Martingale Convergence and the Exponential Interaction in ℝⁿ

Abstract

In this lecture we discuss the exponential interaction in ℝⁿ. We give a proof that the ultraviolet cut-off exponential interaction ${m_{k}}\left( \xi \right) = \int {_{\Lambda }} :{e^{{\alpha {\xi _{k}}\left( x \right)}}}$: dx, where Λ is a fixed bounded region in Rⁿ and k the ultraviolet cut-off parameter, is a martingale in k with respect to the free Euclidean measure µ_O. Moreover ${{\text{e}}^{{{\text{ - }}\lambda {{\text{m}}_{{\text{k}}}}\left( \xi \right)}}}$ is a positive bounded submartingale and we prove that ${{\text{e}}^{{{\text{ - }}\lambda {{\text{m}}_{{\text{k}}}}\left( \xi \right)}}}$ converges pointwise µ_O-almost everywhere and strongly in L¹(dµ_O) as the ultraviolet cut-off k tends to infinity. Especially ${{\text{e}}^{{{\text{ - }}\lambda {{\text{m}}_{{\text{k}}}}\left( \xi \right)}}}{\text{d}}{\mu _{{\text{o}}}}\left( \xi \right)$ converges weakly as k → ∞ which implies that the ultraviolet cut-off Schwinger functions for the exponential interaction in ℝⁿ converge as k → ∞. For n > 4 this limit is the free Euclidean field. Further results concerning the cases n < 3 are also mentioned, as well as applications to the study of the energy representations of groups of mappings from Riemann manifolds into compact Lie groups.