A finite-dimensional canonical formalism in the classical field theory

A canonical formalism based on the geometrical approach to the calculus of variations is given. The notion of multi-phase space is introduced which enables to define whole the canonical structure (physical quantities, Poisson bracket, canonical fields) without use of functional derivatives. All definitions are of pure geometrical (finite dimensional) character.The observable algebra (physical quantities algebra) obtained here is much smaller then the algebra of all (sufficiently smooth) functionals on the space of states, derived from the standard infinite-dimensional formulation. As it is known, the latter is much too large for purposes of quantization. As the examples prove, our algebra could be an adequate start-point for quantization.For simplifying the language the notion of observable-valued distribution is introduced. Many concrete physical examples are given. E.g. it is shown that some problems connected with gauge in electrodynamics are automatically solved in this approach. The introduced language allows to obtain the Noether theorem in a most natural way.     

A generalized Hamiltonian formalism unifying classical and quantum mechanics

A Poisson bracket structure is defined on associative algebras which allows for a generalized Hamiltonian dynamics. Both classical and quantum mechanics are shown to be special cases of the general formalism.     

On the space-time interpretation of classical canonical systems II: Relativistic canonical systems

Relativistic canonical systems and their symmetries are defined and classified within the class of canonical systems treated in a previous paper. Their algebra of variables contains a subset of monotone variables which satisfy a certain uniqueness condition and are later shown to increase strictly in the course of the dynamical evolution of the system on all physically acceptable states. This leads to a unique space-time interpretation of relativistic canonical systems. Finally we study space-time factorizations of such systems and introduce the appropriate notion of states. For a certain simple class of states the theory is shown to describe the motion of relativistic matter in some external gravitational and electromagnetic field.     

Relativistic wave and particle mechanics formulated without classical mass

The fact that the concept of classical mass plays an important role in formulating relativistic theories of waves and particles is well-known. However, recent studies show that Galilean invariant theories of waves and particles can be formulated with the so-called ‘wave mass’, which replaces the classical mass and allows attaining higher accuracy of performing calculations [J.L. Fry and Z.E. Musielak, Ann. Phys. 325 (2010) 1194]. The main purpose of this paper is to generalize these results and formulate fundamental (Poincaré invariant) relativistic theories of waves and particles without the classical mass. In the presented approach, the classical mass is replaced by an invariant frequency that only involves units of time. The invariant frequencies for various elementary particles are deduced from experiments and their relationship to the corresponding classical and wave mass for each particle is described. It is shown that relativistic wave mechanics with the invariant frequency is independent of the Planck constant, and that such theory can attain higher accuracy of performing calculations. The choice of natural units resulting from the developed theories of waves and particles is also discussed.     

Relativistic canonical formalism and the invariant single-particle distribution function in the general theory of relativity

The relativistic canonical formalism is used to construct an eight-dimensional phase space and an invariant distribution function, and integral and differential operations in the phase space and statistical averages, associated with the field of geodesic observers, are introduced. Liouville's theorem is proved.     

Extended canonical vierbein formalism

The canonical vierbein theory, written in the language of differential forms, is extended to include the differentials themselves as independent fields. The resulting theory strongly resembles the BRST extension of the component theory provided the hamiltonian is identified with the BRST charge Q and the differentials dxⁱ with the spatial projections of the ghosts.     

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     

A canonical formalism and its geometric interpretation in the covariant theory of the classical electron

A canonical formalism is constructed for the covariant model of the classical electron proposed in [1, 2, 3]. For the geometric interpretation the notion of the coincidence of the tangent space to the space of outer coordinates with a subspace of the tangent space to the space of inner variables is introduced.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 88–92, June, 1976.In conclusion, the authors wish to thank Yu. G. Borisovich for his constant interest in this work.     

Covariant canonical formalism for gravity

We continue the previous discussion (A. D'Adda, J. E. Nelson, and T. Regge, Ann. Phys. (N.Y.)165) of the covariant canonical formalism for the group manifold and relate it to the standard canonical vierbein formalism as pioneered by Dirac. The form bracket is related to the Poisson bracket of classical mechanics. We utilise systematically the calculus of differential forms and a compound notation which labels Poincaré multiplets. In this way we obtain a particularly clear and compact expression for the Hamiltonian and the constraints algebra of the vierbein formalism.     

Relativistic non-Hamiltonian mechanics

Relativistic particle subjected to a general four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u_μu^μ + c² = 0, where c is the speed of light in vacuum. In the general case, four-forces are non-potential, and the relativistic particle is a non-Hamiltonian system in four-dimensional pseudo-Euclidean space-time. We consider non-Hamiltonian and dissipative systems in relativistic mechanics. Covariant forms of the principle of stationary action and the Hamilton’s principle for relativistic mechanics of non-Hamiltonian systems are discussed. The equivalence of these principles is considered for relativistic particles subjected to potential and non-potential forces. We note that the equations of motion which follow from the Hamilton’s principle are not equivalent to the equations which follow from the variational principle of stationary action. The Hamilton’s principle and the principle of stationary action are not compatible in the case of systems with nonholonomic constraint and the potential forces. The principle of stationary action for relativistic particle subjected to non-potential forces can be used if the Helmholtz conditions are satisfied. The Hamilton’s principle and the principle of stationary action are equivalent only for a special class of relativistic non-Hamiltonian systems.     

Relativistic version of the imaginary-time formalism

A relativistic version of the quasiclassical imaginary-time formalism is developed. It permits calculation of the tunneling probability of relativistic particles through potential barriers, including barriers lacking spherical symmetry. Application of the imaginary-time formalism to concrete problems calls for finding subbarrier trajectories which are solutions of the classical equations of motion, but with an imaginary time (and thus cannot be realized in classical mechanics). The ionization probability of an s level, whose binding energy can be of the order of the rest energy, under the action of electric and magnetic fields of different configuration is calculated using the imaginary-time formalism. Besides the exponential factor, the Coulomb and pre-exponential factors in the ionization probability are calculated. The Hamiltonian approach to the tunneling of relativistic particles is described briefly. Scrutiny of the ionization of heavy atoms by an electric field provides an additional argument against the existence of the “Unruh effect.” Zh. éksp. Teor. Fiz. 114, 798–820 (September 1998)     

Comments on the charge-monopole canonical formalism

A recently published canonical formalism of a charge-monopole system written by means of Clifford algebras is discussed. It is shown that the introduction of the Lorentz force must be accompanied by the removal of the pseudo-scalar terms from the lagrangian. Several conclusions follow.     

Continuity of the temperature and derivation of the Gibbs canonical distribution in classical statistical mechanics

For a classical system of interacting particles we prove, in the microcanonical ensemble formalism of statistical mechanics, that the thermodynamic-limit entropy density is a differentiable function of the energy density and that its derivative, the thermodynamic-limit inverse temperature, is a continuous function of the energy density. We also prove that the inverse temperature of a finite system approaches the thermodynamic-limit inverse temperature as the volume of the system increases indefinitely. Finally, we show that the probability distribution for a system of fixed size in thermal contact with a large system approaches the Gibbs canonical distribution as the size of the large system increases indefinitely, if the composite system is distributed microcanonically.Supported by The British Council and the Universidad Nacional Autónoma de México.     

Monad method and canonical formalism of general relativity

In this note the general monad method is systematically represented, and it is shown how it may be reduced to its two basic special gauges. The last section deals with two kinds of canonical formalism, coordinate and referential ones, based on the kinemetric gauge.     

The operator formalism of quantum mechanics from the viewpoint of short disturbances in nonrelativistic classical motion

The effect of short disturbances on nonrelativistic motion is formulated in terms of operators. Analogies with quantum mechanics are developed and some disparities noted. For the one-dimensional particle we obtain analogues of the de Broglie wave commonly associated with particle motion, Heisenberg's commutation relation, Schrödinger's equation, and the statistical interpretation. Whether these results have any bearing on quantum mechanics itself is left an open question.     

Supersymmetric classical mechanics

The purpose of the paper is to construct a supersymmetric Lagrangian within the framework of classical mechanics which would be regarded as a candidate for passage to supersymmetric quantum mechanics. The authors felicitate Prof. D S Kothari on his eightieth birthday and dedicate this paper to him on this occasion.