Dynamical structure of irregular constrained systems

Abstract

Hamiltonian systems with functionally dependent constraints (irregular systems), for which the standard Dirac procedure is not directly applicable, are discussed. They are classified according to their behavior in the vicinity of the constraint surface into two fundamental types. If the irregular constraints are multilinear (type I), then it is possible to regularize the system so that the Hamiltonian and Lagrangian descriptions are equivalent. When the constraints are power of a linear function (type II), regularization is not always possible and the Hamiltonian and Lagrangian descriptions may be dynamically inequivalent. It is shown that the inequivalence between the two formalisms can occur if the kinetic energy is an indefinite quadratic form in the velocities. It is also shown that a system of type I can evolve in time from a regular configuration into an irregular one, without any catastrophic changes. Irregularities have important consequences in the linearized approximation to nonlinear theories, as well as for the quantization of such systems. The relevance of these problems to Chern-Simons theories in higher dimensions is discussed.

General relativity from a gauged Wess-Zumino-Witten term

“…Generically, as it happens with all higher dimensional Chern-Simons theories, the system will possess degenerate dynamical sectors [26,27,28]. A deeper understanding of this problem would be required prior to any study of the quantum properties of the action (9).…”

New aspects of irregular constrained systems

“…Besides, the notion of equivalent set of constraints is given by [10]: Different sets of constraints are said to be equivalent if they define the same constraint surface.…”

“…Another classification of the regularity conditions (RCs), including the case when this definition is violated, is given below [10]:…”

“…Recently, the irregular systems were discussed and their classification has been performed [10,11]. It has been proved that for irregular systems the Euler-Lagrange equations and Hamilton's equations are not equivalent [10,11] in some cases.…”

“…It has been proved that for irregular systems the Euler-Lagrange equations and Hamilton's equations are not equivalent [10,11] in some cases. Hamilton-Jacobi formalism based on Carathéodory's idea [12,13] is an alternative method of quantization of constrained systems and it was subjected to an intense debate during the last years [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].…”