Five-dimensional theory of interacting scalar,electromagnetic, and gravitational fields

An (n+1) factorization of an (n+1)-dimensional Riemann manifold is performed. For a space permitting a Killing vector, the (n+l)-dimensional Hubert variational principle reduces to the variational principle for the corresponding quantities in an n-dimensional space. Hence, setting n=4 and n=3, versions of a unified theory of gravitational, electromagnetic, and scalar fields and the steady-space theory of general relativity theory, respectively, are constructed. The five-dimensional variational principle for geodesics reduces to the four-dimensional leastaction principle for the test charged particle moving in gravitational, electromagnetic, and scalar fields.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 58–65, November, 1979.     

Hamiltonian models of interacting fermion fields in quantum field theory

We consider Hamiltonian models representing an arbitrary number of spin 1 / 2 fermion quantum fields interacting through arbitrary processes of creation or annihilation of particles. The fields may be massive or massless. The interaction form factors are supposed to satisfy some regularity conditions in both position and momentum space. Without any restriction on the strength of the interaction, we prove that the Hamiltonian identifies to a self-adjoint operator on a tensor product of antisymmetric Fock spaces and we establish the existence of a ground state. Our results rely on new interpolated \(N_\tau \) estimates. They apply to models arising from the Fermi theory of weak interactions, with ultraviolet and spatial cutoffs.

     

Hamiltonian structure of gravitational field theory

Hamiltonian generalizations of Einstein's theory of gravitation introducing a laminar structure of spacetime are discussed. The concepts of general relativity and of quasi-inertial coordinate systems are extended beyond their traditional scope. Not only the metric, but also the coordinate system, if quantized, undergoes quantum fluctuations.     

On the theory of inertial gravitational fields

The generalized equations of the inertial gravitational field are derived from variational principles. It is shown that variational properties of inertial gravitational potentials have important peculiarities which cause peculiarities of equations obtained.     

A Hamiltonian formulation of the Einstein-Cartan-Sciama-Kibble theory of gravity

We postulate the energy-momentum functionE for the ECSK theory of gravity and formulate the functional Hamiltonian equation in terms of the energy-momentum functionE and the symplectic 2-form . The system of partial differential equations which follows from the functional Hamilton equation is equivalent to the system of variational equations of the ECSK theory. The Hamiltonian method gives rise to a natural division of these equations into 10 constraint equations and the set of dynamical equations. We discuss the geometric sense of the constraint equations and their relations to the initial value problem.     

Canonical structure of the theory of gauge fields interacting with matter fields

The dynamics of gauge-field interacting with matter field is discussed. It is shown that the dynamics is a symplectic relation in a certain symplectic space. Momenta canonically conjugate to both gauge-field and matter field are found. It is shown that the gauge invariance of the theory implies the existence of constraints and vice-versa.     

Lagrangian formulation of a general Hamiltonian theory with constraints

A Lagrangian formulation is constructed for a general Hamiltonian theory with constraints. A modification is proposed of the standard procedure of the Hamiltonianization of a Lagrangian theory in the case when the Lagrangian theory has primary constraints. The obtained results are used to establish the Lagrangian form of infinitesimally small canonical transformations in the Hamiltonian formalism.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 104–112, March, 1986.     

On the theory of interacting fields in the Foldy-Wouthuysen representation

The overview is devoted to quantum electrodynamics (QED) and the Standard Model in the Foldy-Wouthuysen representation. The Hamiltonian H _FW in the form of a power series in charge e is obtained as applied to the electromagnetic interaction in the FW representation. Quantum electrodynamics in lowest-order perturbation theory is examined. Calculations of specific QED processes are presented. For external fermion lines (p _f ² = m ²), a possibility to expand the scattering matrix, in powers of the coupling constant with matrix elements, not including fermion propagators, is shown. To take into account particle-antiparticle interaction, a modification of the Foldy-Wouthuysen representation is proposed. Fermions in the modified FW representation can be in two states that are characterized by the sign of a third component of the isotopic spin T _f ³ . The sign of T _f ³ is connected with the sign at mass terms in the modified Hamiltonian H _FW. Real fermions (p _f ² = m _f ² ), as well as antifermions, can interact with each other, while real fermions with a given sign of T _f ³ can only interact with real antifermions with the opposite sign of T _f ³ , and vice versa. The formulation of the Standard Model in the FW representation does not necessarily require an interaction of Higgs bosons with fermions. In this approach, the role of Higgs bosons narrows considerably as they are responsible only for gauge invariance of the theory and interact only with gauge bosons. Quantum electrodynamics in the FW representation is invariant under C, P, and T transformations. Weak interaction does not conserve C and P parity, but conserves combined CP parity. The theory allows a connection of CP violation and total or partial violation of isotopic symmetry in the modified Foldy-Wouthuysen representation.     

Republication of: Quantum theory of weak gravitational fields

This is an English translation of a paper by Matvei Bronstein, first published in German in 1936 in a long-extinct Soviet journal, in which he presented the first attempt at quantizing a weak (linearized) gravitational field, rather modern in its approach. The paper has been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note written by Stanley Deser and Alexei Starobinsky, and Bronstein’s brief biography written by Stanley Deser.     

Some results in the theory of interacting electron systems

The expressions for the longitudinal dielectric function, spin and orbital susceptibilities in the static, long wavelength limit are evaluated by solving the corresponding linearized vertex functions exactly in this limit. The plasma dispersion relation to leading order in the long wave limit is similarly obtained. These are compared with the corresponding results obtained previoulsy by us by a variational solution to the same vertex equations. It is established that the variational method gives the exact results in the static, zero wave vector limit, involving the proper renormalizations. The plasma dispersion relation is found to be the same as in the exact calculation whereas the coefficient of q² in the static density correlation function has an important additional contribution to the variational result. Applications of these results are briefly discussed.     

A complex formulation of generalized Hamiltonian (Birkhoffian) theory

Fundamental analytic, algebraic, and geometric properties of generalized Hamiltonian (Birkhoffian) theory are compared with the properties of a covering unitary phase-space formulation based on complex variables of the form (p+iq). Technical advantages in the unitary phase-space formulation are illustrated by a detailed discussion of the one-dimensional extended damped harmonic oscillator. One advantage is the ability to fully describe nonconservative constraint forces within a globally conservative system. Another advantage is that wider classes of gauge transformations are available to simplify the construction of smooth solution manifolds.     

Quantum theory of electromagnetic fields interacting with atoms and molecules

This paper reviews the recent achievements in nonrelativistic quantum electrodynamics, especially nonlinear and coherent phenomena. The general properties of coupled radiation and matter are presented within simple models in section 1. The following sections treat in some detail three main aspects of the system and can be read independently of each other. Section 2 discusses some experiments with long-wave-length radiation (r.f.) and atoms. Section 3 presents the quantum theory of a laser and the ensuing photon distributions. Section 4 treats the case of strongly correlated emission of radiation called superradiance. The use of statistical ensembles is briefly discussed in Appendix A, whereas Appendices B, C and D present some technical details of the text.     

The covariant Green functions of the photon and electron in weak gravitational fields

By adapting the proper-time method to both the electromagnetic and Dirac fields in interaction with a weak vierbein field, bein gauge is treated on an equal footing with coordinate gauge. The calculations of the local one-photon as well as the second- and first-order-differential one-electron Green functions are developed in terms of the first-order vierbein and auxiliary fields and the weak-field potentials, and are shown to exhibit manifest coordinate- and bein-gauge covariance. The gauge separations of the first-order fields allow the isolation and evaluation of the contributions from both the coordinate- and bein-gauge functions, permit expressions of the first-order contributions to the Green functions in terms of the gauge-invariant weak-field potentials, and lead to the modified versions of the Mandelstam-DeWitt path-dependent line integrals.     

Homogeneous and isotropic solutions of a gravitational theory with scalar fields

We study the homogeneous and isotropic solutions of a gravitational theory with scalar fields. Qualitative characteristics of these solutions are analyzed and important differences with respect to the usual Einstein theory are pointed out.