A Step-Function Approximation in the Theory of Critical Fluctuations

We consider fluctuations near the critical point using the step-function approximation, i.e., the approximation of the order parameter field f(x) by a sequence of step functions converging to f(x). We show that the systematic application of this method leads to a trivial result in the case where the fluctuation probability is defined by the Landau Hamiltonian: the fluctuations disappear because the measure in the space of functions that describe the fluctuations proves to be supported on the single function f0. This can imply that the approximation of the initial smooth functions by the step functions fails as a method for evaluating the functional integral and for defining the corresponding measure, although the step-function approximation proves to be effective in the Gaussian case and yields the same result as alternative methods do.