Quantal time asymmetry: mathematical foundation and physical interpretation

Abstract

For a quantum theory that includes exponentially decaying states and Breit-Wigner resonances, which are related to each other by the lifetime-width relation $\tau =\frac{\hslash }{\Gamma }$, where $\tau$ is the lifetime of the decaying state and $\Gamma$ the width of the resonance, one has to go beyond the Hilbert space and beyond the Schwartz-Rigged Hilbert Space $\Phi \subset \mathcal{H}\subset {\Phi }^{\times }$ of the Dirac formalism. One has to distinguish between prepared states, using a space $Φ_-\subset\mat hcal{H}$, and detected observables, using a space ${\Phi }_{+}\subset \mathcal{H}$, where $-(+)$ refers to analyticity of the energy wave function in the lower (upper) complex energy semiplane. This differentiation is also justified by causality: A state needs to be prepared first, before an observable can be measured in it. The axiom that will lead to the lifetime-width relation is that ${\Phi }_{+}$ and ${\Phi }_{-}$ are Hardy spaces of the upper and lower semiplane, respectively. Applying this axiom to the relativistic case for the variable $\s=p_μp^μ$ leads to semigroup transformations into the forward light cone (Einstein causality) and a precise definition of resonance mass and width.

Plenary talk at the 5th International Symposium on Quantum Theory and Symmetries, July 22-28, 2007, Valladolid, Spain

References in corpus (7)

• Big-Bang nucleosynthesis (Particle Data Group mini-review)
• The Cosmological Parameters 2006
• Relativistic resonances: Their masses, widths, lifetimes, superposition, and causal evolution
• Gamow-Jordan Vectors and Non-Reducible Density Operators from Higher Order S-Matrix Poles
• Symmetry Representations in the Rigged Hilbert Space Formulation of Quantum Mechanics
• Irreversible Quantum Mechanics in the Neutral K-System
• A parametrization of the CKM mixing matrix from a scheme of S3L x S3R symmetry breaking

Cited by in corpus (2)

• Eigenvector continuation for emulating and extrapolating two-body resonances
• Bound State Internal Interactions as a Mechanism for Exponential Decay