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Journal of Cangzhou Normal University

Quantum Mechanics and the Theory of Quantum Field

MA Wen-juan

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

The content, method, function and relationship between quantum mechanics and the theory of quantum field have been discussed to show the inner differences and theoretical problems.

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The quantum-mechanical implications of the inverse-scattering-transform method and its relationship to the structure of Bethe's ansatz are discussed in the context of the nonlinear Schr\"odinger equation (many-body problem) associated with the classical (quantum) field theory $\mathcal{L}=(\frac{i}{2}){\ensuremath{\varphi}}^{*}{\stackrel{}{\ensuremath{\partial}}}_{0}\ensuremath{\varphi}\ensuremath{-}{|{\ensuremath{\partial}}_{1}\ensuremath{\varphi}|}^{2}\ensuremath{-}c{|\ensuremath{\varphi}|}^{4}$. We review the transformation of the classical problem to action and angle variables and the derivation of an infinite number of polynomial conservation laws. The values of the conserved constants are given by the moments of the classical action variable. It is suggested that there exists a corresponding set of conserved polynomial operators in the quantum field theory and that they reflect the conservation of velocity content which characterizes the solution of the many-body scattering problem (Bethe's ansatz). This implies that the quantized action variable is just the occupation-number density operator in the asymptotic momentum-(velocity-) parameter space of Bethe's ansatz, and that Bethe's wave functions are eigenstates of all conserved operators with eigenvalues given by the moments of the NNN-particle distribution in asymptotic momentum space. These statements are verified for the first four operators, including one which has not previously been studied.
Under the present formulation of Quantum Mechanics, there exists a vacuum energy. The function describing an elementary particle, the field function, has random fluctuations superimposed upon it. An existence information measure is proposed to describe the information contained in a particle field. Existence information is the uncertainty in making the decision that a particle field exists, in light of the vacuum fluctuation effect. Relationships between the existence information and basic intuitive ideas of information theory are brought out. Connections are shown between Shannon's formulation of information theory, Wigner's skew information measure, and the proposed, existence information. Plausibility arguments for the appropriateness of existence information as the representation of field information are presented. Interactions between particles, and the self-interference of a particle is discussed and related to distinguishability, choice, correlation and existence information. Electromagnetic scattering is discussed and is associated with the coherence between the two particles vacuum fluctuations. Existence information is calculated and studied as a function of measurements and interactions. The total existence information content of an isolated system, (possibly interacting internally) is found to be independent of time. (Author)
In this chapter, the concept of path integration is generalized to integration over quantum fields. Specifically we will develop an approach to quantum field theory that takes as its starting point an integration over all configurations of a given field, weighted by an appropriate action. To emphasize the importance of the formulation that, methodologically, represents the backbone of the remainder of the text, we have pruned the discussion to focus only on the essential elements. This being so, conceptual aspects stand in the foreground and the discussion of applications is postponed to the following chapters. In this chapter, the concept of path integration will be extended from quantum mechanics to quantum field theory. Our starting point will be from a situation very much analogous to that outlined at the beginning of the previous chapter. Just as there are two different approaches to quantum mechanics, quantum field theory can also be formulated in two different ways: the formalism of canonically quantized field operators, and functional integration. As to the former, although much of the technology needed to efficiently implement this framework – essentially Feynman diagrams – originated in high-energy physics, it was with the development of condensed matter physics through the 1950s, 1960s, and 1970s that this approach was driven to unprecedented sophistication. The reason is that, almost as a rule, problems in condensed matter investigated at that time necessitated perturbative summations to infinite order in the non-trivial content of the theory (typically interactions).