Vật lý Lượng tử Nhất quán: Tái diễn giải Truyền thống bởi Arnold Neumaier

Arnold Neumaier Coherent Quantum Physics www.com Texts and Monographs in Theoretical Physics | Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA www.com Arnold Neumaier Coherent Quantum Physics | A Reinterpretation of the Tradition www.com Mathematics Subject Classification 2010 Primary: 81P15, 81R30, 46E22; Secondary: 17B81, 81T99 Author Prof. Arnold Neumaier Universität Wien Fakultåt für Mathematik Oskar-Morgenstern-Platz 1 1090 Wien Austria Arnold.at ISBN 978-3-11-066729-5 e-ISBN (PDF) 978-3-11-066738-7 e-ISBN (EPUB) 978-3-11-066736-3 ISSN 2627-3934 Library of Congress Control Number: 2019947573 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb. © 2019 Walter de Gruyter GmbH, Berlin/Boston Cover image: Guy N Harris / iStock / Getty Images Plus Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.com | To Maria, in honor of the Creator of our magnificent universe www.com Download Date | 10/31/19 1:08 PM www.com Preface In a statistical description of nature only expectation values or correlations are observable. 2678] One is almost tempted to assert that the usual interpretation in terms of sharp eigenvalues is ‘wrong’, because it cannot be consistently maintained, while the interpretation in terms of expec- tation values is ‘right’, because it can be consistently maintained.

6] What has become known as the quantum measurement problem […] encapsulates many of the fun- damental conceptual difficulties that have to this date prevented us from arriving at a commonly agreed-upon understanding of the physical meaning of the formalism of quantum mechanics and of how this formalism relates to the perceived world around us. VIII] This book introduces mathematicians, physicists, and philosophers to a new, coher- ent approach to theory and interpretation of quantum physics (including quantum mechanics, quantum statistical mechanics, quantum field theory, and their applica- tions), in which classical and quantum thinking live peacefully side by side and jointly fertilize the intuition. An interpretation of quantum mechanics relates its formalism to the actual in- formal practice of using quantum mechanics in our scientific culture. An impeccable interpretation must show that there is a fully consistent relation between theory and practice.

The interpretation may use concepts familiar from our culture to explain the working of quantum physics in practice to everyone’s satisfaction. What are the shortcomings of the current approaches? The minimal statistical interpretation predicts the statistics of outcomes of experiments. It is silent about the interpretation of quantum mechanics in the absence of measurements, and therefore about the interpretation of quantum physics applied to the far past of the universe, be- fore experiments were possible. This constitutes a serious gap—the interpretation is consistent, but incomplete (as it should be for a “minimal” interpretation).

The Copen- hagen interpretation, which claims that nothing can be asserted in the absence of a measurement, is also consistent. But this sounds like the concept that a tree fallen in the wood has fallen only after someone has seen it. This is one of the reasons why quantum mechanics comes across as somewhat strange. In a many-world interpreta- tion, the world splits and splits, completely unnoticed by us, into all possible futures.

This is science fiction by conception. The other known interpretations are either vari- ations of the above or require additional, in principle, unobservable, and hence fic- tional stuff. As a result, much of quantum physics appears to the general public as a kind of quantum magic. Why do physicists live with this? A noteworthy aspect of the standard inter- pretations is that the state vector cannot represent the whole universe, since it must https://doi.com VIII | Preface exclude an observer or measuring device that determines when a measurement has occurred.

This is the so-called Heisenberg cut between the quantum and the classical world. To date, this has not been a problem in making successful experimental predic- tions, so practitioners are often satisfied with the quantum formalism in a standard in- terpretation. Tradition builds the quantum edifice on a time-honored foundation that accounts for essentially all experimental facts. But it takes a “shut-up-and-calculate” attitude towards the interpretation of the foundations.

The traditional presentation of quantum physics is clearly adequate for prediction, but seems not to be suitable for an adequate understanding. A second reason is that a number of popular “quantum magicians”, very experi- enced quantum physics practitioners specializing in quantum optics, like to give their audience the impression that important parts of quantum mechanics are weird. And the general public loves it! Part of the magicians’ art consists of remaining silent about the true reasons why things work rationally, since then the weirdness is gone, and with it the entertainment value. Does quantum mechanics have to be weird? It sells much better to the general public if it is presented that way, and there is a long history of proceeding like this.

But it is an obstacle for everyone who wants to truly understand quantum mechanics, and to physics students, who have to unlearn what they were told as laypersons. When presented in the right way, quantum mechanics is not at all weird, but very close to classical mechanics. Much of the weirdness comes from forcing quantum mechanics into the straightjacket of a particle picture. The particle picture breaks down com- pletely in the subatomic domain, as witnessed by the many weird things that result from such a view.

Coherent quantum physics removes the radical split between classical mechanics and quantum mechanics. This book demonstrates that at any level of detail, Nature can be rationally and objectively understood just by interpreting the traditional, well- established mathematics of quantum physics in an appropriate way. This requires a reinterpretation of the tradition. The interpretation featured in this book succeeds without any change in the theory, and without introducing new counterintuitive fea- tures or new theoretical concepts.

The resulting quantum features then are only those familiar from everyday life. Nature, as we perceive it with our eyes, consists of images—in mathematical terms 2-dimensional fields, with properties (colors) at each point. Our brains interpret im- ages as scenes in a, strictly speaking, not directly perceived 3-dimensional world of objects. The same object seems larger or smaller depending on its distance from us, with a shape that is deduced from images showing the object from different perspec- tives.

All our observations are indirect: We perceive images and other sensory informa- tion and infer the true (theoretical, reproducible, invariant) properties of the objects around us. From the experience of the multitude of such sensory perceptions of many people, our culture created a network of concepts and relations now called science, and in www.com Preface | IX particular physics. Space has become 3-dimensional, represented at each particular time by 3-dimensional fields that tell the spatial properties of the materials present at each point in space. Their boundaries delineate the objects, some sharply, others— such as clouds—only in a fuzzy way.

Space thus becomes equipped with many properties. There are local properties, such as temperature, colors, hardness, stress, and chemical composition. In fluids there are properties like salt concentration, but also pressure, streaming velocity, et cetera. Each of these gives rise to a field that specifies how these properties vary with the position in space.

In addition, there are less tangible invisible properties, such as those described by the electromagnetic field. The latter describes the properties re- sponsible for the electric and magnetic phenomena in Nature, on which much of our modern culture depends. Additionally, there are bilocal properties, such as distances between two points in space. There are also nonlocal, region-dependent properties, such as the diameter, mass, and volume of an extended object, or the surface area of its boundary.

Objects often move. Just like photographs of stars in a long term night exposure, they trace out tracks in an abstract 3-dimensional space. These tracks form curves of a thickness depending on the objects’ size. The theory of special relativity teaches us beyond this 3-dimensional picture of the world a 4-dimensional perspective in a 4-dimensional Minkowski space, whose coordinates represent both space and time.

Due to length contraction and time dilation, shapes look and clocks move differently for observers moving at different velocities relative to each other. In special relativity, moving points are represented by so-called world lines; the curves they trace out in Minkowski space. The objects we see are extended in space, and therefore trace out world tubes—thin or thick tracks in 4 dimensions with boundaries reflecting the sharp or fuzzy, constant or changing shape of the objects. Materials vibrate and produce sound.

The electromagnetic field vibrates and pro- duces light. Both are phenomena characterizing the behavior of waves. These can be decomposed into harmonic waves of specified direction and frequency. The possible frequencies of vibration make up a spectrum.

A small part of these spectra are directly observable by the human ear and eye; a very large part is indirectly observable through various spectroscopic techniques. Fields are representations of the continuum, infinitely divisible space and time. But continuous fields are also the cause of discrete events. Continuous water waves may cause discrete, random damage.

Bullets fired on plexiglass described by the stress fields of continuum mechanics cause visible, discrete random cracks emanating from the center of impact. Casting a die, modeled by the continuous laws of classical me- chanics, results in a random, discrete value—depending on which face it falls. If we compare the motion of the Moon, a car, a leaf falling from a tree, or a pollen corn in water, we realize that light objects move less predictably. This introduces a sec- ond form of randomness into scientific descriptions.

Often, measurements do not pro- duce exactly the same results. Typically, the best empirical approximation to the true www.com X | Preface value of something measured is a simple average of multiple measurements—there is a democracy of measurement results. This insight, a form of the law of large numbers, justifies statistical techniques. They allow one to obtain much useful information from many inaccurate measurements.

This feature of Nature extends down to the smallest scales. On the scale of hu- man experience, unanimated matter is highly predictable. But on the molecular and atomic level, matter is observed to behave mostly in a random way. Therefore, the re- producible information about microscopic events consists mostly of statistical prop- erties, such as chemical reaction rates.

On the subatomic level, Nature’s behavior is so uncertain that even the opinions on what exists are somewhat controversial. From a more technical perspective, the new approach described in this book may be summarized as follows: Coherent quantum physics is physics in terms of a coherent space consisting of a line bundle over a classical phase space and an appropriate coherent product. The kinematical structure of quantum physics and the meaning of the fundamen- tal quantum observables are given by the symmetries of this coherent space, their infinitesimal generators, and associated operators on the quantum space of the co- herent space. The formal, mathematical core of quantum physics is cleanly separated from the interpretational issues.

To achieve this, we need to avoid some of the traditional quan- tum mechanical jargon. In particular, following the convention of Allahverdyan et al. [7], we add the prefix “q-” to all traditional quantum notions that suggest by their name a particular interpretation, and hence might confuse the borderline be- tween theory and interpretation. In particular, the operators usually called1 “ob- servables” will be called “q-observables” to distinguish them from observables in the operational sense of numbers obtainable from observation.

Similarly, we use the terms q-expectation and q-probability for the conventional but formally defined terms expectation and probability. Objective properties, including their uncertainties are given by q-expectations of products of quantum fields and what is computable from these. The dynamics of the universe is given by the Ehrenfest equations for q-expectations, and defines the dy- namics of every physical subsystem by restriction. Particles are approximate effective descriptions of certain extended blops of mass and/or energy, descriptions that make sense only under special conditions.