Fermionization, Convergent Perturbation Theory, and Correlations in the Yang–Mills Quantum Field Theory in Four Dimensions

Abstract

We show that the Yang–Mills quantum field theory with momentum and spacetime cutoffs in four Euclidean dimensions is equivalent, term by term in an appropriately resummed perturbation theory, to a Fermionic theory with nonlocal interaction terms. When a further momentum cutoff is imposed, this Fermionic theory has a convergent perturbation expansion. To zeroth order in this perturbation expansion, the correlation function E(x,y) of generic components of pairs of connections is given by an explicit, finite-dimensional integral formula, which we conjecture will behave as

$$ E(x,y) \sim |x - y|^{-2 - 2 d_G}, $$

for ${|x-y|\gg 0}$, where d _G is a positive integer depending on the gauge group G. In the case where G = SU(N), we conjecture that

$$ d_G = {\rm dim}\;SU(N) - {\rm dim}\;S(U(N-1) \times U(1)), $$

so that the rate of decay of correlations increases as N → ∞.