Introduction to Gauge Fixing
– Gauge fixing is a mathematical procedure in gauge theories to handle redundant degrees of freedom in field variables.
– A gauge theory represents physically distinct configurations as equivalence classes of local field configurations.
– Gauge transformations relate detailed configurations in the same equivalence class, involving shear along unphysical axes.
– Quantitative predictions in gauge theories require suppressing or ignoring these unphysical degrees of freedom.
– Gauge fixing simplifies calculations but becomes more challenging in realistic models, especially in quantum field theory.
Gauge Fixing in Electrodynamics
– Electrodynamics is an archetypical gauge theory, described by the Heaviside-Gibbs formulation.
– The electric and magnetic fields in Maxwell’s equations contain only physical degrees of freedom.
– Field strengths can be expressed in terms of the electric scalar potential and the magnetic vector potential.
– Gauge transformations of the vector potential do not change the magnetic field.
– Gauge transformations of both the vector potential and the scalar potential leave the electric field unchanged.
Different Gauges in Electromagnetism
– Gauge fixing involves choosing a particular scalar and vector potential as a gauge.
– The U(1) gauge freedom in electromagnetism allows for arbitrary numbers of gauge functions.
– The Coulomb gauge is used in quantum chemistry and condensed matter physics, defined by the condition that the divergence of the vector potential is zero.
– The Lorentz gauge is another gauge condition used in classical electromagnetism, characterized by the condition that the divergence of the vector potential plus the time derivative of the scalar potential is zero.
– The Coulomb gauge is minimal and complete, while the Lorentz gauge retains manifest Lorentz invariance but is incomplete.
Gauge Freedom and Gauge Invariance
– Gauge freedom refers to the arbitrariness in choosing the gauge function in the electromagnetic potentials.
– A gauge transformation can be made to change from one gauge to another.
– Pure gauge fields are the derivatives of the gauge function and have no effect on physical observables.
– Gauge invariance refers to quantities or expressions that do not depend on the gauge function.
– All physical observables are required to be gauge invariant.
Comparison and Usage in Quantum Field Theories
– Lorentz covariant gauges, such as the Lorentz gauge, are usually used in relativistic quantum field theories.
– The Coulomb gauge is not Lorentz covariant and requires additional gauge transformations after a Lorentz transformation.
– The choice of gauge depends on the specific requirements of the theory and the physical phenomena being studied.
– Gauges are a generalization of the Lorentz gauge applicable to theories expressed in terms of an action principle with Lagrangian density.
– An equivalent formulation of gauge uses an auxiliary field, a scalar field with no independent dynamics. Source: https://en.wikipedia.org/wiki/Coulomb_Gauge
In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.
Although the unphysical axes in the space of detailed configurations are a fundamental property of the physical model, there is no special set of directions "perpendicular" to them. Hence there is an enormous amount of freedom involved in taking a "cross section" representing each physical configuration by a particular detailed configuration (or even a weighted distribution of them). Judicious gauge fixing can simplify calculations immensely, but becomes progressively harder as the physical model becomes more realistic; its application to quantum field theory is fraught with complications related to renormalization, especially when the computation is continued to higher orders. Historically, the search for logically consistent and computationally tractable gauge fixing procedures, and efforts to demonstrate their equivalence in the face of a bewildering variety of technical difficulties, has been a major driver of mathematical physics from the late nineteenth century to the present.[citation needed]