properties of Cremona transformations, the special transformations are studied which arise in the symmetric scalar and the Chew-Low theory respectively. The emphasis is on the possibility to separate, by means of a finite-order Cremona transformation, the functional equations into a set of uncoupled ones. For the symmetric scalar theory, the separation is trivial. For the Chew-Low theory, a proof of nonseparability is given.     

Electromagnetic solutions of the field equations in presence of zeromass mesons

The field equations of general relativity with electromagnetic stress tensor and zeromass scalar meson field are investigated. The metric coefficients are assumed to be functions of three variables only. It is then shown that, if one assumes a functional relation between some one of the metric coefficients and the electromagnetic potentials, that one can find a solution of the coupled Einstein-Maxwell equations in terms of a solution of the Einstein equations with zeromass scalar meson field as source.     

Approach to statistical equilibrium in a soluble model

The soluble model of an oscillator coupled to a scalar field is used to describe the Brownian motion of an oscillator in a thermal bath. The approach to equilibrium is shown by studying the generating functional that corresponds to a non-equilibrium situation fixed as an initial condition. It is shown how this functional goes over into the functional associated with the equilibrium situation. The solution of the functional equations is discussed.     

Green functions of scalar particles in stochastic fields

A variational method of evaluating functional integrals is proposed. This method is used to investigate the asymptotic behavior of the scalar-particle Green functions in stochastic fields. The equations for the Green functions in Euclidean space in stochastic fields are written. The solutions of these equations are represented in the form of a functional integral and then they are averaged over Gaussian stochastic fields. The variational method formulated above is used to evaluate the asymptotic behavior of these Green functions. The following equations are considered in this paper: a stochastic contribution to the mass of a scalar particle, a gauge stochastic field, and a weak stochastic contribution to the flat metric of Euclidean space.     

New measures for nonrenormalizable quantum field theory

Generic interactions characteristic of so-called nonrenormalizable scalar and spinor quantum field theories are interpreted as discontinuous perturbations in the sense that the theory does not return to the unperturbed theory as the interaction coupling vanishes. To proceed beyond this interpretation specific alternatives to conventional quantization schemes are developed. Solution of a highly idealized (independent-value), nonrenormalizable scalar field theory automatically entails a formally scale-invariant measure (rather than the conventional translation-invariant measure) in a functional integral formulation, and the success of this measure suggests its use more generally. Such a measure can be motivated (by augmented field theory) on heuristic grounds as taking into account the partial hardcore nature of the interaction responsible for its behavior as a discontinuous perturbation. This modification leads generally to what we call scale-covariant quantization, which can be formulated in terms of unconventional functional differential equations, coupled Green's function equations and operator field equations. Use of affine fields establishes equivalence of these various approaches and enables analogous coupled Green's function equations for models with fermions to be most easily obtained. The basic concepts of this program are illustrated with elementary wave-mechanical examples.     

Functional integral formulation of classical wave equations

It is shown in this paper that classical wave equations admit path integral formulations. For this, the evolution of the system is first set-up in terms of a fundamental solution or propagator. We choose this last name because it suggests a connection with functional integrals, which are exploited in this work. A functional integral in terms of non-singular functions is then proposed and shown to converge to the propagator in the appropriate limit for the case of scalar wave equations. One of the advantages of such formulation is that it provides an adequate framework for mesh-free numerical methods. This is demonstrated through a computational implementation that combines a simple second-degree polynomial local approximation of the continuous field and an approximate statement of the exact evolution equations. Numerical simulations of modal analysis and transient dynamics indicate the feasibility of the technique.     

Invariant Solutions of the Yang–Mills Field Equations Coupled to a Scalar Doublet

The Yang–Mills system of field equations which includes coupling to an SU(2) scalar matter doublet is developed. It is shown that an SU(2) current for a scalar matter doublet can be developed. The basic structure which fits the Yang–Mills system is somewhat different from the case of the scalar triplet. Using this form for the scalar current, it is possible to write down the Yang–Mills system which couples to the scalar matter doublet. It is shown that several sets of solutions to this system of equations can be obtained.     

A New Treatment of 2N and 3N Bound States in Three Dimensions

The direct treatment of the Faddeev equation for the three-boson system in 3 dimensions is generalized to nucleons. The one Faddeev equation for identical bosons is replaced by a strictly finite set of coupled equations for scalar functions which depend only on 3 variables. The spin-momentum dependence occurring as scalar products in 2N and 3N forces accompanied by scalar functions is supplemented by a corresponding expansion of the Faddeev amplitudes. After removing the spin degrees of freedom by suitable operations only scalar expressions depending on momenta remain. The corresponding steps are performed for the deuteron leading to two coupled equations.     

Self-gravitating scalar field with cubic nonlinearity

For a self-gravitating massless conformally invariant scalar field a solution is obtained to the Einstein equations for which the geometry of space-time remains arbitrary. For a scalar field with cubic nonlinearity, a static solution to the Einstein equations possessing plane symmetry is found. A cosmological model with nonlinear scalar field in the class of conformally flat Friedmann metrics is investigated. An example is given of an exact solution to the equations of the gravitational field with singularity in the infinite past.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 18–22, December, 1980.     

Simplicial vs. continuum string theory and loop equations

We derive loop equations in a scalar matrix field theory. We discuss their solutions in terms of simplicial string theory—the theory describing embeddings of two-dimensional simplicial complexes into the spacetime of the matrix field theory. This relation between the loop equations and the simplicial string theory gives further arguments that favor one of the statements of the paper hep-th/0407018. The statement is that there is an equivalence between the partition function of the simplicial string theory and the functional integral in a continuum string theory—the theory describing embeddings of smooth two-dimensional world-sheets into the spacetime of the matrix field theory in question.     

The formulation of spontaneous breakdown in the path-integral method

We examine the problem of formulating spontaneous breakdown of symmetries in the path-integral framework; since different solutions of the field equations correspond to different boundary conditions, the possibility of different boundary conditions must be incorporated in the generating functional itself. We propose a modification of the generating functional that provides the required flexibility that can lead to different solutions. We apply our formalism to a model of scalar complex fields with invariance under phase transformations of the first kind and derive Ward-Takahashi identities and the Goldstone theorem.     

Transport equations for chiral fermions to order ? and electroweak baryogenesis: Part I

This is the first in a series of two papers. In this first part, we use the Schwinger-Keldysh formalism to derive semiclassical Boltzmann transport equations, accurate to order ?, for massive chiral fermions, scalar particles, and for the corresponding CP-conjugate states. Our considerations include complex mass terms and mixing fermion and scalar fields, such that CP-violation is naturally included, rendering the equations particularly suitable for studies of baryogenesis at a first order electroweak phase transition. We provide a quantitative criterion in which case the reduction to the diagonal kinetic equations in the mass eigenbasis is justified, leading to a quasiparticle picture even in the case of mixing scalar or fermionic particles. Within the approximations we make, it is possible to first study the Boltzmann equations without the collision term. In a second paper [Ann. Phys. xxx (2004) xxx] we discuss the collision terms and reduce the Boltzmann equations to fluid equations.     

Relativistic Shifted-l Expansion Technique for Dirac and Klein-Gordon Equations

The shifted-l expansion technique (SLET) is extended to solve for Dirac particle trapped in spherically symmetric scalar and/or kvector potentials. A parameter λ = 0, l is introduced in such a way that one can obtain the Klein-Gordon (KG) bound states from Dirac bound states. The 4-vector Coulomb, the scalar linear, and the equally mixed scalar and 4-vector power-law potentials are used in KG and Dirac equations. Exact numerical results are obtained for the Cvector Coulomb potential in both KG and Dirac equations. Highly accurate and fast converging results are obtained for the scalar linear and the equally mixed scalar and 4-vector power-law potentials.     

Relativistic n-body wave equations in scalar quantum field theory

The variational method in a reformulated Hamiltonian formalism of Quantum Field Theory (QFT) is used to derive relativistic n-body wave equations for scalar particles (bosons) interacting via a massive or massless mediating scalar field (the scalar Yukawa model). Simple Fock-space variational trial states are used to derive relativistic n-body wave equations. The equations are shown to have the Schrödinger non-relativistic limits, with Coulombic interparticle potentials in the case of a massless mediating field and Yukawa interparticle potentials in the case of a massive mediating field. Some examples of approximate ground state solutions of the n-body relativistic equations are obtained for various strengths of coupling, for both massive and massless mediating fields.     

On Average Properties of Inhomogeneous Fluids in General Relativity: Perfect Fluid Cosmologies

For general relativistic spacetimes filled with an irrotational perfect fluid a generalized form of Friedmann's equations governing the expansion factor of spatially averaged portions of inhomogeneous cosmologies is derived. The averaging problem for scalar quantities is condensed into the problem of finding an "effective equation of state" including kinematical as well as dynamical "backreaction" terms that measure the departure from a standard FLRW cosmology. Applications of the averaged models are outlined including radiation-dominated and scalar field cosmologies (inflationary and dilaton/string cosmologies). In particular, the averaged equations show that the averaged scalar curvature must generically change in the course of structure formation, that an averaged inhomogeneous radiation cosmos does not follow the evolution of the standard homogeneous-isotropic model, and that an averaged inhomogeneous perfect fluid features kinematical "backreaction" terms that, in some cases, act like a free scalar field source. The free scalar field (dilaton) itself, modelled by a "stiff" fluid, is singled out as a special inhomogeneous case where the averaged equations assume a simple form.     

Accurate integration of scalar equations in multiscalar-tensor theory

The integration of scalar equations in theories generalizing Brans—Dicke—Jordan—Fierz scalar—tensor theory is considered. Conditions under which these equations may be integrated by complete variable separation are established. Under these conditions, the scalar equations take the form of classical equations of motion for a single particle moving in scalar space in an external force field.Institute of High-Power Electronics, Siberian Branch, Russian Academy of Sciences. Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 79–84, February, 1995.