Gauge Fixing - Coulomb Gauge

The Coulomb gauge (also known as the transverse gauge) is much used in quantum chemistry and condensed matter physics and is defined by the gauge condition (more precisely, gauge fixing condition)

It is particularly useful for "semi-classical" calculations in quantum mechanics, in which the vector potential is quantized but the Coulomb interaction is not.

The Coulomb gauge has a number of properties:

  1. The potentials can be expressed in terms of instantaneous values of the fields and densities (in SI units)
    where ρ(r, t) is the electric charge density, R = |r - r'| (where r is any position vector in space and r' a point in the charge or current distribution), the del operates on r and d3r is the volume element at r'. The instantaneous nature of these potentials appears, at first sight, to violate causality, since motions of electric charge or magnetic field appear everywhere instantaneously as changes to the potentials. This is justified by noting that the scalar and vector potentials themselves do not affect the motions of charges, only the combinations of their derivatives that form the electromagnetic field strength. Although one can compute the field strengths explicitly in the Coulomb gauge and demonstrate that changes in them propagate at the speed of light, it is much simpler to observe that the field strengths are unchanged under gauge transformations and to demonstrate causality in the manifestly Lorentz covariant Lorenz gauge described below. Another expression for the vector potential, in terms of the time-retarded electric current density J(r, t), has been obtained to be:
  2. Further gauge transformations that retain the Coulomb gauge condition might be made with gauge functions that satisfy ∇2 ψ = 0, but as the only solution to this equation that vanishes at infinity (where all fields are required to vanish) is ψ(r, t) = 0, no gauge arbitrariness remains. Because of this, the Coulomb gauge is said to be a complete gauge, in contrast to gauges where some gauge arbitrariness remains, like the Lorenz gauge below.
  3. The Coulomb gauge is a minimal gauge in the sense that the integral of A2 over all space is minimal for this gauge: all other gauges give a larger integral. The minimum value given by the Coulomb gauge is
  4. In regions far from electric charge the scalar potential becomes zero. This is known as the radiation gauge. Electromagnetic radiation was first quantized in this gauge.
  5. The Coulomb gauge is not Lorentz covariant. If a Lorentz transformation to a new inertial frame is carried out, a further gauge transformation has to be made to retain the Coulomb gauge condition. Because of this, the Coulomb gauge is not used in covariant perturbation theory, which has become standard for the treatment of relativistic quantum field theories such as quantum electrodynamics. Lorentz covariant gauges such as the Lorenz gauge are used in these theories.
  6. For a uniform and constant magnetic field B the vector potential in the Coulomb gauge is
    which can be confirmed by calculating the div and curl of A. The divergence of A at infinity is a consequence of the unphysical assumption that the magnetic field is uniform throughout the whole of space. Although this vector potential is unrealistic in general it can provide a good approximation to the potential in a finite volume of space in which the magnetic field is uniform.
  7. As a consequence of the considerations above, the electromagnetic potentials may be expressed in their most general forms in terms of the electromagnetic fields as
    where ψ(r, t) is an arbitrary scalar field called the gauge function. The fields that are the derivatives of the gauge function are known as pure gauge fields and the arbitrariness associated with the gauge function is known as gauge freedom. In a calculation that is carried out correctly the pure gauge terms have no effect on any physical observable. A quantity or expression that does not depend on the gauge function is said to be gauge invariant: all physical observables are required to be gauge invariant. A gauge transformation from the Coulomb gauge to another gauge is made by taking the gauge function to be the sum of a specific function which will give the desired gauge transformation and the arbitrary function. If the arbitrary function is then set to zero, the gauge is said to be fixed. Calculations may be carried out in a fixed gauge but must be done in a way that is gauge invariant.

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