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Gauge fixing

In the physics of gauge theories, gauge fixing is a mathematical procedure used to eliminate redundant degrees of freedom in field variables by imposing supplementary conditions that select a unique representative from each equivalence class of gauge-equivalent configurations.^[1] This redundancy arises because gauge theories, such as electromagnetism and quantum chromodynamics, are invariant under local transformations—known as gauge transformations—that do not alter physical observables like electric and magnetic fields. Without gauge fixing, the fields describing these theories would have infinite equivalent solutions, complicating computations and quantization.^[1] Common gauge-fixing conditions include the Coulomb gauge, defined by \nabla \cdot \mathbf{A} = 0 where \mathbf{A} is the vector potential, which simplifies static problems by decoupling scalar and vector potentials and enabling straightforward Fourier expansions.^[1] The Lorenz gauge, \partial_\mu A^\mu = 0, preserves Lorentz invariance and is widely used in relativistic contexts, such as quantum electrodynamics, though it requires careful handling of ghost fields in quantization to maintain unitarity.^[1] Other choices, like the unitary gauge in scalar electrodynamics where the scalar field is chosen real and positive,^[2] provide local and separable descriptions but may be incomplete in cases with unbroken symmetries or negligible matter fields. Gauge fixing plays a crucial role in the quantization of gauge fields, where methods like the Faddeev-Popov procedure introduce a determinant factor in the path integral to account for the volume of the gauge orbit, ensuring that gauge-equivalent configurations are counted once and yielding well-defined propagators for observables.^[1] Importantly, while gauge fixing breaks the explicit gauge symmetry of the Lagrangian, the resulting descriptions remain gauge-invariant for physical quantities, as different gauges yield equivalent results upon appropriate transformations. This framework underpins much of modern particle physics and extends to gravitational theories,^[3] highlighting gauge fixing's essential role in bridging theoretical formalism with computable predictions.^[1]

Gauge symmetry and freedom

Definition and principles of gauge symmetry

Local gauge invariance constitutes a fundamental redundancy in the mathematical description of physical fields, wherein the governing equations of a theory remain unchanged under local transformations that vary from point to point in spacetime. This symmetry principle ensures that physical predictions are independent of the particular choice of field representation, as long as the transformations preserve the form of the Lagrangian or action. In essence, it reflects the idea that certain aspects of the field variables are unobservable and can be redefined without altering measurable outcomes.^[4] The concept originated with Hermann Weyl's 1918 proposal to unify gravitation and electromagnetism through a gauge theory based on local scale (conformal) transformations of the metric tensor. Weyl's framework, though initially unsuccessful as a unified theory due to conflicts with observed atomic spectra, laid the groundwork for modern gauge principles by emphasizing local invariance beyond global symmetries. Subsequently, in the quantum mechanical era, Vladimir Fock in 1926 and Wolfgang Pauli in 1927 refined the notion, reinterpreting gauge invariance as local phase shifts in the wave function of charged particles, thereby linking it directly to electromagnetism within the Schrödinger equation.^[5]^[4] A prototypical example is the U(1) gauge symmetry underlying quantum electrodynamics, where the phase of the matter field (e.g., the electron wave function \psi) undergoes local transformations \psi \to e^{i \alpha(x)} \psi, with \alpha(x) an arbitrary spacetime-dependent function; this requires the electromagnetic four-potential A_\mu to couple to the fields to maintain invariance. In contrast, non-Abelian generalizations appear in Yang-Mills theories, featuring SU(N) internal symmetry groups, as introduced by Chen Ning Yang and Robert Mills in 1954 to describe strong and weak interactions through self-interacting gauge fields.^[4]^[6] Mathematically, for the Abelian U(1) case in electromagnetism, a gauge transformation acts on the vector potential as

A_\mu \to A_\mu + \partial_\mu \Lambda,

where \Lambda(x) is an arbitrary differentiable scalar function, leaving the physical field strength tensor F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu invariant. Physically, gauge symmetries connect to conservation laws: the associated global (constant \Lambda) transformations yield conserved charges, such as electric charge, via Noether's first theorem. However, the local nature introduces redundant degrees of freedom in the Lagrangian, corresponding to unphysical polarizations of the gauge fields that must be addressed through gauge fixing techniques.^[7]^[8]

Implications of gauge freedom in field theories

In gauge theories, the freedom to perform gauge transformations introduces a redundancy in the field variables, where an infinite number of equivalent configurations describe the identical physical situation. This redundancy arises because the equations of motion remain unchanged under local transformations, such as A_\mu \to A_\mu + \partial_\mu \omega for Abelian gauge fields, leading to overparameterization that complicates the formulation of initial and boundary conditions. As a result, solving the dynamics becomes mathematically cumbersome, as the system includes superfluous degrees of freedom that do not correspond to observable effects.^[2] In classical field theories, gauge freedom manifests as redundancy in the description of potentials, where multiple configurations yield the same physical fields. For instance, in electromagnetism, the four-component vector potential includes unphysical longitudinal and timelike degrees of freedom beyond the two transverse modes required for physical electromagnetic propagation. Without gauge fixing, solutions for the potentials are non-unique, complicating computations, though the gauge-invariant electric and magnetic fields remain transverse for radiation and consistent with special relativity. These redundant modes in the potentials do not carry physical energy or propagate signals but obscure the formulation of unique solutions limited to the observable transverse components.^[2] Quantization exacerbates these issues, as gauge freedom generates non-physical states and overcounting in the Hilbert space or path integral formulation. In canonical quantization, the presence of first-class constraints from gauge invariance leads to redundant variables that produce ghost states with negative norms or zero-norm states, necessitating projection onto the physical subspace to ensure unitarity and positive-definiteness. Similarly, in the path integral approach, integrating over all gauge-equivalent field configurations without restriction results in an infinite volume factor, causing divergences and incorrect normalization of amplitudes.^[1]^[2] Physical observables in gauge theories must be constructed to be gauge-invariant, ensuring they remain unchanged under transformations, while intermediate expressions in calculations generally depend on the choice of gauge. Examples include Wilson loops, which trace the parallel transport of fields around closed paths and serve as order parameters for phenomena like confinement in non-Abelian theories, and scattering amplitudes, which encode interaction probabilities and are independent of gauge details despite relying on gauge-dependent propagators in perturbation theory. This distinction underscores that while gauge freedom simplifies the underlying Lagrangian, it demands careful selection of invariant quantities to extract meaningful predictions.^[2] A concrete illustration occurs in quantum electrodynamics (QED), where an unfixed gauge permits spurious solutions that violate causality, such as acausal propagation due to unphysical timelike or longitudinal components in the photon propagator. These artifacts arise from the incomplete constraint of the gauge orbit, allowing non-propagating modes to contribute to Green's functions and potentially leading to inconsistencies in S-matrix elements unless resolved.^[1]

Principles of gauge fixing

Motivation and general methods

Gauge fixing arises primarily from the need to eliminate redundant degrees of freedom in gauge theories, where gauge symmetries introduce equivalences among field configurations that do not correspond to distinct physical states. Without fixing, the equations of motion yield infinite families of solutions related by gauge transformations, complicating the identification of unique physical predictions and leading to divergences in quantization procedures, such as path integrals that overcount gauge-equivalent configurations. This redundancy obscures the true dynamical content, making it essential to impose conditions that select a representative from each equivalence class while preserving observable quantities.^[2]^[9] General methods for gauge fixing involve imposing constraints on the gauge fields, typically algebraic or differential conditions that slice through the space of configurations transversely to the gauge orbits. In classical field theories, these constraints simplify the Hamiltonian formulation by reducing the phase space and enforcing primary constraints like Gauss's law, thereby facilitating solvable dynamics. At the quantum level, more sophisticated techniques are required to maintain unitarity and consistency; the Faddeev-Popov procedure introduces auxiliary ghost fields to compensate for the Jacobian determinant arising from the change of variables in the path integral, ensuring the measure remains well-defined. Additionally, the BRST formalism extends this by embedding a nilpotent symmetry that mimics the original gauge invariance, allowing renormalization while avoiding explicit breaking of the symmetry structure.90065-3)90159-3) Effective gauge choices must preserve the physical content of the theory, ensuring that gauge-invariant observables remain unchanged and that the fixing does not introduce spurious modes or artifacts. A key challenge is avoiding Gribov ambiguities, where multiple field configurations satisfy the same constraint, leading to regions in configuration space where uniqueness fails and potentially affecting non-perturbative aspects like confinement. These criteria guide the selection of gauges suitable for specific computations, balancing computational tractability with theoretical fidelity. Historically, the systematic application of gauge fixing emerged in the 1930s with Enrico Fermi's treatment of quantum electrodynamics, where he first demonstrated how to incorporate gauge conditions into the Hamiltonian to handle the vector and scalar potentials consistently.

An illustrative example in electromagnetism

In classical electrodynamics, Maxwell's equations can be expressed in terms of the scalar potential \phi and the vector potential \mathbf{A}, where the electric field is \mathbf{E} = -\nabla \phi - \frac{1}{c} \frac{\partial \mathbf{A}}{\partial t} and the magnetic field is \mathbf{B} = \nabla \times \mathbf{A}.^[10] These potentials are not unique, as they transform under a gauge transformation \mathbf{A} \to \mathbf{A} + \nabla \chi and \phi \to \phi - \frac{1}{c} \frac{\partial \chi}{\partial t}, where \chi is an arbitrary scalar function; this leaves the physical fields \mathbf{E} and \mathbf{B} invariant.^[10] Without gauge fixing, the equation for the vector potential \mathbf{A} takes the form of a wave equation with an arbitrary source term: \left( \frac{1}{c} \frac{\partial}{\partial t} \right)^2 \mathbf{A} - \nabla^2 \mathbf{A} + \nabla \left( \frac{1}{c} \frac{\partial \phi}{\partial t} + \nabla \cdot \mathbf{A} \right) = \frac{4\pi}{c} \mathbf{j}, where \mathbf{j} is the current density.^[10] The freedom in choosing the gauge term \nabla \left( \frac{1}{c} \frac{\partial \phi}{\partial t} + \nabla \cdot \mathbf{A} \right) allows solutions that include unphysical longitudinal waves propagating at the speed of light, which do not correspond to observable electromagnetic radiation.^[10] To eliminate this redundancy, a simple gauge fixing can be imposed by selecting an appropriate \chi. For instance, choosing \chi such that \phi = 0 (the temporal or Weyl gauge) decouples the scalar potential, simplifying the equations to focus on \mathbf{A} alone in certain static or radiation problems.^[10] Alternatively, setting \nabla \cdot \mathbf{A} = 0 (a Coulomb-like gauge) separates the longitudinal and transverse components of \mathbf{A}, allowing the scalar potential to satisfy Poisson's equation \nabla^2 \phi = -4\pi \rho independently of \mathbf{A}, where \rho is the charge density.^[10] A concrete illustration arises in the electrostatic field of a point charge q at the origin, where \mathbf{B} = 0 and \mathbf{E} = \frac{q}{r^2} \hat{r}. Without fixing, the potentials admit infinitely many solutions, such as \phi = \frac{q}{r} + f(t) and \mathbf{A} = -c f(t) \hat{r}, where f(t) is arbitrary, as the gradient and time derivative cancel to yield the correct \mathbf{E}.^[10] Imposing the Coulomb gauge \nabla \cdot \mathbf{A} = 0 forces \mathbf{A} = 0 and uniquely determines \phi = \frac{q}{r} (in cgs units, up to an additive constant fixed by requiring \phi \to 0 as r \to \infty), eliminating the ambiguity and directly linking the potential to the observable field via \mathbf{E} = -\nabla \phi.^[10] In relativistic notation, a general gauge condition takes the form \partial_\mu A^\mu = f(x), where A^\mu = (\phi/c, \mathbf{A}) is the four-potential and f(x) is a specified function chosen to fix the gauge; this removes the redundancy while preserving the physical content of Maxwell's equations.^[11]

Gauges in electrodynamics

Coulomb gauge

The Coulomb gauge, also known as the transverse or radiation gauge, is defined by the condition \nabla \cdot \mathbf{A} = 0, where \mathbf{A} is the vector potential in the expression for the electromagnetic fields, \mathbf{B} = \nabla \times \mathbf{A} and \mathbf{E} = -\nabla \Phi - \frac{1}{c} \frac{\partial \mathbf{A}}{\partial t}. This condition ensures that the vector potential is purely transverse, eliminating longitudinal components and simplifying the description of electromagnetic waves.^[12] Imposing the Coulomb gauge on Maxwell's equations decouples the scalar and vector potentials. The scalar potential \Phi satisfies the Poisson equation \nabla^2 \Phi = -4\pi \rho, yielding an instantaneous Coulomb potential

\Phi(\mathbf{x}, t) = \int \frac{\rho(\mathbf{x}', t)}{|\mathbf{x} - \mathbf{x}'|} \, d^3\mathbf{x}',

which directly reflects the charge distribution at the same time t. The vector potential \mathbf{A} obeys the inhomogeneous wave equation

\nabla^2 \mathbf{A} - \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} = -\frac{4\pi}{c} \mathbf{J}_\perp,

where \mathbf{J}_\perp is the transverse (divergence-free) component of the current density \mathbf{J}, obtained by subtracting the longitudinal part. This separation highlights the electrostatic nature of \Phi and the radiative dynamics of \mathbf{A}.^[12]^[13] In non-relativistic quantum mechanics, the Coulomb gauge simplifies the Schrödinger equation for particles in an electromagnetic field, as the vector potential enters directly through the minimal coupling \mathbf{p} \to \mathbf{p} - \frac{e}{c} \mathbf{A}, facilitating calculations in atomic physics where transverse photon interactions dominate. It is particularly suited to radiation problems, as the transverse \mathbf{A} naturally describes propagating electromagnetic waves with two polarization states. However, the gauge is not Lorentz covariant, meaning the condition \nabla \cdot \mathbf{A} = 0 does not hold in all reference frames, and it introduces non-localities in relativistic quantum field theory due to the instantaneous \Phi.^[12]^[1] In the Hamiltonian formulation of quantum electrodynamics, the Coulomb gauge leads to a constraint that generates Gauss's law \nabla \cdot \mathbf{E} = 4\pi \rho. Quantally, this is enforced by restricting physical states to those annihilated by the operator \hat{G} = \nabla \cdot \hat{\mathbf{E}} - 4\pi \hat{\rho} = 0, so \hat{G} |\Psi\rangle = 0, ensuring gauge-invariant physical observables. The transverse photons are represented by creation and annihilation operators for two helicity states, with the Hamiltonian expressed as a sum of harmonic oscillators \hat{H} = \int d^3p \, |\mathbf{p}| \sum_{\alpha=1,2} \hat{a}^\dagger_\alpha(\mathbf{p}) \hat{a}_\alpha(\mathbf{p}). This setup is applied in atomic physics for modeling electron-photon interactions and in photon propagation studies, where the transverse gauge captures free-field dynamics without longitudinal artifacts.^[1]^[12]

Lorenz gauge

The Lorenz gauge is a gauge-fixing condition in electrodynamics defined by \partial_\mu A^\mu = 0, where A^\mu = (\phi/c, \mathbf{A}) is the four-potential in Minkowski spacetime with metric signature (+,-,-,-) and c=1 units often adopted for relativity.^[14] This condition, also known as the Lorentz-Lorenz gauge, is named after the Danish physicist Ludvig Valentin Lorenz (1829–1891), distinct from the Lorentz transformation named after Hendrik Lorentz despite the phonetic similarity. Lorenz first proposed this gauge in 1867 as part of his general integral solutions to Maxwell's equations using retarded potentials, demonstrating that it simplifies the description of electromagnetic wave propagation. In this gauge, Maxwell's equations decouple into independent wave equations for the components of the four-potential: \square A^\mu = -J^\mu (in natural units with \mu_0 = 1), where \square = \partial_\mu \partial^\mu is the d'Alembertian operator and J^\mu is the four-current.^[11] This decoupling preserves full Lorentz invariance, allowing the potentials to transform covariantly under Lorentz boosts while maintaining the gauge condition.^[11] The advantages of the Lorenz gauge stem from its covariance, making it particularly suitable for relativistic calculations in classical electrodynamics, such as analyzing radiation fields in vacuum where potentials satisfy homogeneous wave equations far from sources.^[11] In quantum electrodynamics (QED), it corresponds to the Feynman gauge (a special case of covariant gauges with parameter \xi=0), simplifying Feynman diagram computations by yielding a transverse and Lorentz-invariant photon propagator.^[11] The momentum-space photon propagator takes the form

-i \frac{g_{\mu\nu}}{k^2 + i\epsilon},

where g_{\mu\nu} is the Minkowski metric, k^\mu is the four-momentum, and the i\epsilon prescription ensures causality.^[11] This form facilitates perturbative expansions in QED, as it avoids mixing longitudinal and transverse polarizations explicitly.^[11] Despite these benefits, the Lorenz gauge does not fully eliminate gauge freedom; residual transformations remain possible via gauge functions \chi satisfying the homogeneous wave equation \square \chi = 0, which preserve the condition \partial_\mu A^\mu = 0.^[11] These residual degrees of freedom correspond to unphysical modes that must be handled carefully in quantization to ensure physical observables are gauge-invariant.^[11]

Covariant and general gauges

R_xi gauges

The R_ξ gauges form a family of covariant gauge choices parameterized by a real number ξ, introduced by Gerard 't Hooft in the early 1970s to establish the renormalizability of spontaneously broken Yang-Mills theories.^[15] These gauges generalize the Lorenz gauge and are particularly suited for perturbative calculations in quantum field theories with gauge symmetries, ensuring that Ward identities are preserved and divergences can be systematically subtracted.^[15] In these gauges, the gauge-fixing term added to the Lagrangian for a non-Abelian gauge field A_\mu^a (with color index a) is

\mathcal{L}_\text{gf} = -\frac{1}{2\xi} \left( \partial^\mu A_\mu^a \right)^2,

where the parameter ξ controls the weighting of longitudinal modes.^[15] The choice ξ = 1 corresponds to the Feynman-'t Hooft gauge, which simplifies Feynman rules by making the propagator transverse part proportional to the metric tensor g_{\mu\nu}, while ξ = 0 recovers the Landau (or Lorenz) gauge in the appropriate limit, enforcing a stricter transversality condition.^[15] In momentum space, the gauge boson propagator takes the form

-i \frac{g_{\mu\nu} - (1 - \xi) \frac{k_\mu k_\nu}{k^2}}{k^2 + i\epsilon}

for massless fields, highlighting the interpolation between different transverse and longitudinal contributions depending on ξ.^[16] The Faddeev-Popov quantization procedure in R_ξ gauges introduces ghost fields to account for the gauge redundancy, with their Lagrangian derived from the functional determinant of the gauge-fixing condition.^[15] In unbroken gauge theories, the ghosts remain massless, but in spontaneously broken cases like the electroweak sector, the ghost masses become gauge-dependent, scaling as ξ times the square of the vector boson mass, which aids in verifying gauge independence of physical amplitudes.^[15] This flexibility proves advantageous in perturbation theory, as varying ξ can simplify loop integrals or improve convergence, while the physical S-matrix elements remain independent of ξ due to gauge invariance.^[16] R_ξ gauges are standard in perturbative quantum chromodynamics (QCD), where choices like ξ = 1 facilitate gluon self-energy computations, and in electroweak theory, where they ensure consistent treatment of massive W and Z bosons alongside the massless photon.

Temporal gauge

The temporal gauge, also known as the Weyl gauge in some contexts, is defined by the condition A^0 = 0, where A^\mu is the gauge field and the superscript denotes the time component, effectively eliminating the scalar potential \phi. This choice fixes the gauge by removing the temporal degree of freedom, leaving only the spatial components \mathbf{A} to describe the dynamics.^[17]^[1] This gauge is non-covariant under Lorentz transformations, as it privileges a specific time direction, but it simplifies the treatment of time evolution in the Hamiltonian formalism by allowing canonical quantization with straightforward commutation relations for the spatial fields, such as [\hat{A}_j(\mathbf{x}), \hat{\Pi}_{j'}(\mathbf{x}')] = i \delta_{jj'} \delta^3(\mathbf{x} - \mathbf{x}'), where \hat{\Pi}_j = -E_j. However, it retains residual gauge freedom under time-independent spatial transformations, A_i \to A_i + \partial_i \lambda(\mathbf{x}), which must be addressed to fully specify the gauge. In this framework, Gauss's law, \nabla \cdot \mathbf{E} = \rho, emerges as a constraint on the matter fields rather than a dynamical equation, enforced on physical states via \nabla \cdot \hat{\mathbf{E}}(\mathbf{x}) |\psi\rangle = 0. The spatial components A_i then satisfy a wave equation, \partial^2 A_i = 0, but include longitudinal modes due to the absence of transversality conditions.^[1]^[18] The temporal gauge offers advantages in scenarios requiring explicit time-dependent evolution, such as light-front quantization of gauge theories, where a variant like A^+ = 0 (in light-cone coordinates) eliminates unphysical degrees of freedom, yields a trivial vacuum, and simplifies gluon propagators to doubly transverse forms, avoiding ghosts and collinear divergences. It is particularly useful in strong-field quantum electrodynamics (QED), where implementing A^0 = 0 as a time-independent constraint facilitates the analysis of intense laser-matter interactions without introducing the scalar potential. In particle physics contexts, this gauge avoids complications from instantaneous Coulomb interactions, streamlining Hamiltonian approaches.^[19]^[18] Despite these benefits, the temporal gauge suffers from limitations, notably the Gribov ambiguity in non-Abelian theories like QCD, where multiple gauge-equivalent configurations satisfy A^0 = 0, leading to non-unique solutions for gauge-invariant fields and complicating non-perturbative quantization. Its lack of Lorentz invariance also restricts applicability in relativistic calculations requiring manifest covariance. Applications include real-time dynamics in heavy-ion collisions, where A^\tau = 0 (in proper-time coordinates) simplifies numerical solutions of classical Yang-Mills equations on lattices while preserving residual gauge invariance, aiding studies of initial color field evolution in the Color Glass Condensate framework.^[20]^[21]

Abelian and non-perturbative gauges

Maximal abelian gauge

The maximal abelian gauge (MAG) is a partial gauge-fixing procedure in non-Abelian gauge theories, such as SU(N) quantum chromodynamics (QCD), designed to maximize the contribution of the Abelian (Cartan) subgroup by minimizing the norm of the off-diagonal gluon field components. For instance, in SU(2), this involves suppressing the charged (off-diagonal) gluon fields while preserving the residual U(1) symmetry associated with the neutral (diagonal) gluon.^[22] This approach partially fixes the non-Abelian gauge freedom, leaving an unbroken maximal Abelian subgroup intact.^[22] The MAG was introduced by Gerard 't Hooft in 1981 within the framework of Abelian projection, as a means to uncover the dual superconductivity mechanism for quark confinement in QCD.^[22] 't Hooft proposed that, by projecting the non-Abelian theory onto its Abelian subgroup, magnetic monopoles could emerge as relevant degrees of freedom, analogous to the electric charges in standard superconductivity but dualized to explain the linear confinement potential between quarks.^[22] The gauge condition is enforced through a non-linear covariant divergence equation on the off-diagonal fields. Specifically, for the off-diagonal indices i, j = 1, \dots, N^2 - 1 excluding the Cartan directions, it reads

(D_\mu A^\mu)^{ij} = 0,

where D_\mu^{ij} = \delta^{ij} \partial_\mu - i g (T^k)^{ij} A_\mu^k is the covariant derivative in the adjoint representation, restricted to the diagonal (Cartan) gluon fields A_\mu^k with k labeling the Abelian generators. In practice, on the lattice, the MAG is implemented iteratively by maximizing the functional \sum_x \sum_\mu \operatorname{Re} \operatorname{Tr} (P U_\mu(x)), where P projects onto the diagonal part of the link variables U_\mu(x), effectively minimizing the off-diagonal contributions.^[23] A key implication of the MAG is the emergence of Abelian monopoles as topological defects in the diagonal photon fields, which condense in the vacuum to produce a dual Meissner effect, expelling color electric fields and enforcing confinement.^[22] Lattice studies in the MAG confirm Abelian dominance, where the diagonal gluons and monopoles account for nearly the full string tension in the quark-antiquark potential, supporting the dual superconductor picture. This reveals the non-perturbative structure of QCD, linking it to topological phenomena absent in the perturbative regime.^[23] The MAG offers significant advantages in bridging perturbative and non-perturbative analyses of QCD, as the off-diagonal gluons acquire effective masses from interactions with the diagonal sector, resembling a Higgs mechanism that simplifies computations.^[23] It is extensively applied in lattice QCD simulations to probe confinement dynamics, monopole condensation, and the infrared behavior of propagators. However, the gauge suffers from persistent Gribov ambiguities, with multiple configurations (Gribov copies) satisfying the fixing condition due to the non-linear nature of the equation, rendering the choice non-unique and complicating interpretations.

Landau gauge

The Landau gauge is a covariant gauge-fixing condition in gauge theories, defined by the transversality requirement \partial_\mu A^\mu = 0, where A^\mu is the gauge field. This condition corresponds to the limit \xi \to 0 in the general R_\xi covariant gauges, ensuring a Lorentz-invariant formulation without residual gauge freedom in the photon or gluon propagator. In non-perturbative settings, such as lattice gauge theory, the gauge is selected by minimizing the functional F[A_g] = \int d^4x \, (\partial_\mu A^\mu_g(x))^2 over all gauge-equivalent configurations A_g = A^g, where A^g_\mu(x) = g(x) A_\mu(x) g^{-1}(x) + i g(x) \partial_\mu g^{-1}(x) for g \in the gauge group; this chooses a representative closest to the origin in field space, mitigating Gribov copy ambiguities within the first Gribov region.^[24]^[25] Although named after Lev Landau, the term "Landau gauge" was introduced by Bruno Zumino in 1960 while studying propagator properties in quantum electrodynamics, where it was highlighted for its utility in analyzing ultraviolet divergences and gauge invariance. It predates widespread use in quantum chromodynamics but became standard in numerical simulations of lattice gauge theories starting in the 1980s, particularly for SU(3) Yang-Mills theory, due to its compatibility with Monte Carlo methods.^[26] The minimization of the functional yields stationary points satisfying a Laplace equation for the infinitesimal gauge transformation parameter \omega: \partial^2 \omega = \partial_\mu A^\mu, ensuring the transformed field obeys the gauge condition; on the lattice, this is discretized and solved iteratively. This gauge preserves transversality of the fields, making the gluon propagator transverse in momentum space as D_{\mu\nu}^{ab}(p) = \delta^{ab} ( \delta_{\mu\nu} - p_\mu p_\nu / p^2 ) Z(p^2) / p^2, and it is unique up to the Gribov horizon, beyond which multiple copies exist but have measure zero on finite lattices.^[24]^[25] In lattice QCD, the Landau gauge facilitates computations of gluon and ghost propagators, revealing infrared behaviors like scaling (Z(p^2) \sim (p^2)^{2\kappa} with \kappa \approx 0.595) or decoupling solutions that inform confinement mechanisms and the Kugo-Ojima criterion for color confinement. It also supports studies of chiral symmetry breaking, quark-gluon vertices, and renormalization constants through gauge-fixed simulations.^[25] The gauge's advantages include algorithmic implementability via steepest descent, overrelaxation, or evolutionary optimization methods, allowing efficient handling of large lattices and enabling ratios of gauge-dependent quantities that approximate gauge-invariant observables.^[24]

Less common gauge choices

Weyl gauge

The Weyl gauge, also known as the temporal gauge, is a non-covariant gauge-fixing condition in which the temporal component of the gauge field is set to zero, A^0 = 0. This choice, often combined with additional constraints on the spatial components such as the Coulomb condition \partial_i A^i = 0, simplifies the equations of motion for static or time-independent configurations, reducing the dynamics to the spatial vector potential and facilitating Hamiltonian quantization where spatial components serve as coordinates and color electric fields as momenta.^[27] The residual gauge freedom in the Weyl gauge is parameterized by a time-independent spatial function \chi(\mathbf{x}), under which the spatial components transform as A^i \to A^i + \partial^i \chi, preserving A^0 = 0. This leaves a condition on the divergence, \partial_i A^i = f(\chi), where f arises from the transformation, typically requiring further fixation like the Coulomb condition to eliminate remaining ambiguities.^[27] In cosmological applications, the Weyl gauge aids in analyzing metric potentials by aligning the temporal slicing with background expansion, simplifying perturbation equations for scalar modes in Friedmann-Lemaître-Robertson-Walker spacetimes.^[28] Despite its advantages, the Weyl gauge breaks manifest Lorentz invariance by privileging a time direction, complicating relativistic formulations and introducing longitudinal photon modes that demand careful treatment in quantization. It is also susceptible to Gribov ambiguities and interpretive challenges in curved spacetimes, where the preferred frame may conflict with general covariance.^[27] Historically, the Weyl gauge is associated with early explorations in unified theories, bridging classical geometry and field dynamics. In contemporary research, it supports studies in gauge/gravity duality, where non-covariant fixings like A^0 = 0 align bulk gauge fields with boundary conditions in anti-de Sitter spacetimes.^[9]

Axial gauge

The axial gauge is a non-covariant gauge fixing condition in quantum field theory, defined by imposing n \cdot A^a = 0, where A^a is the gauge field, a labels the color or charge index, and n is a fixed spacelike four-vector with n^2 \neq 0.^[29] A special case is the light-cone gauge, where n is light-like (n^2 = 0), often chosen as n = (1, 0, 0, -1) in Minkowski space with metric signature (+, -, -, -).^[30] This gauge choice eliminates one spatial component of the gauge field along the direction specified by n, simplifying the structure of interactions in non-Abelian theories like quantum chromodynamics (QCD).^[29] In the path integral formulation, the Faddeev-Popov determinant for the axial gauge reduces to a form without ghost-gauge boson vertices, as the ghost action becomes S_{FP} = \int \zeta^a (n \cdot \partial) \eta^a, decoupling ghosts from physical processes at tree level and beyond in certain approximations.^[29] This property makes the gauge particularly useful for perturbative calculations, as it removes certain diagrammatic contributions involving ghosts.^[31] The axial gauge offers advantages in high-energy physics by avoiding small denominators that arise in other gauges during perturbation theory, particularly when dealing with collinear singularities.^[32] It aligns naturally with the infinite-momentum frame, facilitating calculations in the parton model by suppressing interactions that would otherwise mix longitudinal and transverse gluon exchanges.^[33] In this frame, the gauge simplifies the evaluation of leading-logarithmic contributions, enhancing computational efficiency for processes involving fast-moving partons.^[32] A key feature is the form of the gauge field propagator in momentum space. For the general axial gauge, it is given by

D_{\mu\nu}^{ab}(p) = -\frac{i \delta^{ab}}{p^2} \left[ g_{\mu\nu} - \frac{n_\mu p_\nu + n_\nu p_\mu}{n \cdot p} + \frac{n^2 p_\mu p_\nu}{(n \cdot p)^2} \right],

which satisfies n^\mu D_{\mu\nu} = 0.^[29] In the light-cone gauge, where n^2 = 0, this simplifies to a two-term expression often used in practice:

d_{\mu\nu}(k) = g_{\mu\nu} - \frac{n_\mu k_\nu}{n \cdot k},

shifting the poles away from the physical region and aiding in the resummation of soft and collinear emissions.^[30] Despite these benefits, the axial gauge introduces singularities when n \cdot k = 0, corresponding to momenta parallel to n, which can lead to spurious poles in loop integrals and require regularization techniques such as the Mandelstam-Leibbrandt prescription.^[34] These singularities complicate higher-order computations and necessitate careful handling to preserve gauge invariance.^[30] Applications of the axial gauge are prominent in deep inelastic scattering (DIS), where it simplifies the computation of structure functions by isolating quark-parton contributions and minimizing gluon exchange effects in the leading twist approximation.^[35] It is also employed in transverse momentum dependent (TMD) physics within hadron physics, enabling the factorization of TMD parton distribution functions in processes like semi-inclusive DIS, though modern treatments often incorporate non-light-like variants to address evolution equations.^[36] These uses highlight its role in bridging perturbative QCD with non-perturbative hadron structure, an area where further developments are ongoing.^[37]

Fock–Schwinger gauge

The Fock–Schwinger gauge is defined by the condition x^\mu A_\mu = 0, where x^\mu represents the spacetime position vector relative to an external source, and A_\mu is the four-potential of the electromagnetic field. This gauge was introduced by Vladimir Fock in 1937 in the context of proper time methods for relativistic quantum mechanics, and further developed by Julian Schwinger in 1951 within quantum electrodynamics to address gauge invariance in vacuum polarization calculations.^[38]^[39] The choice of origin at the external source location makes the gauge particularly suited for theories with prescribed currents or fields, adapting the potential configuration to the source geometry. Key properties of the Fock–Schwinger gauge include its adaptation to inhomogeneous external fields, where it enforces a position-dependent constraint that aligns the potential with the field's spatial structure. In the presence of currents, this condition effectively eliminates the scalar potential component along the position vector from the source, simplifying the vector potential's role in describing transverse field effects.^[40] The gauge preserves certain gauge freedoms in sourced theories but breaks translational invariance due to the fixed origin, rendering propagators non-local in position space. A primary advantage of the Fock–Schwinger gauge lies in its simplification of equations of motion for charged particles interacting with intense external fields, such as those in laser-plasma environments, by expressing the potential directly in terms of field strengths without residual gauge ambiguities. It also ensures gauge invariance for observables like pair production rates in strong-field quantum electrodynamics (QED), facilitating exact non-perturbative computations via proper-time methods. Under a gauge transformation to this condition, the four-potential can be expressed as A_\mu = \int G_{\mu\nu} J^\nu \, d\tau, where G_{\mu\nu} is the Green function satisfying the gauge constraint and J^\nu is the external current, with the integral over proper time \tau incorporating the source distribution. Despite these benefits, the Fock–Schwinger gauge is inherently non-local because the position-dependent condition leads to Green functions that couple distant points via the fixed origin, complicating momentum-space analyses.^[41] It is unsuitable for free-field theories lacking a preferred source position, as the arbitrary origin introduces unphysical artifacts without enhancing calculational efficiency. Applications of the Fock–Schwinger gauge are prominent in strong-field QED, where it simplifies the treatment of electron dynamics in intense laser fields by decoupling longitudinal and transverse components, enabling precise predictions of nonlinear Compton scattering and pair production. In condensed matter physics, it aids modeling of Dirac fermions in graphene subjected to electromagnetic fields, capturing gauge-invariant responses like chiral magnetic effects under strong backgrounds.^[42] These uses highlight its role in bridging fundamental QED with applied scenarios involving external sources.

Dirac gauge

The Dirac gauge is an unconventional gauge-fixing condition in which the electromagnetic four-potential A^\mu satisfies A_\mu A^\mu = -k^2, where k = (m_0 c^2)/e relates the electron rest mass m_0, speed of light c, and charge e, thereby ascribing physical significance to the potential. Proposed by Paul A. M. Dirac in 1951 as part of a classical theory of electrons to resolve ambiguities in the uncharged electromagnetic field, this approach extends to constrained Hamiltonian systems by eliminating redundant degrees of freedom while preserving Lorentz invariance in form.^[43] This gauge quantizes the gauge freedom in a discrete manner by constraining the potential's norm to a fixed value, thereby avoiding the continuous ambiguities inherent in standard gauge choices and facilitating a transition to quantum descriptions. It proves particularly useful in Dirac's Hamiltonian formulation of quantum electrodynamics (QED), where the subsidiary condition simplifies the treatment of charged particle dynamics without invoking infinite self-energies.^[43] Key advantages of the Dirac gauge include its elegant handling of first-class constraints in both QED and general relativity, where it eliminates singularities. This avoids the need for auxiliary fields in perturbation theory and supports Dirac's broader constrained dynamics framework.^[43] Despite these strengths, the Dirac gauge remains abstract and less practical for numerical simulations due to its nonlinear constraint and potential non-causality in relativistic settings, limiting its adoption beyond theoretical explorations. Primarily theoretical, it finds applications in constrained Hamiltonian dynamics for singular Lagrangians, early quantum gravity models via Dirac's quantization procedures, and illuminating Dirac's foundational contributions to gauge-invariant formulations in field theory.^[43]

Multipolar gauge

The multipolar gauge, also referred to as the Poincaré gauge, is a gauge-fixing condition in classical and quantum electrodynamics tailored for multipole expansions in radiation and molecular physics. It sets the vector potential \vec{A}(\vec{r}, t) such that \vec{r} \cdot \vec{A}(\vec{r}, t) = 0 at each point \vec{r}, effectively truncating the longitudinal component beyond the dipole term and expressing the potentials in terms of the physical electric \vec{E} and magnetic \vec{B} fields alone. This formulation emerged in the post-1960s development of quantum optics, building on the Power-Zienau-Woolley (PZW) transformation introduced in 1959 to handle light-matter interactions in atomic systems.^[44]^[45] Key properties of the multipolar gauge include its ability to localize charge distributions within atoms or molecules, separating the interaction into a polarization field \vec{P} and the external electromagnetic field, which simplifies the Hamiltonian to involve only transverse physical modes without spurious longitudinal contributions. In covariant extensions, it maintains gauge invariance for bound systems while facilitating the dipole approximation \vec{P}(\vec{r}) = \vec{d} \delta(\vec{r}) for point-like charges. Unlike the Coulomb gauge, which separates instantaneous Coulomb interactions, the multipolar gauge incorporates radiation aspects by integrating fields along lines from the origin, aiding in the treatment of retarded potentials in scattering processes.^[46]^[47]^[45] The primary advantages lie in simplifying long-wavelength approximations, where the laser wavelength far exceeds molecular sizes, by eliminating non-physical A^2 terms and enabling efficient numerical simulations of electron dynamics. It proves particularly useful for laser-molecule interactions, such as in strong-field ionization, where it enhances convergence in time-dependent Schrödinger equation solvers compared to velocity-gauge formulations.^[48]^[45]^[46] A foundational equation enforcing the gauge is the condition

\vec{r} \cdot \vec{A}(\vec{r}, t) = 0,

derived from a unitary transformation R = \exp\left[-i \int d^3x \, \vec{P}(\vec{x}) \cdot \vec{A}_T(\vec{x})\right] applied to the Coulomb-gauge Hamiltonian, yielding the multipolar form \hat{H}_M = \hat{H}_C + \hat{H}_{\text{int}} with interaction \hat{H}_{\text{int}} = -\int d^3x \, \vec{P}(\vec{x}) \cdot \vec{E}_\perp(\vec{x}, t).^[44]^[46]^[47] Limitations include its approximate nature for finite-sized systems, where higher multipoles become necessary, and potential breakdowns at short distances or high intensities when the long-wavelength assumption \omega d \ll c (with d the system size) fails, leading to non-local effects and gauge inconsistencies in ultrashort pulse regimes.^[48]^[46]^[45] Applications span quantum optics and photoionization processes, including cavity quantum electrodynamics for entanglement in dipole systems and above-threshold ionization in laser-dressed molecules, where it provides a rigorous framework absent in more abstract gauges.^[49]^[45]^[46]

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