Axial GaugeEdit

Axial gauge is a class of gauge-fixing conditions used in gauge theorys, defined by selecting a fixed four-vector n_mu and imposing the constraint n_mu A^mu = 0 on the gauge field. This choice singles out a direction in spacetime and, in doing so, can make the physical content of the gauge field more transparent in certain high-energy or fast-moving settings. In its broad form, axial gauge refers to any gauge fixing that uses a nonzero vector n_mu, while the special case n^2 = 0 corresponds to the popular light-cone gauge often employed in perturbative calculations. Because the gauge condition is not Lorentz-invariant, axial gauge can simplify some problems at the cost of complicating others, especially when one needs results to be manifestly covariant or when dealing with non-perturbative phenomena.

In abelian theories like quantum electrodynamics the axial gauge can be implemented without introducing unphysical degrees of freedom, and in non-abelian theories such as quantum chromodynamics the gauge choice has a more nuanced effect. A notable feature is that the Faddeev–Popov determinant, which encodes the gauge fixing in the quantization procedure, decouples in many axial-gauge formulations, effectively removing the need to introduce ghost fields in perturbation theory. This is appealing for people who prize a theory in which only the physical two transverse polarizations of the gauge boson appear explicitly in calculations. However, axial gauge introduces noncovariant structures and potential singularities that must be handled with care, making the approach a tradeoff between conceptual elegance and technical complexity.

Axial gauge remains a practical choice in a number of contexts, particularly in the analysis of high-energy scattering and jet physics where the focus is on physical, gauge-independent observables and on the simplification of certain operator structures. In these settings, the gauge condition helps to isolate the physical degrees of freedom more directly and can lead to streamlined factorization and power-counting arguments. Yet, because the gauge condition depends on the fixed vector n_mu, results for intermediate quantities can be frame- and gauge-dependent, even while final predictions for measurable quantities are gauge invariant.

Formalism and properties

Gauge condition and field decomposition

In axial gauge the condition n_mu A^mu = 0 is imposed, with n_mu taken as a fixed, nonzero vector. This choice eliminates one component of the gauge field, leaving the remaining components constrained by the dynamics and symmetries of the theory. In practice, calculations separate the transverse physical polarizations from the rest, which can yield a more transparent view of how high-energy quanta propagate along the direction singled out by n_mu. See gauge field and propagator for related background.

Propagator structure and spurious poles

The gauge-fixed propagator in axial gauge takes a form that projects onto the subspace transverse to n_mu, but it contains denominators of the type n·k that can produce unphysical singularities when the momentum k aligns with the fixed direction n. Handling these spurious poles is essential for consistent results in loop calculations. The two standard approaches are the Leibhardt–Mandelstam prescription, designed to preserve causality and gauge invariance of physical observables, and the principal-value prescription, used in some contexts to regulate the pole in a Lorentz-compatible way. See Leibbrandt–Mandelstam prescription and principal value for details.

Ghosts and determinants

A defining practical feature is that, in many axial-gauge implementations, the Faddeev–Popov determinant does not introduce dynamical ghost fields. This can simplify perturbative computations because the path integral does not require additional ghost propagators. Nevertheless, the gauge-fixing condition introduces noncovariant elements into the formalism, which can complicate renormalization and the treatment of boundary terms in nonperturbative contexts. See Faddeev–Popov ghost and renormalization for related discussion.

Benefits in practice

Physical-content clarity: By focusing on the two transverse polarizations, axial gauge can make it easier to interpret certain results in terms of observable gluon or photon degrees of freedom. See physical degree of freedom for context.
Ghost decoupling in perturbation theory: The absence (or simplification) of ghost contributions in many calculations reduces algebraic clutter and can speed up perturbative work. See gauge fixing and gauge theory for background.
Utility in specific kinematic setups: In high-energy or light-front-inspired analyses, axial gauge selections align with particle trajectories and can simplify factorization schemes. See factorization (particle physics) for related concepts.

Limitations and controversies

Lack of manifest covariance: The presence of a fixed vector n_mu means that expressions can depend on the choice of n_mu, complicating cross-checks of Lorentz invariance and demanding careful treatment when comparing results in different frames. See Lorentz invariance and covariance for discussions.
Spurious singularities and regularization: The n·k poles require careful prescription, and different prescriptions can lead to subtle differences in intermediate steps, even if final physical quantities agree. See regularization and prescription (physics).
Residual gauge freedom and boundary conditions: In some implementations, there can be residual gauge freedoms that survive the axial condition, requiring attention to boundary terms and large-gauge transformations. See gauge freedom and boundary conditions.
Nonperturbative challenges: For questions beyond perturbation theory, the noncovariant structure of axial gauge can hinder nonperturbative methods and lattice approaches, which often favor covariant or Coulomb-like gauges. See lattice gauge theory for related considerations.

Comparisons with other gauges

Lorenz/Lorentz gauge (∂_μ A^μ = 0): Covariant and widely used, especially when manifest Lorentz invariance is advantageous. Axial gauge trades covariance for potential simplifications in a particular high-energy context. See Lorenz gauge.
Coulomb gauge (∇·A = 0): Emphasizes instantaneous interactions in certain settings, with a different balance of physical transparency and calculational complexity. See Coulomb gauge.
Light-cone gauge: Special case of axial gauge with n^2 = 0, extensively used in perturbative QCD and in light-front quantization. It shares the spurious-pole issue and requires careful prescriptions, but often aligns well with jet and parton analyses. See light-cone gauge.

Applications in modern theory often reflect a pragmatic blend: axial gauge offers a path to simpler intermediate expressions in some calculations while demanding vigilance about gauge dependence and regularization. The choice of gauge is frequently dictated by the problem at hand, the desired balance between physical transparency and mathematical control, and the available toolkit for handling singularities and renormalization. See gauge fixing and quantum field theory for broader context.