Faddeev–Popov ghost
Quantum field theory |
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History |
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (February 2020) |
In
A more general meaning of the word "ghost" in theoretical physics is discussed in Ghost (physics).
Overcounting in Feynman path integrals
The necessity for Faddeev–Popov ghosts follows from the requirement that
Faddeev–Popov procedure
It is possible, however, to modify the action, such that methods such as Feynman diagrams will be applicable by adding ghost fields which break the gauge symmetry. The ghost fields do not correspond to any real particles in external states: they appear as virtual particles in Feynman diagrams – or as the absence of some gauge configurations. However, they are a necessary computational tool to preserve unitarity.
The exact form or formulation of ghosts is dependent on the particular
Consider for example non-Abelian gauge theory with
The integral needs to be constrained via gauge-fixing via to integrate only over physically distinct configurations. Following Faddeev and Popov, this constraint can be applied by inserting
into the integral. denotes the gauge-fixed field.[3]
Spin–statistics relation violated
The Faddeev–Popov ghosts violate the spin–statistics relation, which is another reason why they are often regarded as "non-physical" particles.
For example, in
In general,
Gauge fields and associated ghost fields
Every gauge field has an associated ghost, and where the gauge field acquires a mass via the
Appearance in Feynman diagrams
In
Ghost field Lagrangian
The Lagrangian for the ghost fields in Yang–Mills theories (where is an index in the adjoint representation of the
The first term is a kinetic term like for regular complex scalar fields, and the second term describes the interaction with the
Footnotes
- ^ Feynman discovered empirically that "boxing" and simply dismissing these diagrams restored unitarity. "Because, unfortunately, I also discovered in the process that the trouble is present in the Yang−Mills theory; and, secondly, I have incidentally discovered a tree−ring connection which is of very great interest and importance in the meson theories and so on. And so I'm stuck to have to continue this investigation, and of course you appreciate that this is the secret reason for doing any work, no matter how absurd and irrational and academic it looks: we all realize that no matter how small a thing is, if it has physical interest and is thought about carefully enough, you're bound to think of something that's good for something else."[4]
References
External links
- Faddeev, Ludwig Dmitrievich (2009). "Faddeev-Popov ghosts". .