Euclidean Quantum Field Research Articles

Stochastic Quantization of Two-Dimensional $$P(\Phi )$$ Quantum Field Theory

AbstractWe give a simple and self-contained construction of the $$P(\Phi )$$ P ( Φ ) Euclidean quantum field theory in the plane and verify the Osterwalder–Schrader axioms: translational and rotational invariance, reflection positivity and regularity. In the intermediate steps of the construction, we study measures on spheres. In order to control the infinite volume limit, we use the parabolic stochastic quantization equation and the energy method. To prove the translational and rotational invariance of the limit measure, we take advantage of the fact that the symmetry groups of the plane and the sphere have the same dimension.

Decay of correlations in stochastic quantization: the exponential Euclidean field in two dimensions

AbstractWe present two approaches to establish the exponential decay of correlation functions of Euclidean quantum field theories (EQFTs) via stochastic quantization (SQ). In particular we consider the elliptic stochastic quantization of the Høegh–Krohn (or $$\exp (\alpha \phi )_2$$ exp ( α ϕ ) 2 ) EQFT in two dimensions. The first method is based on a path-wise coupling argument and PDE apriori estimates, while the second on estimates of the Malliavin derivative of the solution to the SQ equation.

On the variational method for Euclidean quantum fields in infinite volume

On the variational method for Euclidean quantum fields in infinite volume

P-Adic statistical field theory and convolutional deep Boltzmann machines

Abstract Understanding how deep learning architectures work is a central scientific problem. Recently, a correspondence between neural networks (NNs) and Euclidean quantum field theories has been proposed. This work investigates this correspondence in the framework of p-adic statistical field theories (SFTs) and neural networks. In this case, the fields are real-valued functions defined on an infinite regular rooted tree with valence p, a fixed prime number. This infinite tree provides the topology for a continuous deep Boltzmann machine (DBM), which is identified with a statistical field theory on this infinite tree. In the p-adic framework, there is a natural method to discretize SFTs. Each discrete SFT corresponds to a Boltzmann machine with a tree-like topology. This method allows us to recover the standard DBMs and gives new convolutional DBMs. The new networks use O(N) parameters while the classical ones use O(N2) parameters.

Non-local Markovian Symmetric Forms on Infinite Dimensional Spaces

The general framework on the non-local Markovian symmetric forms on weighted lp(p in [1, infty ]) spaces constructed by Albeverio et al. (Commn. Math. Phys. 388, 659–706, 2021 Kagawa) by restricting the situation where p = 2, is applied to probability measure spaces describing the space cut-off P(ϕ)2 Euclidean quantum field, the 2-dimensional Euclidean quantum fields with exponential and trigonometric potentials, and the measure associated with the field describing a system of an infinite number of classical particles. For each measure space, the Markov process corresponding to the non-local type stochastic quantization is constructed.

Grassmannian stochastic analysis and the stochastic quantization of Euclidean fermions

We introduce a stochastic analysis of Grassmann random variables suitable for the stochastic quantization of Euclidean fermionic quantum field theories. Analysis on Grassmann algebras is developed here from the point of view of quantum probability: a Grassmann random variable is an homomorphism of an abstract Grassmann algebra into a quantum probability space, i.e. a C^{*}-algebra endowed with a suitable state. We define the notion of Gaussian processes, Brownian motion and stochastic (partial) differential equations taking values in Grassmann algebras. We use them to study the long time behavior of finite and infinite dimensional Langevin Grassmann stochastic differential equations driven by Gaussian space-time white noise and to describe their invariant measures. As an application we give a proof of the stochastic quantization and of the removal of the space cut-off for the Euclidean Yukawa model.

The microlocal irregularity of Gaussian noise

The study of random Fourier series, linear combinations of trigonometric functions whose coefficients are independent (in our case Gaussian) random variables with polynomially bounded means and standard deviations, dates back to Norbert Wiener in one of the original constructions of Brownian motion. A geometric generalization -- relevant e.g.\ to Euclidean quantum field theory with an infrared cutoff -- is the study of random Gaussian linear combinations of the eigenfunctions of the Laplace-Beltrami operator on an arbitrary compact Riemannian manifold $(M,g)$, Gaussian noise $\Phi$. I will prove that, when our random coefficients are independent Gaussians whose standard deviations obey polynomial asymptotics and whose means obey a corresponding polynomial upper bound, the resultant random $\mathscr{H}^s$-wavefront set $\operatorname{WF}^s(\Phi)$ (defined as a subset of the cosphere bundle $\mathbb{S}^*M$) is either almost surely empty or almost surely the entirety of $\mathbb{S}^*M$, depending on $s \in \mathbb{R}$, and we will compute the threshold $s$ and the behavior of the wavefront set at it. Consequently, the random $C^\infty$-wavefront set $\operatorname{WF}(\Phi)$ is almost surely the entirety of the cosphere bundle. The method of proof is as follows: using Sazonov's theorem and its converse, it suffices to understand which compositions of microlocal cutoffs and inclusions of $L^2$-based fractional order Sobolev spaces are Hilbert-Schmidt (HS), and the answer follows from general facts about the HS-norms of the elements of the pseudodifferential calculus of Kohn and Nirenberg.

Strong uniqueness for Dirichlet operators related to stochastic quantization under exponential/trigonometric interactions on the two-dimensional torus

We consider space-time quantum fields with exponential/trigonometric interactions. In the context of Euclidean quantum field theory, the former and the latter are called the Hoegh-Krohn model and the Sine-Gordon model, respectively. The main objective of the present paper is to construct infinite dimensional diffusion processes which solve modified stochastic quantization equations for these quantum fields on the two-dimensional torus by the Dirichlet form approach and to prove strong uniqueness of the corresponding Dirichlet operators.

Construction of p-Adic Covariant Quantum Fields in the Framework of White Noise Analysis

In this article we construct a large class of interacting Euclidean quantum field theories, over a p-adic space time, by using white noise calculus. We introduce p-adic versions of the Kondratiev and Hida spaces in order to use the Wick calculus on the Kondratiev spaces. The quantum fields introduced here fulfill all the Osterwalder-Schrader axioms, except the reflection positivity.

Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

The perturbative approach to nonlinear Sigma models and the associated renormalization group flow are discussed within the framework of Euclidean algebraic quantum field theory and of the principle of general local covariance. In particular we show in an Euclidean setting how to define Wick ordered powers of the underlying quantum fields and we classify the freedom in such procedure by extending to this setting a recent construction of Khavkine, Melati and Moretti for vector valued free fields. As a by-product of such classification, we prove that, at first order in perturbation theory, the renormalization group flow of the nonlinear Sigma model is the Ricci flow.

Quasiperiodic Hamiltonian motions, scale invariance, harmonic oscillators

The work of Kolmogorov, Arnold and Moser appeared just before the renormalization group approach to statistical mechanics was proposed by Wilson: it can be classified as a multiscale approach which also appeared in works on the convergence of Fourier's series, or construction of Euclidean quantum fields, or the scaling analysis of the short scale behaviour of Navier-Stokes fluids to name a few which originated a great variety of further problems. In this review the proof of the KAM theorem will be presented as a classical renormalization problem with the harmonic oscillator as a `trivial' fixed point.

Global Solutions to Elliptic and Parabolic $${\Phi^4}$$ Φ 4 Models in Euclidean Space

We prove the existence of global solutions to singular SPDEs on $${\mathbb{R}^{\rm d}}$$ with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions d = 4, 5 and in the parabolic setting for d = 2, 3. We prove uniqueness and coming down from infinity for the parabolic equations. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the $${\Phi^{4}_d}$$ Euclidean quantum field theory via Parisi–Wu stochastic quantization, while the elliptic equations are linked to the $${\Phi^{4}_{d-2}}$$ Euclidean quantum field theory via the Parisi–Sourlas dimensional reduction mechanism.

Stochastic Quantization of Massive Fermions

We consider a general solution of the Langevin equation describing massive fermions to an appropriate boundary problem. Assuming existence of such solution we show that its correlators coincide with the Schwinger functions of corresponding Euclidean Quantum Field Theory.

Infinite Volume Limits in Euclidean Quantum Field Theory via Stereographic Projection

We present a general infinite volume limit construction of probability measures obeying the Glimm-Jaffe axioms of Euclidean quantum field theory in arbitrary space-time dimension. In particular, we obtain measures that may be interpreted as corresponding to scalar quantum fields with arbitrary bounded continuous self-interaction. It remains however an open problem whether this general construction contains non-Gaussian measures.

Action principle for OPE

We formulate an “action principle” for the operator product expansion (OPE) describing how a given OPE coefficient changes under a deformation induced by a marginal or relevant operator. Our action principle involves no ad-hoc regulator or renormalization and applies to general (Euclidean) quantum field theories. It implies a natural definition of the renormalization group flow for the OPE coefficients and of coupling constants. When applied to the case of conformal theories, the action principle gives a system of coupled dynamical equations for the conformal data. The last result has also recently been derived (without considering tensor structures) independently by Behan (arXiv:1709.03967) using a different argument. Our results were previously announced and outlined at the meetings “In memoriam Rudolf Haag” in September 2016 and the “Wolfhart Zimmermann memorial symposium” in May 2017.

Risky forward interest rates and swaptions: Quantum finance model and empirical results

Risk free forward interest rates (Diebold and Li, 2006 [1]; Jamshidian, 1991 [2 ]) – and their realization by US Treasury bonds as the leading exemplar – have been studied extensively. In Baaquie (2010), models of risk free bonds and their forward interest rates based on the quantum field theoretic formulation of the risk free forward interest rates have been discussed, including the empirical evidence supporting these models. The quantum finance formulation of risk free forward interest rates is extended to the case of risky forward interest rates. The examples of the Singapore and Malaysian forward interest rates are used as specific cases. The main feature of the quantum finance model is that the risky forward interest rates are modeled both a) as a stand-alone case as well as b) being driven by the US forward interest rates plus a spread – having its own term structure –above the US forward interest rates.Both the US forward interest rates and the term structure for the spread are modeled by a two dimensional Euclidean quantum field. As a precursor to the evaluation of put option of the Singapore coupon bond, the quantum finance model for swaptions is tested using empirical study of swaptions for the US Dollar –showing that the model is quite accurate. A prediction for the market price of the put option for the Singapore coupon bonds is obtained. The quantum finance model is generalized to study the Malaysian case and the Malaysian forward interest rates are shown to have anomalies absent for the US and Singapore case. The model’s prediction for a Malaysian interest rate swap is obtained.

Extraordinary properties of functional integrals and groups of diffeomorphisms

A review of the work of the authors is presented, in which corollaries of the quasi-invariance of functional integrals on the Wiener measure with respect to the action of a group of diffeomorphisms are studied, and the behavior of functional integrals with nonlinear nonlocal change of variables of integration is investigated as well. Using these substitutions, the functional integrals over discontinuous paths can be determined. The simplest models of the (Euclidean) quantum field theory are offered, in which the presence of hidden internal symmetries or the allowance for discontinuous paths in functional integrals leads to a number of paradoxical properties contradicting the conventional view.

Time-sliced perturbation theory for large scale structure I: general formalism

We present a new analytic approach to describe large scale structure formation in the mildly non-linear regime. The central object of the method is the time-dependent probability distribution function generating correlators of the cosmological observables at a given moment of time. Expanding the distribution function around the Gaussian weight we formulate a perturbative technique to calculate non-linear corrections to cosmological correlators, similar to the diagrammatic expansion in a three-dimensional Euclidean quantum field theory, with time playing the role of an external parameter. For the physically relevant case of cold dark matter in an Einstein-de Sitter universe, the time evolution of the distribution function can be found exactly and is encapsulated by a time-dependent coupling constant controlling the perturbative expansion. We show that all building blocks of the expansion are free from spurious infrared enhanced contributions that plague the standard cosmological perturbation theory. This paves the way towards the systematic resummation of infrared effects in large scale structure formation. We also argue that the approach proposed here provides a natural framework to account for the influence of short-scale dynamics on larger scales along the lines of effective field theory.

Reflection Positive Stochastic Processes Indexed by Lie Groups

Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary representation of a symmetric Lie group to a unitary representation of its Cartan dual. In this article we continue our investigation of representation theoretic aspects of reflection positivity by discussing reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations.

Associativity of the operator product expansion

We consider a recursive scheme for defining the coefficients in the operator product expansion (OPE) of an arbitrary number of composite operators in the context of perturbative, Euclidean quantum field theory in four dimensions. Our iterative scheme is consistent with previous definitions of OPE coefficients via the flow equation method or methods based on Feynman diagrams. It allows us to prove that a strong version of the “associativity” condition holds for the OPE to arbitrary orders in perturbation theory. Such a condition was previously proposed in an axiomatic setting in Hollands [SIGMA 5, 090 (2009)] and has interesting conceptual consequences: (1) One can characterise perturbations of quantum field theories abstractly in a sort of “Hochschild-like” cohomology setting, (2) one can prove a “coherence theorem” analogous to that in an ordinary algebra: the OPE coefficients for a product of two composite operators uniquely determine those for n composite operators. We concretely prove our main results for the Euclidean φ44 quantum field theory, covering also the massless case. Our methods are rather general, however, and would also apply to other, more involved, theories such as Yang-Mills theories.