Euclidean Version Research Articles

Counting linearly polarized gluons with lattice QCD

We outline an approach to calculate the transverse-momentum-dependent distribution of linearly polarized gluons inside an unpolarized hadron on the lattice with the help of large momentum effective theory. To achieve this purpose, we propose calculating a Euclidean version of the degree of polarization for a fast-moving hadron on the lattice, which is ultraviolet finite, and no soft function subtraction is needed. It indicates a practical way to explore the distribution of the linearly polarized gluons in a proton and the linearly polarized gluon effects in hadron collisions on the lattice. Published by the American Physical Society 2024

Localization on AdS3× S2. Part I. The 4d/5d connection in off-shell Euclidean supergravity

We begin to develop the formalism of localization for the functional integral of supergravity on AdS3× S2. We show how the condition of supersymmetry in the Euclidean ℍ3/ℤ × S2 geometry naturally leads to a twist of the S2 around the time direction of AdS3. The twist gives us a five-dimensional Euclidean supergravity background dual to the elliptic genus of (0, 4) SCFT2 at the semiclassical level. On this background we set up the off-shell BPS equations for one of the Killing spinors, such that the functional integral of five-dimensional Euclidean supergravity on ℍ3/ℤ × S2 localizes to its space of solutions. We obtain a class of solutions to these equations by lifting known off-shell BPS solutions of 4-dimensional supergravity on AdS2× S2. In order to do this consistently, we construct and use a Euclidean version of the off-shell 4d/5d lift of arXiv:1112.5371, which could be of independent interest.

Saturation phenomena of a nonlocal eigenvalue problem: the Riemannian case

ABSTRACT In this paper we investigate the Riemannian extensibility of saturation phenomena treated first in the Euclidean framework by Brandolini et al. [Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem. Adv Math (N Y). 2011;228(4):2352–2365.]. The saturation problem is formulated in terms of the first eigenvalue of the perturbation of the Laplace-Beltrami operator by the integral of the unknown function: the first eigenvalue increases with the weight affecting the integral up to a finite critical value and then remains constant, i.e. it saturates. Given a Riemannian manifold with certain curvature constraints, by using symmetrization arguments and sharp isoperimetric inequalities, we reduce the general problem to a variational one, formulated on either positively or negatively curved Riemannian model spaces; in addition, the possible scenarios for the optimal domains turn to be either geodesic balls or the union of two disjoint geodesic balls. We then explicitly compute the eigenvalues and eigenfunctions in terms of the radii, curvature and weight. A sufficient condition (incompatibility of a system of nonlinear equations involving special functions) is given that implies similar saturation phenomena to the Euclidean case. Due to its highly nonlinear character of the reduced problem (arising from the presence of curvature and special functions), we provide only partial answers to the original problem. However, both analytical computations and numerical tests suggest that the required incompatibility always persists. In addition, in the limit cases when the curvature tends to zero (for both positive an negative curvature), our results reduce to the Euclidean version.

From Second-Order Differential Geometry to Stochastic Geometric Mechanics

Classical geometric mechanics, including the study of symmetries, Lagrangian and Hamiltonian mechanics, and the Hamilton–Jacobi theory, are founded on geometric structures such as jets, symplectic and contact ones. In this paper, we shall use a partly forgotten framework of second-order (or stochastic) differential geometry, developed originally by L. Schwartz and P.-A. Meyer, to construct second-order counterparts of those classical structures. These will allow us to study symmetries of stochastic differential equations (SDEs), to establish stochastic Lagrangian and Hamiltonian mechanics and their key relations with second-order Hamilton–Jacobi–Bellman (HJB) equations. Indeed, stochastic prolongation formulae will be derived to study symmetries of SDEs and mixed-order Cartan symmetries. Stochastic Hamilton’s equations will follow from a second-order symplectic structure and canonical transformations will lead to the HJB equation. A stochastic variational problem on Riemannian manifolds will provide a stochastic Euler–Lagrange equation compatible with HJB one and equivalent to the Riemannian version of stochastic Hamilton’s equations. A stochastic Noether’s theorem will also follow. The inspirational example, along the paper, will be the rich dynamical structure of Schrödinger’s problem in optimal transport, where the latter is also regarded as a Euclidean version of hydrodynamical interpretation of quantum mechanics.

Progress in the numerical studies of the type IIB matrix model

The type IIB matrix model, also known as the IKKT model, has been proposed as a promising candidate for a non-perturbative formulation of superstring theory. Based on this proposal, various attempts have been made to explain how our four-dimensional space-time can emerge dynamically from superstring theory. In this article, we review the progress in numerical studies on the type IIB matrix model. We particularly focus on the most recent results for the Euclidean and Lorentzian versions, which are obtained using the complex Langevin method to overcome the sign problem. We also review the earlier results obtained using conventional Monte Carlo methods and clarify the relationship among different calculations.

Quantum cosmology of pure connection general relativity

We study homogeneous cosmological models in formulations of general relativity with cosmological constant based on a (complexified) connection rather than a spacetime metric, in particular in a first order theory obtained by integrating out the self-dual two-forms in the chiral Plebański formulation. Classical dynamics for the Bianchi IX model are studied in the Lagrangian and Hamiltonian formalism, where we emphasise the reality conditions needed to obtain real Lorentzian solutions. The solutions to these reality conditions fall into different branches, which in turn lead to different real Hamiltonian theories, only one of which is the usual Lorentzian Bianchi IX model. We also show the simpler case of the flat Bianchi I model, for which both the reality conditions and dynamical equations simplify considerably. We discuss the relation of a real Euclidean version of the same theory to this complex theory. Finally, we study the quantum theory of homogeneous and isotropic models, for which the pure connection action for general relativity reduces to a pure boundary term and the path integral is evaluated immediately, reproducing known results in quantum cosmology. An intriguing aspect of these theories is that the signature of the effective spacetime metric, and hence the interpretation of the cosmological constant, are intrinsically ambiguous.

A Multi-Phase Method for Euclidean Traveling Salesman Problems

The Traveling Salesman Problem (TSP) aims to find the shortest tour for a salesman who starts and ends in the same city and visits the remaining n−1 cities exactly once. There are a number of common generalizations of the problem including the Multiple Traveling Salesman Problem (MTSP), where instead of one salesman, there are k salesmen and the same amount of individual tours are to be constructed. We consider the Euclidean version of the problem where the distances between the cities are calculated in two-dimensional Euclidean space. Both general the TSP and its Euclidean version are strongly NP-hard. Hence, approximation algorithms with a good practical behavior are of primary interest. We describe a general method for the solution of the Euclidean versions of the TSP (including MTSP) that yields approximation algorithms with a favorable practical behavior for large real-life instances. Our method creates special types of convex hulls, which serve as a basis for the constructions of our initial and intermediate partial solutions. Here, we overview three algorithms; one of them is for the bounded version of the MTSP. The proposed novel algorithm for the Euclidean TSP provides close-to-optimal solutions for some real-life instances.

Parameterized Approximation Algorithms for K-center Clustering and Variants

k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.93, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm yields a 2^{O((klog k)/{epsilon})}dn-time (1+epsilon)-approximation for Euclidean k-center, where d is the dimension. In this work, we give a faster algorithm for small dimensions: roughly speaking an O^*(2^{O((1/epsilon)^{O(d)} k^{1-1/d} log k)})-time (1+epsilon)-approximation. In particular, the running time is roughly O^*(2^{O((1/epsilon)^{O(1)}sqrt{k}log k)}) in the plane. We complement our algorithmic result with a matching hardness lower bound. We also consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a 2^{O(klog k)}n^2 time 3-approximation for NUkC, and a 2^{O((klog k)/epsilon)}dn time (1+\epsilon)-approximation for Euclidean NUkC. The latter time bound matches the bound for k-center.

Tracking and Regret Bounds for Online Zeroth-Order Euclidean and Riemannian Optimization

We study numerical optimization algorithms that use zeroth-order information to minimize time-varying geodesically convex cost functions on Riemannian manifolds. In the Euclidean setting, zeroth-order algorithms have received a lot of attention in both the time-varying and time-invariant cases. However, the extension to Riemannian manifolds is much less developed. We focus on Hadamard manifolds, which are a special class of Riemannian manifolds with global nonpositive curvature that offer convenient grounds for the generalization of convexity notions. Specifically, we derive bounds on the expected instantaneous tracking error, and we provide algorithm parameter values that minimize the algorithm's performance. Our results illustrate how the manifold geometry in terms of the sectional curvature affects these bounds. Additionally, we provide dynamic regret bounds for this online optimization setting. To the best of our knowledge, these are the first regret bounds even for the Euclidean version of the problem. Lastly, via numerical simulations, we demonstrate the applicability of our algorithm on an online Karcher mean problem.

Loss of ergodicity in a quantum hopping model of a dense many body system with repulsive interactions

In this work we report on a loss of ergodicity in a simple hopping model, motivated by the Hubbard Hamiltonian, of a many body quantum system at zero temperature, quantized in Euclidean time. We show that this quantum system may lose ergodicity at high densities on a large lattice, as a result of both Pauli exclusion and strong Coulomb repulsion. In particular we study particle hopping susceptibilities and the tendency towards particle localization. It is found that the appearance and existence of quantum phase transitions in this model, in the case of high density and strong Coulomb repulsion, depends on the starting configuration of particle trajectories in the numerical simulation. We argue that this breakdown may be the Euclidean time version of a breakdown of the eigenstate thermalization hypothesis in real time quantization.

General massless field equations for higher spin in dimension 4

Massless field equations are fundamental in particle physics. In Clifford analysis, the Euclidean version of these equations has been dealt with, but it is not clear, even in dimension 4, what should be the right analogue of massless field equations for fields with values in a general irreducible Spin(4)‐module. The main aim of the paper is to explain that a good possibility is to take the so‐called generalized Cauchy–Riemann equations proposed a long time ago by Stein and Weiss. For this choice of the equations, we show that their polynomial solutions form different irreducible Spin(4)‐modules. This is an important step in developing the corresponding function theory.

A Superspace Dirac Operator in NCG and the “Factorization” of the Ordinary Dirac Operator

We review a procedure of factorizing the Minkowski space Dirac operator over a~suitable superspace, discuss its Euclidean space version and apply the worked out formalism in the case od an almost-commutative Dirac operator. The presented framework is an attempt to reconcile non-commutative geometry and supersymmetry.

Quaternionic analysis, representation theory and physics II

We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula for the second order pole, in addition to the one studied in the first paper with the same title. We also realize the doubly regular functions as a subspace of the quaternionic-valued functions satisfying a Euclidean version of Maxwell's equations for the electromagnetic field. Then we return to the study of the original quaternionic analogue of Cauchy's second order pole formula and its relation to the polarization of vacuum. We find the decomposition of the space of quaternionic-valued functions into irreducible components that include the spaces of doubly left and right regular functions. Using this decomposition, we show that a regularization of the vacuum polarization diagram is achieved by subtracting the component corresponding to the one-dimensional subrepresentation of the conformal group. After the regularization, the vacuum polarization diagram is identified with a certain second order differential operator which yields a quaternionic version of Maxwell equations. Next, we introduce two types of quaternionic algebras consisting of spaces of scalar-valued and quaternionic-valued functions. We emphasize that these algebra structures are invariant under the action of the conformal Lie algebra. This uses techniques from our study of the vacuum polarization diagram. These algebras are not associative, but we can define an infinite family of n-multiplications, and we conjecture that they have structures of weak cyclic A-infinity algebras. We also conjecture the relation between the multiplication operations of the scalar and non-scalar quaternionic algebras with the n-photon Feynman diagrams in the scalar and ordinary conformal QED.

Complex Langevin analysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model

The IKKT matrix model is a promising candidate for a nonperturbative formulation of superstring theory. In this model, spacetime is conjectured to emerge dynamically from the microscopic matrix degrees of freedom in the large-N limit. Indeed in the Lorentzian version, Monte Carlo studies suggested the emergence of (3+1)-dimensional expanding spacetime. Here we study the Euclidean version instead, and investigate an alternative scenario for dynamical compactification of extra dimensions via the spontaneous symmetry breaking (SSB) of 10D rotational symmetry. We perform numerical simulations based on the complex Langevin method (CLM) in order to avoid a severe sign problem. Furthermore, in order to avoid the singular-drift problem in the CLM, we deform the model and determine the SSB pattern as we vary the deformation parameter. From these results, we conclude that the original model has an SO(3) symmetric vacuum, which is consistent with previous results obtained by the Gaussian expansion method (GEM). We also apply the GEM to the deformed matrix model and find consistency with the results obtained by the CLM.

Liar’s dominating set problem on unit disk graphs

In this paper, we consider Euclidean versions of the 2-tuple dominating set problem and the liar’s dominating set problem. For a given set P={p1,p2,…,pn} of n points in R2, the objective of the Euclidean 2-tuple dominating set problem is to find a minimum size set D⊆P such that |N[pi]∩D|≥2 for each pi∈P, where N[pi]={pj∈P∣δ(pi,pj)≤1} and δ(pi,pj) is the Euclidean distance between pi and pj. The objective of the Euclidean liar’s dominating set problem is to find a set D(⊆P) of minimum size satisfying the following two conditions: (i) D is a 2-tuple dominating set of P, and (ii) for every distinct pair of points pi and pj in P, |(N[pi]∪N[pj])∩D|≥3.We first propose a simple O(nlogn) time 632-factor approximation algorithm for the Euclidean liar’s dominating set problem. Next, we propose approximation algorithms to improve the approximation factor to 732α for 3≤α≤183, and 846α for 3≤α≤282. The running time of both the algorithms is O(nα+1Δ), where Δ=max{|N[p]|:p∈P}. Finally, we propose a PTAS for the Euclidean 2-tuple dominating set problem.

Simple Constructive, Insertion, and Improvement Heuristics Based on the Girding Polygon for the Euclidean Traveling Salesman Problem

The Traveling Salesman Problem (TSP) aims at finding the shortest trip for a salesman, who has to visit each of the locations from a given set exactly once, starting and ending at the same location. Here, we consider the Euclidean version of the problem, in which the locations are points in the two-dimensional Euclidean space and the distances are correspondingly Euclidean distances. We propose simple, fast, and easily implementable heuristics that work well, in practice, for large real-life problem instances. The algorithm works on three phases, the constructive, the insertion, and the improvement phases. The first two phases run in time O ( n 2 ) and the number of repetitions in the improvement phase, in practice, is bounded by a small constant. We have tested the practical behavior of our heuristics on the available benchmark problem instances. The approximation provided by our algorithm for the tested benchmark problem instances did not beat best known results. At the same time, comparing the CPU time used by our algorithm with that of the earlier known ones, in about 92% of the cases our algorithm has required less computational time. Our algorithm is also memory efficient: for the largest tested problem instance with 744,710 cities, it has used about 50 MiB, whereas the average memory usage for the remained 217 instances was 1.6 MiB.

No smooth beginning for spacetime

In this paper, I will review an obstruction for theories of the beginning of the universe which can be formulated as semiclassical path integrals. Hartle and Hawking’s no boundary proposal and Vilenkin’s tunneling proposal are examples of such theories. Each may be formulated as the quantum amplitude for obtaining a final 3-geometry by integrating over 4-geometries. The result is obtained using a new mathematical tool — Picard–Lefschetz theory — for defining the semiclassical path integral for gravity. The Lorentzian path integral for quantum cosmology with a positive cosmological constant is mathematically meaningful in this approach, but the Euclidean version is not. Framed in this way, the resulting framework and predictions are unique. Unfortunately, the outcome is that primordial gravitational wave fluctuations are unsuppressed.

The Doppler effect and similar triangles

Abstract The Euclidean version of Doppler effect will be explained using the properties of similar triangles. These properties will serve to explain the relationship between frequencies of propagated wavefronts, number of wavefronts and the time of emitting and receiving of these wavefronts. This explanation of Doppler effect represents not only a new method of obtaining equations which deal with Doppler effect, but using this geometric construction we will have an easier and deep understanding of some problems related to this effect, such as the general formulas of the Doppler effect and time Doppler effect, which are replaced by the new proposed formulas.

Quantizations on Nilpotent Lie Groups and Algebras Having Flat Coadjoint Orbits

For a connected simply connected nilpotent Lie group $$\textsf {G}$$ with Lie algebra $${{\mathfrak {g}}}$$ and unitary dual $$\widehat{{\textsf {G}}}$$ one has (a) a global quantization of operator-valued symbols defined on $$\textsf {G}\times \widehat{{\textsf {G}}}$$ , involving the representation theory of the group, (b) a quantization of scalar-valued symbols defined on $$\textsf {G}\times {{\mathfrak {g}}}^*$$ , taking the group structure into account and (c) Weyl-type quantizations of all the coadjoint orbits $$\big \{\Omega _\xi \mid \xi \in \widehat{{\textsf {G}}}\big \}$$ . We show how these quantizations are connected, in the case when flat coadjoint orbits exist. This is done by a careful new analysis of the composition of two different types of Fourier transformations, interesting in itself. We also describe the concrete form of the operator-valued symbol quantization, by using Kirillov theory and the Euclidean version of the unitary dual and Plancherel measure. In the case of the Heisenberg group, this corresponds to the known picture, presenting the representation theoretical pseudo-differential operators in terms of families of Weyl operators depending on a parameter. For illustration, we work out a couple of examples and put into evidence some specific features of the case of Lie algebras with one-dimensional center. When $$\textsf {G}$$ is also graded, we make a short presentation of the symbol classes $$S^m_{\rho ,\delta }$$ , transferred from $$\textsf {G}\times \widehat{{\textsf {G}}}$$ to $$\textsf {G}\times {{\mathfrak {g}}}^*$$ by means of the connection mentioned above.

Relativistic Black-Scholes Model

Black-Scholes equation, after a certain coordinate transformation, is equivalent to the heat equation. On the other hand the relativistic extension of the latter, the telegraphers equation, can be derived from the Euclidean version of the Dirac equation. Therefore the relativistic extension of the Black-Scholes model follows from relativistic quantum mechanics quite naturally.We investigate this particular model for the case of European vanilla options. Due to the notion of locality incorporated in this way one finds that the volatility frown-like effect appears when comparing to the original Black-Scholes model.