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 $f\equiv 0$. 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.