Wilsonian Renormalization Group Approach Research Articles

Wilson renormalization group study of inverse symmetry breaking.

For a large class of field theories there exist portions of parameter space for which the loop expansion predicts increased symmetry breaking at high temperature. Even though this behavior would clearly have far reaching implications for cosmology such theories have not been fully investigated in the literature. This is at least partially due to the counter intuitive nature of the result, which has led to speculations that it is merely an artifact of perturbation theory. To address this issue we study the simplest model displaying high temperature symmetry breaking using a Wilson renormalization group approach. We find that although the critical temperature is not reliably estimated by the loop expansion the total volume of parameter space which leads to the inverse phase structure is not significantly different from the perturbative prediction. We also investigate the temperature dependence of the coupling constants and find that they run approximately according to their one-loop $\beta$-functions at high temperature. Thus, in particular, the quartic coupling of $phi^4$ theory is shown to increase with temperature, in contrast to the behavior obtained in some previous studies.

Wilson renormalization group approach to light-front Tamm-Dancoff scalar field theory.

A program to utilize the Tamm-Dancoff approximation, on the light front, to solve relativistic quantum field theories, is presented. We present a well-defined renormalization program for the Tamm-Dancoff approximation. This renormalization program utilizes a Minkowski space version of Wilson's renormalization group. We study light-front ${\mathrm{\ensuremath{\varphi}}}^{4}$ field theory in (3+1) dimensions, within a two-particle truncation of Fock space. We further simplify our calculations by considering only one marginal operator and one irrelevant operator. The renormalization procedure requires no more marginal or relevant operators. We derive an effective, renormalized, Hamiltonian. These techniques may be germane to the effort to find an effective, low-energy, light-front Hamiltonian for quantum chromodynamics.

The d-dimensional sine-Gordon model in the presence of random fields

We study the properties of a d-dimensional sine-Gordon model in the presence of a random field that couples linearly to the sine-Gordon field using the Wilson renormalization group approach via the replica trick. No stable fixed point is found for dimensionalities d<4, corresponding to the absence of long-range order. Such a situation seems to occur in experiments on impurity-pinned charge-density-wave systems in which a “glassy behaviour” appears to be induced by arbitrarily weak symmetry-breaking randomness.

The critical exponents η and z to second order in ϵ=2− d for a Bose system at T=0

The critical exponents η and z for a d-dimensional Bose system at T=0 are calculated to second order in ϵ=2− d via Wilson's renormalization group approach. The result seems to confirm that a cross-over ( d, n=2)→( d+2, n=−2) occurs for T→0.

Conformal covariance in the framework of Wilson's renormalization group approach

The author constructs a conformal operator in analogy to the generating operator of Wilson's incomplete-integration renormalization group. The invariance of the partition function with respect to that conformal operation yields identities among the cumulants. Evaluating these identities he finds a generalized and corrected form of the selection rule which determines those two-point cumulants which show a long-range tail. A general equation which governs the asymptotic form of the three-point cumulants is established. It is solved for several examples which involve operators of vector- or tensor-type. It is found that surface effects cannot be excluded a priori. However, the asymptotic expressions for the cumulants are consistent with a neglect of surface effects.

On the critical thermodynamics of inhomogeneous systems

Wilson's renormalization group approach, in the original wave packet approximation, is used to discuss the critical properties of inhomogeneous systems within the paramagnetic region, starting with a position-dependent n-component classical hamiltonian corresponding to Ising, XY, and Heisenberg spin systems, respectively. Both spatially periodic and aperiodic inhomogeneities are considered. The possibilities of local or global critical behaviour are discussed. For periodically inhomogeneous systems, the critical behaviour is the same as for homogeneous systems. For aperiodically inhomogeneous systems with n < 4, a new “random” fixed point is obtained with a specific heat exponent α = 0. Finally, stable quasi homogeneous critical behaviour is possible for n 4, where α < 0. For n = 4, the critical exponents depend continuously on the disorder.

The Jost-Schroer theorem and Wilson's fixed point condition for the renormalized coupling constant

On the basis of the Jost-Schroer-Pohlmeyer-Strocchi theorems it is proved that Wilson's renormalization group approach leads to the fixed point condition α fix Λ = 0, unless the skeleton theory is a nonlocal theory with a non-analytic dependence on mass and coupling parameter. This result resolves a paradoxon in connection with the Adler-Schwinger-Bell-Jackiw anomaly.