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Journal of Physics A Mathematical and General

Reflections on the quantum Zeno paradox

Dipankar Home, M. A. B. Whitaker

Published Jul 11, 1986• 44 citations
Publisher

Abstract

The authors clarify some of the controversial aspects involved in the treatment of the quantum Zeno paradox. An argument for the paradox is given on the basis of the uncertainty principle, and the conditions for the paradox are analysed. Fleming's rule is proved in a simple way. Connection with the standard time-dependent perturbation theory is discussed. They compare the Zeno paradox with the other well known paradoxes of quantum mechanics and point out that the quantum Zeno paradox is immune to the use of the ensemble interpretation which has some success in tackling the other paradoxes. Remarks are made concerning the broader significance of the quantum Zeno paradox.

DOI
10.1088/0305-4470/19/10/025

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The authors clarify some of the controversial aspects involved in the treatment of the quantum Zeno paradox. An argument for the paradox is given on the basis of the uncertainty principle, and the conditions for the paradox are analysed. Fleming's rule is proved in a simple way. Connection with the standard time-dependent perturbation theory is discussed. They compare the Zeno paradox with the other well known paradoxes of quantum mechanics and point out that the quantum Zeno paradox is immune to the use of the ensemble interpretation which has some success in tackling the other paradoxes. Remarks are made concerning the broader significance of the quantum Zeno paradox.
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.
We revisited the quantum Zeno paradox, which claims that a generic quantum system prepared in a state which is not an eigenstate of the Hamiltonian operator and is continuously observed never decays. Since any perfectly isolated quantum system always interact with a vacuum field, we analyze the possibility of using this fact to solve the above mentioned conceptual problem. Therefore we discuss a two-level system or qubit-Bose field interaction Hamiltonians. We consider the quantum dynamics of this two-level system, prepared in the excited state interacting with a Bose field prepared in the Poincare invariant vacuum state. Using a first-order approximation in time-dependent perturbation theory, we evaluate the probability of spontaneous decay of the two-level system driven by the vacuum field. This probability is evaluated for a finite time interval. Using the standard argument to obtain the quantum Zeno paradox, we consider N measurements where N {yields} {infinity} and we obtain that the non-decay probability law is a pure exponential, recovering the classical behavior. (author)
There is a very reason to consider that to solve Zeno's paradoxes is to propose the theory of mechanical world view. We believe that this is not only our opinion but also most philosophers' opinion. Recently, in order to justify Heisenberg`s uncertainty principle (cf. Rep. Math. Phys Vol. 29, No. 3, 1991) more firmly. we proposed the linguistic interpretation of quantum mechanics (called quantum and classical measurement theory), which was characterized as the metaphysical and linguistic turn of the Copenhagen interpretation. This turn from physics to language does not only extend quantum mechanicsto classical systems but also yield the (quantum and classical) mechanical world view (and therefore, establish the method of science). If it be so, we may assert that Zeno's paradoxes (Flying Arrow Paradox, Achilles and the tortoise, etc.) were already solved in measurement theory. The purpose of this paper is to examine this assertion.
A discussion of the quantum Zeno effect and paradox is given. The quantum Zeno paradox claims that a continuously observed system, prepared in a state which is not an eigenstate of the Hamiltonian operator, never decays. To recover the classical behavior of unstable systems we consider a two-level system interacting with a Bose field, respectively prepared in the excited state and in the Poincare invariant vacuum state. Using time-dependent perturbation theory, we evaluate for a finite time interval the probability of spontaneous decay of the two-level system. Using the standard argument to obtain the quantum Zeno paradox, we consider N measurements where N goes to infinity and we obtain that the non-decay probability law is a pure exponential, therefore recovering the classical behavior.
The article discusses the new approaches to the four Zeno’s Paradoxes: “The Ar­row”, “The Achilles”, “The Dichotomy” and “The Moving Rows”. Paradoxes are analyzed in relation to the current research in the field of elementary particle Physics and promising areas of the quantum theory of gravity development. Pro­vided with the necessary philosophical interpretation Physics concepts are used to clarify Zeno’s Paradoxes and to search for answers to them. The involvement of modern approaches is shown to be ineffective in solving the paradoxes due to their scope losses when analyzed by the microworld Physics on ultra-small scales. The work’s main part is devoted to demonstrating this circumstance – the questions posed by paradoxes are deprived of the possibility of getting the answers (at least, in their classical interpretation). As a possible explanation stands the ground of mixing the properties of the macroworld and the microworld in the paradoxes statements (which is historically justified given the intuitive homogeneity of big and small, and the origin of non-classical Physics – quantum mechanics – only in the 20th century), meaning the transition from observation of the large physical objects to the infinitely small in terms of discreteness and continuity. However, the principles of space organization on ultra-small scales are beginning to be clarified in general terms only now, and, perhaps, these prin­ciples may turn out to be quite far from classical ideas on fundamental physical reality.