Ghost (physics)

In quantum field theory, a ghost, ghost field, ghost particle, or gauge ghost refers to an unphysical state in a gauge theory. These ghosts are introduced to maintain gauge invariance in theories where the local field components exceeds the number of physical degrees of freedom. Ghosts ensure mathematical consistency in gauge theories.

If a given theory is self-consistent by the introduction of ghosts, these states are labeled "good". Good ghosts are virtual particles that are introduced for regularization, like Faddeev–Popov ghosts. Otherwise, "bad" ghosts admit undesired non-virtual states in a theory, like Pauli–Villars ghosts that introduce particles with negative kinetic energy.

An example of the need of ghost fields is the photon, which is usually described by a four-component vector potential A_μ, even if light has only two allowed polarizations in the vacuum. To remove the unphysical degrees of freedom, it is necessary to enforce some restrictions; one way to do this reduction is to introduce some ghost field in the theory. While it is not always necessary to add ghosts to quantize the electromagnetic field, ghost fields are strictly needed to consistently and rigorously quantize non-Abelian Yang–Mills theory, such as done with BRST quantization.^[1][2]

A field with a negative ghost number (the number of ghosts excitations in the field) is called an anti-ghost.

Some theories quantum gravity are usually discarded for having the wrong kind of ghosts, like in quadratic gravity.^[3] Theories to make sense of these ghosts also exist.^[3]