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Zeitschrift für Naturforschung A

On the Lattice-cell-volume Dependence in the Calculation of the Two-point Function in Quantum Mechanics by the Method of Functional Integration

M. Hoffmann

Published Aug 1, 1967• 0 citations
Publisher

Abstract

It is shown that the transition from lattice space, where the two-point functions were calculated by means of functional integration, to continuous space Ɛ → 0, which requires in field theories some kind of renormalization, is practicable in quantum mechanics after reordering of series without renormalization. An ambiguity, which occurs, is discussed.

DOI
10.1515/zna-1967-0809

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It is shown that the transition from lattice space, where the two-point functions were calculated by means of functional integration, to continuous space Ɛ → 0, which requires in field theories some kind of renormalization, is practicable in quantum mechanics after reordering of series without renormalization. An ambiguity, which occurs, is discussed.
The difference between the ordinary method of renormalization of Dyson and that of the present authors developed in the previous paper is emphasized. In order to clarify this difference, the renormalization of the two-electron Green-function is investigated. The proof of the finiteness of the renormalized Green-function seems to be given in quite a brief way. It is the essential point in this proof that there is no ambiguity in the order of integration appearing in course of calculation of Green-functions.
A real-space renormalization scheme for calculating the thermodynamic properties of models with continuous variables on a lattice having phase transitions in the Ising universality class is described. The first renormalization transformation maps the initial model onto a spin-\textonehalf{} Ising model with generalized interactions, which is then analyzed by an Ising renormalization transformation or by other means. Two versions of this procedure are used to calculate the phase diagram of a double-well model for structural phase transitions, which is identical with the ${\ensuremath{\varphi}}^{4}$ field theory on a lattice, in two dimensions. The results are in excellent agreement with the molecular-dynamics studies of Schneider and Stoll.
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.
In this work, we discuss a new method for calculation of extremal eigenvectors and eigenvalues in systems or regions of parameter space where direct calculation is problematic. This technique relies on the analytic continuation of the power series expansion for the eigenvector around a point in the complex plane. We start this document by introducing the background material relevant to understand the basics of quantum mechanics and quantum field theories on the lattice, how we perform our numerical simulations, and how this relates to the nuclear physics we probe. We then move to the mathematical formalism of the eigenvector continuation, which is rooted in analytic function theory and linear algebra. We then discuss how these techniques are implemented numerically, with a discussion about the computational costs. Finally, we discuss applications of this method to full, quantum many-body systems. These include neutron matter, the Bose-Hubbard model, the Lipkin model, and the Coulomb interaction in light nuclei with LO chiral forces. These systems cover two categories of interest to the field: systems with a substantial sign problem, or systems that exhibit quantum phase transitions.
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 is extended from quantum mechanics to quantum field theory. Our starting point is 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).