Time Asymmetric Quantum Mechanics (original) (raw)
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Mathematical Foundations of Time Asymmetric Quantum Mechanics
Journal of Physics: Conference Series, 2017
We review the mathematical tools that are suitable for a formulation of time asymmetry in quantum mechanics. In particular, Hardy functions on a half plane and rigged Hilbert spaces constructed with a subclass of Hardy functions. This time asymmetry often appears in quantum scattering and, in particular, in resonance scattering. We review the construction of Gamow vectors, often considered Gamow states for resonances. A brief summary of the fundamental ideas of time asymmetric quantum mechanics is presented in a last section.
Resonance Phenomena and time asymmetric quantum mechanics
This article is a review of time asymmetric quantum theory and its consequences applied to the resonance and decay phenomena. We first give some phenomenological results about resonances and decaying states to support the popular idea that resonances characterized by a width Γ and decaying states characterized by a lifetime τ are different appearances of the same physical entity. Based on Weisskopf-Wigner (WW) methods, one obtains approximately τ ≈ Γ . However, using standard axioms of quantum physics it is not possible to establish a rigorous theory to which the various WW methods can be considered as approximations. In standard quantum theory, the set of states and the set of observables are mathematically identified and described by the same Hilbert space H . Modifying this Hilbert space axiom to a Hardy space axiom one distinguishes the prepared (in) states and detected (out) observables. This leads to semi-group time evolution and to beginnings of time for individual microsystems. As a consequence of this time asymmetric theory one derives τ = Γ as an exact relation, and this unifies resonances and decaying states. Finally, we show that this unification can also be extended to the relativistic regime.
A time-symmetric of quantum mechanics
2010
That quantum mechanics is a probabilistic theory was, by 1964, an old but still troubling story. The fact that identical measurements of identically prepared systems can yield different outcomes seems to challenge a basic tenet of science and philosophy. Frustration with the indeterminacy intrinsic to quantum mechanics was famously expressed in Albert Einstein's assertion that "God doesn't play dice." By 1964 most physicists had abandoned the struggle and taken a more pragmatic view. The theory seemed to answer all questions in the workaday world of calculating ground states, energy levels, and scattering cross sections. Asking what actually happens at a measurement played no role in understanding, say, the properties of condensed matter or nuclei. The wavefunction and its evolution seemed to be all that was needed. The puzzle of indeterminism hadn't gone away, but it was safely marginalized. But 1964 brought a reversal of fortune. Indeterminacy, until then an unpleasant feature of an indispensible theory, suddenly became an open door to new freedoms implicit in the theory. One such freedom, the possibility of nonlocal correlations, was discovered by John Bell. 1 Another is the freedom to impose independent initial and final conditions on the evolution of a quantum system. The inquiry into that latter freedom, started by one of us (Aharonov), Peter Bergmann, and Joel Lebowitz 2 (ABL), is the subject of this article. Our inquiry evolved slowly at first, but by now it has led to a new approach to quantum mechanics, to the discovery of a number of new quantum effects, to a powerful amplification method, and to an admittedly controversial new view of the nature of time.
On the problem of time in quantum mechanics
European Journal of Physics, 2017
The problem of time in quantum mechanics concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time-energy uncertainty relation analogue to the positionmomentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of electron's relativistic quantum mechanics (RQM) does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of RQM.
New Developments in the Study of Time as a Quantum Observable
International Journal of Modern Physics B - IJMPB, 2008
Some results are briefly reviewed and developments are presented on the study of Time in quantum mechanics as an observable, canonically conjugate to energy. Operators for the observable Time are investigated in particle and photon quantum theory. In particular, this paper deals with the hermitian (more precisely, maximal hermitian, but non-selfadjoint) operator for Time which appears: (i) for particles, in ordinary non-relativistic quantum mechanics; and (ii) for photons (i.e., in first-quantization quantum electrodynamics).
Time and Quantum Theory: A History and a Prospectus
Studies in History and Philosophy of Modern Physics, 2013
The historical part of this paper analyzes in detail how ideas and expectations regarding the role of time in quantum theory arose and evolved in the early years of quantum mechanics (from 1925 to 1927). The general theme is that expectations which seemed reasonable from the point of view of matrix mechanics and Dirac's q-number formalism became implausible in light of Dirac–Jordan transformation theory, and were dashed by von Neumann's Hilbert space formalism which came to replace it. Nonetheless, I will identify two concerns that remain relevant today, and which blunt the force of Hilgevoord's (2005) claim that the demand that time feature as an observable arose as the result of a simple conceptual error. First, I advocate the need for event time observables, which provide a temporal probability distribution for the occurrence of a particular event. Second, I claim that Dirac's use of the extended phase space to define time and (minus the) energy as conjugates is not subject to ‘Pauli's Theorem,’ the result that rules out time observables in von Neumann's formalism. I also claim that the need to define these event time observables leads to a novel motivation for considering Dirac's extended state space.
Time's Arrow and Irreversibility in Time‐Asymmetric Quantum Mechanics
International Studies in the Philosophy of Science, 2005
The aim of this paper is to analyze time-asymmetric quantum mechanics with respect to the problems of irreversibility and of time's arrow. We begin with arguing that both problems are conceptually different. Then, we show that, contrary to a common opinion, the theory's ability to describe irreversible quantum processes is not a consequence of the semigroup evolution laws expressing the non-time-reversal invariance of the theory. Finally, we argue that time-asymmetric quantum mechanics, either in Prigogine's version or in Bohm's version, does not solve the problem of the arrow of time because it does not supply a substantial and theoretically founded criterion for distinguishing between the two directions of time.
The challenging concept of time in quantum mechanics
2012
Rarely, the authors of quantum mechanics books have discussed Dirac – Jordan transformation theory in abstract and pure form . Mostly, in the topics of mathematical tools, quantum mechanics assimilates in a manner with matrices theory that and its operability and ability differentiates as a pure theory is difficult. The subject of this study is that to show the ability of this theory in different discussions and particular differences of its solution methods with other theories. Principally, applied mathematics in Dirac – Jordan transformation theory is particular and differs with the mathematics present in theorems and relations of wave and matrices theories . Encountering with wave and matrices theories maybe implies, at least, applied mathematics in these theories gives certain relation between them, but there is not the case of Dirac – Jordan transformation theory. Quantum state of a particle in a given time, in Schrodinger’s wave theory, was defined by wave function . Probabili...
Regaining time symmetry in the generalized quantum mechanics of the Brussels School
Despite the fact that the fundamental physical laws are symmetric in time, most observed processes do not show this symmetry. The phenomenon of decay, however, seems to involve some kind of irreversibility, making the definition of a microscopic arrow of time imaginable. Such an intrinsic irreversibility of decaying systems is incorporated within the generalized quantum mechanics of the Brussels School, contrasting to the statements of standard quantum mechanics. As shown in this paper, the formlism bears significant advantages describing systems involving decay, however the breaking of time symmetry can be avoided using a different mathematical framework.
Bardon/A Companion to the Philosophy of Time, 2013
First, I briefly review the different conceptions of time held by three rival interpretations of quantum theory: the collapse of the wave-packet, the pilotwave interpretation, and the Everett interpretation (Section 2).