Geometry of canonical variables in gravity theories

We show how to pass from an SL(2, C) covariant to an SU(2) covariant formulation of the theories of gravity. Our construction determines the canonical and gauge variables of the theory and establishes an appropriate framework for a hamiltonian picture.     

Spinning fluids in the Einstein-Cartan theory: A variational formulation

In this work we propose a lagrangian for spinning fluids in the Einstein-Cartan theory. The basic characteristic of the model is to consider each infinitesimal volume element as replicas of micro-rigid bodies. The theory obtained represents the thermodynamical equilibrium limit of a more general situation where dissipative processes due to spin take place. We outline the extension of such processes to the Einstein-Cartan theory.     

Field equations and contracted Bianchi identities in the generalized Einstein-Cartan theory

The generalized Palatini variational principle for the gravitational field interacting with a matter field is considered. The contracted Bianchi identities are proved and the reduction of the generalized Einstein-Cartan equations to the set of 6 dynamical equations and 4 constraints is presented.     

Gauge group,degeneration, and degrees of freedom in the ECSK theory of gravity. I. The symplectic structure of the ECSK theory

This paper is the first of a two-article series in which the connections between the gauge group, degeneration, and degrees of freedom in the ECSK theory coupled to an arbitrary tensor field are discussed. In this paper a multisymplectic formulation of the ECSK theory is presented and the symplectic 2-form, which plays a leading role in our considerations, is found.     

Embedding variables in the canonical theory of gravitating shells

A thin shell of light-like dust with its own gravitational field is studied in the special case of spherical symmetry. The action functional for this system due to Louko, Whiting, and Friedman is reduced to Kuchař form: the new variables are embeddings, their conjugate momenta, and Dirac observables. The concepts of background manifold and covariant gauge fixing, that underlie these variables, are reformulated in a way that implies the uniqueness and gauge invariance of the background manifold. The reduced dynamics describes motion on this background manifold.     

Conjugate variables in quantum field theory and natural symplectic structures

Within standard quantum field theory we establish relations which operators conjugate to the energy–momentum operator of the theory would have. They thus can be understood as representing the effect of coordinate operators. The non-trivial commutation relations we derive constitute natural symplectic structures in the theory. The example which is based on the energy–momentum tensor of the theory is constructed to all orders of perturbation theory. The reference theory is massless ?⁴${?}^4$. The extension to other theories is indicated.     

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.     

The dynamical structure of the Einstein-Cartan-Sciama-Kibble theory of gravity

It is shown that in the SO(3)-covariant Hamiltonian formulation the system of the ECSK equations can be reduced to 7 gravitational constraints, 18 gravitational dynamical equations, and a system of matter field equations. The geometric meaning of the canonical (symplectic) and gauge variables is also explained. Moreover, a general method of how to analyse degenerate matter field lagrangians in the framework of the ECSK theory is discussed. The exposition is given in the language of SO(3)-covariant differential operators on 3-dimensional slices of spacetime.     

力学、热力学及电磁波导中的正则变换和辛描述

从正则变换入手,讨论经典力学中正则变换的辛描述,得到了正则变换的条件,给出了热力学系统满足正则变换条件的证明,同时给出了在辛体系下电磁波导的正则变换的物理意义.通过正则变换给出在辛体系下力学、热力学、电磁学公式的相似性,在统一理论方面做出有益探讨.     

A partly reduced system of Hamiltonian equations and the initial value problem in the Ecsk theory of gravity

It is shown that by solving 12 of the field equations with respect to the connection components and , the quantities used to describe the geometry of space-time can be divided into two sets. In the first set we have the canonical variables the time evolution of which is determined by the dynamical equations. The second set contains ten gauge variables N, N^k, , n ⁽ⁱ⁾ which can be given arbitrarily on space-time. This partial reduction of the Hamiltonian equations enabls us to discuss the initial value problem in the ECSK theory of gravity coupled to matter tensor fields. Such an analysis is performed for the phenomenological ECSK theory and for the ECSK theory coupled to: a covector matter field, the generalized Maxwell electrodynamics, and the generalized Fermi-Dirac electrodynamics. The Poisson brackets of the seven Hamiltonian constraints, which have to be satisfied by the canonical variables, are found. It is proved that they are first class.     

Cosmology in the Einstein-Cartan theory of gravitation

A number of new solutions to Einstein-Cartan equations for homogeneous isotropic models of the universe are obtained. A catalog of these and earlier found solutions is given, in comparison with typical General Relativity models.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 25–29, March, 1988.The author wants to thank Prof. V. N. Ponomarev for his supervision and help in this work, and I. S. Nurgaliev for the discussion of the results and valuable comments.     

广义经典力学中Hamilton-Tabarrok-Leech正则方程的对称性理论

提出广义Hamilton-Tabarrok-Leech正则方程的对称性理论.列写系统的运动方程.研究系统的Noether对称性、形式不变性和Lie对称性，并求出相应的守恒量.举例说明结果的应用. 关键词： 广义经典力学 H-T-L 正则方程 对称性 守恒量     

Reversibility and irreversibility from an initial value formulation

From a time evolution equation for the single particle distribution function derived from the N-particle distribution function (A. Muriel, M. Dresden, Physica D 101 (1997) 297), an exact solution for the 3D Navier–Stokes equation – an old problem – has been found (A. Muriel, Results Phys. 1 (2011) 2). In this Letter, a second exact conclusion from the above-mentioned work is presented. We analyze the time symmetry properties of a formal, exact solution for the single-particle distribution function contracted from the many-body Liouville equation. This analysis must be done because group theoretic results on time reversal symmetry of the full Liouville equation (E.C.G. Sudarshan, N. Mukunda, Classical Mechanics: A Modern Perspective, Wiley, 1974). no longer applies automatically to the single particle distribution function contracted from the formal solution of the N-body Liouville equation. We find the following result: if the initial momentum distribution is even in the momentum, the single particle distribution is reversible. If there is any asymmetry in the initial momentum distribution, no matter how small, the system is irreversible.     

On a generalization of the Einstein-Cartan theory and the Klein-Kaluza theory

The Klein-Kaluza theory with a nonvanishing torsion is developed. The torsion is associated with the spin and polarization of an electromagnetic field. The electromagnetic polarization is considered as a source of additional components of a torsion connected with a fifth dimension. It is proved that the new effects are 10³⁶ times bigger than the effects from the Einstein-Cartan theory. Partially supported by Polish Ministry of Science, Higher Education and Technology, project No. Mr 17.     

Phase topology of one system with separated variables and singularities of the symplectic structure

We consider an example of a system with two degrees of freedom admitting separation of variables but having a subset of codimension 1 on which the 2-form defining the symplectic structure degenerates. We show how to use separation of variables to calculate the topological invariants of non-degenerate singularities and singularities appearing due to the symplectic structure degeneration. New types of non-orientable 3-atoms are found.