Some generalizations of the Marcinkiewicz theorem and its implications to certain approximation schemes in many-particle physics

The Marcinkiewicz theorem states that the characteristic function of a probability distribution function cannot be an exponential of a polynomial of degree larger than 2. This theorem is generalized in the present paper to (i) probability distribution functions of many variables and to (ii) probability distribution functionals when the stochastic variables are both commuting (Bose) and anticommuting (Fermi). The consequences of this theorem to certain approximation schemes in many-particle physics, involving truncation of hierarchical equations, are pointed out. These follow when one observes that the hierarchical equations such as those of many-particle Green's functions can be generated from a single equation for a Green's functional whose structure is that of a characteristic functional of a probability distribution functional. The theorem implies that this would exhibit a nonpositive behavior when certain truncation schemes are employed. Specific examples illustrating our results are drawn from the theory of an electron gas, turbulence theory, and quantum optics.