[Paper Review] Generalized Wavefunctions for Correlated Quantum Oscillators II: Geometry of the Space of States

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

Quantum chaos and dynamical systems18 references3 citations

TL;DR

This paper develops a generalized formalism for correlated quantum oscillators using complex symplectic transformations and a rigged Hilbert space with distributional states, enabling the description of non-unitary, irreversible dynamics such as Gamow vectors with complex eigenvalues. The key contribution is a mathematically consistent framework for resonant quantum systems—especially Breit-Wigner resonances—by replacing the complex field with the commutative real algebra ℂ(1,i), allowing dynamical semigroups and non-Hermitian Hamiltonians while preserving well-defined geometry of the state space.

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{ö}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{ö}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.

Motivation & Objective

• To construct a mathematically rigorous formalism for nontrivial quantum dynamics, including resonances, beyond standard Hilbert space quantum mechanics.
• To address the challenge of representing complex spectra and exponential decay/growth in quantum systems using distributional solutions and non-unitary transformations.
• To replace the standard complex field with the real algebra ℂ(1,i) to ensure well-defined dynamics for non-Hermitian Hamiltonians and semigroup time evolution.
• To establish the Hamiltonian as the generator of canonical time translations via parallel transport in a generalized state space, extending the Schrödinger equation to rigged Hilbert spaces.
• To unify classical symplectic dynamics with quantum dynamics through analytic continuation of symplectic transformations, enabling a quantum probability framework rooted in classical action-angle variables.

Proposed method

• The formalism uses the rigged Hilbert space (RHS) to include distributional solutions (Gamow vectors) as generalized states, extending beyond standard Hilbert space vectors.
• Complex symplectic transformations—specifically from the group SL(2,ℂ)—are used to analytically continue free oscillator systems into coupled, resonant systems, generating non-unitary dynamics.
• The ring of scalars is replaced by the commutative real algebra ℂ(1,i), which allows consistent treatment of complex spectra without violating the real structure of SU(1,1) representations.
• The Hamiltonian is treated as the generator of canonical transformations in the space of generalized states, with the Schrödinger equation interpreted as the equation of parallel transport for energy eigenvectors.
• The SU(1,1) algebra is extended to its complexification 𝔰𝔩(2,ℂ), allowing eigenvalues of iY to become purely imaginary and complex, enabling Gamow vector solutions with exponential decay or growth.
• The time evolution is restricted to semigroup domains: ψ⁺ᴳ is defined for t ≤ 0, ψ⁻ᴳ for t ≥ 0, reflecting irreversible decay and growth.

Experimental results

Research questions

• RQ1 How can a consistent quantum formalism be constructed to describe resonant systems with complex energy eigenvalues, such as Breit-Wigner resonances?
• RQ2 What mathematical structure allows non-unitary, irreversible time evolution while preserving well-defined dynamics and symplectic geometry?
• RQ3 How can the standard Hilbert space formalism be extended to include Gamow vectors with exponential decay/growth and complex eigenvalues?
• RQ4 What role does the real algebra ℂ(1,i) play in replacing the complex field to maintain consistency in non-Hermitian quantum systems?
• RQ5 How does the Hamiltonian generate time translations in a generalized state space, and how does this relate to the Schrödinger equation as a parallel transport condition?

Key findings

• The generalized wavefunctions, including Gamow vectors, are constructed as solutions in the dual space Φ× of a rigged Hilbert space, allowing for complex eigenvalues and exponential time evolution.
• The Hamiltonian H = H₀ + H₁ has complex eigenvalues due to the action of iY, which is essentially Hermitian in the Hilbert space but acquires complex eigenvalues in the generalized state space.
• The time evolution of Gamow states ψᴳ±(t) exhibits pure exponential decay or growth: ψᴳ±(t) = e∓i(π/2)X |j,m⟩ e^−2iΩjt ± (Γ/2)(2m+1)t, with t restricted to t ≥ 0 or t ≤ 0.
• The transformation e^i(π/2)X maps SU(1,1) representations into the complex algebra 𝔰𝔩(2,ℂ), allowing the Hamiltonian to act on states with complex eigenvalues.
• The use of the real algebra ℂ(1,i) instead of ℂ ensures mathematical consistency in representing non-unitary dynamics and complex spectra, avoiding contradictions in real Lie group representations.
• The Schrödinger equation is reinterpreted as the equation of parallel transport for generalized energy eigenvectors under the Hamiltonian as a generator of canonical transformations, establishing a geometric dynamics framework.