[Paper Review] The Rigged Hilbert Space Formulation of Quantum Mechanics and its Implications for Irreversibility

Christoph Schulte, Reidun Twarock+1 more|ArXiv.org|May 12, 1995

Quantum Mechanics and Applications5 references3 citations

TL;DR

This paper formulates quantum mechanics using the rigged Hilbert space (RHS) formalism to rigorously describe quasistationary states via Gamow vectors, which exhibit intrinsic quantum irreversibility through a fundamental arrow of time. It derives an exact golden rule for the decay of a pure Gamow state into a mixture of decay products, providing a mathematically consistent framework for irreversible processes in quantum theory.

ABSTRACT

Quantum mechanics in the Rigged Hilbert Space formulation describes quasistationary phenomena mathematically rigorously in terms of Gamow vectors. We show that these vectors exhibit microphysical irreversibility, related to an intrinsic quantum mechanical arrow of time, which states that preparation of a state has to precede the registration of an observable in this state. Moreover, the Rigged Hilbert Space formalism allows the derivation of an exact golden rule describing the transition of a pure Gamow state into a mixture of interaction-free decay products.

Motivation & Objective

• To provide a mathematically rigorous description of quasistationary quantum states using the rigged Hilbert space (RHS) formalism.
• To address the problem of irreversibility in quantum mechanics, particularly the asymmetry between state preparation and measurement.
• To derive an exact expression for the decay of a pure Gamow state into a mixture of decay products.
• To establish a fundamental quantum mechanical arrow of time based on the causal order of preparation and registration.
• To extend the standard quantum formalism to include unstable states with exponential decay behavior in a consistent Hilbert space framework.

Proposed method

• Utilizes the rigged Hilbert space (RHS) formalism to extend the standard Hilbert space framework to include generalized eigenvectors for resonant states.
• Introduces Gamow vectors as generalized eigenvectors of the Hamiltonian with complex eigenvalues, representing decaying states.
• Applies the Dirac delta function and complex-conjugate spectral resolution to describe the time evolution of Gamow states.
• Derives the exact golden rule for decay by analyzing the time evolution of a Gamow state in the RHS framework.
• Imposes boundary conditions corresponding to outgoing waves to model irreversible decay processes.
• Uses the time-asymmetric structure of the RHS to define a fundamental arrow of time in quantum mechanics.

Experimental results

Research questions

• RQ1 How can quasistationary quantum states be rigorously described within a Hilbert space framework?
• RQ2 What is the mathematical origin of irreversibility in quantum mechanics, and how is it related to the time-ordering of preparation and measurement?
• RQ3 Can an exact golden rule for decay be derived for a pure Gamow state using the RHS formalism?
• RQ4 How does the rigged Hilbert space formalism account for the transition from a pure state to a mixed state during decay?
• RQ5 What is the role of complex eigenvalues and Gamow vectors in modeling irreversible quantum processes?

Key findings

• Gamow vectors in the rigged Hilbert space formalism exhibit intrinsic quantum mechanical irreversibility due to their complex energy eigenvalues and time-asymmetric boundary conditions.
• The formalism establishes a fundamental arrow of time in quantum mechanics, where state preparation must precede observation, reflecting physical causality.
• An exact golden rule for the decay of a pure Gamow state into a mixture of decay products is derived, providing a rigorous alternative to the standard perturbative golden rule.
• The time evolution of Gamow states is described by an exponential decay law that is consistent with the RHS structure and the outgoing wave boundary condition.
• The RHS framework allows for a mathematically consistent treatment of unstable quantum systems, including resonances and decays, beyond the standard Dirac formalism.
• The formalism shows that irreversibility is not an emergent feature but an intrinsic property of the quantum state space when generalized functions are properly included.