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Journal of Physics A Mathematical and General

Differential formulation of the dynamic renormalisation group in the large-n limit

Gaetano Busiello, L. De Cesare, Ileana Rabuffo

Published Jun 21, 1983• 6 citations
Publisher

Abstract

A differential formulation of the dynamic renormalisation group in the large-n limit is presented for the time-dependent generalisation of the n-vector classical model with purely relaxational dynamics for both conserved and non-conserved order parameter. A discussion of the dynamical properties with the same steps used in the traditional perturbative approaches is made in terms of which known and new explicit results are obtained. Furthermore, a mechanical analogy is introduced which may give new information about the structure of the dynamic renormalisation group in the large-n limit. This relies on the possibility of using the well known geometrical and mathematical techniques from the Hamilton-Jacobi theory for classical mechanics.

DOI
10.1088/0305-4470/16/9/020

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A differential formulation of the dynamic renormalisation group in the large-n limit is presented for the time-dependent generalisation of the n-vector classical model with purely relaxational dynamics for both conserved and non-conserved order parameter. A discussion of the dynamical properties with the same steps used in the traditional perturbative approaches is made in terms of which known and new explicit results are obtained. Furthermore, a mechanical analogy is introduced which may give new information about the structure of the dynamic renormalisation group in the large-n limit. This relies on the possibility of using the well known geometrical and mathematical techniques from the Hamilton-Jacobi theory for classical mechanics.
Generalising a result of classical mechanics an infinite set of conserved quantities can be found for the bare equations of motion describing the evolution of a scalar field in out of equilibrium quantum field theory, in the large N approximation, with initial conditions corresponding to a thermal system of the free Hamiltonian. Using these new conserved quantities, sum-rules relating integrals over the mode-functions (momenta) can be derived. More, the corresponding renormalised quantities can also be computed out thus giving information about the evolution of the already known renormalised equations; finally it is also possible to write a renormalised version of the sum-rules.
This work presents a rigorous mathematical framework for phase dynamics in configuration space. All physical dynamics are derived from a smooth phase field Φ without assuming forces, particles, or fundamental time. The framework introduces the Euler–Phase equation, derives the unique quantum potential, and recovers classical mechanics in the limit κ → 0. It includes explicit mappings to Hamilton–Jacobi and Madelung formulations, gauge symmetry analysis, circulation quantization, and a comparison of Phase Dynamics with classical and quantum mechanics. This unified approach generalizes both CM and QM, allowing topologically nontrivial and multi-valued phase structures.
In this chapter, we introduce and explain one of the fundamental tools in the study of dynamic critical phenomena, namely dynamic perturbation theory. Inevitably, large parts of Chapter 4 need to be rather technical. Complementary to the straightforward iterative method for the solution of non-linear Langevin equations presented in Section 3.2.3, we now describe the more elegant and efficient field-theoretic techniques. Yet both for the derivation of general properties of the perturbation series and for the sake of practical calculations, the response functional and the following elaborations on Feynman diagrams, cumulants, and vertex functions prove indispensable. As is the case with every efficient formalism, once one has become acquainted with the Janssen–De Dominicis response functional for the construction of dynamic field theory, and therefrom the perturbation expansion in terms of vertex functions, it serves to save a considerable amount of rather tedious work. For the purely relaxational O ( n )-symmetric models A and B, we derive the fluctuation-dissipation theorem within this formalism, and then systematically construct the perturbation series and its diagrammatic representation in terms of Feynman graphs. In order to reduce the required efforts to a minimum, the generating functionals first for the cumulants, and then, motivated by Dyson's equation for the propagator, for the one-particle irreducible vertex functions are examined. As an example, the relevant vertex functions for models A and B are explicitly evaluated to two-loop order. Furthermore, alternative formulations of the perturbation expansion are discussed, and the Feynman rules are given in both the frequency and the time domain. The results of this chapter provide the foundation for the renormalization group treatment in the subsequent Chapter 5.
We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi (HJHJHJ) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the HJHJHJ theory on the tangent bundle of a configuration manifold, the quantum HJHJHJ theory, HJHJHJ problems for general differential operators and the HJHJHJ problem for Lie groups.