The question whether quantum discontinuity can or cannot provide an answer to Zeno's Paradoxes is reopened. It is observed that what is usually understood by the term "discontinuity", namely, Einstein's conception of the photon as described by himself and all others, is unsuitable to the task because, essentially,it reduces to the trivial 'discontinuity' of objects scattered in space. By contrast, quantization of energy levels, which are not in space but can only alternate in time, provide the right sort of discontinuity required. Discrete quantized orbits, corresponding to eigen-frequencies, are irreducible, and nothing is allowed to stand in-between them in satisfaction of the quantum postulate, furnishing the requisite, and so far missing, immediate nextness of a point to a certain other. In this way, Zeno's Runner need not postpone his first step indefinitely, always waiting upon an infinity of preceding steps, before it can be taken. There is now a point that is next to a point and so a step on that point, which is the first step. It follows that, if one kind of discontinuity, Einstein's, is incapable of offerring an answer to Zeno, while another kind can, the two are discrepant. One of them, the former, is not a kind of discontinuity properly so called at all, though evidently the consequence of one.
The question whether quantum discontinuity can or cannot provide an answer to Zeno's Paradoxes is reopened. It is observed that what is usually understood by the term "discontinuity", namely, Einstein's conception of the photon as described by himself and all others, is unsuitable to the task because, essentially,it reduces to the trivial 'discontinuity' of objects scattered in space. By contrast, quantization of energy levels, which are not in space but can only alternate in time, provide the right sort of discontinuity required. Discrete quantized orbits, corresponding to eigen-frequencies, are irreducible, and nothing is allowed to stand in-between them in satisfaction of the quantum postulate, furnishing the requisite, and so far missing, immediate nextness of a point to a certain other. In this way, Zeno's Runner need not postpone his first step indefinitely, always waiting upon an infinity of preceding steps, before it can be taken. There is now a point that is next to a point and so a step on that point, which is the first step. It follows that, if one kind of discontinuity, Einstein's, is incapable of offerring an answer to Zeno, while another kind can, the two are discrepant. One of them, the former, is not a kind of discontinuity properly so called at all, though evidently the consequence of one.
G.H. von Wright validly remarks that Zeno’s arrow “is neither moving nor at rest”. Then he invalidly proceeds to turn this into “both moving and at rest ” eo ipso. Hegel does the exact same thing and so, it seems, does everybody else. A violat-ion of the Law of the Excluded Middle (LEM), in the form of –A and --A, is equated (eo ipso) with a violation of the Law of Non-Contradiction (LNC), as A and –A. The move is both, circular (it employs double negation) and contradictory. When it is as-serted that neither A nor –A, the last thing which follows is that both A and –A. If these two are both false, contradicting LEM, they cannot also be both true. Unexpect-edly, a violation of LEM protects LNC. In accordane [a] I argue that a system violating LEM eo ipso satisfies LNC, contrary to what von Wright, Hegel, paraconsistent logicians and everybody else seems to think. [b] Zeno’s paradoxes produce an antinomy, iff motion is continuous. (Ancient Atomists only reacted to Zeno’s infinite divisibility of processes.) [c] A comparable quantum model for discontinuous transitions, including motion, displays properties identical to those specified in [a] and [b]. If a body moves discontinuously from A to B, it is nowhere at all in-between A and B. And therefore cannot be where it is not, offerring an alternative to Zeno’s antinomies. [d] Zeno’s paradoxes, if handled by quantum discontinuity, lead to a 3-valued but consistent system. Discontinuity eliminates all possible descriptions of a system’s state. Hence, if nothing can be truthfully said about the system during a discontinuous transition, nothing self-contraditory can be said about it either. Discontinuous motion is nonbivalent but consistent.
As of October 2006, there were approximately 535 citations to the seminal 1977 paper of Misra and Sudarshan that pointed out the quantum Zeno paradox (more often called the quantum Zeno effect). In simple terms, the quantum Zeno effect refers to a slowing down of the evolution of a quantum state in the limit that the state is observed continuously. 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 experiment of Itano, Heinzen, Bollinger, and Wineland, published in 1990, has been cited around 347 times and seems to be the one most often called a demonstration of the quantum Zeno effect. Given that there is disagreement as to what the quantum Zeno effect is , there naturally is disagreement as to whether that experiment demonstrated the quantum Zeno effect. Some differing perspectives regarding the quantum Zeno effect and what would constitute an experimental demonstration are discussed.
This article reconsiders the double-slit experiment from the nonrealist or, in terms of this article, "reality-without-realism" (RWR) perspective, grounded in the combination of three forms of quantum discontinuity: (1) "Heisenberg discontinuity", defined by the impossibility of a representation or even conception of how quantum phenomena come about, even though quantum theory (such as quantum mechanics or quantum field theory) predicts the data in question strictly in accord with what is observed in quantum experiments); (2) "Bohr discontinuity", defined, under the assumption of Heisenberg discontinuity, by the view that quantum phenomena and the data observed therein are described by classical and not quantum theory, even though classical physics cannot predict them; and (3) "Dirac discontinuity" (not considered by Dirac himself, but suggested by his equation), according to which the concept of a quantum object, such as a photon or electron, is an idealization only applicable at the time of observation and not to something that exists independently in nature. Dirac discontinuity is of particular importance for the article's foundational argument and its analysis of the double-slit experiment.
In 1927 at the fifth Solvay Council, that reunited all the aristocracy of theoretical physics, Einstein, regarding with solicitude the new-born “quantum mechanics” of Louis de Broglie, Schrödinger, Heisenberg and Dirac, discerned with his usual sagacity an indelible mark that was destined to become, with time, a subject of passionate discussion among those whose vocation is to adulate this enigmatic and capricious personality. In 1926 Born had given the prophetic stroke to the portrait. Turning to probability as to the official factotum of the reconciliation of the continuous and the discontinuous-here, the associated wave and particle-he transmuted the waves of de Broglie and Schr6dinger into an undulatory calculus o f probabilities, deducing, from a surprising principle, consequences that were even more surprising but always verified through experiment. Parting from the idea that the intensity of the wave is the probability of the detection of the particle at a given point and time, Born replaced the classic principle of addition of partial probabilities with his “principle of the addition of partial amplitudes” that are, as in classical optics, represented by “complex” dimensions, with one real part and one imaginary part. In general, the square of the module of the sum of amplitudes will be the probability. This expression contains, of course, the terms “square” and “rectangular.” The first, if they were alone, would give the former law; as for the second, they express the existence of phenomena of interference that are at the origin of the thousand and one well verified paradoxes of the “new mechanics “-the one thousand and first being the one under consideration here.
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 argument of this chapter is governed by the double meaning of its title: (a) the relationships between the concepts of discreteness and continuity in modern mathematics and physics; and (b) the relationships, both continuous and discontinuous, between mathematics and physics, as, from Galileo on, a mathematical-experimental science, with mathematics coming first in this conjunction. The project of modern physics was, thus, defined by an essential continuity between mathematics and physics, as a mathematical representation of nature, principally by means of continuous functions and then calculus, until the rise of quantum theory, specifically as quantum mechanics (QM) and quantum field theory (QFT). With QM, at least in certain interpretations, such as the one adopted in this chapter, quantum theory disconnected the ultimate constitution of nature from any mathematical representation, thus, making physics discontinuous with mathematics. As this chapter will argue, however, this new epistemological situation did not disconnect quantum physics from mathematics, but, on the contrary, led to relating abstract mathematics, such as that of Hilbert spaces over ℂ and operator algebra there, to discrete physical phenomena in terms of probabilities, thus reestablishing the connection, which was no longer representational, between mathematics and physics. This chapter will also argue that the development of the concepts of continuity and discontinuity, and the relationships between them, has acquired a great richness and complexity through the nineteenth and twentieth centuries in mathematics itself, richness and complexity that found their way into physics, in particular relativity and quantum theory. One of the intriguing aspects of this development is the idea, advanced by, among others, A. Grothendieck, following B. Riemann's comment on the subject, that the continuous may serve as an approximation of the discontinuous, rather than seeing, as is more common, the discontinuous as a mode of technical approach to the continuous. This chapter will discuss this idea, and the relationships between continuity and discontinuity in, and between, mathematics and physics in terms of two new concepts: reality without realism (RWR), applicable in both mathematics and physics, and the relationships between them, and ideality without idealism (IWI), the version of the concept of RWR, applicable specifically to mathematics.