[Paper Review] Generalized Wavefunctions for Correlated Quantum Oscillators I: Basic Formalism and Classical Antecedants

S. Maxson|ArXiv.org|Jan 21, 2003

Quantum Mechanics and Applications14 references3 citations

TL;DR

This paper introduces a generalized quantum dynamics framework based on quasi-invariant measures and distributional probability amplitudes on classical phase space, enabling functorial analytic continuation to a rigged Hilbert space (RHS) formalism. The approach naturally incorporates Breit-Wigner resonances and Gamow vectors, yielding a field-theoretic quantum description with intrinsic irreversibility and hyperbolic dynamics, distinct from standard Hilbert space quantum mechanics.

ABSTRACT

In this first of a series of four articles, it is shown how a hamiltonian quantum dynamics can be formulated based on a generalization of classical probability theory using the notion of quasi-invariant measures on the classical phase space for our description of dynamics. This is based on certain distributions rather than invariant Gibbs measures. The first quantization is by functorial analytic continuation of real probability amplitudes, mathematically effecting the introduction of correlation between otherwise independent subsystems, and whose physical consequence is the incorporation of Breit-Wigner resonances associated to Gamow vectors into our description of dynamics. The resulting quantum dynamics admits a natural field theory interpretation. This basic formalism will be employed in subsequent installments of the series.

Motivation & Objective

• To develop a Hamiltonian quantum dynamics formalism based on generalized probability amplitudes and distributional measures, extending classical statistical mechanics beyond Gibbs measures.
• To demonstrate that analytic continuation of real probability amplitudes induces a symplectic structure, thereby effecting a functorial first quantization process.
• To show that the resulting quantum dynamics is mathematically equivalent to the rigged Hilbert space (RHS) formulation, enabling weak (distributional) solutions and resonant states such as Gamow vectors.
• To lay the foundational formalism for subsequent papers in the series, focusing on irreversibility, hyperbolic dynamics, and gauge structure emergence in field theories.
• To establish a natural field theory interpretation of the dynamics, particularly for interacting, correlated quantum oscillators, via phase space distributions and Lie-Poisson structures.

Proposed method

• Formulates Hamiltonian dynamics using quasi-invariant measures on classical phase space, generalizing Gibbs measures to allow for non-equilibrium and correlated dynamics.
• Introduces real probability amplitudes as distributional measures, which are analytically continued to complex functions to generate a symplectic structure and induce quantization.
• Employs the rigged Hilbert space (RHS) formalism of Bohm and Gadella to accommodate generalized eigenvectors (e.g., Gamow states) and weak solutions to the Schrödinger equation.
• Applies the generalized Gel’fand-Maurin (nuclear spectral) theorem to construct a rigged Hilbert space Φ ⊂ H ⊂ Φ× with a unique positive measure μ on the spectrum Λ, enabling spectral decomposition.
• Uses the Lie-Poisson bracket on phase space to define a Poisson structure on distribution densities, enabling a field-theoretic interpretation of the dynamics.
• Establishes a correspondence between classical ensemble dynamics and quantum-like behavior via analytic continuation, with complex symplectic correlations modeling oscillatory and resonant behavior.

Experimental results

Research questions

• RQ1 How can classical Hamiltonian dynamics on phase space be generalized to include non-Gibbsian, quasi-invariant measures that support correlation and irreversibility?
• RQ2 What is the role of real probability amplitudes in the analytic continuation process that leads to a quantum-like formalism with symplectic structure?
• RQ3 How does the rigged Hilbert space formalism allow for the inclusion of Gamow vectors and Breit-Wigner resonances as physically meaningful solutions?
• RQ4 In what way does the proposed formalism naturally lead to a field theory interpretation of correlated quantum oscillators?
• RQ5 What is the mathematical and physical significance of extending the scalar field to a complex structure that avoids reliance on C, particularly in the context of dynamical hyperbolicity?

Key findings

• The analytic continuation of real probability amplitudes induces a symplectic structure, thereby effecting a functorial first quantization that generalizes standard quantum mechanics.
• The resulting quantum dynamics is mathematically indistinguishable from the rigged Hilbert space (RHS) formulation, allowing for generalized eigenvectors and weak solutions, including Gamow states.
• The formalism naturally incorporates Breit-Wigner resonances as solutions to the Schrödinger equation, with self-consistent decay and time evolution, consistent with microphysical irreversibility.
• A positive measure μ on the spectrum Λ ensures the spectral decomposition of operators via generalized eigenvectors |Fλ⟩ in Φ×, satisfying the generalized Gel’fand-Maurin theorem.
• The space of distribution densities on phase space inherits a Poisson structure, enabling a Hamiltonian formulation of Vlasov-type equations and a field-theoretic interpretation of many-body dynamics.
• The framework supports a field theory interpretation of correlated quantum oscillators, with potential for deriving Yang-Mills-like gauge structures in later installments, independent of specific field content.