Generalized Wavefunctions for Correlated Quantum Oscillators II: Geometry of the Space of States

TL;DR

This paper develops a mathematical formalism for representing nontrivial quantum dynamics of coupled oscillators, including resonances, using complex symplectic transformations and rigged Hilbert spaces, advancing the understanding of quantum resonances and energy transfer.

Contribution

It introduces a formalism transforming free oscillators into resonant systems via algebraic analytic continuation, incorporating distributional solutions and complex spectra within a rigged Hilbert space framework.

Findings

Representation of Breit-Wigner resonances with Gamow vectors

Hamiltonian as generator of dynamical time translations

Accommodation of complex spectra in a consistent formalism

Abstract

In this second in a series of four articles we create a mathematical formalism sufficient to represent nontrivial hamiltonian quantum dynamics, including resonances. Some parts of this construction are also mathematically necessary. The specific construction is the transforming of a pair of quantized free oscillators into a resonant system of coupled oscillators by analytic continuation which is performed algebraically by the group of complex symplectic transformations, thereby creating dynamical representations of numerous semi-groups. The quantum free oscillators are the quantum analogue of classical action angle variable solutions for the coupled oscillators and quantum resonances, including Breit-Wigner resonances. Among the exponentially decaying Breit-Wigner resonances represented by Gamow vectors are hamiltonian systems in which energy transfers from one oscillator to the other. There are significant mathematical constraints in order that complex spectra be accommodated in a well defined formalism which represents dynamics, and these may be met by using the commutative real algebra $\C (1,i)$ as the ring of scalars in place of the field of complex numbers. These mathematical constraints compel use of fundamental (spinor) representations rather than UIR's. By including distributional solutions to the Schr{\"o}dinger equation, placing us in a rigged Hilbert space, and by using the Hamiltonian as a generator of canonical transformation of the space of states, the Schr{\"o}dinger equation is the equation for parallel transport of generalized energy eigenvectors, explicitly establishing the Hamiltonian as the generator of dynamical time translations in this formalism.