Gauge Fixing Research Articles

Impact of gauge fixing precision on the continuum limit of nonlocal quark-bilinear lattice operators

We analyze the gauge fixing precision dependence of some nonlocal quark-bilinear lattice operators interesting in computing parton physics for several measurements, using five lattice spacings ranging from 0.032 to 0.121 fm. Our results show that gauge-dependent nonlocal measurements are significantly more sensitive to the precision of gauge fixing than anticipated. The impact of imprecise gauge fixing is significant for fine lattices and long distances. For instance, even with the typically defined precision of Landau gauge fixing of 10−8, the deviation caused by imprecise gauge fixing can reach 12%, when calculating the trace of Wilson lines at 1.2 fm with a lattice spacing of approximately 0.03 fm. Similar behavior has been observed in ξ gauge and Coulomb gauge as well. For both quasi-parton-distribution functions (quasi-PDFs) and quasi-transverse-momentum-dependent PDFs operators renormalized using the regularization independent momentum subtraction (RI/MOM) scheme, convergence for different lattice spacings at long distance is only observed when the precision of Landau gauge fixing is sufficiently high. To describe these findings quantitatively, we propose an empirical formula to estimate the required precision. Published by the American Physical Society 2024

BRST covariant phase space and holographic Ward identities

This paper develops an enlarged BRST framework to treat the large gauge transformations of a given quantum field theory. It determines the associated infinitely many Noether charges stemming from a gauge fixed and BRST invariant Lagrangian, a result that cannot be obtained from Noether’s second theorem. The geometrical significance of this result is highlighted by the construction of a trigraded BRST covariant phase space, allowing a BRST invariant gauge fixing procedure. This provides an appropriate framework for determining the conserved BRST Noether current of the global BRST symmetry and the associated global Noether charges. The latter are found to be equivalent with the usual classical corner charges of large gauge transformations. It allows one to prove the gauge independence of their physical effects at the perturbative quantum level. In particular, the underlying BRST fundamental canonical relation provides the same graded symplectic brackets as in the classical covariant phase space. A unified Lagrangian Ward identity for small and large gauge transformations is built. It consistently decouples into a bulk part for small gauge transformations, which is the standard BRST-BV quantum master equation, and a boundary part for large gauge transformations. The boundary part provides a perturbation theory origin for the invariance of the Hamiltonian physical -matrix under asymptotic symmetries. Holographic anomalies for the boundary Ward identity are studied and found to be solutions of a codimension one Wess-Zumino consistency condition. Such solutions are studied in the context of extended BMS symmetry. Their existence clarifies the status of the 1-loop correction to the subleading soft graviton theorem.

Classical Dynamical r-matrices for the Chern–Simons Formulation of Generalized 3d Gravity

AbstractClassical dynamical r-matrices arise naturally in the combinatorial description of the phase space of Chern–Simons theories, either through the inclusion of dynamical sources or through a gauge fixing procedure involving two punctures. Here we consider classical dynamical r-matrices for the family of Lie algebras which arise in the Chern–Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical r-matrices in this case and show that they can be viewed as generalized complexifications, in a sense which we define, of the equations governing dynamical r-matrices for $$\mathfrak {su}(2)$$ su ( 2 ) and $$\mathfrak {sl}(2,{\mathbb {R}})$$ sl ( 2 , R ) . We obtain explicit families of solutions and relate them, via Weierstrass factorization, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models.

Looking for Carroll Particles in the Two-Time Spacetime

We make an attempt to describe Carroll particles with a non-vanishing value of energy (i.e., the Carroll particles which always stay in rest) in the framework of two-time physics, developed in the series of papers by I. Bars and his co-authors. In the spacetime with one additional time dimension and one additional space dimension, where one can localize the symmetry which exists between generalized coordinates and their conjugate momenta. Such a localization implies the introduction of the gauge fields, which, in turn, implies the appearance of some first-class constraints. Choosing different gauge-fixing conditions and solving the constraints, we obtain different time parameters, Hamiltonians, and, generally, physical systems in the standard one-time spacetime. We find a set of gauge fixing conditions which gives the description of a Carroll particle in the one-time world. We construct the quantum theory of such a particle using an unexpected correspondence between our parametrization and that obtained by Bars for the hydrogen atom in 1999.

General mass formulas for charged Kerr-AdS black holes

It is well-known that the mass of a non-asymptotically flat spacetime cannot be uniquely defined. Some mass formulas for the Kerr-AdS black hole have been found and used in studying black hole thermodynamics. However, the derivations usually need a background subtraction to eliminate the divergence at infinity. It is also unknown whether the mass depends on the choice of coordinates. In this paper, we provide a more straightforward derivation for the mass formula, only demanding that the first law of black hole thermodynamics and Smarr formula are satisfied. We first make use of the Iyer-Wald formalism to derive a first law which avoids the divergence at infinity. Then we apply this formula to charged Kerr-AdS black hole expressed in the coordinates rotating at infinity. However, the first law associated with the timelike Killing vector field ∂∂t is not integrable. Then, by making use of the gauge freedom of t, we find a favorite parameter t′ which just makes the mass integrable. Applying the scaling argument, we show that the mass satisfies the Smarr formula and takes the form M/Ξ3/2. Moreover, applying the conformal method with ∂∂t′ , we obtain the same mass. By applying the first law to the coordinates which is not rotating at infinity, we find a preferred time T that makes the first law integrable and the mass is just the familiar mass M/Ξ2 in the literature. This mass is also confirmed by the conformal method. We find that the two mass formulas correspond to different families of observers and the preferred Killing times. So our work clarifies the ambiguities of mass in Kerr-AdS spacetimes.

On the conformal limit of a QED-inspired model

A conformal invariant QED-inspired model is solved for a general covariant linear gauge using the Dyson–Schwinger equations for the propagators assuming a pure vector like interaction. The leading corrections to the asymptotic solutions and the exponents, that characterize the corrections to each of the two fermion propagator functions, are computed as a function of the coupling and gauge fixing parameter ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document}. For the scalar component of the fermion propagator our findings generalizes for linear covariant gauges previous results found in the literature and reproduce the outcome of the perturbative analysis of quenched QED in the Landau gauge. Our solution for the exponent associated with vector component of the fermion propagator is new and, in the weak coupling regime, agrees with the estimation based on the perturbative analysis of quenched QED. Of the two critical exponents describing the conformal limit of the vector interaction, one of them is, in QED, associated with the regime where chiral symmetry is broken dynamically, which demands one mass scale, namely the Miransky scaling. A second mass scale has to be introduced at larger coupling constants and is associated with a change on the nature of the fermion wave function. This provides one example, that it is possible to find two interwoven cycles in Quantum Field Theory, albeit in a truncated framework, as it is known in the quantum few-body problem in the limit of a zero-range interaction.

Making Peace with Random Phases: Ab Initio Conical Intersection Quantum Dynamics in Random Gauges.

Ab initio modeling of conical intersection wave packet dynamics is crucial for various photochemical, photophysical, and biological processes. However, adiabatic electronic states obtained from electronic structure computations involve random phases, or more generally, random gauge fixings, which cannot be directly used in the modeling of nonadiabatic wave packet simulations. Here we develop a random-gauge local diabatic representation that allows an exact modeling of conical intersection dynamics directly using the adiabatic electronic states with phases randomly assigned during the electronic structure computations. Its utility is demonstrated by an exact ab initio modeling of the two-dimensional Shin-Metiu model with and without an external magnetic field. Our results provide a simple approach to integrating the electronic structure computations into nonadiabatic quantum dynamics, thus paving the way for ab initio modeling of conical intersection dynamics.

Dimensional reduction gauge and effective dimensional reduction in four-dimensional Yang-Mills theory

Motivated by one-dimensional color-electric flux-tube formation in four-dimensional (4D) QCD, we investigate a possibility of effective dimensional reduction in the 4D Yang-Mills (YM) theory. We propose a new gauge fixing of “dimensional reduction (DR) gauge” defined so as to minimize RDR≡∫d4sTr[Ax2(s)+Ay2(s)], which has a residual gauge symmetry for the gauge function Ω(t,z) like 2D QCD on the t−z plane. We investigate effective dimensional reduction in the DR gauge using SU(3) quenched lattice QCD at β=6.0. The amplitude of Ax(s) and Ay(s) are found to be strongly suppressed in the DR gauge. We consider “tz-projection” of Ax,y(s)→0 for the gauge configuration generated in the DR gauge, in a similar sense to Abelian projection in the maximally Abelian gauge. By the tz-projection in the DR gauge, the interquark potential is not changed, and At(s) and Az(s) play a dominant role in quark confinement. In the DR gauge, we calculate a spatial correlation ⟨TrA⊥(s)A⊥(s+ra⊥)⟩(⊥=x,y) and estimate the spatial mass of A⊥(s)(⊥=x,y) as M≃1.7 GeV. It is conjectured that this large mass makes A⊥(s) inactive and realizes the dominance of At(s) and Az(s) in infrared region in the DR gauge. We also calculate the spatial correlation of two temporal link-variables and find that the correlation decreases as exp(−mr) with m≃0.6 GeV. Using a crude approximation, the 4D YM theory is reduced into an ensemble of 2D YM systems with the coupling of g2D=gm. Published by the American Physical Society 2024

Universal framework for simultaneous tomography of quantum states and SPAM noise

We present a general denoising algorithm for performing simultaneous tomography of quantum states and measurement noise. This algorithm allows us to fully characterize state preparation and measurement (SPAM) errors present in any quantum system. Our method is based on the analysis of the properties of the linear operator space induced by unitary operations. Given any quantum system with a noisy measurement apparatus, our method can output the quantum state and the noise matrix of the detector up to a single gauge degree of freedom. We show that this gauge freedom is unavoidable in the general case, but this degeneracy can be generally broken using prior knowledge on the state or noise properties, thus fixing the gauge for several types of state-noise combinations with no assumptions about noise strength. Such combinations include pure quantum states with arbitrarily correlated errors, and arbitrary states with block independent errors. This framework can further use available prior information about the setting to systematically reduce the number of observations and measurements required for state and noise detection. Our method effectively generalizes existing approaches to the problem, and includes as special cases common settings considered in the literature requiring an uncorrelated or invertible noise matrix, or specific probe states.

Substructures of the Weyl group and their physical applications

We study substructures of the Weyl group of conformal transformations of the metric of (pseudo)Riemannian manifolds. These substructures are identified by differential constraints on the conformal factors of the transformations which are chosen such that their composition is associative. Mathematically, apart from rare exceptions, they are partial associative groupoids, not groups, so they do not have an algebra of infinitesimal transformations, but this limitation can be partially circumvented using some of their properties cleverly. We classify and discuss the substructures with two-derivatives differential constraints, the most famous of which being known as the harmonic or restricted Weyl group in the physics literature, but we also show the existence of a lightcone constraint which realizes a proper subgroup of the Weyl group. We then show the physical implications that come from invariance under the two most important substructures, concentrating on classical properties of the energy-momentum tensor and a generalization of the quantum trace anomaly. We also elaborate further on the harmonic substructure, which can be interpreted as partial gauge fixing of full Weyl invariance using BRST methods. Finally, we discuss how to construct differential constraints of arbitrary higher-derivative order and present, as examples, generalizations involving scalar constraints with four and six derivatives.

Analytical calculation of Kerr and Kerr-Ads black holes in f(R) theory

In this paper, we extend Chandrasekhar’s method of calculating rotating black holes into f(R) theory. We show that the solution with constant Ricci scalar always exists in a general f(R) gravity and derive the Kerr and Kerr-Ads metric by using the analytical mathematical method. Suppose that the spacetime is a 4-dimensional Riemannian manifold with a general stationary axisymmetric metric, we calculate Cartan’s equation of structure and derive the Einstein tensor. In order to reduce the solving difficulty, we fix the gauge freedom to transform the metric into a more symmetric form. We solve the field equations in the two cases of the Ricci scalar R=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=0$$\\end{document} and R≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\ e 0$$\\end{document}. In the case of R=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R=0$$\\end{document}, the Ernst’s equations are derived. We give the elementary solution of Ernst’s equations and show the way to obtain more solutions including Kerr metric. In the case of R≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\ e 0$$\\end{document}, we reasonably assume that the solution to the equations consists of two parts: the first is Kerr part and the second is introduced by the Ricci scalar. Giving solution to the second part and combining the two parts, we obtain the Kerr-Ads metric. The calculations are carried out in a general f(R) theory. Furthermore, the whole solving process can be treated as a standard calculation procedure to obtain rotating black holes, which can be applied to other modified gravities.

Symmetry, gauge freedoms, and the interpretability of sequence-function relationships.

Quantitative models that describe how biological sequences encode functional activities are ubiquitous in modern biology. One important aspect of these models is that they commonly exhibit gauge freedoms, i.e., directions in parameter space that do not affect model predictions. In physics, gauge freedoms arise when physical theories are formulated in ways that respect fundamental symmetries. However, the connections that gauge freedoms in models of sequence-function relationships have to the symmetries of sequence space have yet to be systematically studied. Here we study the gauge freedoms of models that respect a specific symmetry of sequence space: the group of position-specific character permutations. We find that gauge freedoms arise when model parameters transform under redundant irreducible matrix representations of this group. Based on this finding, we describe an "embedding distillation" procedure that enables analytic calculation of the number of independent gauge freedoms, as well as efficient computation of a sparse basis for the space of gauge freedoms. We also study how parameter transformation behavior affects parameter interpretability. We find that in many (and possibly all) nontrivial models, the ability to interpret individual model parameters as quantifying intrinsic allelic effects requires that gauge freedoms be present. This finding establishes an incompatibility between two distinct notions of parameter interpretability. Our work thus advances the understanding of symmetries, gauge freedoms, and parameter interpretability in sequence-function relationships.

Gauge fixing for sequence-function relationships.

Quantitative models of sequence-function relationships are ubiquitous in computational biology, e.g., for modeling the DNA binding of transcription factors or the fitness landscapes of proteins. Interpreting these models, however, is complicated by the fact that the values of model parameters can often be changed without affecting model predictions. Before the values of model parameters can be meaningfully interpreted, one must remove these degrees of freedom (called "gauge freedoms" in physics) by imposing additional constraints (a process called "fixing the gauge"). However, strategies for fixing the gauge of sequence-function relationships have received little attention. Here we derive an analytically tractable family of gauges for a large class of sequence-function relationships. These gauges are derived in the context of models with all-order interactions, but an important subset of these gauges can be applied to diverse types of models, including additive models, pairwise-interaction models, and models with higher-order interactions. Many commonly used gauges are special cases of gauges within this family. We demonstrate the utility of this family of gauges by showing how different choices of gauge can be used both to explore complex activity landscapes and to reveal simplified models that are approximately correct within localized regions of sequence space. The results provide practical gauge-fixing strategies and demonstrate the utility of gauge-fixing for model exploration and interpretation.

Unconstrained Lagrangian formulation for bosonic continuous spin theory in flat spacetime of arbitrary dimension

We have discovered two unconstrained forms of free Lagrangian for continuous spin(CS) theory in arbitrary flat spacetime dimension for bosonic case. These Lagrangians, unlike that by Schuster and Toro, do not include delta functions and are conventional. The first form consists of five kinds of totally symmetric helicity fields and one kind of gauge parameter. By introducing auxiliary creation and annihilation operators, each is combined into a state vector in Fock space, including all ranks one by one. The Lagrangian imposes no constraints, such as trace conditions, on these fields or the gauge parameter field. Additionally, the Lagrangian does not contain higher-order derivative terms. In the limit as CS parameter μ approaches zero, it naturally reproduces a Lagrangian for helicity fields in higher spin(HS) theory, known as unconstrained quartet formulation. Permitting third-order derivatives, we also obtain the second unconstrained form of Lagrangian that can be written in terms of three kinds of fields, including μ, similar to the formulation by Francia and Sagnotti. Partial gauge fixing and partial use of equations of motion (EOM) on this Lagrangian yield a Fronsdal-like Lagrangian with a single double-traceless field, including μ. By imposing further gauge fixing on the field in the EOM with respect to divergence and trace, we confirm the reproduction of the modified Wigner equations already known in literature.

Gribov problem in the eletroweak and gluon confined theories

We show that gauge fields after a spontaneous symmetry breaking (SSB) mechanism do not have a Gribov problem along the broken directions. In order to make this proof, we describe a gauge fixing procedure inspired on Morse’s theory leading to the concept of a gauge fixing generating functional. This approach is specially suited in order to understand the unitary gauges of ’t Hooft. The conclusion is that after a SSB process, the generalized Faddeev-Popov operator acquires a positive definite value along the broken directions. We show how this works in the eletroweak SU(2)XU(1) theory, where such a result is in fact expected as in this case the gauge fields acquire masses after the SSB. Then we apply this development to the SL(3,c) confining model of Amaral [A path to confine gluons and fermions through complex gauge theory, ]. The final result explains why such confining mechanism is actually a solution to the Gribov problem for the confined gauge fields. These cases show that although confinement is not a general effect of the solution of the Gribov problem, as we can infer from the eletroweak example, its solution indeed seems to be necessary in order to achieve confinement. In the end, these conclusions allow us to speculate that the strong interaction confinement may be the result of some hidden SSB mechanism yet to be described. Published by the American Physical Society 2024

Hamiltonian for Weyl transverse gravity

In this short note we determine Hamiltonian for Weyl transverse gravity. We find primary, secondary and tertiary constraints and calculate Poisson brackets between them. We also show that gauge fixing in Weyl transverse gravity leads to the Hamiltonian for unimodular gravity.

Unimodular proca theory: breaking the U(1) gauge symmetry of unimodular gravity via a mass term

We study the Hamiltonian structure of unimodular-like theories, where the cosmological constant (or other supposed constants of nature) are demoted from fixed parameters to classical constants of motion. No new local degrees of freedom are present as a result of a U(1) gauge invariance of the theory. Hamiltonian analysis of the action reveals that the only possible gauge fixing that can be enforced is setting the spatial components of the four-volume time vector Ti≈0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{T}^{i}\\approx 0$$\\end{document}. As a consequence of this, the gauge-fixed unimodular path integral is equivalent to the minisuperspace unimodular path integral. However, should we break the U(1) gauge invariance, two things happen: a massless propagating degree of freedom appears, and the (gauge-invariant) zero-mode receives modified dynamics. The implications are investigated, with the phenomenology depending crucially on the target “constant”.

Closed string vertex operators with various ghost number

We construct closed string vertex operators with various ghost numbers in addition to the conventional ones, using the Faddeev-Popov procedure for the gauge fixing of the conformal Killing group, from matter primary fields. We find that these operators give solutions to the descent equations in the framework of the BRST formalism. Similarly, we also construct solutions to the descent equations for the dilaton vertex operator with the Lorentz covariant form. Using the unintegrated vertex operator of the dilaton with the ghost number three, we obtain the correct result of the tadpole amplitude on the disk, including a non-zero contribution from a BRST exact term which comes from a conformal transformation.

Scattering of spinning compact objects from a worldline EFT

We study the EFT of a spinning compact object and show that with appropriate gauge fixing, computations become amenable to worldline quantum field theory techniques. We use the resulting action to compute Compton and one-loop scattering amplitudes at fourth order in spin. By matching these amplitdes to solutions of the Teukolsky equations, we fix the values of Wilson coefficients appearing in the EFT such that it reproduces Kerr black hole scattering. We keep track of the spin supplementary condition throughout our computations and discuss alternative ways to ensure its preservation.

The partial Bondi gauge: Gauge fixings and asymptotic charges

In the companion paper [SciPost Phys. 13, 108 (2022)] we have studied the solution space at null infinity for gravity in the partial Bondi gauge. This partial gauge enables to recover as particular cases and among other choices the Bondi-Sachs and Newman-Unti gauges, and to approach the question of the most general boundary conditions and asymptotic charges in gravity. Here we compute and study the asymptotic charges and their algebra in this partial Bondi gauge, by focusing on the flat case with a varying boundary metric \delta q_{AB}≠0δqAB≠0. In addition to the super-translations, super-rotations, and Weyl transformations, we find two extra asymptotic symmetries associated with non-vanishing charges labelled by free functions in the solution space. These new symmetries arise from a weaker definition of the radial coordinate and switch on traces in the transverse metric. We also exhibit complete gauge fixing conditions in which these extra asymptotic symmetries and charges survive. As a byproduct of this calculation we obtain the charges in Newman-Unti gauge, in which one of these extra asymptotic charges is already non-vanishing. We also apply the formula for the charges in the partial Bondi gauge to the computation of the charges for the Kerr spacetime in Bondi coordinates.