Abelian gauge field equations

We present a gauge and Lorentz invariant effective field theory model for the interaction of a charged scalar matter field with a magnetic monopole source, described by an external magnetic current. The quantum fluctuations of the monopole field are described effectively by a strongly coupled “dual” Ud(1) gauge field, which is independent of the electromagnetic Uem(1) gauge field. The effective interactions of the charged matter with the monopole source are described by a gauge invariant mixed Chern-Simons-like (Pontryagin-density) term between the two U(1) gauge fields. The latter interaction coupling is left free, and a lattice study of the system is performed with the aim of determining the phase structure of this effective theory. Our study shows that, in the spontaneously broken-symmetry phase, the monopole source triggers, via the mixed Chern-Simons term, which is nontrivial in its presence, the generation of a dynamical singular configuration (magnetic-monopolelike) for the respective gauge fields. The scalar field also behaves in the broken phase in a way similar to that of the scalar sector of the ’t Hooft-Polyakov monopole. Moreover, we show that the modest size of the lattices involved does not have significant effects on the main conclusions of our analysis. Published by the American Physical Society 2024

We consider SU(2) gauge potentials over a space with a compactified\ndimension. A non-Abelian Fourier transform of the gauge potential in the\ncompactified dimension is defined in such a way that the Fourier coefficients\nare (almost) gauge invariant. The functional measure and the gauge field\nstrengths are expressed in terms of these Fourier coefficients. The emerging\nformulation of the non-Abelian gauge theory turns out to be an Abelian gauge\ntheory of a set of fields defined over the initial space with the compactified\ndimension excluded. The Abelian theory contains an Abelian gauge field, a\nscalar field, and an infinite tower of vector matter fields, some of which\ncarry Abelian charges. Possible applications of this formalism are discussed\nbriefly.\n

The possibility of nontrivial representations of the gauge group on wave functionals of a gauge invariant quantum field theory leads to a generation of mass for intermediate vector and tensor bosons. The mass parameters m show up as central charges in the algebra of constraints, which then become of second-class nature. The gauge group coordinates acquire dynamics outside the null-mass shell and provide the longitudinal field degrees of freedom that massless bosons need to form massive bosons. This is a non-Higgs mechanism that could provide new clues for the best understanding of the symmetry breaking mechanism in unified field theories. A unified quantization of massless and massive non-Abelian Yang–Mills, linear gravity and Abelian two-form gauge field theories are fully developed from this new approach, where a cohomological origin of mass is pointed out.

In this chapter we construct a number of models that exhibit the mechanism of spontaneous symmetry breaking for gauge field theories. These models do not yet contain “real” particles but they exhibit the important Higgs mechanism that appears when a local symmetry is spontaneously broken.

The Higgs mechanics is very powerful, it furnishes a description of the electroweak theory in the standard Model which has a convincing experimental verification. But although the Higgs mechanism had been applied successfully, the conceptual background is not clear. The Higgs mechanism is often presented as spontaneous breaking of local gauge symmetry. But local gauge symmetry is rooted in redundancy of description, gauge transformation connect states that cannot be physically distinguished. Gauge symmetry is therefore not symmetry of nature, but of our description of nature. The spontaneous breaking of such symmetry cannot be expected to have physical effects since asymmetries are not reflected in the physics. If spontaneous gauge symmetry breaking cannot have physical effects, this causes conceptual problems for the Higgs mechanism, if taken to be described as spontaneous gauge symmetry breaking. In a gauge invariant theory, gauge fixing is necessary to retrieve the physics from the theory. This means that also in a theory with spontaneous gauge symmetry breaking, a gauge should be fixed. But gauge fixing itself breaks the gauge symmetry, and thereby obscures the spontaneous breaking of the symmetry. It suggests that spontaneous gauge symmetry breaking is not part of the physics, but an unphysical artefact of the redundancy in description.

Intrinsic geometry of D-branes

We investigate the phenomenology of spontaneous symmetry breaking (SSB) and spontaneous antisymmetry breaking (SASB) in the framework of a system of coupled nonlinear Schrödinger equations(NLSEs) with a combination of linear and nonlinear double-well potentials (DWPs), including self- and cross-phase-modulation (SPM and XPM) nonlinear terms. The DWP structure is represented by a symmetric pair of delta functions. This setup can be realized in optics and for matter waves in Bose-Einstein condensates. It is commonly known that SSB states are completely stable in the single one-dimensional NLSE with a combination of DWP and attractive SPM nonlinearity. The objective of the present work is to elaborate on the SSB and SASB phenomenology in the two-component system. One factor that affects the phenomenology (actually, the stability of stationary states) in this context is the relative sign of the two components, the states with identical/opposite signs defined as ones with in-phase/out-of-phase components. We demonstrate that the SSB/SASB, with both components originating from spatially symmetric or antisymmetric modes, of the out-of-phase type, in the coupled NLSEs including the SPM and XPM terms with opposite signs, are similar to their counterparts in the single NLSE. On the other hand, the SSB/SASB of in-phase states in the coupled NLSEs, including the SPM and XPM terms with opposite signs, are quite different from the ones in the single NLSE. In particular, we find that the in-phase SSB/SASB states are unstable in a part of the intervals of their propagation constantif the system includes SPM and XPM terms with opposite signs. As another feature that is possible only in the two-component system, we address mixed-parity states, originally built of spatially symmetric and antisymmetric components, and find that such states, featuring SSB or SASB, are stable in a part of their interval of the propagation constant. We clarify that the spatially antisymmetric component is a dominant one in determining the asymmetry of the established state if it is coupled to the symmetric one, in the case of the repulsive SPM. On the other hand, the symmetric component dominates if it is coupled to the antisymmetric one in the case of the attractive SPM.

This paper describes CosmoLattice, a modern package for lattice simulations of the dynamics of interacting scalar and gauge fields in an expanding universe. CosmoLattice incorporates a series of features that makes it very versatile and powerful: i) it is written in C++ fully exploiting the object oriented programming paradigm, with a modular structure and a clear separation between the physics and the technical details, ii) it is MPI-based and uses a discrete Fourier transform parallelized in multiple spatial dimensions, which makes it specially appropriate for probing scenarios with well-separated scales, running very high resolution simulations, or simply very long ones, iii) it introduces its own symbolic language, defining field variables and operations over them, so that one can introduce differential equations and operators in a manner as close as possible to the continuum, iv) it includes a library of numerical algorithms, ranging from O(δt2) to O(δt10) methods, suitable for simulating global and gauge theories in an expanding grid, including the case of ‘self-consistent’ expansion sourced by the fields themselves. Relevant observables are provided for each algorithm (e.g. energy densities, field spectra, lattice snapshots) and we note that, remarkably, all our algorithms for gauge theories (Abelian or non-Abelian) always respect the Gauss constraint to machine precision. Program summaryProgram Title::CosmoLatticeCPC Library link to program files:https://doi.org/10.17632/44vr5xssc6.1Developer's repository link:http://github.com/cosmolattice/cosmolatticeLicensing provisions: MITProgramming language: C++, MPINature of problem: The phenomenology of high energy physics in the early universe is typically characterized by non-linear dynamics, which cannot be captured accurately with analytical techniques. In order to fully understand the non-linearities developed in a given scenario, one needs to carry out lattice simulations. A number of public packages for lattice simulations have appeared over the years, but most of them are only capable of simulating scalar fields. However, realistic models of particle physics do contain other kind of field species, such as (Abelian or non-Abelian) gauge fields, whose non-linear dynamics can also play a relevant role in the early universe. Tensor modes representing gravitational waves are also naturally expected in many scenarios.Solution method:CosmoLattice represents a modern code for lattice simulations of scalar-gauge field theories in an expanding universe. It allows for the simulation of the evolution of interacting (singlet) scalar fields, charged scalar fields under U(1) and/or SU(2) gauge groups, and the corresponding associated Abelian and/or non-Abelian gauge fields. From version 1.1 onward, CosmoLattice also allows to simulate the production of gravitational waves. Simulations can be done either in a flat space-time background, or in a homogeneous and isotropic (spatially flat) expanding FLRW background. CosmoLattice provides symplectic integrators, with accuracy ranging from O(δt2) up to O(δt10), to simulate the non-linear dynamics of the appropriate fields in comoving three-dimensional lattices. The code is parallelized with MPI, and uses a discrete Fourier Transform parallelized in multiple spatial dimensions, which makes it a very powerful code for probing physical problems with well-separated scales. Moreover, the code has been designed as a ‘platform’ to implement any system of dynamical equations suitable for discretization on a lattice.

For the theory with spontaneously broken phase symmetry, the Ward-Takahashi-type identities, which are qualitatively different from the corresponding ones in the unbroken theory, are proved in the framework of quantum field theory. On the other hand, for the spontaneously broken gauge theory, the Ward-Takahashi identities are shown to be the same as those in the unbroken theory, as far as the abelian gauge field is concerned. § I. Introduction The W ard-Takahashi (W -T) identities are representations of the current conservation law in terms of vacuum r-functions*l or Green's functions.**l In a theory with spontaneously broken symmetry, the current conservation law re­ mains true and therefore W-T identities are expected to hold. However, since the vacuum in such a theory is no longer invariant under the symmetry trans­ formations, the W-T identities can be in general different from those in the cor­ responding unbroken theory. This remark, however,' has not been clearly rec­ ognized until recently, and many authors 1l uncritically used the W-T identities of the unbroken theory in the spontaneously broken theory. Very recently, Matsumoto, Papastamatiou and Umezawa2l (M-P-U) have given an important warning to the persons who work in the functional integral formalism of the theory with spontaneously broken symmetry. Since the functional integral formalism directly deals with Green's functions only, they must contain the bound­ ary condition expressing the spontaneous breakdown of symmetry in themselves. M-P-U have found that the appropriate boundary condition can be embodied by modifying the infinitesimal, purely imaginary term of the action integral which produces the Feynman -ie prescription. Using this modified functional integral formalism, M-P-U have derived new W-T-type identities for the theory with spon­ taneously broken phase symmetry. Let ¢ be a scalar field and suppose that the Lagrangian is invariant under

Spontaneous chiral symmetry breaking in QCD with massless quarks at infinite volume can be seen in a finite box by studying, for instance, the dependence of the chiral condensate from the volume and the quark mass. We perform a feasibility study of this program by computing the quark condensate on the lattice in the quenched approximation of QCD at small quark masses. We carry out simulations in various topological sectors of the theory at several volumes, quark masses and lattice spacings by employing fermions with an exact chiral symmetry, and we focus on observables which are infrared stable and free from mass-dependent ultraviolet divergences. The numerical calculation is carried out with an exact variance-reduction technique, which is designed to be particularly efficient when spontaneous symmetry breaking is at work in generating a few very small low-lying eigenvalues of the Dirac operator. The finite-size scaling behaviour of the condensate in the topological sectors considered agrees, within our statistical accuracy, with the expectations of the chiral effective theory. Close to the chiral limit we observe a detailed agreement with the first Leutwyler-Smilga sum rule. By comparing the mass, the volume and the topology dependence of our results with the predictions of the chiral effective theory, we extract the corresponding low-energy constant.

Symmetry breaking has emerged as a novel strategy to enhance energy conversion and storage performance, which refers to changes in the atomic configurations within a material reducing its internal symmetry. According to the location of the symmetry breaking, it can be classified into spontaneous symmetry breaking within the material, local symmetry breaking on the surface of the material, and symmetry breaking caused by external fields outside the material. However, there are currently few summaries in this field, so it is necessary to summarize how symmetry breaking improves energy conversion and storage performance. In this review, the fundamentals of symmetry breaking are first introduced, which allows for a deeper understanding of its meaning. Then the applications of symmetry breaking in energy conversion and storage are systematically summarized, providing various mechanisms in energy conversion and storage, as well as how to improve energy conversion performance and storage efficiency. Last but not least, the current applications of symmetry breaking are summarized and provide an outlook on its future development. It is hoped that this review can provide new insights into the applications of symmetry breaking and promote its further development.

Building a universal quantum computer is a central goal of emerging quantum technologies and it is expected to revolutionize science and technology. Unfortunately, this future does not seem very close, however, quantum computers built for a special purpose, i.e., quantum simulators, are currently being developed in many leading laboratories. Numerous schemes for quantum simulation have been proposed and realized using, e.g., ultracold atoms in optical lattices, ultracold trapped ions, atoms in arrays of cavities, atoms/ions in arrays of traps, quantum dots or superconducting circuits. The progress in experimental implementations is more than spectacular.Particularly interesting are those systems that simulate quantum matter evolving in artificial, or synthetic, Abelian or even non-Abelian gauge fields. Abelian gauge fields are analogues to the standard magnetic field and lead to fascinating effects such as the integer or fractional quantum Hall effects (IQHE, FQHE) and vortex lattices. Non-Abelian gauge fields couple the motional states of the particles to their internal degrees of freedom (such as hyperfine states for atoms or ions, electronic spins for electrons, etc). In this sense, external non-Abelian fields extend the concept of spin–orbit coupling, which is familiar from AMO and condensed matter physics. They lead to yet another variety of fascinating novel phenomena such as the quantum spin Hall effect (QSHE), 3D topological insulators, topological superconductors and superfluids of various kinds.Even more fascinating is the possibility of generating synthetic gauge fields that are dynamical, i.e., that evolve in time according to the corresponding lattice gauge theory (LGT). These dynamical gauge fields can also couple to matter fields, allowing the quantum simulation of such complex systems (notoriously hard to simulate using 'traditional' computers), which are particularly relevant for modern high-energy physics. So far there are only theoretical proposals for simulating Abelian LGTs, but many groups are working on extensions to the non-Abelian scenarios.The scope of this special issue of Journal of Physics B: Atomic, Molecular and Optical Physics is on all of these developments, with particular emphasis on the non-Abelian case. We invite the leading theory and experimental groups to contribute to this very special issue of the journal in order to provide a reference collection for quantum simulations of gauge fields.To summarize the key features should be Synthetic spin–orbit coupling and the physics of topological insulating phases Strongly correlated phases in non-Abelian gauge potentials Dynamical non-Abelian gauge fields and the simulation of lattice gauge theories Spin–orbit coupled BEC and vortex physics Simulators of Abelian LGTs Simulators of non-Abelian LGTsYou are invited to submit your article by 15 December 2012. Expected publication: Summer 2013.Corrections were made to this article on 7 November 2012. A change was made to the affiliations.

We show the derivation of the self-duality relation of abelian higher-form gauge field strength in the topologically nontrivial spacetime background. The so-called Pasti-Sorokin-Tonin action for the self-dual abelian gauge field assumes that the spacetime topology is trivial to derive the self-duality relation using a gauge transformation of the action. In this paper we find a new gauge transformation of the same theory and show that this new gauge transformation enables us to derive the self-duality relation even in the topologically nontrivial spacetime background.

We consider U(1)-symmetric scalar quantum field theories at zero temperature. At nonzero charge densities, the ground state of these systems is usually assumed to be a superfluid phase, in which the global symmetry is spontaneously broken along with Lorentz boosts and time translations. We show that, in d > 2 spacetime dimensions, this expectation is always realized at one loop for arbitrary non-derivative interactions, confirming that the physically distinct phenomena of nonzero charge density and spontaneous symmetry breaking occur simultaneously in these systems. We quantify this result by deriving universal scaling relations for the symmetry breaking scale as a function of the charge density, at low and high density. Moreover, we show that the critical value of μ above which a nonzero density develops coincides with the pole mass in the unbroken, Poincaré invariant vacuum of the theory. The same conclusions hold non-perturbatively for an O(N) theory with quartic interactions in d = 3 and 4, at leading order in the 1/N expansion. We derive these results by computing analytically the zero-temperature, finite-μ one-loop effective potential, paying special attention to subtle points related to the iε terms. We check our results against the one-loop low-energy effective action for the superfluid phonons in λϕ4 theory in d = 4 previously derived by Joyce and ourselves, which we further generalize to arbitrary potential interactions and arbitrary dimensions. As a byproduct, we find analytically the one-loop scaling dimension of the lightest charge-n operator for the λϕ6 conformal superfluid in d = 3, at leading order in 1/n, reproducing a numerical result of Badel et al. For a λϕ4 superfluid in d = 4, we also reproduce the Lee-Huang-Yang relation and compute relativistic corrections to it. Finally, we discuss possible extensions of our results beyond perturbation theory.

It is argued that a spontaneous breaking of local symmetry for a symmetrical gauge theory without gauge fixing is impossible. The argument is demonstrated in a simple system of Abelian gauge fields on a lattice.