Generators Of Gauge Transformations Research Articles

A note on the Hamiltonian structure of transgression forms

By incorporating two gauge connections, transgression forms provide a generalization of Chern-Simons actions that are genuinely gauge-invariant on bounded manifolds. In this work, we show that, when defined on a manifold with a boundary, the Hamiltonian formulation of a transgression field theory can be consistently carried out without the need to implement regularizing boundary terms at the level of first-class constraints. By considering boundary variations of the relevant functionals in the Poisson brackets, the surface integral in the very definition of a transgression action can be translated into boundary contributions in the generators of gauge transformations and diffeomorphisms. This prescription systematically leads to the corresponding surface charges of the theory, reducing to the general expression for conserved charges in (higher-dimensional) Chern-Simons theories when one of the gauge connections in the transgression form is set to zero.

Stability and moduli space of generalized Ricci solitons

The generalized Einstein–Hilbert action is an extension of the classic scalar curvature energy and Perelman’s F-functional which incorporates a closed three-form. The critical points are known as generalized Ricci solitons, which arise naturally in mathematical physics, complex geometry, and generalized geometry. Through a delicate analysis of the group of generalized gauge transformations, and implementing a novel connection, we give a simple formula for the second variation of this energy which generalizes the Lichnerowicz operator in the Einstein case. As an application we show that all Bismut-flat manifolds are linearly stable critical points, and admit nontrivial deformations arising from Lie theory. Furthermore, this leads to extensions of classic results of Koiso [1–4] and Podesta, Spiro, Kröncke [5–8] to the moduli space of generalized Ricci solitons. To finish we classify deformations of the Bismut-flat structure on S3 and show that some are integrable while others are not.

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.

Action for N D0-branes invariant under gauged Galilean transformations

In this short note we formulate an action for N D0-branes that is manifestly invariant under gauged Galilean transformations. We also find its canonical form and determine first class constraints that are generators of gauge transformations.

Generalized gauge transformation and the corresponding Hermitian counterparts of SU(1, 1), SU(2) pseudo-Hermitian Hamiltonians

We study in this paper both the stationary and time-dependent pseudo-Hermitian Hamiltonians consisting respectively of SU(1, 1), SU(2) generators. The pseudo-Hermitian Hamiltonians can be generated from kernel Hermitian-Hamiltonians by a generalized gauge transformation with a non-unitary but Hermitian operator. The metric operator of the biorthogonal sets of eigenstates is simply the square of the transformation operator, which is formulated explicitly. The exact solutions of pseudo-Hermitian Hamiltonians are obtained in terms of the eigenststates of the Hermitian counterparts. We observe a critical point G c of coupling constant, where all eigenstates of the stationary Hamiltonians are degenerate with a vanishing eigenvalue. This critical point is modified as G c (ω) in the time-dependent case including the frequency of external field. Returning to the original gauge we obtain analytically the wave functions and associated non-adiabatic Berry phase, which diverges at the critical point for the SU(2)Hamiltonian. Beyond the critical point Berry phase becomes a complex domain.

The Significance of Generalized Gauge Transformation across Fundamental Interactions

The author of this paper has put forward a unified program of gauge field from the mathematical and physical picture of the principal associated bundles: thinking that our universe may have more fundamental interactions than the four fundamental interactions, and these basic interaction gauge fields are only the projection components to the base manifold, that is our universe, from a unified gauge potential or connection of the principal associated bundle manifold on the base manifold. These components can satisfy the transformation of gauge potential, and can even be transformed from one basic interaction gauge potential to another basic interaction gauge potential, and can be summarized into a unified equation, that is, the generalized gauge Equation (GGE), but the gauge potential or connection on the principal bundle is invariant, corresponding to the invariance of gauge transformation [1]. In this paper, we will continue to discuss this aspect concretely, and specifically construct a spatiotemporal model with the frame bundle as the principal bundle, and the tensor bundle as the associated bundle, so that the four fundamental interactions, especially the electromagnetic interaction and the gravitational interaction, can be reflected in the bottom manifold, that is, the regional distributions in our universe. Furthermore, this paper studies the existence of gauge transformation across basic interactions by establishing a model of gauge transformation of basic interaction field; it is found that the unified expression formula is GGE and the expression relation on the curvature of space-time. Therefore, the author discusses the feasibility of the generalized gauge transformation across the basic electromagnetic interaction and the basic gravitational interaction, and on this basis, specifically determines a method or way to find the generalized gauge transformation, so as to try to realize the last step of the “unification” of the four fundamental interactions in physics, that is, the “unification” of electromagnetism and gravity.

Twisted C‐Brackets

We consider the double field formulation of the closed bosonic string theory, and calculate the Poisson bracket algebra of the symmetry generators governing both general coordinate and local gauge transformations. Parameters of both of these symmetries depend on a double coordinate, defined as a direct sum of the initial and T-dual coordinate. When no antisymmetric field is present, the C-bracket appears as the Lie bracket generalization in a double theory. With the introduction of the Kalb-Ramond field, the B-twisted C-bracket appears, while with the introduction of the non-commutativity parameter, the θ-twisted C-bracket appears. We present the derivation of these brackets and comment on their relations to analogous twisted Courant brackets and T-duality.

Gauge transformations of two-component KP and mKP hierarchies

Based on generalized Lax equations and the Sato theory on KP and mKP hierarchies derived from the basic algebra of pseudo-differential operators, Miura- and auto-Bäcklund transformations can be derived from the gauge transformations of the Lax operators. In this paper, we study the gauge transformations of two-component KP and mKP hierarchies using the two-component generalized gauge transformations of the original operators.

Superselection, boundary algebras, and duality in gauge theories

We consider the generators of gauge transformations with test functions which do not vanish on the boundary of a spacelike region of interest. These are known to generate the edge degrees of freedom in a gauge theory. In this paper, we augment these by introducing the dual or magnetic analog of such operators. We then study the algebra of these operators, focusing on implications for the superselection sectors of the gauge theory. A manifestly duality-invariant action is also considered, from which alternate descriptions which are $SL(2,\mathbb{Z})$ transforms of each other can be obtained. We also comment on a number of issues related to local charges, definition of confinement and the appearance of interesting mathematical structures such as the Drinfel'd double and the Manin triple.

Generalized Gauge Transformation with PT$PT$‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System

AbstractThis paper demonstrates that the parity‐time ()‐symmetric non‐Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time‐dependent transformation operator has to be symmetric and non‐unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of the non‐Hermitian Hamiltonian. The wave functions and associated non‐adiabatic Berry phase for the nth eigenstate are obtained analytically. The classical version of the non‐Hermitian Hamiltonian becomes a complex function of canonical variables and time. The corresponding kernel Hamiltonian is derived with symmetric canonical‐variable transfer in the classical gauge transformation. Moreover, with the change of position‐momentum to angle‐action variables it is revealed that the non‐adiabatic Hannay's angle and Berry phase satisfy precisely the quantum‐classical correspondence, .

The spatial gauge-dependence of single-field inflationary bispectra

In single-field inflationary models the bispectra are usually given in the ζ-gauge, because its temporal part leads to the super-horizon conservation of fluctuations. However, this property is independent of the choice of spatial gauge, so in this letter we explore this freedom. We compute the variation of the bispectra under the most general spatial gauge transformation that is globally defined and privileges no point, direction or scale. In the squeezed configuration we then obtain a generalization of the classic ζ-gauge consistency relations, which we also derive through the ‘large diffeomorphism’ approach for all four bispectra. At leading order in the long wave-number the transformation only affects the case where the long mode is a scalar. The first effect is a shift of the tilt factor, so that one can significantly reduce the amplitude of that contribution. Secondly, there is now an extra term depending on the triangle shape, the same as in solid inflation, which is due to the fact that the 3-metric has a scalar anisotropy in generic spatial gauge. At next-to-leading order there is no variation, so the conformal consistency relations of the ζ-gauge are preserved.

Color-kinematics relation from the Feynman diagram perspective

Feynman diagrams for gluon tree amplitudes are studied in the Feynman gauge and in any number of spacetime dimensions. The color-kinematics combinations $\mathrm{\ensuremath{\Delta}}={n}_{s}\ensuremath{-}{n}_{t}\ensuremath{-}{n}_{u}$ of numerators are explicitly calculated for $N=4$, 5, 6 gluons to see whether the color-kinematics relation $\mathrm{\ensuremath{\Delta}}=0$ is satisfied. This is a tedious task because of the presence of four-gluon vertices, and the large number of Feynman diagrams, numerators, and $\mathrm{\ensuremath{\Delta}}$ combinations involved, especially when $N=6$. For on-shell amplitudes, it is found that $\mathrm{\ensuremath{\Delta}}=0$ for $N=4$, but $\mathrm{\ensuremath{\Delta}}\ensuremath{\ne}0$ for $N=5$ and $N=6$ owing to the presence of the four-gluon vertex. However, a local generalized gauge transformation can bring about $\mathrm{\ensuremath{\Delta}}=0$ for $N=5$, but not for $N=6$. This raises the question whether gluon amplitudes satisfying the color-kinematics relation contain nonlocal interactions.

Generalized Gauge Transformation with Non-Unitary Operator and Realization of Hermitian Counterparts of Time-Dependent Pseudo-Hermitian Hamiltonians

Generalized Gauge Transformation with Non-Unitary Operator and Realization of Hermitian Counterparts of Time-Dependent Pseudo-Hermitian Hamiltonians

Note on the bundle geometry of field space, variational connections, the dressing field method, & presymplectic structures of gauge theories over bounded regions

In this note, we consider how the bundle geometry of field space interplays with the covariant phase space methods so as to allow to write results of some generality on the presymplectic structure of invariant gauge theories coupled to matter. We obtain in particular the generic form of Noether charges associated with field-independent and field-dependent gauge parameters, as well as their Poisson bracket. We also provide the general field-dependent gauge transformations of the presymplectic potential and 2-form, which clearly highlights the problem posed by boundaries in generic situations. We then conduct a comparative analysis of two strategies recently considered to evade the boundary problem and associate a modified symplectic structure to a gauge theory over a bounded region: namely the use of edge modes on the one hand, and of variational connections on the other. To do so, we first try to give the clearest geometric account of both, showing in particular that edge modes are a special case of a differential geometric tool of gauge symmetry reduction known as the “dressing field method”. Applications to Yang-Mills theory and General Relativity reproduce or generalise several results of the recent literature.

On the double copy for spinning matter

We explore various tree-level double copy constructions for amplitudes including massive particles with spin. By working in general dimensions, we use that particles with spins s ≤ 2 are fundamental to argue that the corresponding double copy relations partially follow from compactification of their massless counterparts. This massless origin fixes the coupling of gluons, dilatons and axions to matter in a characteristic way (for instance fixing the gyromagnetic ratio), whereas the graviton couples universally reflecting the equivalence principle. For spin-1 matter we conjecture all-order Lagrangians reproducing the interactions with up to two massive lines and we test them in a classical setup, where the massive lines represent spinning compact objects such as black holes. We also test the amplitudes via CHY formulae for both bosonic and fermionic integrands. At five points, we show that by applying generalized gauge transformations one can obtain a smooth transition from quantum to classical BCJ double copy relations for radiation, thereby providing a QFT derivation for the latter. As an application, we show how the theory arising in the classical double copy of Goldberger and Ridgway can be naturally identified with a certain compactification of mathcal{N} = 4 Supergravity.

Three-Loop Color-Kinematics Duality: A 24-Dimensional Solution Space Induced by New Generalized Gauge Transformations.

We obtain full-color three-loop, three-point form factors of the stress-tensor supermultiplet and also of a length-3 half-Bogomol'nyi-Prasad-Sommerfield operator in N=4 supersymmetric Yang-Mills (SYM) theory based on the color-kinematics duality and on-shell unitarity. The integrand results are verified by all planar and nonplanar unitarity cuts, and they satisfy the minimal power counting of loop momenta and diagrammatic symmetries. Interestingly, these three-loop solutions, while manifesting all dual Jacobi relations, contain a large number of free parameters; in particular, there are 24 free parameters for the form factor of the stress-tensor supermultiplet. Such degrees of freedom are due to a new type of generalized gauge transformation associated with the operator insertion for form factors. We also perform numerical integration and obtain consistent full-color infrared divergences and the known planar remainder. The form factors we obtain can be understood as the N=4 SYM counterparts of three-loop Higgs plus three-gluon amplitudes in QCD and are expected to provide the maximally transcendental parts of the latter.

Bundle geometry of the connection space, covariant Hamiltonian formalism, the problem of boundaries in gauge theories, and the dressing field method

We take advantage of the principal bundle geometry of the space of connections to obtain general results on the presymplectic structure of two classes of (pure) gauge theories: invariant theories, and non-invariant theories satisfying two restricting hypothesis. In particular, we derive the general field-dependent gauge transformations of the presymplectic potential and presymplectic 2-form in both cases. We point-out that a generalisation of the standard bundle geometry, called twisted geometry, arises naturally in the study of non-invariant gauge theories (e.g. non-Abelian Chern-Simons theory). These results prove that the well-known problem of associating a symplectic structure to a gauge theory over bounded regions is a generic feature of both classes. The edge modes strategy, recently introduced to address this issue, has been actively developed in various contexts by several authors. We draw attention to the dressing field method as the geometric framework underpinning, or rather encompassing, this strategy. The geometric insight afforded by the method both clarifies it and clearly delineates its potential shortcomings as well as its conditions of success. Applying our general framework to various examples allows to straightforwardly recover several results of the recent literature on edge modes and on the presymplectic structure of general relativity.

Study of gauge symmetry in the velocity phase space through Lagrangian formulation: some case studies

Gauge symmetry of some models is investigated through Lagrangian formulation in the velocity phase space. It is found that the Lagrangian formulation is a useful instrument to verify whether a given model has gauge symmetry or not. It can derive the complete set of gauge transformation generators of a Lagrangian. The prescription based on Lagrangian formulation has ability to show a direction towards the extension of phase space introducing auxiliary fields to restore the gauge symmetry. It has been found that this prescription works well in the usual phase space as well as extended phase space.

More on the classical double copy in three spacetime dimensions

It is well-known that General Relativity (GR) in three spacetime dimensions (3D) has no well-defined Newtonian limit. Recently, a static solution mimicking the behaviour of the expected Newtonian potential has been found in arXiv:1904.11001 by studying the classical double copy of a point charge in gauge theory. This is the first example where the vacuum solution in the gauge theory leads to a non-vacuum solution on the gravity side. The resulting energy-momentum tensor was attributed to a free scalar ghost field; however, alternatively, the source can be seen as one resulting from a spacelike perfect fluid. In this paper, we first give an alternative derivation of the solution where there is no need to perform a generalized gauge transformation to obtain a quadratic Lagrangian without propagating ghost fields. Then, we present a stationary version of the solution and show that the scalar field interpretation of the source does not survive in this case, leaving the spacelike fluid as the only possibility. We give the gauge theory single copy of our solution and comment on the implications of our results on the validity of the classical double copy in 3D. The effect of the cosmological constant is also discussed.

Covariant momentum map for non-Abelian topological BF field theory

We analyze the inherent symmetries associated to the non-Abelian topological BF theory from the geometric and covariant perspectives of the Lagrangian and the multisymplectic formalisms. At the Lagrangian level, we classify the symmetries of the theory as natural and Noether symmetries and construct the associated Noether currents, while at the multisymplectic level the symmetries of the theory arise as covariant canonical transformations. These transformations allowed us to build within the multisymplectic approach, in a complete covariant way, the momentum maps which are analogous to the conserved Noether currents. The covariant momentum maps are fundamental to recover, after the space plus time decomposition of the background manifold, not only the extended Hamiltonian of the BF theory but also the generators of the gauge transformations which arise in the instantaneous Dirac–Hamiltonian analysis of the first-class constraint structure that characterizes the BF model under study. To the best of our knowledge, this is the first non-trivial physical model associated to general relativity for which both natural and Noether symmetries have been analyzed at the multisymplectic level. Our study shed some light on the understanding of the manner in which the generators of gauge transformations may be recovered from the multisymplectic formalism for field theory.