_{1}

^{*}

After a brief reference to the quantum Zeno effect, a quantum Zeno paradox is formulated. Our starting point is the usual version of Time Dependent Perturbation Theory. Although this theory is supposed to account for transitions between stationary states, we are led to conclude that such transitions cannot occur. Paraphrasing Zeno, they are nothing but illusions. Two solutions to the paradox are introduced. The first as a straightforward application of the postulates of Orthodox Quantum Mechanics; the other is derived from a Spontaneous Projection Approach to quantum mechanics previously formulated. Similarities and differences between both solutions are highlighted. A comparison between the two versions of quantum mechanics, supporting their corresponding solutions to the paradox, shines a new light on quantum weirdness. It is shown, in particular, that the solution obtained in the framework of Orthodox Quantum Mechanics is defective.

The Greek philosopher Zeno of Elea (ca. 490-430 BC) supported Parmenide’s doctrine. This philosophy states that, contrary to the evidence of our senses, the belief in plurality and change is mistaken; in particular motion is nothing but an illusion.

The most popular Zeno paradoxes concerning motion are “Achilles and the Tortoise” and the “Arrow Paradox”. In the latter it is assumed that for motion to occur, an object must change the position which it occupies. In the case of an arrow in flight, Zeno argues that “the flying arrow is at rest, which result follows from the assumption that time is composed of moments… he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always in a now, the flying arrow is therefore motionless. (Aristotle Physics, 239b. 30) Zeno abolishes motion, saying ‘What is in motion moves neither in the place it is nor in one in which it is not’. (Diogenes Laertius Lives of Famous Philosophers, ix.72)” [

In 1977 Baidyanath Misra and George Sudarshan studied the behavior of an unstable particle continuously observed to see whether it decays or not [

In their 1977 paper, Misra and Sudarshan referred to the behavior of a quantum system subjected to frequent ideal measurements. They considered the process of continuing observation as the limiting case of successions of (practically) instantaneous measurements as the intervals between successive measurements approach zero. They argued that, “since there does not seem to be any principle, internal to quantum theory, that forbids the duration of a single measurement or the dead time between successive measurements from being arbitrarily small, the process of continuous observation seems to be an admissible process in quantum theory” [

Misra and Sudarshan article stimulated a great deal of theoretical and experimental work. The possibility that the decay of an unstable particle could be prevented by continued observation was, however, considered an alarming result by some physicists. In particular, as early as in 1983, Mario Bunge and Andrés Kálnay explicitly dealt with the suspicion that the quantum Zeno paradox must be a fraud [

In 1990 Wayne Itano et al. published a paper entitled Quantum Zeno effect. In its abstract they assert: “The quantum Zeno effect is the inhibition of transitions between quantum states by frequent measurements of the state. The inhibition arises because the measurement causes a collapse (reduction) of the wave function. If the time between measurements is short enough, the wave function usually collapses back to the initial state. We have observed this effect in an rf transition… Short pulses of light, applied at the same time as the rf field, made the measurements” ( [

In 2009, Itano published a revision of different opinions regarding the quantum Zeno effect. He acknowledges that “there has been much disagreement as to how the quantum Zeno effect should be defined and as to whether it is really a paradox, requiring new physics, or merely a consequence of ‘ordinary’ quantum mechanics” [

The theoretical and experimental work dealing with the quantum Zeno effect is exciting. But its relation with Zeno’s arrow paradox is questionable: Zeno’s purpose was not to stop the flying arrow; it was to show that motion is an illusion. By contrast, both Turing’s argument and Misra and Sudarshan’s contribution aim to stop transitions between quantum states by frequent measurements; let alone the experiment by Itano et al. (and many others we have not mention for brevity) where transitions between quantum states seem to have been truly inhibited, at least partially.

Differing from other references to the quantum Zeno effect, the present paper highlights a True Quantum Zeno paradox (TQZ paradox for short): we show that the usual version of Time Dependent Perturbation Theory (TDPT) leads to the conclusion that transitions between stationary states cannot happen. They are nothing but illusions.

The outlook of this paper is as follows: In Section 2, we formulate the TQZ paradox. In Section 3 we introduce and compare two different solutions to the paradox: an orthodox solution results from a straightforward application of the postulates of Orthodox (Ordinary, Standard) Quantum Mechanics (OQM); the other is derived from a Spontaneous Projection Approach to quantum mechanics (SPA) previously formulated. Section 4 contrasts the main traits of SPA and OQM. In particular, similarities and differences between both solutions to TQZ paradox are highlighted. Section 5 sums up the conclusions of the present work.

The aim of TDPT is to calculate the transition probability between stationary states induced by a time dependent perturbation. In the following we sketch the essential features of TDPT. For more details see for instance: D. R. Bes ( [

Consider a system with Hamiltonian

where

We shall suppose that at initial time

where ħ is Planck’s constant divided by _{j}.

A system in a stationary state (i.e. an eigenstate of the unperturbed Hamiltonian

If at

The perturbation

where

Let us underline the difference between the state vector at time t when no time dependent perturbation is applied and the state vector at time t resulting from the application of

At this point the probability of a transition taking place from state

For

TDPT deals with processes having two clearly different stages [

A collapse at t implies that the process is discontinuous at this instant. Since the sum of probabilities of a transition from

there is no room for a non-null probability corresponding to a process continuous at time t.

Let us now consider the following argument:

(a) A system initially in the state

(b) The interval

(c) Taking into account the validity of Equations (5) and (6) during the interval

With the noticeable exception of Albert Messiah, neither Dirac nor any other author known to us imposes any particular condition on the interval

Let us review what happens in a small time interval

where

For

the sum of these probabilities for all

and the probability of no transition taking place at all would be

Always assuming that the process is a Schrödinger evolution during the interval

We have shown, nevertheless, that according to the usual version of TDPT the system cannot follow a Schrödinger evolution during any time interval. Therefore, the state vector at time t cannot be

First solution: While remaining in the framework of OQM, Messiah version of TDPT differs somewhat from the usual one. In his words: “Supposons qu’à l’instant initial

In Section 2 we pointed out that, except Messiah, neither Dirac nor any other author known to us imposes any particular condition on the interval

Even if the notion of instantaneous measurement is questionable ( [_{j} and E_{k}; we suppose both of them non-degenerate. In the first stage no measurement is per- formed. As a consequence, during the time interval

leads the state vector from

Second solution: SPA provides another solution to TQZ paradox. According to this approach [

(i) Two kinds of processes, irreducible to one another, occur in nature: the strictly continuous and causal ones; and those implying discontinuities, where the system’s state

where

with probability

or

with probability

Here

and

Changes (13) are projections to one of the preferential states with probabilities given by Equation (14). As

jumps like

Except (v), all these points have been introduced and discussed in previous papers [

For simplicity we assume

where

If

_{j} and E_{k}; we suppose both of them non- degenerate. If no spontaneous projection happens in the time interval

Differing from what happens in the framework of OQM there is always room for a Schrödinger evolution in SPA. There is, however, a complete agreement between SPA solution and orthodox solution to TQZ paradox in which concerns the ratio of probabilities corresponding to jumps to

cases it takes on the value

OQM was first formulated by Dirac in 1930 [

conflict with the Schrödinger equation; and it implies a kind of action-at-a-dis- tance ( [

The presence in parallel of two different, irreducible to one another laws accounting for the change of the state vector

OQM marvelous success in the area of experimental predictions is mostly based on TDPT. It is agreed that the method provided by this theory must be used to solve all problems involving time, including time dependent spontaneous processes. Should TDPT be discarded, OQM and many of its extensions would lose almost completely their power of explanation and prediction [

SPA, a version of quantum mechanics previously introduced [

Other approaches aiming to confront quantum weirdness are close to, but different from OQM. By contrast, SPA does not introduce substantial changes into the theory. It does not modify the Schrödinger equation and recovers a version of Born’s postulate where no reference to measurement is made. The exponential decay law is obtained in cases where the Hamiltonian does not depend explicitly on time [

It has been pointed out that some theories of spontaneous state reduction are incompatible with the attainment of equilibrium [

The orthodox solution to TQZ paradox obtained in Section 3 results from a straightforward application of the postulates of OQM. But let us perform a close examination of this solution in the particular case where the perturbation ap- plied at

where

where

According to the postulates of OQM the only possible result of the measure- ment of a physical quantity is one of the eigenvalues of the operator which represents it. So a measurement of the energy performed at

By contrast, SPA solution to TQZ paradox makes no reference to measure- ments. Transitions between eigenstates of

In the framework of OQM, there are no projections without measurements. So it is necessary to invoke measurements even in spontaneous processes where measurements should obviously be absent. This is v.g. the case of absorption and emission of light and of processes occurring in semiconductors.

Both our Critical Review of TDPT [

We are indebted to Professor J. C. Centeno for many fruitful discussions. We are grateful to Professors F. G. Criscuolo and Marco Ortiz Palanques for some useful comments. We thank Carlos Valero for his assistance with the figures and the transcription of formulas into Math Type.

Burgos, M.E. (2017) Zeno of Elea Shines a New Light on Quantum Weirdness. Journal of Modern Physics, 8, 1382-1397. https://doi.org/10.4236/jmp.2017.88087

Let

and

the state vector

In addition to the problems referred to in Section 4, OQM conflicts with con- servation laws. Let us briefly review this issue which has been largely ignored [

In Schrödinger evolutions the validity of Equation (A1) and (A2) ensures that ^{ }

It has been shown that in processes involving projections (like OQM mea- surement processes) the mean value _{0}

and ending at t_{f} which yields

many times is close to zero [

Let us consider a set of

where

for the state

If there is a unique set of

where

we shall say that

The concept of preferential states plays a paramount role in SPA for

(a) The simplest case is that where the Hamiltonian

Since

is valid, the requirement (III) established in Appendix A is satisfied for every_{1} is

where

(b) We assume that the operators

and

are satisfied for

condition (III) stated in Appendix A is fulfilled for every state of the system. In the particular case where the state vector at t_{0} is

where

(c) We assume, as in case (b), that

while in the basis of the latter we have

Collapses to the vectors of the basis