Continuous sample paths in quantum field theory

Abstract

Nelson's free Markoff field on ℝ^l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ℝ¹, mapping a class of distributions φ(x,t) on ℝ^l×ℝ¹ to mean zero Gaussian random variables φ with covariance given by the inner product$\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2$. The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl>0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ℝ^l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.