Quantum Mechanics: Knocking at the Gates of Mathematical Foundations (original) (raw)

"In the language of mathematics": On mathematical foundations of quantum foundations

Conference lecture (slides), Quantum Information and Probability: from Foundations to Engineering (QIP24), Linnaeus University, Växjö, Sweden11-14 June , 2024

Building on Martin Heidegger’s insight that, beginning with René Descartes and Galileo Galilei, “modern [physics] is experimental because of its mathematical project” [Heidegger 1967, p. 93], I argue that the advancement of modern physics has been defined by the invention of new mathematical schemes (possibly borrowing them from mathematics itself). Among the greatest such inventions, all using differential equations, are: *Classical physics, based on calculus; *Maxwell’s electromagnetic theory, based on the ideal of (classical) field and its mathematization, as represented by Maxwell’s equations; *Relativity, SR and especially GR, based on Riemannian geometry; *Quantum mechanics (QM) and quantum field theory (QFT), based on the mathematics of Hilbert spaces over C, and the operator algebras. I further argue that QM and QFT (to either of which the term quantum theory will refer hereafter) gave this thesis a radically new meaning: Quantum phenomena are defined physically, as essentially different from all previous physics, as is manifested in paradigmatic experiments such as the double-slit experiment or those dealing with quantum correlations. On the other hand, quantum theory, at least QM or QFT, is defined as different from classical physics on the basis of purely mathematical postulates, which connect it to quantum phenomena strictly in terms of probabilities, without, at least in the interpretation adopted here, representing or otherwise relating to how these phenomena come about.

2 4 Ju n 20 01 Quantum Mechanics : Structures , Axioms and Paradoxes ∗

1999

We present an analysis of quantum mechanics and its problems and paradoxes taking into account the results that have been obtained during the last two decades by investigations in the field of ‘quantum structures research’. We concentrate mostly on the results of our group FUND at Brussels Free University. By means of a spin 1 2 model where the quantum probability is generated by the presence of fluctuations on the interactions between measuring apparatus and physical system, we show that the quantum structure can find its origin in the presence of these fluctuations. This appraoch, that we have called the ‘hidden measurement approach’, makes it possible to construct systems that are in between quantum and classical. We show that two of the traditional axioms of quantum axiomatics are not satisfied for these ‘in between quantum and classical’ situations, and how this structural shortcoming of standard quantum mechanics is at the origin of most of the quantum paradoxes. We show that ...

On the foundations of quantum physics

Physics Essays, 2012

Some aspects of the interpretation of quantum theory are discussed. It is emphasized that quantum theory is formulated in a Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutator relations between 'canonically conjugated' coordinate and momentum operators leads to a wrong version of quantum mechanics. In this connection the Feynman integral formalism is also discussed. In this formalism the measure is not well-defined and there is no idea how to distinguish between the true version of quantum mechanics and an incorrect one; it is rather a mnemonic rule to generate perturbation series from an undefined zero order term. The origin of time is analyzed in detail by the example of atomic collisions. It is shown that the time-dependent Schrödinger equation for the closed three-body (two nuclei + electron) system has no physical meaning since in the high impact energy limit it transforms into an equation with two independent time-like variables; the time appears in the stationary Schrödinger equation as a result of extraction of a classical subsystem (two nuclei) from a closed three-body system. Following the Einstein-Rosen-Podolsky experiment and Bell's inequality the wave function is interpreted as an actual field of information in the elementary form. The relation between physics and mathematics is also discussed.

“In Mathematical Language”: On Mathematical Foundations of Quantum Foundations

Entropy 26, 989., 2024

The argument of this article is threefold. First, the article argues that from its rise in the sixteenth century to our own time, the advancement of modern physics as mathematical-experimental science has been defined by the invention of new mathematical structures. Second, the article argues that quantum theory, especially following quantum mechanics, gives this thesis a radically new meaning by virtue of the following two features: on the one hand, quantum phenomena are defined as essentially different from those found in all previous physics by purely physical features; and on the other, quantum mechanics and quantum field theory are defined by purely mathematical postulates, which connect them to quantum phenomena strictly in terms of probabilities, without, as in all previous physics, representing or otherwise relating to how these phenomena physically come about. While these two features may appear discordant, if not inconsistent, I argue that they are in accord with each other, at least in certain interpretations (including the one adopted here), designated as “reality without realism”, RWR, interpretations. This argument also allows this article to offer a new perspective on a thorny problem of the relationships between continuity and discontinuity in quantum physics. In particular, rather than being concerned only with the discreteness and continuity of quantum objects or phenomena, quantum mechanics and quantum field theory relate their continuous mathematics to the irreducibly discrete quantum phenomena in terms of probabilistic predictions while, at least in RWR interpretations, precluding a representation or even conception of how these phenomena come about. This subject is rarely, if ever, discussed apart from previous work by the present author. It is, however, given a new dimension in this article which introduces, as one of its main contributions, a new principle: the mathematical complexity principle.

Philosophical Foundations of Quantum Mechanics

In this dissertation we propose to investigate, after a thorough presentation of what the basic structures of quantum mechanics are, the main philosophical problem which arises as the theory is formulated in its non-relativistic manner. This problem has yielded physicists since the conception of the theory to develop "interpretations" which experimentally are all equivalent or underdetermined. These questions over how to interpret quantum mechanics are philosophical by their very nature. The American physicist Richard Feynman once famously stated: "I think I can safely say that nobody understands quantum mechanics." Whether or not this statement is an extrapolation of reality can be decided based upon the interpretation one has of the theory altogether. We here shall argue that of all of the interpretations concerning the philosophical problem of non-relativistic quantum mechanics, the most satisfying is the reformulation of the theory in terms of a pilot-wave dynamics. This reformulation was first given by Louis de Broglie and later on perfected by David Bohm and J.S. Bell. It will become clear throughout the dissertation that it is the most satisfying solution to what has become known as the measurement problem.

On foundation of quantum physics

Physics of Atomic Nuclei, 2009

Quantum Mechanics in a New Light

Foundations of Science, 2016

Although the present paper looks upon the formal apparatus of quantum mechanics as a calculus of correlations, it goes beyond a purely operationalist interpretation. Having established the consistency of the correlations with the existence of their correlata (measurement outcomes), and having justified the distinction between a domain in which outcome-indicating events occur and a domain whose properties only exist if their existence is indicated by such events, it explains the difference between the two domains as essentially the difference between the manifested world and its manifestation. A single, intrinsically undifferentiated Being manifests the macroworld by entering into reflexive spatial relations. This atemporal process implies a new kind of causality and sheds new light on the mysterious nonlocality of quantum mechanics. Unlike other realist interpretations, which proceed from an evolving-states formulation, the present interpretation proceeds from Feynman's formulation of the theory, and it introduces a new interpretive principle, replacing the collapse postulate and the eigenvalueeigenstate link of evolving-states formulations. Applied to alternatives involving distinctions between regions of space, this principle implies that the spatiotemporal differentiation of the physical world is incomplete. Applied to alternatives involving distinctions between things, it warrants the claim that, intrinsically, all fundamental particles are identical in the strong sense of numerical identical. They are the aforementioned intrinsically undifferentiated Being, which manifests the macroworld by entering into reflexive spatial relations.

A new approach toward the quantum foundation and some consequences

Academia Quantum, 2024

A general theory based on six postulates is introduced. The basic notions are theoretical variables that are associated with an observer or with a group of communicating observers. These variables may be accessible or inaccessible. From these postulates, the ordinary formalism of quantum theory is derived. The mathematical derivations are not given in this article, but I refer to the recent articles. Three possible applications of the general theory can be given as follows: (1) the variables may be decision variables connected to the decisions of a person or a group of persons, (2) the variables may be statistical parameters or future data, and (3) most importantly, the variables are physical variables in some context. The last application gives a completely new foundation of quantum mechanics, a foundation which in my opinion is much easier to understand than ordinary formalism. So-called paradoxes like that of Schrödinger’s cat can be clarified under the theory. Explanations of the outcomes of David Bohm’s version of the EPR (Einstein–Podolsky–Rosen) experiment and the Bell experiment are provided. Finally, references to links toward relativity theory and quantum field theory are given. The concluding remarks point to further possible developments.

Quantum Mechanics: Knocking at the Gates of Mathematical Foundations (original) (raw)

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