Problems in the quantization of quadratic Hamiltonians

Adherents to the Lie-admissible formulation of strongly interacting systems are interested in the possible non-locality of the strong field, and in the need of an algebraic covering of canonical quantization. From a general field theory with non-localized action, linearization via the introduction of supplementary conditions produces a model which may be analyzed in terms of independent subsystems with finite degrees of freedom and general quadratic Hamiltonian. This paper is largely devoted to a review of the quantization of arbitrary quadratic Hamiltonians. This subject has had a long history, but problems still remain. A system with general quadratic Hamiltonian can either be reduced to a collection of independent harmonic oscillators, or (a) the classical system possesses an unstable equilibrium point; and (b) canonical quantization in the Heisenberg picture leads to mode operators whose formal mode space, unlike the Fock space for Boson annihilation operators cannot have a positive definite inner product; and (c) the Segal approach to construction of a vacuum representation of the C.C.R. is inapplicable; and (d) closely associated with the quadratic Hamiltonian H(q,p) is a system of Bosons with quadratic Hamiltonians H'(a,a/sup f/) which cannot be reduced by a Gogoliubov transformation to a system of free quasi-particles; and (e) from well-known examples, the Hamiltonian will not have a discrete semi-bounded spectrum in the Schroedinger representation. In this paper, a formal presentation is avoided, except on those occasional instances where applicable reference sources are not known to the author. Missing details can be retrieved from the listed references, which make contact with the vast subject area outlined above