Canonical quantisation

by R. F. Streater; Commun. Mathematical Phys., 2, 354-374, 1966.

ABSTRACT

The dynamical variables of a classical system from a Lie algebra dG, where the Lie multiplication is given by the Poisson bracket. Following the ideas of SOURIAU and SEGAL, but with some modifications, we show that it is possible to realise dG as a concrete algebra of smooth transformations of the functionals Phi on the manifold M 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 realisation, constant functionals are realised 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 H, the Koopman space, by using the invariant Liouville volume element; the dynamical variables F (functions on phase space) become operators F^ on this space. We prove that for any Hamiltonian H(p,q) quadratic in the canonical variables q(1),...q(n),p(1),...p(n) there exists a subspace H(1) of H which is invariant under the action of p(j)^, q(k)^ and the Hamiltonian H^, and such that the restriction of p(j)^,q(k)^ to H(1) form an irreducible set of operators. Therefore, SOURIAU'S quantisation rule agrees with the usual one for quadratic Hamiltonians. In fact, it gives the Bargmann-Segal holomorphic function realisation. For non-linear problems in general, however, the operators p(j)^, q(k)^ form a reducible set on any subspace of H invariant under the action of the Hamiltonian. In particular, this happens for H(p,q)=p² /2 + lambda q^4. Therefore SOURIAU'S rule cannot agree with the usual quantisation procedure for general non-linear systems.

The method can be applied to the quantisation 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 (Box+m²)phi(x)=lambda phi^3(x), we show heuristically that the interacting field may be defined as a first order operator acting on C-infinity 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; it is noted that the Hilbert space structure defined by the asymptotic states is not adequate for this.

This work was later made rigorous by R. Racka. G.G. Emch was able to explain the lack of positivity of the Hamiltonian in terms of KMS states.

The Souriau formula solves the "Dirac Problem"; see my polemic article XVI for more comments.