Dynamics of totally constrained systems I. Classical theory’ Prog.Theor.Phys. 94

This is the first of a series of papers in which a new formulation of quantum theory is developed for totally constrained systems, that is, canonical systems in which the hamiltonian is written as a linear combination of constraints ha with arbitrary coefficients. The main purpose of the present paper is to clarify that classical dynamics of a totally constrained system is nothing but the foliation of the constraint submanifold in phase space by the involutive system of infinitesimal canonical transformations Ya generated by the constraint functions. From this point of view it is shown that statistical dynamics for an ensemble of a totally constrained system can be formulated in terms of a relative distribution function without gauge fixing or reduction. There the key role is played by the fact that the canonical measure in phase space and the vector fields Ya induce natural conservative measures on acausal submanifolds, which are submanifolds transversal to the dynamical foliation. Further it is shown that the structure coefficients c,?;p defined by {h., hp}=2:.,c~ph7 should weakly commute with ha, 2:,,{h 7, c,?;p};,;Q, in order that the description in terms of the relative distribution function is consistent. The overall picture on the classical dynamics given in this paper provides the basic motivation for the quantum formulation developed in the subsequent papers.