Euclidean Field Research Articles

From Euclidean field theory to hyperkähler Floer theory via regularized polysymplectic geometry

Hamiltonian Floer theory plays an important role for finding periodic solutions of Hamilton’s equation, which can be seen as a generalization of Newton’s equation. Generalizing Newton’s equation to Laplace’s equation with nonlinearity, we show, building on the work of Ginzburg and Hein, that this role is taken over by the hyperkähler Floer theory of Hohloch, Noetzel, and Salamon. Apart from establishing [Formula: see text]-bounds in order to be able to deal with noncompact hyperkähler manifolds, the core ingredient is a regularization scheme for the polysymplectic formalism due to Bridges, which allows us to link Euclidean field theory with hyperkähler Floer theory. As a concrete result, we prove a cuplength estimate.

Axion Mass and the Ground State of Deconfining SU(2) Yang–Mills Thermodynamics

For the deconfinement phase of an SU(2) Yang–Mills theory, we compute the axion mass mA by appealing to the Veneziano–Witten formula. The topological susceptibility χ arises (i) from a precisely computable thermal ground-state contribution due to a center of a relevant (anti)caloron, and (ii) from contributions due to free thermal quasi-particles in the effective theory. Both (i) and (ii) are derived by using standard Euclidean thermal field theory techniques. While contribution (i) is positive and ∝T4, contribution (ii) is negative, as demanded by reflection positivity, but negligible compared to contribution (i). As a consequence, practically from the critical temperature Tc onward, a real-valued axion mass mA(T)=23πT2MP emerges when the Peccei–Quinn scale is assumed to be the Planck mass MP, independently of the Yang–Mills scale that the axion associates with. We discuss why our results deviate from those found in the dilute instanton gas and interacting instanton liquid approximations, and from results obtained in lattice simulations. Assuming the universe is dark sector to be based on such ultralight axion species, which are nonrelativistic for T≪MP, we investigate the cosmological conditions for their global Bose condensation as the very early universe cooled to temperatures of the order of 109eV.

A Dynamical Yukawa2 Model

We prove local (in space and time) well-posedness for a mildly regularised version of the stochastic quantisation of the Yukawa2 Euclidean field theory with a self-interacting boson. Our regularised dynamic is still singular but avoids non-local divergences, allowing us to use a version of the Da Prato–Debussche argument (Da Prato and Debussche in Ann Probab 31(4):1900–1916, 2003. https://doi.org/10.1214/aop/1068646370). This model is a test case for a non-commutative probability framework for formulating the kind of singular SPDEs arising in the stochastic quantisation of field theories mixing both bosons and fermions.

Magnetic braneworlds: cosmology and wormholes

We construct 4D flat Big Bang-Big Crunch cosmologies and Anti-de Sitter (AdS) planar eternally traversable wormholes using braneworlds embedded in asymptotically AdS5 spacetimes. The background geometries are the AdS5 magnetic black brane and the magnetically charged AdS5 soliton, respectively. The two setups arise from different analytic continuations of the same saddle of the gravitational Euclidean path integral, in which the braneworld takes the form of a Maldacena-Maoz Euclidean wormhole. We show the existence of a holographic dual description of this setup in terms of a microscopic Euclidean boundary conformal field theory (BCFT) on a strip. By analyzing the BCFT Euclidean path integral, we show that the braneworld cosmology is encoded in a pure excited state of a CFT dual to a black brane microstate, whereas the braneworld wormhole is encoded in the ground state of the BCFT. The latter confines in the IR, and we study its confining properties using holography. We also comment on the properties of bulk reconstruction in the two Lorentzian pictures and their relationship via double analytic continuation. This work can be interpreted as an explicit, doubly-holographic realization of the relationship between cosmology, traversable wormholes, and confinement in holography, first proposed in arXiv:2102.05057, arXiv:2203.11220.

Sums of squares in function fields over henselian discretely valued fields

Let n∈N and let K be a field with a henselian discrete valuation of rank n with hereditarily euclidean residue field. Let F/K be a function field in one variable. It is known that every sum of squares is a sum of 3 squares. We determine the order of the group of nonzero sums of 3 squares modulo sums of 2 squares in F in terms of equivalence classes of certain discrete valuations of F of rank at most n. In the case of function fields of hyperelliptic curves of genus g, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by 2n(g+1). We show that this bound is optimal. Moreover, in the case where n=1, we show that if F/K is a hyperelliptic function field such that the order of this quotient group is 2g+1, then F is nonreal.

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.

Wormhole inducing exponential expansion in R2 gravity

Wormholes are considered both from the Wheeler deWitt equation, as well as from the field equations in the Euclidean background of Robertson Walker mini-superspace in [Formula: see text] gravity. Quantum wormhole satisfies Hawking Page wormhole boundary condition in the Euclidean background of mini-superspace, however, in the Lorentzian background wave functional turns to the usual oscillatory function. The Euclidean field equations for [Formula: see text] lead to the wormhole configuration, as well as oscillating universe in Euclidean time [Formula: see text]. The oscillating universe in Euclidean time [Formula: see text] transforms to an expanding universe only for [Formula: see text] in Lorentz time [Formula: see text] under analytic continuation [Formula: see text] and asymptotically leads to an exponential solution. An Euclidean wormhole in the very early era evolves to an oscillating universe in [Formula: see text], thereafter crossing deSitter radius transition to an inflationary era is evident at later epoch only for [Formula: see text].

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.

The Euclidean $\phi^{4}_{2}$ theory as a limit of an interacting Bose gas

We prove that the complex Euclidean field theory with local quartic self-interaction in two dimensions arises as a limit of an interacting Bose gas at positive temperature, when the density of the gas becomes large and the range of the interaction becomes small. The field theory is supported on distributions of negative regularity, which requires a renormalization by divergent mass and energy counterterms. We obtain convergence of the relative partition function and uniform convergence of the renormalized reduced density matrices. The proof is based on three main ingredients: (a) a quantitative analysis of the infinite-dimensional saddle point argument for the functional integral introduced by Fröhlich et al. (2022) using continuity properties of Brownian paths, (b) a Nelson-type estimate for a general nonlocal field theory in two dimensions, and (c) repeated Gaussian integration by parts in field space to obtain uniform control on the renormalized correlation functions. As a byproduct of our proof, in two and three dimensions we also extend the results on the mean-field limit by Fröhlich et al. (2022) and Lewin et al. (2021) to unbounded interaction potentials satisfying the optimal integrability conditions proposed by Bourgain (1997).

Scaled affine quantization of φ312 is nontrivial

We prove through path integral Monte Carlo that the covariant Euclidean scalar field theory, [Formula: see text], where [Formula: see text] denotes the power of the interaction term and [Formula: see text] with [Formula: see text] as the spatial dimension and 1 adds imaginary time, such that [Formula: see text], [Formula: see text] can be acceptably quantized using scaled affine quantization (AQ) and the resulting theory is nontrivial, unlike what happens using canonical quantization which finds it trivial.

On the variational method for Euclidean quantum fields in infinite volume

On the variational method for Euclidean quantum fields in infinite volume

Accelerating Cosmology from a Holographic Wormhole.

We consider cosmological models in which the cosmology is related via analytic continuation to a Euclidean asymptotically AdS planar wormhole geometry defined holographically via a pair of three-dimensional Euclidean conformal field theories (CFTs). We argue that these models can generically give rise to an accelerating phase for the cosmology due to the potential energy of scalar fields associated with relevant scalar operators in the CFT. We explain how cosmological observables are related to observables in the wormhole spacetime and argue that this leads to a novel perspective on naturalness puzzles in cosmology.

Vacuum decay and Euclidean lattice Monte Carlo

The decay rate of a metastable vacuum is usually calculated using a semiclassical approximation to the Euclidean path integral. The extension to a complete Euclidean lattice Monte Carlo computation, however, is hampered by analytic continuations that are ill-suited to numerical treatment, and the nonequilibrium nature of a metastable state. In this paper we develop a new methodology to compute vacuum decay rates from Monte Carlo simulations of Euclidean lattice theories. To test the new method, we consider simple quantum mechanical systems systems with metastable vacua. This work can be extended to Euclidean field theories, which we discuss in the Conclusions.

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.

Scaled affine quantization of varphi ^4_4 in the low temperature limit

We prove through Monte Carlo analysis that the covariant Euclidean scalar field theory, varphi ^r_n, where r denotes the power of the interaction term and n = s + 1 where s is the spatial dimension and 1 adds imaginary time, such that r = n = 4 can be acceptably quantized using scaled affine quantization and the resulting theory is nontrivial and renormalizable even at low temperatures in the highly quantum regime.

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.

Cosmic scenario of early universe in context of Euclidean wormhole in dilatonic Einstein–Gauss–Bonnet gravity

In this paper, we present some wormhole configurations using analytic and numerical solutions of the Euclidean field equations considering Einstein–Gauss–Bonnet dilaton coupled interaction in the Robertson–Walker metric. Analytic solutions invoking slow-roll type approximation for some potentials yield identical wormholes in the early era. Eventually, in later era, the universe enters to an expanding era much faster than usual inflation accompanied by oscillations with analytic continuation [Formula: see text]. The numerical solutions of wormholes in terms of the scale factor [Formula: see text] are almost identical for some standard potentials of the dilaton field, except for the evolution of potentials. The Hubble parameter and deceleration parameter obtained by curve fit of [Formula: see text] of the numerical solution of wormhole yield an inflationary era away from the throat of the wormhole using [Formula: see text]. A sharp decay of the potential is also observed.

Neither Fast nor Slow: How to Fly Through Narrow Tunnels

Nowadays, multirotors are playing important roles in abundant types of missions. During these missions, entering confined and narrow tunnels that are barely accessible to humans is desirable yet extremely challenging for multirotors. The restricted space and significant ego airflow disturbances induce control issues at both fast and slow flight speeds, meanwhile bringing about problems in state estimation and perception. Thus, a smooth trajectory at the proper speed is necessary for safe tunnel flights. To address these challenges, in this letter, a complete autonomous aerial system that can achieve smooth flights through tunnels with dimensions as narrow as to 0.6 m is presented. The system contains a motion planner that generates smooth mini-jerk trajectories along the tunnel center lines, which are extracted according to the map and Euclidean distance field (EDF), and its practical speed range is obtained through computational fluid dynamics (CFD) and flight data analyses. Extensive flight experiments on the quadrotor are conducted inside multiple narrow tunnels to validate the planning framework as well as the robustness of the whole system.

Factoring isometries of quadratic spaces into reflections

Let V be a vector space endowed with a non-degenerate quadratic form Q. If the base field F is different from F2, it is known that every isometry can be written as a product of reflections. In this article, we detail the structure of the poset of all minimal length reflection factorizations of an isometry. If F is an ordered field, we also study factorizations into positive reflections, i.e., reflections defined by vectors of positive norm. We characterize such factorizations, under the hypothesis that the squares of F are dense in the positive elements (this includes Archimedean and Euclidean fields). In particular, we show that an isometry is a product of positive reflections if and only if its spinor norm is positive. As a final application, we explicitly describe the poset of all factorizations of isometries of the hyperbolic space.

Quantum field theories, Markov random fields and machine learning

The transition to Euclidean space and the discretization of quantum field theories on spatial or space-time lattices opens up the opportunity to investigate probabilistic machine learning within quantum field theory. Here, we will discuss how discretized Euclidean field theories, such as the ϕ 4 lattice field theory on a square lattice, are mathematically equivalent to Markov fields, a notable class of probabilistic graphical models with applications in a variety of research areas, including machine learning. The results are established based on the Hammersley-Clifford theorem. We will then derive neural networks from quantum field theories and discuss applications pertinent to the minimization of the Kullback-Leibler divergence for the probability distribution of the ϕ 4 machine learning algorithms and other probability distributions.