[Paper Review] Forward and Backward time observables for quantum evolution and quantum stochastic processes-I: The time observables

Y. Strauss|ArXiv.org|Jun 2, 2007

Spectral Theory in Mathematical Physics25 references3 citations

TL;DR

This paper introduces unique positive operators, $\mathbf{T}_F$ and $\mathbf{T}_B$, as forward and backward time observables for quantum evolution under the condition that the Hamiltonian's spectrum is the positive real line. Using the Sz.-Nagy–Foias theory of contraction operators and Hardy space functional models, the authors construct these observables via semigroup decomposition formalism, proving their uniqueness and showing they serve as time observables in quantum stochastic processes within Hudson-Parthasarathy calculus.

ABSTRACT

Given a Hamiltonian $H$ on a Hilbert space $\mathcal H$ it is shown that, under the assumption that $σ(H)=σ_{ac}(H)=R^+$, there exist unique positive operators $T_F$ and $T_B$ registering the Schrödinger time evolution generated by $H$ in the forward (future) direction and backward (past) direction respectively. These operators may be considered as time observables for the quantum evolution. Moreover, it is shown that the same operators may serve as time observables in the construction of quantum stochastic differential equations and quantum stochastic processes in the framework of the Hudson-Parthasarathy quantum stochastic calculus. The basic mechanism enabling for the definition of the time observables originates from the recently developed semigroup decomposition formalism used in the description of the time evolution of resonances in quantum mechanical scattering problems.

Motivation & Objective

• To define unique positive time observables $\mathbf{T}_F$ and $\mathbf{T}_B$ for quantum evolution governed by a Hamiltonian $\mathbf{H}$ with $\sigma(\mathbf{H}) = \sigma_{ac}(\mathbf{H}) = \mathbb{R}^+$.
• To establish that these time observables are compatible with the framework of quantum stochastic processes in Hudson-Parthasarathy calculus.
• To demonstrate that the time observables emerge from the semigroup decomposition formalism used in resonance dynamics.
• To prove the uniqueness and spectral equivalence of the time observables through characteristic function analysis in the Sz.-Nagy–Foias theory framework.

Proposed method

• Utilizes the semigroup decomposition formalism based on the Sz.-Nagy–Foias theory of $C_{\cdot 0}$ contraction semigroups and strongly contractive semigroups.
• Constructs time observables via compression of the Toeplitz semigroup $T_u^+(t)$ on Hardy space $\mathscr{H}^+_{\mathcal{N}}(\mathbb{R})$ to an invariant subspace $\hat{\mathscr{K}} = \mathscr{H}^+_{\mathcal{N}}(\mathbb{R}) \ominus \Theta_T(\cdot)\mathscr{H}^+_{\mathcal{N}}(\mathbb{R})$, where $\Theta_T$ is an inner function related to the scattering matrix.
• Defines $\mathbf{T}_F$ and $\mathbf{T}_B$ as the inverses of the generators of the forward and backward semigroups, respectively, ensuring positivity and self-adjointness.
• Applies the functional model of isometric dilations and characteristic functions to relate the spectral properties of $\mathbf{T}_F^{-1}$ and $\hat{T}_F^{-1}$ via the intertwining operator $\Omega_f$.
• Uses the characteristic function $\Theta_T(z)$ to prove spectral equivalence between $\mathbf{T}_F^{-1}$ and $\hat{T}_F^{-1}$, relying on the Sz.-Nagy–Foias theorem that the characteristic function determines the spectrum uniquely.
• Establishes unitary equivalence between the time observable operators and the semigroup generators through the isometric dilation framework, ensuring consistency in quantum stochastic processes.

Experimental results

Research questions

• RQ1 Can unique positive time observables be defined for quantum evolution when the Hamiltonian has a purely absolutely continuous spectrum on $\mathbb{R}^+$?
• RQ2 How can forward and backward time observables be constructed using semigroup decomposition and Hardy space functional models?
• RQ3 What is the relationship between the characteristic functions of the time observable generators and their spectral equivalence?
• RQ4 To what extent can these time observables be embedded into the Hudson-Parthasarathy quantum stochastic calculus framework?
• RQ5 How do the intertwining operators $\Omega_f$ and $\Omega_f^*$ preserve spectral and functional properties between the time observable systems?

Key findings

• Under the condition $\sigma(\mathbf{H}) = \sigma_{ac}(\mathbf{H}) = \mathbb{R}^+$, unique positive operators $\mathbf{T}_F$ and $\mathbf{T}_B$ exist as time observables for forward and backward quantum evolution.
• $\mathbf{T}_F$ and $\mathbf{T}_B$ are constructed as inverses of generators of Lax-Phillips type semigroups derived from the Toeplitz semigroup on Hardy space via compression to an invariant subspace.
• The characteristic functions $\Theta_{\mathbf{T}_F^{-1}}(z)$ and $\Theta_{\hat{T}_F^{-1}}(z)$ satisfy the intertwining relation $\Omega_f^*\Theta_{\hat{T}_F^{-1}}(z) = \Theta_{\mathbf{T}_F^{-1}}(z)\Omega_f^*$, proving spectral equivalence.
• The spectra of $\mathbf{T}_F^{-1}$ and $\hat{T}_F^{-1}$ are identical, as guaranteed by the Sz.-Nagy–Foias theorem, which ensures that the characteristic function determines the spectrum uniquely.
• The time observables $\mathbf{T}_F$ and $\mathbf{T}_B$ are compatible with the Hudson-Parthasarathy quantum stochastic calculus, enabling their use in quantum stochastic differential equations.
• The construction relies on the contractive nesting of Hilbert spaces and the isometric dilation of $C_{\cdot 0}$ semigroups, providing a rigorous functional-analytic foundation for time observables in quantum dynamics.