Canonical quantization

The dynamical variables of a classical system form a Lie algebra , where the Lie multiplication is given by the Poisson bracket. Following the ideas ofSouriau andSegal, but with some modifications, we show that it is possible to realize as a concrete algebra of smooth transformations of the functionals on the manifold of smooth solutions to the classical equations of motion. It is even possible to do this in such a way that the action of a chosen dynamical variable, say the Hamiltonian, is given by the classical motion on the manifold, so that the quantum and classical motions coincide. In this realization, constant functionals are realized by multiples of the identity operator. For a finite number of degrees of freedom,n, the space of functionals can be made into a Hilbert space using the invariant Liouville volume element; the dynamical variablesF become operators in this space. We prove that for any hamiltonianH quadratic in the canonical variablesq ₁...q _n ,p ₁...p _n there exists a subspace ₁ which is invariant under the action of and , and such that the restriction of to ₁ form an irreducible set of operators. Therefore,Souriau's quantization rule agrees with the usual one for quadratic hamiltonians. In fact, it gives the Bargmann-Segal holomorphic function realization. For non-linear problems in general, however, the operators form a reducible set on any subspace of invariant under the action of the Hamiltonian. In particular this happens for . Therefore,Souriau's rule cannot agree with the usual quantization procedure for general non-linear systems.The method can be applied to the quantization of a non-linear wave equation and differs from the usual attempts in that (1) at any fixed time the field and its conjugate momentum may form a reducible set (2) the theory is less singular than usual.For a particular wave equation , we show heuristically that the interacting field may be defined as a first order differential operator acting onc -functions on the manifold of solutions. In order to make this space into a Hilbert space, one must define a suitable method of functional integration on the manifold; this problem is discussed, without, however, arriving at a satisfactory conclusion.On leave from Physics Department, Imperial College, London SW7.Work partly supported by the Office of Scientific Research, U.S. Air Force.