Effective potentials and Bogoliubov's quasiaverage

The effective potential method used in quantum field theory to study spontaneous symmetry violation is discussed from the point of view of Bogoliubov's quasiaveraging procedure. It is shown that the effective potential method is a disguised type of this procedure. The catastrophe theory approach to the study of phase transitions is discussed. The existence of the potentials used in this approach is proved from the statistical point of view. It is shown that in the case of a broken symmetry, the nonconvex effective potential is not a Legendre transform for the connected generating functional Green functions. Instead, it is the part of the potential used in catastrophe theory. The relation between the effective potential and the Legendre transform of generating functional for the connected Green functions is given by Maxwell's rule. A rigorous rule for evaluating quasiaveraged quantities in the framework of the effective potential method is established.