Ghosts lurking around in advanced physics

Jack Dikian

February 2012

Lurking deep at the core of Gauge Quantum Field Theory are two Russian ghosts called Faddeev and Popov - somewhat reminiscent of the ghost monsters in the old video game; Munchkin. Luckily for us, these ghosts are also thought to be “good ghosts”. The bad ghosts, which apparently also exist in the strange and schizophrenic world of quantum mechanics are another thing altogether.

As the theory goes, these ghosts where send for so as to avoid the over-counting in the complex area of theoretical physics, to fix gauges and make path integrals right.

In high school, we were taught classical mechanics (remember fiction, momentum, force, gravitation, etc) and one of the things we learned was that the path of say a moving object (think of a ball thrown across the room) was that for which the action is a single unique trajectory, i.e., a stationary action [formulation].

In the world of the quantum mechanics, however, the stationary action formulation of classical mechanics extends to the path integral formulation, where a physical system follows simultaneously an infinity of possible trajectories with associated probability amplitudes for each path being determined by the action for the path. Now, path integral formulation should yield unambiguous, non-singular solutions; which they don’t. To modify the action such that these calculations yield applicable results, the good ghosts are used to break the gauge symmetries and make things right.

So every gauge field has an associated ghost, and where the gauge field acquires a mass, the associated ghost field acquires the same mass in some cases.