научная статья по теме INTERFERENCE IN TIME Физика

Текст научной статьи на тему «INTERFERENCE IN TIME»

ЯДЕРНАЯ ФИЗИКА, 2008, том 71, № 5, с. 918-924

= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

INTERFERENCE IN TIME

© 2008 A. Gozdz, M. Deebicki, K. Stefanska

Department of Mathematical Physics, Institute of Physics University of M. Curie-Sklodowska, Lublin, Poland

Received August 20, 2007

A series of experiments has shown a possibility of existence of interference in time, a phenomenon similar to interference in space. A possible description of this phenomenon based on the hypothesis of the quantum projection evolution is presented.

PACS:03.65.-w, 03.65.Ta

1. INTRODUCTION

The notion of time was being discussed for centuries. The nonrelativistic physics introduces the time as a kind of the "absolute" parameter which orders physical events. The special relativity shows that the time have some features similar to the spatial coordinates and should be treated on the same footing as positions. The time and space are even more related one to another within the general relativity, e.g., at the vicinity of black-hole horizon, where time and spatial coordinates can even interchange.

In spite of this, classical observables are always functions of time which parametrizes subsequent events ordered by the causality relations.

In nonrelativistic quantum mechanics the traditional way of thinking follows the above idea. Especially that the Pauli's "theorem" about time operator in quantum mechanics [1—3] supports this considerations. The Pauli's conclusion that "...the introduction of an operator t must fundamentally be abandoned . . . " has been generally regarded as a "no-go theorem" for the possibility to treat the time as an observable within the quantum theory. The Pauli's "theorem" has been criticized as it turned out that it was only a formal argument; no mathematical rigor was used in the original formulation. This point was discussed by Galapon [4] who has shown that no attention was paid in the "Pauli's theorem" to the domains of the operators involved and that one can construct a self-adjoint time operator conjugate to the Hamiltonian. Also, within the POVM (Positive Operator-Valued Measure) approach to the quantum measurement theory the Pauli's argument can be easily overcome [5, 6]. However, there is still a problem of physical meaning of such operators.

A good historical review and a collection of references on these problems can be found in Chap. 1 of [7].

Within the relativistic quantum mechanics the problem of time becomes more complicated. However, a very simple, nearly naive argument saying that the Lorentz transformations mix time and spatial coordinates, suggests that the corresponding observables should be teated on the same footing. It means that the time cannot be teated as a parameter while the spatial coordinates are variables.

It is also obvious that in many physical situations it is required the physical time to be a dynamical variable.

First example is the problem of the time—energy uncertainty relations Chap. 3 of [7]. We are able to write a lot of relations among characteristic times, defined in more or less clear way, and the uncertainty of energy, like known Mandelstam—Tamm relation [8]. However, there is no relation between uncertainty of energy and time of occurrence of physical events, i.e., there is no analog to the relation between position and spatial linear momentum.

Another question about time as a variable is the problem of time of transitions or quantum jumps which originally comes from Heisenberg discussion with Pauli [9].

The practical problem to measure time of occurrence of quantum events, e.g., when a particle reaches a detector [6] (known as a time-of-arrival problem) has till now no satisfactory solution [7].

The tunneling time (see also superluminality, Hartman effect) or the dwell time which are not well defined within the standard approach are also arguments that time should be treated as a variable. More generally, one can notice that the standard approach has difficulties to address the question "how long does a particle take to traverse a spatial region".

Another class of problems which require time as a variable started in the fifties. In 1952 Moshinsky

has written a paper entitled "Diffraction in time" [10]. This intriguing title could suggest a nonlocality of the Schrodinger mechanics in time. However, it turns out that there is only a formal similarity of solutions of the Schro dinger equation with the diffraction theory results. On the other hand, independently from theoretical considerations, there were experimental investigations about possibility of finding quantum effects related to the time itself.

One of the more interesting experiments was done more than thirty years ago by Houser, Neuwirth, and Thesen [11]. They considered a beam of Meóssbauer quanta emitted from excited 56Fe nuclei, having the mean energy E0 = 14.4 keV and a lifetime t = = 141 ns. The particles were sent through the fast rotating chopper modulating the beam. The observed count rate was about 3000 events per second which compared with a lifetime gave, on average, one photon within 3000/t w 2000 lifetimes passed the device. It means that the detectors observed individual photons. The standard conclusion from this experiment was that the interference pattern obtained in this measurements demonstrates the "nonobjectivity" of the time of passage of the photon through the chopper. Another possibility to explain the energy spectrum obtained in this experiment is to consider a photon which interferes with itself being in different moments of time. In fact, the calculations were done according to the second proposal, see p. 95 of [7], and they are in a very good agreement with the experimental data. There was a series of similar experiments which confirmed the phenomenon [12].

Another group of experiments is related to laser physics. The recent experiment by Lindner et al. [13], see also references therein, can be considered as a two-slit Young experiment but in the temporal domain. Within the experiment, the double slit is realized by electromagnetic field. The electron can be ionized only within short time windows of maximal values of the electric field, but the electron "does not know" which time window of attosecond duration leads to ionization. This gives an interference fringes in the energy pattern of ionized electrons. To explain this direct realization of the temporal double-slit experiment the authors assumed that time and energy are conjugated variables.

The considerations given above require the time to be an observable similar to the position observables. It seams that the most natural way to treat time on the same footing as spatial coordinates is a reasonable modification of the evolution postulate of quantum mechanics.

The main purpose of this paper is to consider a hypothesis about the projection evolution approach in a context of time interference.

2. PROJECTION EVOLUTION

We want to construct the quantum evolution which treats the time and spatial variables on the same footing. For this purpose one has to include events in the evolution process, i.e., the time when an event happens should be a random variable. It implies, the evolution should not be driven by the physical time but it should be based on causally related physical quantum events. For this purpose we assume by definition existence of a real, growing parameter t which enumerates subsequent causally related events. This parameter has the meaning of ordering parameter. The domain for the parameter t does not need even to have a metric or more reach structure. However, for practical reason one can always treat it as a subset of real numbers. We will call t the evolution parameter.

It means that the quantum evolution process should be understood not in terms of the time variable but as the subsequent, causally ordered by the evolution parameter, "screen-shoots" of events in the space—time. We treat here the causality as a basic, fundamental notion.

The projection evolution was proposed in a series of papers [14—16]. In fact, it is independent of whether we consider relativistic or nonrelativistic quantum mechanics, though time as an independent dynamical variable fits better to the former case. The projection evolution approach requires only the so-called "minimal interpretation" of quantum mechanics [17]. It means that quantum states are given by density operator (with possibility of extension to functionals) and the rules for calculating probabilities are quite standard. Obviously, the time-evolution postulate is replaced by the projection evolution procedure.

The replacement of quantum-evolution postulate is now described in a few points [ 16]:

1. The states of a quantum system are described by the density operators and the term "quantum evolution" we understand as a sequence of state changes, ordered by the evolution parameter t which parametrizes the subsequent "causally" related events. It means that the quantum evolution is regarded as a kind of stochastic process parametrized by t .

2. For every quantum system there exists a set of orthogonal resolutions of unity, consisted of the projection operators E(t; v), which for each fixed t fulfills usual conditions in respect to the sets of quantum numbers represented by v:

E(t ; v) E(t ; v' ) = Svy Rt ; v), (1)

E ?(t ; v) = iv

The operators |(t; v) represent a set of alternatives for physical system under consideration, available at the step of evolution labeled by t. The evolution operators |(t; v) force changes of the physical system they describe. In this sense they are similar to standard unitary evolution.

3. There exists a mechanism (chooser) which chooses randomly the next state of the quantum system, in a way dependent on the physical structure of this quantum system. A new state is chosen randomly from the following states:

n( lA = §{T-y)p{T - dr-v)§{T-y)

where v' runs over all allowed by the projection evolution operator sets of quantum numbe

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