Zeno of Elea Shines a New Light on Quantum Weirdness

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.


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-thing 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)" [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 observable N 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 N 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 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" [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 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 [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 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" ( [7]; emphases added).
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" [8]. For instance, according to Asher Peres, the quantum Zeno effect 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 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.

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  We shall suppose that at initial time A system in a stationary state (i.e. an eigenstate of the unperturbed Hamiltonian E ) will remain in that state forever. Nevertheless, TDPT establishes that by applying a time dependent perturbation, transitions between different eigenstates of E can be induced and determines the probability corresponding to every particular transition.
The perturbation ( ) 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 ( ) t W during the time interval ( ) 0,t . In the former case, the state vector coincides with the stationary state j φ ; see Equation (2). In the latter, the state vector will in general not be stationary but a linear superposition of several (at least two) stationary states; one of them being j φ .
At this point the probability of a transition taking place from state j φ to state k φ during the time interval ( ) 0,t is introduced. In the words of Paul Dirac, "at time t the ket corresponding to the state in Schrödinger's picture will be ( ) according to Equation (4). The probability of the n E 's then is the probability of a transition taking place from is the probability of no transition taking place at all. The sum of ( ) for all k is, of course, unity" ([12], pp. 172-173; emphases added).
TDPT deals with processes having two clearly different stages [16]. In the first-during the time interval ( ) 0,t -a Schrödinger evolution leads the state vector from given by Equation (4) with certitude. In the second an instantaneous projection of ( ) t ψ to a stationary state k φ is ruled by probability laws [16].
can collapse either to a state k φ where k j ≠ , or to the initial state j φ . According to Dirac, in this last case no transi-tion takes place at all. This does not mean that the system stays in the initial state j φ during the whole process. It means: during the interval ( ) 0,t the system follows a Schrödinger evolution and at instant t, when the state vector is given by Equation (4), it jumps to j φ .
A collapse at t implies that the process is discontinuous at this instant. Since the sum of probabilities of a transition from j φ to k φ with k j ≠ , plus the probability of no transition takes place at all during the time interval ( ) 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 (c) Taking into account the validity of Equations (5) and (6) during the interval ( ) 0,t′ we are forced to conclude that the probability corresponding to a process continuous at time t′ is null. But if the state vector is not continuous at t , the system cannot follow a Schrödinger evolution during the interval ( ) This conclusion contradicts (a).
With the noticeable exception of Albert Messiah, neither Dirac nor any other author known to us imposes any particular condition on the interval ( ) 0,t ; the condition imposed by Messiah will be discussed in the next section. In the usual version of TDPT it is assumed that Equations (5) and (6) where I is the identity operator ( [11], p. 309). Should the system follow a Schrödinger evolution during the interval ( ) For k j ≠ the probability of a transition taking place from state the sum of these probabilities for all k j ≠ would be (10) and the probability of no transition taking place at all would be Always assuming that the process is a Schrödinger evolution during the interval ( ) 0, dt , we see that if 0 dt → the probability for a transition between different stationary states becomes negligible while the probability for no transition taking place at all approaches unity. In colloquial speech we would conclude: as soon as the state vector becomes different from the initial state j φ a projection forces it to return to the starting point.
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 ( )  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 ( ) 0,t . This is why we could say that both Equation (5) and Equation (6)  Even if the notion of instantaneous measurement is questionable ( [5]; [17], p. 200), Messiah successfully eludes TQZ paradox through this concept. We shall call it the orthodox solution to TQZ paradox. Figure 1 illustrates the case where the non-perturbed Hamiltonian E has only two eigenvalues: E j and E k ; we suppose both of them non-degenerate. In the first stage no measurement is performed. As a consequence, during the time interval ( ) 0,t a Schrödinger evolution

Solving TQZ Paradox
with probability Here ( ) ( ) and Changes (13) are projections to one of the preferential states with probabilities given by Equation (14). As Except (v), all these points have been introduced and discussed in previous papers [18] [19] [20]. For more details on points (ii) and (iii) leading to the definition of preferential states see Appendix A; for examples of the determination of preferential states see Appendix B.
For simplicity we assume E spectrum to be entirely discrete and non-degenerate. The state vector can be written every stationary state is a preferential state, and not every preferential state is a stationary state [18], in the particular case where E has discrete non-degenerate spectrum, every preferential state is a stationary state; see Appendix B.
, the dominant process is the Schrödinger evolution [18]. As soon as the condition . During the small interval ( ) can undergo the following changes: either it jumps to one of its preferential states ( k φ and j φ ), or it follows a Schrödinger evolution which leads it from . The probabilities are, respectively, ; and 1 .
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 k φ and to j φ : in both cases it takes on the value ( ) ( )

SPA versus OQM
OQM was first formulated by Dirac in 1930 [12]. It refers to individual systems and imposes two laws of change of the state vector . Spontaneous processes are governed by the Schrödinger equation, a deterministic law. Measurement processes are ruled by probability laws through Born's postulate and the projection postulate. Measurement processes require either the intervention of an observer or the interaction of the quantum system with a macroscopic object playing the role of measuring device [20]. It has been pointed out that the projection postulate introduces a subjective element into the theory; it , or it jumps to one of its preferential states: conflict with the Schrödinger equation; and it implies a kind of action-at-a-distance ( [17], pp. 191-205; [20]). The presence in parallel of two different, irreducible to one another laws accounting for the change of the state vector ( ) t ψ calls for a rule to decide which one should be applied in every particular case. But OQM does not provide such a rule. Concerning this issue John Bell complains: "during 'measurement' the linear Schrödinger evolution is suspended and an ill-defined 'wave-function collapse' takes over. There is nothing in the mathematics to tell what is 'system' and what is 'apparatus,' nothing to tell which natural processes have the special status of 'measurements.' Discretion and good taste, born from experience, allow us to use quantum theory with marvelous success, despite the ambiguity of the concepts named above in quotation marks'' ( [21], p.160; emphasis added).
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 [16]. At the same time, TDPT is a good example of the ambiguities OQM confronts. In Section 2 we pointed out that TDPT deals with processes having two clearly different stages [16]. In the first-during the time interval ( ) 0,t -a Schrödinger evolution leads the system's state from In the second an instantaneous projection of ( ) t ψ to a stationary state k φ is ruled by probability laws [16]. Both laws are necessary for TDPT to work, but the fact that TDPT requires the application of postulates concerning measurements to account for processes supposedly spontaneous is at the very heart of OQM incoherence [16].
SPA, a version of quantum mechanics previously introduced [18] [19] [20], deals with these issues. Our motivation to formulate this approach is the restoration of philosophical realism as the basis of quantum mechanics. Albert Einstein was right when he proclaimed: "the belief in an external world independent of the perceiving subject is the basis of all natural science" [22]. We have also taken into account Bunge's notion of epistemological realism: "The main epistemological problem about quantum theory is whether it represents real (autonomously existing) things, and therefore whether it is compatible with epistemological realism. The latter is the family of epistemologies which assume that (a) the world exists independently of the knowing subject, and (b) the task of science is to produce maximally true conceptual models of reality…" ( [17], pp. 191-192).
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 [18]. Differing from OQM, SPA yields an expression for the probability of transitions to the continuum which is valid for every time and, except for some minimal restrictions, for every added potential. This prediction could be tested by experiment [19].
It has been pointed out that some theories of spontaneous state reduction are incompatible with the attainment of equilibrium [23]. This is obviously not the case of SPA where stationary states are not only possible: they play a fundamental role. We should also stress the radical difference between SPA and theories of quantum measurement based in the concept of decoherence. According to these theories, the off-diagonal elements of the density matrix should progressively vanish; it is not clear, however, why all diagonal elements but one should vanish [24]. By contrast, SPA states that a spontaneous projection to a preferential state instantaneously deletes as well the off-diagonal elements of the density matrix as all diagonal elements but one, as established by OQM when a measurement is performed.
The orthodox solution to TQZ paradox obtained in Section 3 results from a

Conclusions
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 [16] and the orthodox solution to TQZ paradox introduced in Section 3 highlight that in OQM the notion of measurement and consequent projections are ad-hoc. By contrast, in SPA projections are not surreptitious but explicitly included in the formalism. The same is true of the rule necessary to decide whether the system will forcibly follow a Schrödinger evolution or not. This is why SPA enjoys of a coherence which is absent from OQM [16] [20]. In previous papers we have introduced this requirement as a postulate ensuring the statistical sense of conservation laws [18] [19] [20].