Internal Gauge Symmetries Research Articles

Non-trivial thick brane realisations with 3-forms

We explore the construction of four-dimensional thick branes supported by massless 3-forms in a five-dimensional bulk space. The required residual Poincaré symmetry on the brane is realised as a combination of the bulk symmetries and the internal gauge symmetry of the 3-form. We show the presence of a gradient instability for the perturbations orthogonal to the brane when its profile decays slowly in the asymptotic regions. In particular, we ascertain that the prevailing profiles found in the literature are susceptible to such instabilities. We confirm our results by transitioning to the dual formulation in terms of a shift-symmetric scalar. In this formulation, the residual Poincaré group is trivially realised without internal symmetries, and there is a diagonal translational invariance in the fifth dimension reminiscent of the invariance under translations orthogonal to the brane of the dual 3-form. We demonstrate the extension of our results to the cases of de Sitter and Anti de Sitter branes where the gradient instabilities of asymptotically slowly decaying branes persist. Finally, we briefly comment on the construction of thick branes with massive 3-forms and their 1-form duals.

Backgrounds in gravitational effective field theory

Effective field theories describing gravity coupled to matter are investigated, allowing for operators of arbitrary mass dimension. Terms violating local Lorentz and diffeomorphism invariance while preserving internal gauge symmetries are included. The theoretical framework for violations of local Lorentz and diffeomorphism invariance and associated conceptual issues are discussed, including transformations in curved and approximately flat spacetimes, the treatment of various types of backgrounds, the implications of symmetry breaking, and the no-go constraints for explicit violation in Riemann geometry. Techniques are presented for the construction of effective operators, and the possible terms in the gravity, gauge, fermion, and scalar sectors are classified and enumerated. Explicit expressions are obtained for terms containing operators of mass dimension six or less in the effective action for General Relativity coupled to the Standard Model of particle physics. Special cases considered include Einstein-Maxwell effective field theories and the limit with only scalar coupling constants.

The gauge symmetries of gravity with torsion in the Cartan formalism

First-order general relativity in n dimensions () has an internal gauge symmetry that is the higher-dimensional generalization of three-dimensional local translations. We report the extension of this symmetry for n-dimensional gravity with torsion in the Cartan formalism. The new symmetry arises from the direct application of the converse of Noether’s second theorem to the action principle of gravity with torsion. We show that infinitesimal diffeomorphisms can be written as a linear combination of the new internal gauge symmetry, local Lorentz transformations, and terms proportional to the variational derivatives of the action. It means that the new internal symmetry together with local Lorentz transformations can be used to describe the full gauge symmetry of gravity with torsion, and thus diffeomorphisms become a derived symmetry in this setting.

Metric formulation of the simple theory of 3d massive gravity

We have recently introduced a new and very simple action for three-dimensional massive gravity. This action is written in a first order formulation where the triad and the connection play a manifestly symmetric role, but where internal Lorentz gauge symmetry is broken. The absence of Lorentz invariance, which in this model is the mechanism underlying the propagation of a massive graviton, does however prevent from writing a purely metric non-linear action for the theory. Nonetheless, in this letter, we explain how to disentangle, at the non-linear level, the metric and non-metric degrees of freedom in the equations of motion. Focusing on the metric part, we show that it satisfies modified Einstein equations with higher derivative terms. As a particular case, these equations reproduce a well-studied model known as minimal massive gravity. In the general case, we obtain new metric field equations for massive gravity in three dimensions starting from the simple first order action. These field equations are consistent through a mechanism known as "third way consistency", which our theory therefore provides a new example of.

Hilbert series and plethystics: paving the path towards 2HDM- and MLRSM-EFT

Effective Field Theory (EFT) technique is one of the most elegant ways to capture the impact of high scale theory, if any, at some low energy by incorporating higher mass dimensional (≥5) effective operators ( mathcal{O} i). The low energy EFT is described in terms of only light degrees of freedom (DOF) which can appear on-shell. An essential task while developing the EFT framework is to compute these mathcal{O} i’s. Hilbert Series (HS) is a novel and mathematically robust method to construct the complete set of gauge invariant independent, effective operators. The HS requires the knowledge of the transformation properties of the light DOF and the covariant derivatives under the internal gauge symmetries and conformal groups. The Hilbert Series method, by its virtue, automatically takes care of the redundancies in the operator set due to the Equations of Motion (EOMs) of fields and Integration by Parts (IBPs) with impeccable accuracy.In this paper, we have adopted this methodology to construct the complete set of independent operators up to dimension-6 in the “Warsaw”-like basis for two different Beyond Standard Model scenarios — Two Higgs Doublet Model (2HDM) and Minimal Left-Right Symmetric Model (MLRSM). For both these cases, we have calculated the corrections to the scalar, gauge boson and fermion mass spectra due to the dimension-6 operators. The additional contributions to all the Feynman vertices are computed and their impact on different observables, namely Weak mixing angle, Fermi constant, ρ and oblique (S, T, U) parameters. We have further discussed how the magnetic moments of charged leptons and production and decay of the massive BSM particles, e.g., charged scalar and different rare processes are affected in the presence of effective operators. We have also constructed the effective scalar four-point interactions and commented on the possible reinvestigation of the theoretical constraints, e.g., unitarity and vacuum stability within these frameworks.

What Are Observables in Hamiltonian Einstein–Maxwell Theory?

Is change missing in Hamiltonian Einstein-Maxwell theory? Given the most common definition of observables (having weakly vanishing Poisson bracket with each first-class constraint), observables are constants of the motion and nonlocal. Unfortunately this definition also implies that the observables for massive electromagnetism with gauge freedom (Stueckelberg) are inequivalent to those of massive electromagnetism without gauge freedom (Proca). The alternative Pons-Salisbury-Sundermeyer definition of observables, aiming for Hamiltonian-Lagrangian equivalence, uses the gauge generator G, a tuned sum of first-class constraints, rather than each first-class constraint separately, and implies equivalent observables for equivalent massive electromagnetisms. For General Relativity, G generates 4-dimensional Lie derivatives for solutions. The Lie derivative compares different space-time points with the same coordinate value in different coordinate systems, like 1 a.m. summer time vs. 1 a.m. standard time, so a vanishing Lie derivative implies constancy rather than covariance. Requiring equivalent observables for equivalent formulations of massive gravity confirms that G must generate the 4-dimensional Lie derivative (not 0) for observables. These separate results indicate that observables are invariant under internal gauge symmetries but covariant under external gauge symmetries, but can this bifurcated definition work for mixed theories such as Einstein-Maxwell theory? Pons, Salisbury and Shepley have studied G for Einstein-Yang-Mills. For Einstein-Maxwell, both the electromagnetic field strength F and the metric g are invariant under electromagnetic gauge transformations and covariant (changing by a Lie derivative) under 4-dimensional coordinate transformations. Using the bifurcated definition, these quantities count as observables, as one would expect on non-Hamiltonian grounds.

What Are Observables in Hamiltonian Einstein\u2013Maxwell Theory?

Is change missing in Hamiltonian Einstein–Maxwell theory? Given the most common definition of observables (having weakly vanishing Poisson bracket with each first-class constraint), observables are constants of the motion and nonlocal. Unfortunately this definition also implies that the observables for massive electromagnetism with gauge freedom (‘Stueckelberg’) are inequivalent to those of massive electromagnetism without gauge freedom (‘Proca’). The alternative Pons–Salisbury–Sundermeyer definition of observables, aiming for Hamiltonian–Lagrangian equivalence, uses the gauge generator G, a tuned sum of first-class constraints, rather than each first-class constraint separately, and implies equivalent observables for equivalent massive electromagnetisms. For General Relativity, G generates 4-dimensional Lie derivatives for solutions. The Lie derivative compares different space-time points with the same coordinate value in different coordinate systems, like 1 a.m. summer time versus 1 a.m. standard time, so a vanishing Lie derivative implies constancy rather than covariance. Requiring equivalent observables for equivalent formulations of massive gravity confirms that G must generate the 4-dimensional Lie derivative (not 0) for observables. These separate results indicate that observables are invariant under internal gauge symmetries but covariant under external gauge symmetries, but can this bifurcated definition work for mixed theories such as Einstein–Maxwell theory? Pons, Salisbury and Shepley have studied G for Einstein–Yang–Mills. For Einstein–Maxwell, both F_{mu nu } and g_{mu nu } are invariant under electromagnetic gauge transformations and covariant (changing by a Lie derivative) under 4-dimensional coordinate transformations. Using the bifurcated definition, these quantities count as observables, as one would expect on non-Hamiltonian grounds.

Constraints on Symmetries from Holography.

In this Letter we show that a set of old conjectures about symmetries in quantum gravity hold within the anti-de Sitter/conformal field theory correspondence. These conjectures are that no global symmetries are possible, that internal gauge symmetries must come with dynamical objects that transform in all irreducible representations, and that internal gauge groups must be compact. These conjectures are not obviously true from a bulk perspective, they are nontrivial consequences of the nonperturbative consistency of the correspondence. More details of and background for these arguments are presented in an accompanying paper.

The gauge symmetries of first-order general relativity with matter fields

In n-dimensional spacetimes (n > 3), there exists an internal gauge symmetry of the Palatini action with a cosmological constant that is the natural generalization of the so-called ‘local translations’ of three-dimensional general relativity. We report the extension of this symmetry to include the minimal coupling of Yang–Mills and fermion fields to the Palatini action with a cosmological constant. We show that, as in the case of three-dimensional local translations, the extended symmetry depends on the energy–momentum tensor of the corresponding matter field and, for fermions, it contains an additional term that in four dimensions is proportional to the axial fermion current. We also report the extension of the analog of this internal gauge symmetry for the Holst action with a cosmological constant by incorporating minimally coupled scalar and Yang–Mills fields, as well as a non-minimally coupled fermion field. In the last case, the extended symmetry is affected by both the Immirzi parameter and the energy–momentum tensor and, for fermions, it also depends on the axial fermion current.

Searching for Supersolidity in Ultracold Atomic Bose Condensates with Rashba Spin-Orbit Coupling.

We developed a functional integral formulation for the stripe phase of spinor Bose-Einstein condensates with Rashba spin-orbit coupling. The excitation spectrum is found to exhibit double gapless band structures, identified to be two Goldstone modes resulting from spontaneously broken internal gauge symmetry and translational invariance symmetry. The sound velocities display anisotropic behavior with the lower branch vanishing in the direction perpendicular to the stripe in the x-y plane. At the transition point between the plane-wave phase and the stripe phase, physical quantities such as fluctuation correction to the ground-state energy and quantum depletion of the condensates exhibit discontinuity, characteristic of the first-order phase transition. Despite strong quantum fluctuations induced by Rashba spin-orbit coupling, we show that the supersolid phase is stable against quantum depletion. Finally, we extend our formulation to finite temperatures to account for interactions between excitations.

Neutrinos propagating in curved spacetimes

In the Dirac-Weyl equation that describes massless neutrino propagation in the minimal Standard Model, the ($2\times 2$ equivalence of the) gamma matrices convert Weyl spinors into spacetime tensors, and vice versa. They can thus be regarded as amalgamations of three different types of mappings, one that connects particle spinors directly forming representations to internal gauge symmetries to their spacetime counterparts that are embodied by null flags, another that translates the spacetime spinors into their corresponding tensors expressed in an orthonormal tetrad, and finally a purely tensorial transformation into the coordinate tetrad. The splitting of spinors into particle and spacetime varieties is not usually practised, but we advocate its adoption for better physical clarity, in terms of distinguishing internal and spacetime transformations, and also for understanding the scattering of neutrinos by spacetime curvature. We construct the basic infrastructure required for this task, and provide a worked example for the Schwarzschild spacetime. Our investigation also uncovers a possible under-determinacy in the flavoured Dirac-Weyl equation, which could serve as a new incision point for introducing flavour oscillation mechanisms.

Reformulation of the symmetries of first-order general relativity

We report a new internal gauge symmetry of the n-dimensional Palatini action with cosmological term () that is the generalization of three-dimensional local translations. This symmetry is obtained through the direct application of the converse of Noether’s second theorem on the theory under consideration. We show that diffeomorphisms can be expressed as linear combinations of it and local Lorentz transformations with field-dependent parameters up to terms involving the variational derivatives of the action. As a result, the new internal symmetry together with local Lorentz transformations can be adopted as the fundamental gauge symmetries of general relativity. Although their gauge algebra is open in general, it allows us to recover, without resorting to the equations of motion, the very well-known Lie algebra satisfied by translations and Lorentz transformations in three dimensions. We also report the analog of the new gauge symmetry for the Holst action with cosmological term, finding that it explicitly depends on the Immirzi parameter. The same result concerning its relation to diffeomorphisms and the open character of the gauge algebra also hold in this case. Finally, we consider the non-minimal coupling of a scalar field to gravity in n dimensions and establish that the new gauge symmetry is affected by this matter field. Our results indicate that general relativity in dimension greater than three can be thought of as a gauge theory.

Gravity as an SU(1, 1) gauge theory in four dimensions

We start with the Hamiltonian formulation of the first order action of pure gravity with a full internal gauge symmetry. We make a partial gauge-fixing which reduces to its sub-algebra . This case corresponds to a splitting of the space-time where inherits an arbitrary Lorentzian metric of signature (−, +, +). Then, we find a parametrization of the phase space in terms of an commutative connection and its associated conjugate electric field. Following the techniques of loop quantum gravity, we start the quantization of the theory and we consider the kinematical Hilbert space on a given fixed graph whose edges are colored with unitary representations of . We compute the spectrum of area operators acting on the kinematical Hilbert space: we show that space-like areas have discrete spectra, in agreement with usual loop quantum gravity, whereas time-like areas have continuous spectra. We conclude on the possibility to make use of this formulation of gravity to construct a holographic description of black holes in the framework of loop quantum gravity.

Dijkgraaf–Witten type invariants of Seifert surfaces in 3-manifolds

We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf–Witten theory. The appropriate initial data for the construction are certain three object categories, with coefficients satisfying a partially degenerate cocycle condition.

An alternative derivation of the Dirac operator generating intrinsic Lagrangian local gauge invariance

This paper introduces an alternative formalism for deriving the Dirac operator and equation. The use of this formalism concomitantly generates a separate operator coupled to the Dirac operator. When operating on a Clifford field, this coupled operator produces field components which are formally equivalent to the field components of Maxwell's electromagnetic field tensor. Consequently, the Lagrangian of the associated coupled field exhibits internal local gauge symmetry. The coupled field Lagrangian is seen to be equivalent to the Lagrangian of Quantum Electrodynamics.
 Received: 8 November 2016, Accepted: 4 January 2017; Edited by: D. Gomez Dumm; DOI: http://dx.doi.org/10.4279/PIP.090002
 Cite as: B J Wolk, Papers in Physics 9, 090002 (2017)
 This paper, by B J Wolk, is licensed under the Creative Commons Attribution License 3.0.

Constraining the physical state by symmetries

After reviewing the hole argument and its relations with initial value problem and general covariance, we shall discuss how much freedom one has to define the physical state in a generally covariant field theory (with or without internal gauge symmetries).Our analysis relies on Cauchy problems, thus it is restricted to globally hyperbolic spacetimes. We shall show that in generally covariant theories on a compact space (as well as for internal gauge symmetries on any spacetime) one has no freedom and one is forced to declare as physically equivalent two configurations which differ by a global spacetime diffeomorphism (or by an internal gauge transformation) as it is usually prescribed.On the contrary, when space is not compact, the result does not hold true and one may have different options to define physically equivalent configurations, still preserving determinism.

Solution phase space and conserved charges: A general formulation for charges associated with exact symmetries

We provide a general formulation for calculating conserved charges for solutions to generally covariant gravitational theories with possibly other internal gauge symmetries, in any dimensions and with generic asymptotic behaviors. These solutions are generically specified by a number of exact (continuous, global) symmetries and some parameters. We define "parametric variations" as field perturbations generated by variations of the solution parameters. Employing the covariant phase space method, we establish that the set of these solutions (up to pure gauge transformations) form a phase space, the \emph{solution phase space}, and that the tangent space of this phase space includes the parametric variations. We then compute conserved charge variations associated with the exact symmetries of the family of solutions, caused by parametric variations. Integrating the charge variations over a path in the solution phase space, we define the conserved charges. In particular, we revisit "black hole entropy as a conserved charge" and the derivation of the first law of black hole thermodynamics. We show that the solution phase space setting enables us to define black hole entropy by an integration over any compact, codminesion-2, smooth spacelike surface encircling the hole, as well as to a natural generalization of Wald and Iyer-Wald analysis to cases involving gauge fields.

Exact Space-Time Gauge Symmetry of Gravity, Its Couplings and Approximate Internal Symmetries in a Total-Unified Model

The gravitational field is the manifestation of space-time translational (T_4) gauge symmetry, which enables the gravitational interaction to be unified with the strong and the electroweak interactions. Such a total-unified model is based on a generalized Yang-Mills framework in flat space-time. Following the idea of Glashow-Salam-Ward-Weinberg, we gauge the groups T_4 × (SU_3)_(color) × SU_2 × U_1 × U_(1b) on equal-footing, so that we have the total-unified gauge covariant derivative δ_μ = ∂μ - igϕ _μ^νpv + ig_sG _μ^a (λ^a/2) + ifW_μ^kt^k + if'U_μt_o + ig_bB_μ. The generators of the external T_4 group have the representation p_μ = i∂_μ, which differs from the other generators of all the other internal groups, which have constant matrix representations. Consequently, the total-unified model leads to the following new results: (a) All internal (SU_3)_(color), SU_2, U_1, and baryonic U_(1b) gauge symmetries have extremely small violations due to the gravitational interaction. (b) The T_4 gauge symmetry remains exact and dictates the universal coupling of gravitons. (c) Such a gravitational violation of internal gauge symmetries leads to modified eikonal and Hamilton-Jacobi type equations, which are obtained in the geometric-optics limit and involve effective Riemann metric tensors. (d) The rules for Feynman diagrams involving new couplings of the photon-graviton, gluon-graviton, and quark-graviton are obtained.

Translating the Concept of Five-Dimensional Tangent Vectors in Space-Time to Internal Gauge Symmetries in Elementary Particle Physics

This paper discusses the analogs of space-time five-dimensional tangent vectors for the case of internal symmetry vector spaces associated with gauge groups in elementary particle physics and their possible application in the Standard Model of quarks and leptons.

A NOTE ON THE LOCALLY LINEAR EMBEDDING ALGORITHM

The paper presents mathematical underpinnings of the locally linear embedding technique for data dimensionality reduction. It is shown that a cogent framework for describing the method is that of optimization on a Grassmann manifold. The solution delivered by the algorithm is characterized as a constrained minimizer for a problem in which the cost function and all the constraints are defined on such a manifold. The role of the internal gauge symmetry in solving the underlying optimization problem is illuminated.