The elliptic stochastic quantization of some two dimensional Euclidean QFTs

Abstract

We study a class of elliptic SPDEs with additive Gaussian noise on R 2 × M , with M a ddimensional manifold equipped with a positive Radon measure, and a real-valued non linearity given by the derivative of a smooth potential V , convex at infinity and growing at most exponentially. For quite general coefficients and a suitable regularity of the noise we obtain, via the dimensional reduction principle discussed in [11], the identity between the law of the solution to the SPDE evaluated at the origin with a Gibbs type measure on the abstract Wiener space L 2 (M ). The results are then applied to the elliptic stochastic quantization equation for the scalar field with polynomial interaction over T 2 , and with exponential interaction over R 2 (known also as Høegh-Krohn or Liouville model in the literature). In particular for the exponential interaction case, the existence and uniqueness properties of solutions to the elliptic equation over R 2+2 is derived as well as the dimensional reduction for all values of the "charge parameter" σ = α 2 √ π < √ 8π for which the model has an Euclidean invariant measure (hence also permitting to get the corresponding relativistic invariant model on the two dimensional Minkowski space).

Construction of a non-Gaussian and rotation-invariant $Φ^4$-measure and associated flow on ${\mathbb R}^3$ through stochastic quantization

“…This paper is concerned with the construction of solutions of a certain stochastic partial differential equation (SPDE) of parabolic type, with singular coefficients, known as stochastic quantization equation (SQE) for the ϕ 4 3 -model of relativistic quantum fields on the Euclidean 3-dimensional space-time R 3 . Particular attention is given to the construction of an invariant probability measure for the process associated with the SQE, that is stationary (i.e.…”

“…invariance under the full Poincaré group) of the associated relativistic quantum fields. In this sense the construction of solutions of the SQE and their invariant measures provides an alternative construction of the φ 4 3 -Euclidean and relativistic fields by other methods than those used in the constructive quantum field theory.…”

“…This work was extended to other types of interaction terms V , namely an exponential or trigonometric term, for σ = 1 (see [119]). For the case where σ = 2 these Hamiltonian methods were extended to the case of V of a fourth power (the ϕ 4 3 -model) in [94]. But, major further developments, both for σ = 1 and σ = 2 came after "Euclidean methods" were introduced in the area of constructive quantum field theory.…”

“…in a bounded region for the space variable in R 2 ) the existence of a Hilbert space and a densely defined symmetric Hamiltonian, the Hilbert space being different from the original Hilbert-Fock space for the interaction free case. Other Hamiltonian methods were used in the study of canonical commutation relations for the ϕ 4 3 -model in a bounded spatial region in [72], where independence (in the sense of unitary equivalence) of the representations with respect to the space cut-off has been proven. For other work on the ϕ 4 3 -model by Hamiltonian methods see also [117].…”

“…Other Hamiltonian methods were used in the study of canonical commutation relations for the ϕ 4 3 -model in a bounded spatial region in [72], where independence (in the sense of unitary equivalence) of the representations with respect to the space cut-off has been proven. For other work on the ϕ 4 3 -model by Hamiltonian methods see also [117]. The non-locally Fock property of the (time zero) canonical commutation relation is proven in [73] (see also [117]).…”