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arXiv:hep-ph/0201182 · Discrete Dynamics in Nature and Society

Symmetries, variational principles, and quantum dynamics

J. Manjavidze, A. N. Sissakian

Published Jan 1, 2004• 0 citations

Abstract

We describe the role of symmetries in formation of quantum dynamics. A quantum version of d′Alembert′s principle is proposed to take into account the symmetry constrains more exact. It is argued that the time reversibility of quantum process, as the quantum analogy of d′Alembert′s principle, makes the measure of the corresponding path integral δ‐like. The argument of this δ‐function is the sum of all classical forces of the problem under consideration plus the random force of quantum excitations. Such measure establishes the one‐to‐one correspondence with classical mechanics and, for this reason, allows a free choice of the useful dynamical variables. The analysis shows that choosing the action‐angle variables, one may get to the free‐from‐divergences quantum field theory. Moreover, one can try to get an independence from necessity to extract the degrees of freedom constrained by the symmetry. These properties of new quantization scheme are vitally essential for such theories as the non‐Abelian Yang‐Mills gauge theory and quantum gravity.

DOI: 10.1155/s1026022604310022

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We describe the role of symmetries in formation of quantum dynamics. A quantum version of d′Alembert′s principle is proposed to take into account the symmetry constrains more exact. It is argued that the time reversibility of quantum process, as the quantum analogy of d′Alembert′s principle, makes the measure of the corresponding path integral δ‐like. The argument of this δ‐function is the sum of all classical forces of the problem under consideration plus the random force of quantum excitations. Such measure establishes the one‐to‐one correspondence with classical mechanics and, for this reason, allows a free choice of the useful dynamical variables. The analysis shows that choosing the action‐angle variables, one may get to the free‐from‐divergences quantum field theory. Moreover, one can try to get an independence from necessity to extract the degrees of freedom constrained by the symmetry. These properties of new quantization scheme are vitally essential for such theories as the non‐Abelian Yang‐Mills gauge theory and quantum gravity.

The review paper presents generalization of d’Alembert’s variational principle: the dynamics of a quantum system for an external observer is defined by the exact equilibrium of all acting in the system forces, including the random quantum force ħj , ∀ ħ . Spatial attention is dedicated to the systems with (hidden) symmetries. It is shown how the symmetry reduces the number of quantum degrees of freedom down to the independent ones. The sin-Gordon model is considered as an example of such field theory with symmetry. It is shown why the particles S -matrix is trivial in that model.

Even though its classical equations of motion are then left invariant, when an action is redefined by an additive total derivative or divergence term (in time, in the case of a mechanical system) such a transformation induces non-trivial consequences for the system’s canonical phase space formulation. This is even more true and then in more subtle ways for the canonically quantized dynamics, with, in particular, an induced transformation in the unitary configuration space representation of the Heisenberg algebra being used for the quantum system. When coupled to a background gauge field, such considerations become crucial for a proper understanding of the consequences for the system’s quantum dynamics of gauge transformations of that classical external background gauge field, while under such transformations, the system’s degrees of freedom, abstract quantum states and quantum dynamics are certainly strictly invariant. After a detailed analysis of these different points in a general context, these are then illustrated specifically in the case of the quantum Landau problem with its classical external background magnetic vector potential for which the most general possible parametrized gauge choice is implemented herein. The latter discussion aims as well to clarify some perplexing statements in the literature regarding the status of gauge choices to be made for the magnetic vector potential for that quantum system. The role of the global spacetime symmetries of the Landau problem and their gauge invariant Noether charges is then also emphasized.

We show that non-relativistic Quantum Mechanics can be faithfully represented in terms of a classical diffusion process endowed with a gauge symmetry of group Z_4. The representation is based on a quantization condition for the realized action along paths. A lattice regularization is introduced to make rigorous sense of the construction and then removed. Quantum mechanics is recovered in the continuum limit and the full U(1) gauge group symmetry of electro-magnetism appears. Anti-particle representations emerge naturally, albeit the context is non-relativistic. Quantum density matrices are obtained by averaging classical probability distributions over phase-action variables. We find that quantum conditioning can be described in classical terms but not through the standard notion of sub sigma-algebras. Delicate restrictions arise by the constraint that we are only interested in the algebra of gauge invariant random variables. We conclude that Quantum Mechanics is equivalent to a theory of gauge invariant classical stochastic processes we call Stochastic Mechanics.

A general path integral analysis of the separable Hamiltonian of Liouville-type is reviewed. The basic dynamical principle used is the Jacobi's principle of least action for given energy which is reparametrization invariant, and thus the gauge freedom naturally appears. The choice of gauge in path integral corresponds to the separation of variables in operator formalism. The gauge independence and the operator ordering are closely related. The path integral in this formulation sums over orbits in space instead of space-time. An exact path integral of the Green's function for the hydrogen atom in parabolic coordinates is ilustrated as an example, which is also interpreted as one-dimensional quantum gravity with a quantized cosmological constant.