# On the stochastic Sine-Gordon model: an interacting field theory approach

November 02, 2023

We investigate the massive Sine-Gordon model in the finite ultraviolet regime
on the two-dimensional Minkowski spacetime $(\mathbb{R}^2,\eta)$ with an
additive Gaussian white noise. In particular we construct the expectation value
and the correlation functions of a solution of the underlying stochastic
partial differential equation (SPDE) as a power series in the coupling
constant, proving ultimately uniform convergence. This result is obtained
combining an approach first devised in [11] to study SPDEs at a perturbative
level with the one discussed in [4] to construct the quantum sine-Gordon model
using techniques proper of the perturbative, algebraic approach to quantum
field theory (pAQFT). At a formal level the relevant expectation values are
realized as the evaluation of suitably constructed functionals over
$C^\infty(\mathbb{R}^2)$. In turn, these are elements of a distinguished
algebra whose product is a deformation of the pointwise one, by means of a
kernel which is a linear combination of two components. The first encompasses
the information of the Feynmann propagator built out of an underlying Hadamard,
quantum state, while the second encodes the correlation codified by the
Gaussian white noise. In our analysis, first of all we extend the results
obtained in [3,4] proving the existence of a convergent modified version of the
S-matrix and of an interacting field as elements of the underlying algebra of
functionals. Subsequently we show that it is possible to remove the
contribution due to the Feynmann propagator by taking a suitable $\hbar\to
0^+$-limit, hence obtaining the sought expectation value of the solution and of
the correlation functions of the SPDE associated to the stochastic Sine-Gordon
model.

Keywords:

stochastic Sine-Gordon model, pAQFT