General solution to the quantum master equation in the finite-dimensional case

The general solution to the quantum master equation (and its Sp(2) symmetric counterpart) is explicity constructed in the case of a finite number of variables. It is shown that the finite-dimensional solution is physically trivial and, therefore, cannot be directly extended to a local field theory. Thus, the locality condition is important in obtaining nontrivial physical results when quantizing gauge field theories in the field-antifield formalism. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 114. No. 2, pp. 250–270, February 1998.     

Equivalence of Lagrangian and HamiltonianSp(2)-symmetric BRST quantizations

The existence of a local solution to the quantum Sp(2) master equation and the equivalence of the Lagrangian and Hamiltonian Sp(2) quantizations are proved in the framework of perturbation theory. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 137–148, January, 1997.     

NEW NON-NATURALLY REDUCTIVE EINSTEIN METRICS ON Sp(n)

In this paper,we consider a class of left invariant Riemannian metrics on Sp(n),which is invariant under the adjoint action of the subgroup Sp(n-3)×Sp(1)×Sp(1)×...     

Canonical formulation of the embedded theory of gravity equivalent to Einstein’s general relativity

We study the approach in which independent variables describing gravity are functions of the space-time embedding into a flat space of higher dimension. We formulate a canonical formalism for such a theory in a form that requires imposing additional constraints, which are a part of Einstein’s equations. As a result, we obtain a theory with an eight-parameter gauge symmetry. This theory becomes equivalent to Einstein’s general relativity either after partial gauge fixing or after rewriting the metric in the form that is invariant under the additional gauge transformations. We write the action for such a theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 271–288, November, 2007.     

引力规范理论中的一类引力波方程

该文给出了Vierbein表述的局域Lorentz群引力规范理论中的一类引力波方程。证明了Bondi平面波方程和引力孤立波方程均被该类方程所包含,这些方程的解均为该类方程在一定条件下的特解。因而这些解是与量子场论协调一致的。     

Non-Abelian tensor gauge fields

A recently proposed extension of Yang-Mills theory contains non-Abelian tensor gauge fields. The Lagrangian has quadratic kinetic terms, as well as cubic and quartic terms describing nonlinear interaction of tensor gauge fields with the dimensionless coupling constant. We analyze the particle content of non-Abelian tensor gauge fields. In four-dimensional space-time the rank-2 gauge field describes propagating modes of helicity 2 and 0. We introduce interaction of the non-Abelian tensor gauge field with fermions and demonstrate that the free equation of motion for the spinor-vector field correctly describes the propagation of massless modes of helicity 3/2. We have found a new metric-independent gauge invariant density which is a four-dimensional analog of the Chern-Simons density. The Lagrangian augmented by this Chern-Simons-like invariant describes the massive Yang-Mills boson, providing a gauge invariant mass gap for a four-dimensional gauge field theory.     

On a field equation generating a new class of particular solutions to the Yang-Mills equations

In a pseudo-Euclidean space, a field equation (system of equations) is considered that is invariant under orthogonal (from the group O(p, q)) coordinate transformations and invariant under gauge transformations from the spinor group Pin(p, q). The solutions to the field equation are connected with a class of new particular solutions to the Yang-Mills equations.     

Finite-Energy Global Well-Posedness of the Maxwell–Klein–Gordon System in Lorenz Gauge

It is known that the Maxwell–Klein–Gordon system (M–K–G), when written relative to the Coulomb gauge, is globally well-posed for finite-energy initial data. This result, due to Klainerman and Machedon, relies crucially on the null structure of the main bilinear terms of M–K–G in Coulomb gauge. It appears to have been believed that such a structure is not present in Lorenz gauge, but we prove here that it is, and we use this fact to prove finite-energy global well-posedness in Lorenz gauge. The latter has the advantage, compared to Coulomb gauge, of being Lorentz invariant, hence M–K–G in Lorenz gauge is a system of nonlinear wave equations, whereas in Coulomb gauge the system has a less symmetric form, as it contains also an elliptic equation.     

The holomorphic effective action inN=2D=4 supergauge theories with various gauge groups

We consider the theory of hypermultiplets in arbitrary representations of arbitrary semisimple gauge groups coupled to gauge superfields. Using the N=2 harmonic superspace formulation of these models, we find the general structure of the holomorphic effective action depending on the gauge superfield with values in the Cartan subalgebra of the gauge algebra. We find explicit expressions for the effective actions in the cases where the hypermultiplets are in the fundamental and adjoint representations of SU(n), SO(n), and Sp(2n). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 444–455, April, 2000.     

Local well-posedness for the Maxwell-Schrödinger equation

Time local well-posedness for the Maxwell-Schrödinger equation in the Coulomb gauge is studied in Sobolev spaces by the contraction mapping principle. The Lorentz gauge and the temporal gauge cases are also treated by the gauge transform.     

Local variational differential operators in field theory

We develop a new calculus for local variational differential operators where the action of higher-order operators on local functionals does not lead to indefinite quantities like δ(0). We apply this formalism to the Batalin-Vilkovisky formulation of local general gauge field theory and to its Sp(2)-symmetrical generalization. Its relation to a semiclassical expansion is also discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 2, pp. 256–276, August, 1999.     

Local and global solutions of the Chern-Simons-Higgs system

We study low regularity solutions of the Chern-Simons-Higgs equations. The Lorentz gauge condition makes them hyperbolic equations with the null form. Under the Coulomb gauge condition they are formulated in the hyperbolic equation coupled with elliptic equation. The div-curl decomposition is used in the temporal gauge.     

Space‐time as a structured relativistic continuum

There are various models of gravitation: the metrical Hilbert‐Einstein theory, a wide class of intrinsically Lorentz‐invariant tetrad theories (generally covariant in the space‐time sense), and many gauge models based on various internal symmetry groups (Lorentz, Poincare, , etc). The gauge models are usually preferred but nevertheless it is an interesting idea to develop the class of ‐invariant (or rather ‐invariant) tetrad (n‐leg) generally covariant models. This is done below and motivated by our idea of bringing back to life the Thales of Miletus concept of affine symmetry. Formally, the obtained scheme is a generally covariant tetrad (n‐leg) model, but it turns out that generally covariant and intrinsically affinely invariant models must have a kind of nonaccidental Born‐Infeld‐like structure. Let us also mention that they, being based on tetrads (n‐legs), have many features common with continuous defect theories. It is interesting that they possess some group‐theoretical solutions and more general spherically symmetric solutions, discussion of which is the main new result presented in this paper, including the applications of the 't Hooft‐Polyakov monopoles in the generally covariant theories, which enables us to find some rigorous solutions of our strongly nonlinear equations. It is also interesting that within such a framework, the normal‐hyperbolic signature of the space‐time metric is not introduced by hand but appears as a kind of solution, rather integration constants, of differential equations. Let us mention that our Born‐Infeld scheme is more general than alternative tetrad models. It may be also used within more general schemes, including also the gauge ones.     

Analytic complexity: Gauge pseudogroup,its orbits,and differential invariants

All characteristics of analytic complexity of functions are invariant under a certain natural action (gauge pseudogroup G). For the action of the pseudogroup G, differential invariants are constructed and the equivalence problem is solved. Functions of two as well as of a greater number of variables are considered. Questions for further analysis are posed.     

One-parameter family of solitons from minimal surfaces

In this paper, we discuss a one parameter family of complex Born–Infeld solitons arising from a one parameter family of minimal surfaces. The process enables us to generate a new solution of the B–I equation from a given complex solution of a special type (which are abundant). We illustrate this with many examples. We find that the action or the energy of this family of solitons remains invariant in this family and find that the well-known Lorentz symmetry of the B–I equations is responsible for it.     

Group invariant Colombeau generalized functions

Colombeau generalized functions invariant under smooth (additive) one-parameter groups are characterized. This characterization is applied to generalized functions invariant under orthogonal groups of arbitrary signature, such as groups of rotations or the Lorentz group. Further, a one-dimensional Colombeau generalized function with two (real) periods is shown to be a generalized constant, when the ratio of the periods is an algebraic nonrational number. Finally, a nonstandard Colombeau generalized function invariant under standard translations is shown to be constant. Supported by research grants M949 and Y237 of the Austrian Science Foundation (FWF). Author’s address: Institut für Grundlagen der Bauingenieurwissenschaften, Technikerstra?e 13, 6020 Innsbruck, Austria     

Analysis in space-time bundles IV. Natural bundles deforming into and composed of the same invariant factors as the spin and form bundles

We study two interesting new bundles over the universal cosmos M̃ (or maximal isotropic space-time), which may be physically applicable. The treatment is from a homogeneous vector bundle point of view and uses the notation and some of the results of the treatment in Papers I–III (S. M. Paneitz and I. E. Segal, J. Funct. Anal. 47 (1982), 78–142; 49 (1982), 335–414; 54 (1983), 18–22)) of conventional bundles over M̃. The “spannor” bundle deforms into essentially the usual spinor bundle as a conformally invariant parameter that may be interpreted as the space curvature becomes arbitrarily small. From a Minkowski space standpoint, however, the spannors involve a nontrivial action of space-time translations that deforms into a trivial action in the spinor limit and also have more complex transformation properties under discrete symmetries.Also studied are the “plyors,” consisting of the dual to the bundle product of the spannors with themselves. Composition series for the spannor and plyor section spaces are treated, relative to the conformal group, and irreducible subquotients are identified with certain that occur in conventional bundles. In particular, factors corresponding to the Maxwell and massless Dirac equations, and which may represent certain of the observed elementary particles, are determined. A gauge and conformally invariant nonlinear coupling between spannors and plyors, constituting essentially a generalization of that used in quantum electrodynamics, is developed, and an associated invariant nonlinear partial differential equation is derived. Covariant and causal quantization for spannors (as fermions) and plyors (as bosons) is formulated algebraically.The present treatment is basically mathematical, but physical motivations and possible interpretations are briefly noted.     

Renormalization equations in gravitational theory with higher derivatives

The structure of renormalization equations in gravitational theories with higher derivatives is considered. The gauge dependence of invariant divergences of the effective action is found to be nontrivial. The external source technique is used to construct a consistent Green's function renormalization. One- and two-loop divergences of the effective action are explicitly calculated for an arbitrary parametrization and gauge. These calculations fit the general structure of the obtained renormalization equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 3, pp. 387–411, December, 1999.     

Krein space representation and Lorentz groups of analytic Hilbert modules

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H~2(D~2). A closed subspace M in H~2(D~2) is called a submodule if ziM ? M(i = 1, 2). An associated integral operator(defect operator) CM captures much information about M. Using a Kre??n space indefinite metric on the range of CM, this paper gives a representation of M. Then it studies the group(called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup(called little Lorentz group) which turns out to be a finer invariant for M.