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arXiv:2001.05869

Density Formalism for Quantum Theory

Roderick I. Sutherland

Published Jan 14, 2020• 3 citations
arXivHTMLPDF

Abstract

A simple mathematical extension of quantum theory is presented. As well as opening the possibility of alternative methods of calculation, the additional formalism implies a new physical interpretation of the standard theory by providing a picture of an external reality. The new formalism, developed first for the single-particle case, has the advantage of generalizing immediately to quantum field theory and to the description of relativistic phenomena such as particle creation and annihilation.

DOI
10.48550/arxiv.2001.05869

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A simple mathematical extension of quantum theory is presented. As well as opening the possibility of alternative methods of calculation, the additional formalism implies a new physical interpretation of the standard theory by providing a picture of an external reality. The new formalism, developed first for the single-particle case, has the advantage of generalizing immediately to quantum field theory and to the description of relativistic phenomena such as particle creation and annihilation.
We expound an alternative to the Copenhagen interpretation of the formalism of nonrelativistic quantum mechanics. The basic difference is that the new interpretation is formulated in the language of epistemological realism. It involves a change in some basic physical concepts. The ψ function is no longer interpreted as a probability amplitude of the observed behavior of an elementary particle but as an objective physical field representing the particle itself. The particles are thus extended objects whose extension varies in time according to the variation of ψ. They are considered as fundamental regions of space with no internal structure. This implies some kind of nonlocality. Symmetrization of the configuration space wave function is interpreted as a mathematical description of a physical process, which also leads to nonlocal effects. Special consideration
Despite its name, Quantum Field Theory (QFT) has been built to describe interactions between localizable particles. For this reason the actual formalism of QFT is partly based on a suitable generalization of the one already used for systems of point particles. This circumstance gives rise to a number of conceptual problems, stemming essentially from the fact that the existence within QFT of non-equivalent representations implies the existence of field theories allowing, within the same theory, different, inequivalent, descriptions of particles. This led some authors to claim that in QFT the concept itself of particle should be abandoned. In this paper we will shortly discuss the validity of this claim, as well as the possibilities, so far existing, of building alternative versions of QFT, not designed in advance to allow some kind of particle representation. We will also spend some words about the generalizations of the concept itself of particle which could grant for a better cohabitation of particles and fields within a wider formulation of QFT. The latter is indispensable if we want to extend the range of application of QFT from particle physics or simple condensed matter physics to other domains of scientific research.
In this paper, we try to give a new approach to the quantum mechanics(QM) on the framework of quantum field theory(QFT). Firstly, we make a detail study on the (non-relativistic) Schrödinger field theory, obtaining the Schrödinger equation as a field equation, after field quantization, the Heisenberg equations for the momentum and position operators of the particles excited from the (Schrödinger) field and the Feynman path integral formula of QM are also obtained. We then give the probability concepts of quantum mechanics in terms of a statistical ensemble, realizing the ensemble(or statistical) interpretation. With these, we make a series of conceptual modifications to the standard quantum mechanics, especially propose a new assumption about the quantum measurement theory which can solve the EPR paradox from the view of the QFT. Besides, a field theoretical description to the double-slit interference experiment is developed, obtaining the required particle number distribution. In the end, we extend all the above concepts to the relativistic case so that the ensemble interpretation is still proper. Two extra topics are added, in the first one, an operable experiment is proposed to distinguish the Copenhagen interpretation from the ensemble one via very different experimental results. While the second topic concerns with the extensions of the concept of coherent state to both the Bosonic and Fermionic field cases, to obtain the corresponding classical fields. And in the concluding section, we make some general comparisons between the standard QM and the one derived from the QFT, from which we claim that the QFT is the fundamental theory.
Quantum field theory (QFT) provides us with one and almost only suitable language (or mathematical tool) for describing not only the motion and interaction of particles but also their “annihilation” and “creation” out of a field considered a priori in a sophisticated way, whose view seems to be suited for describing dislocations, as a particle or a string embedded within a crystalline ordered field. This chapter concisely overviews the method of QFT, emphasizing distinction from the quantum mechanics, conventionally used for a single and/or many particle problems, and its equivalence to the statistical mechanics. The alternative formalism based on Feynman path integral and its imaginary time representation are reviewed, as the foundation for our use in Chapter 10.
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
The additivity of classical probabilities is only the first in a hierarchy of possible sum rules, each of which implies its successor. The first and most restrictive sum rule of the hierarchy yields measure theory in the Kolmogorov sense, which is appropriate physically for the description of stochastic processes such as Brownian motion. The next weaker sum rule defines a generalized measure theory which includes quantum mechanics as a special case. The fact that quantum probabilities can be expressed "as the squares of quantum amplitudes" is thus derived in a natural manner, and a series of natural generalizations of the quantum formalism is delineated. Conversely, the mathematical sense in which classical physics is a special case of quantum physics is clarified. The present paper presents these relationships in the context of a "realistic" interpretation of quantum mechanics.