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Time Asymmetric Quantum Theory — I Modifying an Axiom of Quantum Physics

A. R. Bohm, Mark Loewe, Bryan Van De Ven

Published Jan 1, 2008• 23 citations
OpenAlex

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. 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

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Time asymmetric quantum theory – I. Modifying an axiom of quantum physics

A.R. Bohm, M. Loewe, B. Van de Ven

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.

arXiv:quant-ph/0212130•Fortschritte der Physik•May 13, 2003•21 citations

Time asymmetric quantum theory and its applications in non-relativistic and relativistic quantum systems

Hai Viet Bui

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.

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Resonance and decay phenomena lead to quantum mechanical time asymmetry

Arno Böhm, Hai Viet Bui

The states (Schrödinger picture) and observables (Heisenberg picture) in the standard quantum theory evolve symmetrically in time, given by the unitary group with time extending over − ∞ < t < + ∞ . This time evolution is a mathematical consequence of the Hilbert space boundary condition for the dynamical differential equations. However, this unitary group evolution violates causality. Moreover, it does not solve an old puzzle of Wigner: How does one describe excited states of atoms which decay exponentially, and how is their lifetime τ related to the Lorentzian width Γ? These question can be answered if one replaces the Hilbert space boundary condition by new, Hardy space boundary conditions. These Hardy space boundary conditions allow for a distinction between states (prepared by a preparation apparatus) and observables (detected by a registration apparatus). The new Hardy space quantum theory is time asymmetric, i.e, the time evolution is given by the semigroup with t 0 ≤ t < + ∞ , which predicts a finite "beginning of time" t 0 , where t 0 is the ensemble of time at which each individual system has been prepared. The Hardy space axiom also leads to the new prediction: the width Γ and the lifetime τ are exactly related by τ = ℏ/Γ .

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Irreversibility in quantum mechanics

Robert C. Bishop, Arno Böhm, Manuel Gadella

Time asymmetry and irreversibility are signal features of our world. They are the reason of our aging and the basis for our belief that effects are preceded by causes. These features have many manifestations called arrows of time. In classical physics, some of these arrows are described by the increase of entropy or probability, and others by time‐asymmetric boundary conditions of time‐symmetric equations (e.g., Maxwell or Einstein). However, there is some controversy over whether probability or boundary conditions are more fundamental. For quantum systems, entropy increase is usually associated with the effects of an environment or measurement apparatus on a quantum system and is described by the von Neumann‐Liouville equation. But since the traditional (von Neumann) axioms of quantum mechanics do not allow time‐asymmetric boundary conditions for the dynamical differential equations (Schrödinger or Heisenberg), there is no quantum analogue of the radiation arrow of time. In this paper, we review consequences of a modification of a fundamental axiom of quantum mechanics. The new quantum theory is time asymmetric and accommodates an irreversible time evolution of isolated quantum systems.

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Time Asymmetric Quantum Mechanics

Arno Böhm

The meaning of time asymmetry in quantum physics is discussed.On the basis of a mathematical theorem, the Stone-von Neumann theorem, the solutions of the dynamical equations, the Schrödinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary group.Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution.Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)).If one requires that scattering resonances of width Γ and exponentially decaying states of lifetime τ = Γ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS's of Hardy functions and connected with it, to a semigroup time evolution t 0 ≤ t < ∞, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system.The decay of quasi-stable particles is used to illustrate this quantum mechanical time asymmetry.From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics.

arXiv:1109.0598•Symmetry Integrability and Geometry Methods and Applications•Sep 3, 2011•12 citations

Complex Energies and Beginnings of Time Suggest a Theory of Scattering and Decay

Arno Böhm, Piotr Kielanowski, Sujeewa Wickramasekara

Many useful concepts for a quantum theory of scattering and decay (like Lippmann-Schwinger kets, purely outgoing boundary conditions, exponentially decaying Gamow vectors, causality) are not well defined in the mathematical frame set by the conventional (Hilbert space) axioms of quantum mechanics. Using the Lippmann-Schwinger equations as the takeoff point and aiming for a theory that unites resonances and decay, we conjecture a new axiom for quantum mechanics that distinguishes mathematically between prepared states and detected observables. Suggested by the two signs $\\pm i\\epsilon$ of the Lippmann-Schwinger equations, this axiom replaces the one Hilbert space of conventional quantum mechanics by two Hardy spaces. The new Hardy space theory automatically provides Gamow kets with exponential time evolution derived from the complex poles of the $S$-matrix. It solves the causality problem since it results in a semigroup evolution. But this semigroup brings into quantum physics a new concept of the semigroup time $t=0$, a beginning of time. Its interpretation and observations are discussed in the last section.

•Oct 8, 2005•11 citations
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
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.
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.
The states (Schrödinger picture) and observables (Heisenberg picture) in the standard quantum theory evolve symmetrically in time, given by the unitary group with time extending over − ∞ < t < + ∞ . This time evolution is a mathematical consequence of the Hilbert space boundary condition for the dynamical differential equations. However, this unitary group evolution violates causality. Moreover, it does not solve an old puzzle of Wigner: How does one describe excited states of atoms which decay exponentially, and how is their lifetime τ related to the Lorentzian width Γ? These question can be answered if one replaces the Hilbert space boundary condition by new, Hardy space boundary conditions. These Hardy space boundary conditions allow for a distinction between states (prepared by a preparation apparatus) and observables (detected by a registration apparatus). The new Hardy space quantum theory is time asymmetric, i.e, the time evolution is given by the semigroup with t 0 ≤ t < + ∞ , which predicts a finite "beginning of time" t 0 , where t 0 is the ensemble of time at which each individual system has been prepared. The Hardy space axiom also leads to the new prediction: the width Γ and the lifetime τ are exactly related by τ = ℏ/Γ .
Time asymmetry and irreversibility are signal features of our world. They are the reason of our aging and the basis for our belief that effects are preceded by causes. These features have many manifestations called arrows of time. In classical physics, some of these arrows are described by the increase of entropy or probability, and others by time‐asymmetric boundary conditions of time‐symmetric equations (e.g., Maxwell or Einstein). However, there is some controversy over whether probability or boundary conditions are more fundamental. For quantum systems, entropy increase is usually associated with the effects of an environment or measurement apparatus on a quantum system and is described by the von Neumann‐Liouville equation. But since the traditional (von Neumann) axioms of quantum mechanics do not allow time‐asymmetric boundary conditions for the dynamical differential equations (Schrödinger or Heisenberg), there is no quantum analogue of the radiation arrow of time. In this paper, we review consequences of a modification of a fundamental axiom of quantum mechanics. The new quantum theory is time asymmetric and accommodates an irreversible time evolution of isolated quantum systems.
The meaning of time asymmetry in quantum physics is discussed.On the basis of a mathematical theorem, the Stone-von Neumann theorem, the solutions of the dynamical equations, the Schrödinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary group.Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution.Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)).If one requires that scattering resonances of width Γ and exponentially decaying states of lifetime τ = Γ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS's of Hardy functions and connected with it, to a semigroup time evolution t 0 ≤ t < ∞, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system.The decay of quasi-stable particles is used to illustrate this quantum mechanical time asymmetry.From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics.
Many useful concepts for a quantum theory of scattering and decay (like Lippmann-Schwinger kets, purely outgoing boundary conditions, exponentially decaying Gamow vectors, causality) are not well defined in the mathematical frame set by the conventional (Hilbert space) axioms of quantum mechanics. Using the Lippmann-Schwinger equations as the takeoff point and aiming for a theory that unites resonances and decay, we conjecture a new axiom for quantum mechanics that distinguishes mathematically between prepared states and detected observables. Suggested by the two signs pmiepsilon\\pm i\\epsilonpmiepsilon of the Lippmann-Schwinger equations, this axiom replaces the one Hilbert space of conventional quantum mechanics by two Hardy spaces. The new Hardy space theory automatically provides Gamow kets with exponential time evolution derived from the complex poles of the SSS-matrix. It solves the causality problem since it results in a semigroup evolution. But this semigroup brings into quantum physics a new concept of the semigroup time t=0t=0t=0, a beginning of time. Its interpretation and observations are discussed in the last section.