## Journal of Geometry and Symmetry in Physics

- J. Geom. Symmetry Phys.
- Volume 11 (2008), 1-22.

### Conjecturing The Mathematical Axiom That Provides a Unified Theory of Resonance and Decay and Connects It to Causal Time Evolution

#### Abstract

Combining the ideas of causality and the phenomenology of resonances and decaying states, we modify standard quantum theory by changing one of its axioms. The first step was taken decades ago when Dirac kets were given a mathematical meaning as functionals on a Schwartz space, which led to a Gelʼfand triplet in which observables were represented by an algebra of continuous operators. The second step, motivated by the analytic continuation to complex energies for the S-matrix and the two Lippmann-Schwinger kets, distinguishes between prepared in-states-vectors and detected out-vectors. This leads to the pair of Gelʼfand triplets realized by Hardy functions of the upper (for prepared states) and lower (for detected observables) complex energy plane.

Replacing the Hilbert space axiom of quantum mechanics by the Hardy space axiom, one obtains a mathematically consistent theory that unifies resonance and decay phenomena and that has a causal, asymmetric time evolution.

#### Article information

**Source**

J. Geom. Symmetry Phys., Volume 11 (2008), 1-22.

**Dates**

First available in Project Euclid: 24 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.jgsp/1495591300

**Digital Object Identifier**

doi:10.7546/jgsp-11-2008-1-22

**Mathematical Reviews number (MathSciNet)**

MR2433993

**Zentralblatt MATH identifier**

1221.81016

#### Citation

Bohm, Arno; Bryant, Peter. Conjecturing The Mathematical Axiom That Provides a Unified Theory of Resonance and Decay and Connects It to Causal Time Evolution. J. Geom. Symmetry Phys. 11 (2008), 1--22. doi:10.7546/jgsp-11-2008-1-22. https://projecteuclid.org/euclid.jgsp/1495591300