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Journal of Modern Physics, 2017, 8, 1382-1397

http://www.scirp.org/journal/jmp

ISSN Online: 2153-120X

ISSN Print: 2153-1196

DOI: 10.4236/jmp.2017.88087

July 21, 2017

Zeno of Elea Shines a New Light on

Quantum Weirdness

María Esther Burgos

Department of Physics, University of Los Andes, Mérida, Venezuela

Abstract

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 Pe

turbation 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 pos

ulates of Orthodox Quantum Mechanics; the other is derived from a Spont

neous Projection Approach to quantum mechanics previously formulated.

Similarities and differences between both solutions are highlighted. A co

parison 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 fram

work of Orthodox Quantum Mechanics is defective.

Keywords

Quantum Weirdness, Quantum Measurements, Spontaneous Quantum

Jumps, Time Dependent Perturbation Theory, Quantum Zeno Paradox

1. Introduction and Outlook

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 every-

How to cite this paper:

Burgos, M.E.

(201

Zeno of Elea Shines a New Light on

Quantum Weirdness

Journal of Modern

Physics

, 1382-1397.

https://doi.org/10.4236/jmp.2017.88087

Received:

April 17, 2017

Accepted:

July 18, 2017

Published:

July 21, 2017

7 by author and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution International

License (CC BY

4.0).

http://creativecommons.org/licens

es/by/4.0/

Open Access

M. E. Burgos

1383

thing when it occupies an equal space is at rest, and if that which is in locomo-

tion is always in a now, the flying arrow is therefore motionless. (Aristotle

Phys-

ics

, 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)” [1].

In 1977 Baidyanath Misra and George Sudarshan studied the behavior of an

unstable particle continuously observed to see whether it decays or not [2]. The

resulting effect have previously been described by Alan Turing in the following

terms: “it is easy to show using standard [quantum] theory that if a system starts

in an eigenstate of some observable, and measurements are made of that ob-

servable

times a second, then, even if the state is not a stationary one, the

probability that the system will be in the same state after, say, one second, tends

to one as

tends to infinity; that is, that continual observations will prevent

motion…” [3]. Initially this argument received the name of Turing paradox.

In their 1977 paper, Misra and Sudarshan referred to the behavior of a quan-

tum system subjected to frequent ideal measurements. They considered the

process of continuing observation as the limiting case of successions of (practi-

cally) instantaneous measurements as the intervals between successive mea-

surements 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 mea-

surement or the dead time between successive measurements from being arbi-

trarily small, the process of continuous observation seems to be an admissible

process in quantum theory” [2]. They concluded that an unstable particle which

is continuously observed to see whether it decays or not will never be found to

decay and named this phenomenon the quantum Zeno paradox [2].

Misra and Sudarshan article stimulated a great deal of theoretical and experi-

mental work. The possibility that the decay of an unstable particle could be pre-

vented 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 [4] [5] [6].

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 func-

tion. 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” ([7]; emphases added).

In 2009, Itano published a revision of different opinions regarding the quan-

tum 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” [8]. For instance, according to Asher Peres, the quantum Zeno effect

M. E. Burgos

1384

has nothing paradoxical: “What happens simply is that the quantum system is

overwhelmed by the meters which continuously interact with it” ([9], p. 394).

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 illu-

sion. By contrast, both Turing’s argument and Misra and Sudarshan’s contribu-

tion 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 mechan-

ics (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.

2. Formulation of TQZ Paradox

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 ([10],

Chapter IX); C. Cohen-Tannoudji

et al

. ([11], Chapter XIII); P. A. M. Dirac

([12], Chapter VII); W. Heitler ([13], Chapter IV); E. Merzbacher ([14], Chapter

XIX); and/or A. Messiah ([15], Chapitre XVII).

Notes

Symbols used by these

authors may have been changed for homogeneity. All the states referred to in

this paper are normalized.

Consider a system with Hamiltonian

which does not depend explicitly on

time. It will be called the

unperturbed Hamiltonian

of the system. Its eigenvalue

equations are

n nn

φφ

(1)

where

( )

1, 2,

En= 

are the eigenvalues of

and

the corresponding

eigenstates. For simplicity we assume

spectrum to be entirely discrete and

non-degenerate.

We shall suppose that at initial time

0t =

the system is in the stationary

state

. If for

0t ≥

the Hamiltonian were

, the state vector at time t

would be

( ) ( )

e 0e

iE t iE t

ψψ φ

−−

= =



(2)

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