Adiabatic elimination in stochastic systems. III. Application to renormalization-group transformations of the time-dependent Ginsburg-Landau model

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Abstract
We present a stochastic theory of the renormalization-group transformations of the time-dependent Ginzburg-Landau model, describing a system which exhibits the characteristics of a second-order phase transition. We eliminate a certain range of the Fourier components of the magnetic spin variables via a projection-operator method. This effectively changes the scale of the system and transforms the coupling constants. The procedure is shown to be equivalent to the integration over short-wavelength modes as in the renormalization-group transformation performed by Wilson and Kogut. This equivalence shows that the projection-operator method is a valid procedure for scaling critical systems, and in particular it indicates that the treatment of fluctuations is systematic. We suggest that such a method should also provide a straightforward approach to the dynamic renormalization group.