Abstract
A positive representation for a set of complex densities is constructed. In particular, complex measures on a direct product ofU(1) groups are studied. After identifying general conditions which such representations should satisfy, several concrete realizations are proposed. Their utility is illustrated in few concrete examples representing problems in abelian lattice gauge theories.
Highlights
A positive representation for a set of complex densities is constructed
The Langevin approach is a popular way to replace averaging over given positive probability distribution ρ(x) = e−S (x) by an average over the suitably constructed stochastic process
Under general conditions one can prove that in the infinite τ limit this distribution tends to the original ρ(x), which explains the whole idea
Summary
The Langevin approach is a popular way to replace averaging over given positive probability distribution ρ(x) = e−S (x) by an average over the suitably constructed stochastic process. Given a real action S (x), one constructs/generates the stochastic process x(τ), according to the corresponding Langevin equation. For non-positive, or in general complex, weights ρ ≡ e−S the approach can still be straightforwardly extended by introducing the complex stochastic process z(τ). S (x) z(τ)=−−∂→zS +η(τ) z(τ) −→ P(x, y, τ) , z ∈ C, η ∈ R This offers a promising possibility to replace an "average" over a complex weight by a statistical average over a positive probability [1, 2] - the task needed badly in many applications of Lattice Field theory:. For recent review see Ref.[9]
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