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arXiv:hep-th/0312027

Is a quantum theory of resonances really time asymmetric?

Frieder Kleefeld

Published Dec 2, 2003• 1 citation
arXivHTMLPDF

Abstract

The title "Time Asymmetric Quantum Theory: the Theory of Resonances" (without questionmark) of the CFIF workshop (23.-26.7.2003, Lisbon, Portugal) implies that the theoretical description of resonances is uniquely described by the formalism of A. Bohm et al. reflecting the title of the workshop. Our presentation in this workshop tries to introduce an apparently inequivalent, alternative feasible relativistic formalism provided by the author under the name "(Anti)Causal Quantum Theory" which is compared to the former.

DOI
10.48550/arxiv.hep-th/0312027

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The title "Time Asymmetric Quantum Theory: the Theory of Resonances" (without questionmark) of the CFIF workshop (23.-26.7.2003, Lisbon, Portugal) implies that the theoretical description of resonances is uniquely described by the formalism of A. Bohm et al. reflecting the title of the workshop. Our presentation in this workshop tries to introduce an apparently inequivalent, alternative feasible relativistic formalism provided by the author under the name "(Anti)Causal Quantum Theory" which is compared to the former.
Recently, several authors have criticized time-symmetrized quantum theory originated by the work of Aharonov et al. (1964). The core of this criticism was the proof, which appeared in various forms, showing that counterfactual interpretation of time-symmetrized quantum theory cannot be reconciled with the standard quantum theory. I argue here that the apparent contradiction appears due to inappropriate usage of traditional time asymmetric approach to counterfactuals, and that the contradiction disappears when the problem is analyzed in terms of time-symmetric counterfactuals. I analyze various aspects of time-symmetry of quantum theory and defend the time-symmetrized formalism.
The exact relation τ = ~/Γ between the width Γ of a resonance and the lifetime\nτ for a decay of this resonance could not be obtained in the conventional quantum\ntheory based on the Hilbert space as well as on the Schwartz space in which the spaces\nof the states and of the observables are described by the same space. Furthermore,\nboth dynamical evolution of the states (in Schrodinger picture) and observables (in\nHeisenberg picture) are symmetrically in time, given by an unitary group with time\nextending over −∞ < t < +∞. This time symmetric evolution is a mathematical\nconsequence of Von Neumann theorem for the dynamical differential equations,\nSchrodinger equation for the state or Heisenberg equation for the observable, under\neither the Hilbert space or Schwartz space boundary condition. However, this unitary\ngroup evolution violates causality. In order to get a quantum theory in which the\nexact relation and causality are obtained, one has to replace the Hilbert space or the\nSchwartz space boundary condition by a new boundary condition based on the Hardy\nspace axioms in which the space of the states and the space of the observables are\ndescribed by two distinguishable Hardy spaces. As a consequence of the new Hardy\nspace axiom, one obtains, instead of the time symmetric evolution for the states and\nthe observables, time asymmetrical evolutions for the states and observables which\nare described by two semi-groups. The time asymmetrical evolution predicts an finite\nbeginning of time t0(= 0) for quantum system which can be experimental observed\nby directly measuring the lifetime of the decaying state as the beginning of time of\nthe ensemble in scattering experiments. A resonance obeying the exponential time\nevolution can then be described by a Gamow vector, which is defined as superposition\nof the exact out-plane wave states with exact Breit-Wigner energy distribution.
Abstract A slight modification of one axiom of quantum theory changes a reversible theory into a time asymmetric theory. Whereas the standard Hilbert space axiom does not distinguish mathematically between the space of states (in‐states of scattering theory) and the space of observables (out‐“states” of scattering theory) the new axiom associates states and observables to two different Hardy subspaces which are dense in the same Hilbert space and analytic in the lower and upper complex energy plane, respectively. As a consequence of this new axiom the dynamical equations (Schrödinger or Heisenberg) integrate to a semigroup evolution. Extending this new Hardy space axiom to a relativistic theory provides a relativistic theory of resonance scattering and decay with Born probablilities that fulfill Einstein causality and the exponential decay law.
A slight modification of one axiom of quantum theory changes a reversible theory into a time asymmetric theory. Whereas the standard Hilbert space axiom does not distinguish mathematically between the space of states (in-states of scattering theory) and the space of observables (out-“states ” of scattering theory) the new axiom associates states and observables to two different Hardy subspaces which are dense in the same Hilbert space and analytic in the lower and upper complex energy plane, respectively. As a consequence of this new axiom the dynamical equations (Schrödinger or Heisenberg) integrate to a semigroup evolution. Extending this new Hardy space axiom to a relativistic theory provides a relativistic theory of resonance scattering and decay with Born probablilities that fulfill Einstein causality and the exponential decay law. 1 Introduction—Time Asymmetry Time asymmetry, irreversibility, time reversal non-invariance are different concepts and they are (probably) not (all) related to each other [1], [2], [3]. These concepts are usually called arrows-of-time. Time asymmetry comes from time asymmetric boundary conditions of time symmetric equations; its most prominent consequence is causality. The radiation arrow of time is its example from classical physics. Irreversibility
09 April 1996, “Randomness” I don't know if you remember me, but we met at the Santa Fe Institute Workshop on the Physics of Information two summers ago. I spoke to you about your old papers on randomness and quantum mechanics. In any case, I am very much interested in them again. (Presently, in particular, the ones titled “Finite and Infinite Measurement Sequences in Quantum Mechanics and Randomness: The Everett Interpretation” JMP 18 (1977) 2289, and “A Note on the Everett Interpretation of Quantum Mechanics” Found. Phys . 8 (1978) 709.) Carl Caves and I are writing a book on Quantum Information and are trying to get a feel for how far these sorts of considerations really go toward deriving the standard probability expression of quantum theory. Last night I reread the latter of the two papers mentioned above. The main question on my mind now is about Section 4 “The Asymptotic Description.” At the end you state, “Thus it is an open question whether or not there is any possible state description of the asymptotic situation that includes component states each corresponding to a possible universe as perceived by an observer with memory trace ν.” Has there been any progress in this direction since then?? If there has, could you please send me a reference or outline the solution?