Gauge Transformations Research Articles

Wavelength-division multiplexing optical Ising simulator enabling fully programmable spin couplings and external magnetic fields.

Recently various physical systems have been proposed for modeling Ising spin Hamiltonians appealing to solve combinatorial optimization problems with remarkable performance. However, how to implement arbitrary spin-spin interactions is a critical and challenging problem in unconventional Ising machines. Here, we propose a general gauge transformation scheme to enable arbitrary spin-spin interactions and external magnetic fields as well, by decomposing an Ising Hamiltonian into multiple Mattis-type interactions. With this scheme, a wavelength-division multiplexing spatial photonic Ising machine (SPIM) is developed to show the programmable capability of general spin coupling interactions. We exploit the wavelength-division multiplexing SPIM to simulate three spin systems: ±J models, Sherrington-Kirkpatrick models, and only locally connected J1-J2 models and observe the phase transitions. We also demonstrate the ground-state search for solving Max-Cut problem with the wavelength-division multiplexing SPIM. These results promise the realization of ultrafast-speed and high-power efficiency Boltzmann sampling to a generalized large-scale Ising model.

W-symmetries, anomalies and heterotic backgrounds with SU holonomy

We show that the sigma models with target spaces supersymmetric heterotic backgrounds with SU(2) and SU(3) holonomy are invariant under a W-symmetry algebra generated by the covariantly constant forms of these backgrounds. The closure of the W-algebra requires additional generators that we identify. We prove that the chiral anomalies of all these symmetries are consistent at one-loop in perturbation theory. We also demonstrate that these anomalies cancel at the same loop level either by adding suitable finite local counterterms in the sigma model effective action or by assuming a plausible quantum correction to the transformations. Such a correction is compatible with both the cancellation mechanism for spacetime frame rotation and gauge transformation anomalies, and the correction to the heterotic supergravity up to two loops in the sigma model perturbation theory.

Restrictions of Pfaffian systems for Feynman integrals

This work studies limits of Pfaffian systems, a class of first-order PDEs appearing in the Feynman integral calculus. Such limits appear naturally in the context of scattering amplitudes when there is a separation of scale in a given set of kinematic variables. We model these limits, which are often singular, via restrictions of D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{D} $$\\end{document}-modules. We thereby develop two different restriction algorithms: one based on gauge transformations, and another relying on the Macaulay matrix. These algorithms output Pfaffian systems containing fewer variables and of smaller rank. We show that it is also possible to retain logarithmic corrections in the limiting variable. The algorithms are showcased in many examples involving Feynman integrals and hypergeometric functions coming from GKZ systems. This work serves as a continuation of [1].

Soft particles and infinite-dimensional geometry

In the sigma model, soft insertions of moduli scalars enact parallel transport of S-matrix elements about the finite-dimensional moduli space of vacua, and the antisymmetric double-soft theorem calculates the curvature of the vacuum manifold. We explore the analogs of these statements in gauge theory and gravity in asymptotically flat spacetimes, where the relevant moduli spaces are infinite-dimensional. These models have spaces of vacua parameterized by (trivial) flat connections on the celestial sphere, and soft insertions of photons, gluons, and gravitons parallel transport S-matrix elements about these infinite-dimensional manifolds. We argue that the antisymmetric double-soft gluon theorem in d + 2 bulk dimensions computes the curvature of a connection on the infinite-dimensional space Map, where G is the global part of the gauge group. The analogous metrics in abelian gauge theory and gravity are flat, as indicated by the vanishing of the antisymmetric double-soft theorems in those models. In other words, Feynman diagram calculations not only know about the vacuum manifold of Yang–Mills theory, they can also be used to compute its curvature. The results have interesting implications for flat space holography.

A new technique to incorporate multiple fermion flavors in tensor renormalization group method for lattice gauge theories

We propose a new technique to incorporate multiple fermion flavors in the tensor renormalization group method for lattice gauge theories, where fermions are treated by the Grassmann tensor network formalism. The basic idea is to separate the site tensor into multiple layers associated with each flavor and to introduce the gauge field in each layer as replicas, which are all identified later. This formulation, after introducing an appropriate compression scheme in the network, enables us to reduce the size of the initial tensor with high efficiency compared with a naive implementation. The usefulness of this formulation is demonstrated by investigating the chiral phase transition and the Silver Blaze phenomenon in 2D Abelian gauge theories with Nf flavors of Wilson fermions up to Nf = 4.

Extended phase space in general gauge theories

In a recent paper, it was shown that in diffeomorphism-invariant theories, the symplectic vector fields induced by spacetime diffeomorphisms are integrable if one introduces an extended phase space. In this paper we extend the notion of extended phase space to all gauge theories with arbitrary combinations of internal and spacetime local symmetries. We formulate this in terms of a corresponding Atiyah Lie algebroid, a geometric object derived from a principal bundle which features internal symmetries and diffeomorphisms on an equal footing. In this language, gauge transformations are understood as morphisms between Atiyah Lie algebroids that preserve the geometric structures encoded therein. The extended configuration space of a gauge theory can subsequently be understood as the space of pairs (φ,Φ), where φ is a Lie algebroid morphism and Φ is a field configuration in the non-extended sense. Starting from this data, we outline a very powerful, manifestly geometric approach to the extended phase space. Using this approach, we find that the action of the group of gauge transformations and diffeomorphisms on the symplectic geometry of any covariant theory is integrable. We motivate our construction by carefully examining the need for extended phase space in Chern-Simons gauge theories and display its usefulness by re-computing the charge algebra. We also describe the implementation of the configuration algebroid in Einstein-Yang-Mills theories.

A generalized 4d Chern-Simons theory

A generalization of the 4d Chern-Simons theory action introduced by Costello and Yamazaki is presented. We apply general arguments from symplectic geometry concerning the Hamiltonian action of a symmetry group on the space of gauge connections defined on a 4d manifold and construct an action functional that is quadratic in the moment map associated to the group action. The generalization relies on the use of contact 1-forms defined on non-trivial circle bundles over Riemann surfaces and mimics closely the approach used by Beasley and Witten to reformulate conventional 3d Chern-Simons theories on Seifert manifolds. We also show that the path integral of the generalized theory associated to integrable field theories of the PCM type, takes the canonical form of a symplectic integral over a subspace of the space of gauge connections, turning it a potential candidate for using the method of non-Abelian localization. Alternatively, this new quadratic completion of the 4d Chern-Simons theory can also be deduced in an intuitive way from manipulations similar to those used in T-duality. Further details on how to recover the original 4d Chern-Simons theory data, from the point of view of the Hamiltonian formalism applied to the generalized theory, are included as well.

Spin effects from Four Bosons EM

Electromagnetism is the theory of electric charge and spin. Our study is on a spin-valued four bosons electromag- netism. An EM under the charge exchange {+, 0, −} intermediated by four bosons {AμI } ≡ {Aμ, Uμ, V ± μ } where Aμ means the usual photon, Uμ a massive photon, V ± μ charged photons. EM should express electric charge and spin together. Understand from first principles on the spin role in the electric and magnetic properties of particles. Theoretically, the spin is a space-time physical entity derived from Lorentz group. Phenomenologically, it appears as a vectorial entity inserted in the magnetic moment and electric dipole. A theoretical closure between them is expected. A spin-valued four bosons EM is constituted by introducing Lorentz group Lie Algebra valued fields. Consider the quadruplet fields as A I μ = A I μ,κλ(Σκλ)αβ where (Σκλ)αβ is the Lorentz generator. It provides spin as an intrinsec entity compatible with relativity and group theory. Similarly to the non-abelian gauge theory, where Aμa = Aμata, one incorporates the spin valued field through a Lie algebra. From first principles. Electric charge and spin are unified under a constructivist Lagrangian. Spin effects arestudied through equations of motion and Bianchi identities. Enlarging the EM for interactions beyond electric charge. Four types are derived. Usual electric charge interaction, neutral interaction, electric charge and spin, neutral and spin. A formalism is expressed. The spin valued performance is related through a Lagrangian. Spin interactions are derived. The magnetic moment and electric dipole are expressed by vectors S~ and ~s respectivity. They are able to couple spin with granular and collective fields strengths. Developing interacting terms constitutive as B~ · S~, E~ · ~s, ~e · S~ and so on. Faraday interaction between magnetic field and photon is reproduced from first principles.

Lump solutions for a (2+1)-dimensional generalized KP-type equation

A new ([Formula: see text])-dimensional generalized KP-type equation and its Lax pair are established. Based on gauge transformation between spectral problems, a Darboux transformation for the ([Formula: see text])-dimensional generalized KP-type equation is obtained. Some exact solutions are derived by utilizing the formula of Darboux transformation, including soliton solutions, trigonometric function solution, lump solutions, lump-soliton solution. In order to analyze the dynamical behavior of the solutions, the plots of solutions are depicted by Mathematical software.

Modelling coupled electro-mechanical phenomena in elastic dielectrics using local conformal symmetry

The local scaling symmetry of the Lagrange density is exploited to investigate the myriad electro-mechanical coupling effects observed in the elastic dielectrics. In contrast to most known flexoelectricity theories, this approach has also explicated on the geometric underpinnings of the induced polarization and electric potential. Due to the inhomogeneous scaling of the metric, the invariance of the Lagrange density will be lost. To restore the gauge invariance of the Lagrange density, we introduce the notion of minimal replacement, viz., define a gauge covariant operator in place of the ordinary partial derivative. Minimal replacement introduces gauge compensating one form field. We relate the exact part of the gauge compensating one form field with electric potential. However, the gauge compensating one form field's anti-exact part is correlated with the electric polarization vector. We constructed different components of the gauge invariant energy density using the geometric objects like scale invariant gauge curvature. We appropriately contracted the skew symmetric gauge curvature tensor with the metric to construct the scale invariant energy associated with the gauge field. Finally, we derive the governing coupled equations employing Hamilton's principle. In order to assess how the theory performs, we carry out a few numerical simulations and validation. Model prediction of the pressurized flexoelectric disk with a central hole shows good agreement with the analytical solution. We also investigated flexoelectric cantilever beam subjected to a point load and different earthing conditions. Explorations of this kind of coupling may have notable implications in many industrial and laboratory applications.

Integrable coupled massive Thirring model with field values in a Grassmann algebra

A coupled massive Thirring model of two interacting Dirac spinors in 1 + 1 dimensions with fields taking values in a Grassmann algebra is introduced, which is closely related to a SU(1) version of the Grassmannian Thirring model also introduced in this work. The Lax pair for the system is constructed, and its equations of motion are obtained from a zero curvature condition. It is shown that the system possesses several infinite hierarchies of conserved quantities, which strongly confirms its integrability. The model admits a canonical formulation and is invariant under space-time translations, Lorentz boosts and global U(1) gauge transformations, as well as discrete symmetries like parity and time reversal. The conserved quantities associated to the continuous symmetries are derived using Noether’s theorem, and their relation to the lower-order integrals of motion is spelled out. New nonlocal integrable models are constructed through consistent nonlocal reductions between the field components of the general model. The Lagrangian, the Hamiltonian, the Lax pair and several infinite hierarchies of conserved quantities for each of these nonlocal models are obtained substituting its reduction in the expressions of the analogous quantities for the general model. It is shown that, although the Lorentz symmetry of the general model breaks down for its nonlocal reductions, these reductions remain invariant under parity, time reversal, global U(1) gauge transformations and space-time translations.

Particle realization of Bondi-Metzner-Sachs symmetry in 2 + 1 space-time

We construct a Lorentz invariant massive particle model in (2+1) space-time with an enlarged set of symmetries which includes Bondi-Metzner-Sachs (BMS) translations (supertranslations), using the non-linear realization framework. The Hamiltonian formalism for the resulting Lagrangian is constructed, and the infinite phase-space constraints and the set of gauge transformations are analysed. We also compute the massless limit of the theory in phase-space. After eliminating the gauge degrees of freedom, the physical reduced space is left only with the degrees of freedom of a standard Poincaré particle but with a residual set of symmetries that we prove to be BMS. A similar result for the massless limit, including in this case superrotations, is pointed out.

A new method that automatically regularizes scattering amplitudes

We present a new regularization procedure called autoregularization. The new procedure regularizes the divergences, encountered previously in a scattering process, using the intrinsic scale of the process. We use autoregularization to calculate the amplitudes of several scattering processes in QED and compare the calculations with experimental measurements over a broad range of center-of-momentum energies (≲MeV to ≳200 GeV). The calculated amplitudes are found to be in good agreement with experimental data. 1 1 Specifically, the O(α) correction to electron’s gyromagnetic ratio predicted by autoregularization agrees with experimental measurement to within 0.06% (Section ), which is to be compared to Schwinger’s O(α) correction which agrees with experimental measurement to within 0.15%; the O(α) estimate of the Lamb shift predicted by autoregularization agrees with the experimental measurements to within 0.33% (section ); the running fine structure constant predicted by autoregularization at O(α) agrees with the prediction of cutoff regularization to within 0.8% over one to four orders of magnitude above the electron’s mass scale (section ); the tree-level prediction of autoregularization for Compton scattering is in better agreement with experimental data than the prediction of the well-known Klein-Nishina formula by about 4.02% (section ); the tree-level predicition of autoregularization for pair annihilation at center-of-momentum energy of 206.671 GeV agrees with the experimental data about 0.67 % better than the prediction of the standard QED (section ). To test autoregularization in a non-Abelian gauge theory, we calculate the QCD coupling constant at 1-loop and show that, like the known regularization schemes, autoregularization also predicts asymptotic freedom in QCD. Finally, we show that the vacuum energy density of the free fields in the Standard Model, calculated using autoregularization, is smaller than the current estimate of the cosmic critical density.

A GAUGE-INVARIANT LAGRANGIAN DETERMINED BY THE <i>n</i>-POINT PROBABILITY DENSITY FUNCTION OF VORTICITY FIELD OF THE WAVE OPTICAL TURBULENCE

The geometry methods for Yang–Mills fields of the gauge transformations are applied to finding an invariant Lagrangian in fiber bundle of the configuration \(2d\) space \(X\) of the turbulent flow defined by the \(n\)-point probability density function \({{f}_{n}}\) (PDF). The two-dimensional wave optical turbulence is considered in the case of the inverse cascade of energy. The n-point PDF of the vorticity field satisfies the \({{f}_{n}}\)-equation from the Landgren–Monin–Novikov (LMN) hierarchy. The basic result reads: we construct the Lagrangian which is invariant under a subgroup \(H \subset G\) – the group of the gauge transformations in fiber bundles of the space X and the conserved currents.

Quantum covariant derivative: a tool for deriving adiabatic perturbation theory to all orders

The covariant derivative suitable for differentiating and parallel transporting tangent vectors and other geometric objects induced by a parameter-dependent adiabatic quantum eigenstate is introduced. It is proved to be covariant under gauge and coordinate transformations and compatible with the quantum geometric tensor. For a quantum system driven by a Hamiltonian depending on slowly-varying parameters , , the quantum covariant derivative is used to derive a recurrence relation that determines an asymptotic series for the wave function to all orders in . This adiabatic perturbation theory provides an efficient tool for calculating nonlinear response properties.

Quantum and Classical Spin-Network Algorithms for q-Deformed Kogut-Susskind Gauge Theories.

Treating the infinite-dimensional Hilbert space of non-Abelian gauge theories is an outstanding challenge for classical and quantum simulations. Here, we employ q-deformed Kogut-Susskind lattice gauge theories, obtained by deforming the defining symmetry algebra to a quantum group. In contrast to other formulations, this approach simultaneously provides a controlled regularization of the infinite-dimensional local Hilbert space while preserving essential symmetry-related properties. This enables the development of both quantum as well as quantum-inspired classical spin-network algorithms for q-deformed gauge theories. To be explicit, we focus on SU(2)_{k} gauge theories with k∈N that are controlled by the deformation parameter q=e^{2πi/(k+2)}, a root of unity, and converge to the standard SU(2) Kogut-Susskind model as k→∞. In particular, we demonstrate that this formulation is well suited for efficient tensor network representations by variational ground-state simulations in 2D, providing first evidence that the continuum limit can be reached with k=O(10). Finally, we develop a scalable quantum algorithm for Trotterized real-time evolution by analytically diagonalizing the SU(2)_{k} plaquette interactions. Our work gives a new perspective for the application of tensor network methods to high-energy physics and paves the way for quantum simulations of non-Abelian gauge theories far from equilibrium where no other methods are currently available.

Yang–Mills–Stueckelberg theories, framing and local breaking of symmetries

We consider Yang–Mills theory with a compact structure group [Formula: see text] on a Lorentzian 4-manifold [Formula: see text] such that gauge transformations become identity on a submanifold [Formula: see text] of [Formula: see text] (framing over [Formula: see text]). The space [Formula: see text] is not necessarily a boundary of [Formula: see text] and can have dimension [Formula: see text]. Framing of gauge bundles over [Formula: see text] demands introduction of a [Formula: see text]-valued function [Formula: see text] with support on [Formula: see text] and modification of Yang–Mills equations along [Formula: see text]. The fields [Formula: see text] parametrize non-equivalent flat connections mapped into each other by a dynamical group [Formula: see text] changing gauge frames over [Formula: see text]. It is shown that the charged condensate [Formula: see text] is the Stueckelberg field generating an effective mass of gluons in the domain [Formula: see text] of space [Formula: see text] and keeping them massless outside [Formula: see text]. We argue that the local Stueckelberg field [Formula: see text] can be responsible for color confinement. We also briefly discuss local breaking of symmetries in gravity. It is shown that framing of the tangent bundle over a subspace of spacetime makes gravitons massive in this subspace.

Geometric tool kit for higher spin gravity (Part II): An introduction to Lie algebroids and their enveloping algebras

These notes provide a self-contained introduction to Lie algebroids, Lie-Rinehart algebras and their universal envelopes. This review is motivated by the speculation that higher-spin gauge symmetries should admit a natural formulation as enveloping algebras of Lie algebroids since rigid higher-spin algebras are enveloping algebras of Lie algebras. Nevertheless, the material covered here may be of general interest to anyone interested in the description of gauge symmetries, connections and covariant derivatives, in terms of Lie algebroids. In order to be self-contained, a concise introduction to the algebraic characterization of vector bundles as projective modules over the algebra of functions on the base manifold is provided.

S-matrix path integral approach to symmetries and soft theorems

We explore a formulation of the S-matrix in terms of the path integral with specified asymptotic data, as originally proposed by Arefeva, Faddeev, and Slavnov. In the tree approximation the S-matrix is equal to the exponential of the classical action evaluated on-shell. This formulation is well-suited to questions involving asymptotic symmetries, as it avoids reference to non-gauge/diffeomorphism invariant bulk correlators or sources at intermediate stages. We show that the soft photon theorem, originally derived by Weinberg and more recently connected to asymptotic symmetries by Strominger and collaborators, follows rather simply from invariance of the action under large gauge transformations applied to the asymptotic data. We also show that this formalism allows for efficient computation of the S-matrix in curved spacetime, including particle production due to a time dependent metric.

Darboux transformation and explicit solutions for the Kaup-Kupershmidt equation

In this paper, the N-fold Darboux transformation of the Kaup-Kupershmidt equation associated with 3×3 order spectral problem is constructed by means of the gauge transformation. By applying the Darboux transformation, some soliton solutions of this equation are derived.