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arXiv:2103.05374

The foundations of quantum theory and its possible generalizations

Valentin Al'fredovich Franke

Published Mar 9, 2021• 0 citations
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Abstract

Possible generalizations of quantum theory permitting to describe in a unique way the development of the quantum system and the measurement process are discussed. The approach to the problem based on the Lindblad's equation for the statistical operator is reviewed. The Tomonaga-Schwinger like equation of this type is introduced to establish Lorentz invariance. The application of tachyonic field to overcome divergences arising in this equation is analyzed. Other approaches to the problem are shortly discussed.

DOI
10.48550/arxiv.2103.05374

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Possible generalizations of quantum theory permitting to describe in a unique way the development of the quantum system and the measurement process are discussed. The approach to the problem based on the Lindblad's equation for the statistical operator is reviewed. The Tomonaga-Schwinger like equation of this type is introduced to establish Lorentz invariance. The application of tachyonic field to overcome divergences arising in this equation is analyzed. Other approaches to the problem are shortly discussed.
According to our current understanding of quantum mechanics, a `measurement' violates unitarity. In other words as the act of measurement `resets' the probabilities described by the Schrödinger equation, in the very `moment' of the collapse of the wavefunction, conservation of probability does not hold. To make progress in our understanding of the measurement problem it is thus expected that one must encompass in a single equation both otherwise mutually-exclusive processes of measurement and unitary evolution. In this light, turning to the continuity equation, we realize the possibility that if we allow for existence of sources and sinks in the `fluid of probability' we get closer to such a theory by arriving at nonlinear generalizations of Schrödinger and Klein-Gordon equations. The generalized equations derived are then shown to satisfy all conditions that are expected from a proper generalization: simplification to their linear counterparts by a well-defined dynamical condition, and Galilean and Lorentz invariance.
This dissertation is divided into two main topics. The first is the generalization of quantum dynamics when the Schrodinger partial differential equation is not defined even in the weak mathematical sense because the potential function itself is a distribution in the spatial variable, the same variable that is used to define the kinetic energy operator, i.e. the Laplace operator. The procedure is an extension and broadening of the distributional calculus and offers spectral results as an alternative to the only other two known methods to date, namely a) the functional calculi; and b) non-standard analysis. Furthermore, the generalizations of quantum dynamics presented within give a resolution to the time asymmetry paradox created by multi-particle quantum mechanics due to the time evolution still being unitary. A consequence is the randomization of phases needed for the fundamental justification Pauli master equation. The second topic is foundations of the quantum theory of fields. The title is phrased as ``foundations'' to emphasize that there is no claim of uniqueness but rather a proposal is put forth, which is markedly different than that of constructive or axiomatic field theory. In particular, the space of fields is defined as a space of generalized functions with involutive symmetry maps (the CPT invariance) that affect the topology of the field space. The space of quantum fields is then endowed the Fr'{e}chet property and interactions change the topology in such a way as to cause some field spaces to be incompatible with others. This is seen in the consequences of the Haag theorem. Various examples and discussions are given that elucidate a new view of the quantum theory of fields and its (lack of) mathematical structure.
The two-body Dirac (2BD) equations of constraint theory are of special interest not only in view of applications for phenomenological calculations of mesonic spectra but also because they avoid no-go theorems about relativistic interactions. Furthermore, they provide a quantum mechanical description in a manifestly Lorentz invariant way using the concept of a multi-time wave function. In this paper, we place them into the context of the multi-time formalism of Dirac, Tomonaga and Schwinger for the first time. A general physical and mathematical framework is outlined and the mechanism which permits relativistic interaction is identified. The main requirement derived from the general framework is the existence of conserved tensor currents with a positive component which can play the role of a probability density. We analyze this question for a general class of 2BD equations thoroughly and comprehensively. While the free Dirac current is not conserved, it is possible to find replacements. Improving on previous research, we achieve definite conclusions whether restrictions of the function space or of the interaction terms can guarantee the positive definiteness of the currents—and whether such restrictions are physically adequate. The consequences of the results are drawn, with respect to both applied and foundational perspectives.
A relativistic configuration space (many-time) presentation of quantum electrodynamics is developed, which, being equivalent to the Tomonaga-Schwinger formalism, may be expected to have the advantage of its direct applicability to bound-state problems of multi-electron systems. The main purpose of this paper is therefore to get an evidently relativistically invariant description of the interactions between individual electrons, in such a way, however, as to avoid those divergence difficulties which may be accounted for by the use of electron plane waves for building up the actual state of the system. The order of the presentation is as follows: a general relativistically invariant proof of equivalence of the Tomonaga-Schwinger formalism and the many-time formalism of Dirac, Fock, and Podolski given in Sec. I will provide a necessary starting point for further generalization in the form of Eq. (53). This equation, when eventually solve in Sec. II, gives rise to a formal extension of the multi-electron wave function concept into regions for which the latter remained hitherto undefined. In Sec. III a method of elimination of virtual processes is outlined and invariant expressions for the multi-electron interactions are derived. All explicit derivations in this paper are carried out to terms of the order of e2{e}^{2}e2. An example, a relativistic two-electron wave equation, is given. This equation accounts for the Coulomb and Breit interactions.
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.
Abstract Much attention has been directed to the philosophical implications of quantum field theory (QFT) in recent years; this paper attempts a survey in low-technical terms. First the relations of QFT to other kinds of theory, classical and quantum, particle and field, are discussed. Then various formulations of QFT are introduced, along with related interpretations. Finally a review is made of some of the most interesting foundational problems. For our discussion of the foundations of quantum field theory (QFT) it will be useful to have a map showing where QFT stands in relation to other physical theories. Figure 1 locates theories as points of a three-dimensional lattice: describing particles or fields, formulated in different spacetimes and either classical or quantum. Consider the classical plane of Figure