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Physica A Statistical Mechanics and its Applications

Variational principles in renormalization theory

Wim van Saarloos, J. M. J. van Leeuwen, Adrianus M. M. Pruisken

Published Jul 1, 1978• 20 citations
Publisher
DOI
10.1016/0378-4371(78)90135-8

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A quantum mechanical time-dependent variational principle is generalized using the classical theory of fluids to obtain a variational principle suitable for the fluid dynamical description of mixed state quantum mechanics. A newly derived set of moment equations, in both standard and renormalized form, can be derived with the aid of this principle through minimization of the error in expressing the total derivative of the Wigner function. Coupled systems are studied in the time-dependent Hartree (TDH) approximation using a novel variational principle, and the renormalization procedure used earlier in the examination of single particle dynamics is extended to the TDH analysis. Use of a local Maxwellian ansatz for each particle results in a particularly simple ‘‘two-fluid’’ theory, the TDH/LM approximation, which does not violate the standard and renormalized energy conservation theorems derived earlier for the single particle equations. The fluid dynamical TDH/LM approximation is shown to possess a simple semiclassical interpretation.
The Ising model has been employed in a study of the variational principles which are associated with renormalized diagram expansions in statistical mechanics. For this model the variational functional is expressed in terms of the renormalized semi-invariants, which play the role of the one-particle density or the one-particle Green's function in quantum statistics. We review the derivation of this functional and discuss some of its properties. In order to examine the mathematical content of the variational principle in more detail, we specialize to the exactly soluble infinite-dimensional (infinite-range) model. We find that, by virtue of the Dyson relation, the variational theorem is not valid over all possible values of the renormalized semi-invariants but applies only within a restricted domain. Within this domain the variational functional is multiple valued. The renormalized expansion is asymptotically convergent to the branch of this functional which describes a single phase with uniform magnetization.