Poisson Distributions for Sharp-Time Fields: Antidote for Triviality

Abstract

Standard lattice-space formulations of quartic self-coupled Euclidean scalar quantum fields become trivial in the continuum limit for sufficiently high space-time dimensions, and in particular the moment generating functional for space-time smeared fields becomes a Gaussian appropriate to that of a (possibly generalized) free field. For sharp-time fields this fact implies that the ground-state expectation functional also becomes Gaussian in the continuum limit. To overcome these consequences of the central limit theorem, an auxiliary, nonclassical potential is appended to the original lattice form of the model and parameters are tuned so that a generalized Poisson field distribution emerges in the continuum limit for the ground-state probability distribution. As a consequence, the sharp-time expectation functional is infinitely divisible, but the Hamiltonian operator is such, in the general case, that the generating functional for the space-time smeared field is not infinitely divisible in Minkowski space. This feature permits the models in question to escape a manifestly trivial scattering matrix imposed on all infinitely divisible covariant Minkowski fields. Two sequentially related proposals for an alternative lattice formulation of interacting covariant models in four and more space-time dimensions are analyzed in some detail.