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Physica Scripta

Quantum field theory of many-body systems: a functional formulation

Klaus Dietz

Published May 1, 1988• 8 citations
Publisher

Abstract

The Schrödinger picture of Quantum Field Theory is taken as the general framework for the formulation of a relativistic many-body theory. The main emphasis is put on many-electron systems within the scope of Quantum Electrodynamics. Relativistic mean-field theories are discussed and a perturbative scheme is introduced. Hamilton's principle is proposed as a criterion for the construction of mean-fields which allow for optimisation. Criteria for the choice of the latter are discussed.

DOI
10.1088/0031-8949/37/5/001

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The Schrödinger picture of Quantum Field Theory is taken as the general framework for the formulation of a relativistic many-body theory. The main emphasis is put on many-electron systems within the scope of Quantum Electrodynamics. Relativistic mean-field theories are discussed and a perturbative scheme is introduced. Hamilton's principle is proposed as a criterion for the construction of mean-fields which allow for optimisation. Criteria for the choice of the latter are discussed.
Relativistic quantum dynamics requires a unitary representation of the Poincare group on the Hilbert space of states. The dynamics of many-body systems must satisfy cluster separability requirements. In this paper we formulate an abstract framework of four dimensional Euclidean Green functions that can be used to construct relativistic quantum dynamics of N-particle systems consistent with these requirements. This approach should be useful in bridging the gap between few-body dynamics based on phenomenological mass operators and on quantum field theory.
We discuss the Legendre conjugate of the Hohenberg-Kohn energy-density functional, i.e., the total energy of an inhomogeneous many-electron system, considered as a functional of the external Coulomb potential (the nuclear Coulomb skeleton of a molecule, for instance) as the starting point of an alternative formulation of a theory of electronic correlations. We then relate this functional to the nonrelativistic or relativistic microscopic many-body theory. The essential bridge between the two theories is many-body perturbation theory perturbing around a mean field. We then point out that a particular choice for the latter, the g-Hartree mean field, leads to a transparent physical picture: The relevant functional of the external field, g0{g}_{0}g0​-1, representing electronic correlations, is interpreted as a polarization charge density induced by the latter. This picture, in turn, leads to a Clausius-Mosotti type of equation for this correlation functional. Applications to atomic- and molecular-structure calculations are discussed.
In this article a survey is given of the present status of the quantum theory of many-particle systems with special emphasis on the underlying principles. Very little is said about applications. A brief introduction to some models, which are of interest for an approximate or qualitative description of actual physical systems, is followed by a discussion of two important phenomena in many-particle systems: quasi-particles and collective excitations. Next, an outline is given of mathematical methods which have been used for a precise treatment of such phenomena. A rather thorough discussion of the Bogoliubov model for interacting bosons, and the BCS model for fermions, serves as an illustration. Some new features, typical of many-particle systems, are discussed on the basis of these models. Also another approach to the many-body problem is discussed which is more logical and mathematically more rigorous. This approach differs from standard procedures in that the limit Ω-> infinity, N -> infinity is not taken at the end of the calculations but in the beginning. This quantum theory of infinite systems leads, among other things, to a better understanding of the models mentioned above.
Abstract In this work, we establish a novel approach to the foundations of relativistic quantum theory, which is based on generalizing the quantum-mechanical Born rule for determining particle position probabilities to curved spacetime. A principal motivator for this research has been to overcome internal mathematical problems of relativistic quantum field theory (QFT) such as the ‘problem of infinities’ (renormalization), which axiomatic approaches to QFT have shown to be not only of mathematical but also of conceptual nature. The approach presented here is probabilistic by construction, can accommodate a wide array of dynamical models, does not rely on the symmetries of Minkowski spacetime, and respects the general principle of relativity. In the analytical part of this work, we consider the 1-body case under the assumption of smoothness of the mathematical quantities involved. This is identified as a special case of the theory of the general-relativistic continuity equation. While related approaches to the relativistic generalization of the Born rule assume the hypersurfaces of interest to be spacelike and the spacetime to be globally hyperbolic, we employ prior contributions by C. Eckart and J. Ehlers to show that the former condition is naturally replaced by a transversality condition and that the latter one is obsolete. We discuss two distinct formulations of the 1-body case, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. We provide a comprehensive treatment of both. The main contribution of this work to the mathematical physics literature is the development of the Lagrangian picture. The Langrangian picture shows how one can address the ‘problem of time’ in this approach and, therefore, serves as a blueprint for the generalization to many bodies and the case that the number of bodies is not conserved. We also provide an example to illustrate how this approach can in principle be employed to model particle creation and annihilation.