I have a general question about a statement from Wikipedia about ghost states as occuring in gauge theory:
"In the terminology of quantum field theory, a ghost, ghost field, or gauge ghost is an unphysical state in a gauge theory."
I learned gauge theory up to now with mainly mathematical beckground. My main reference is this https://arxiv.org/abs/1607.03089 paper by A. Marsh.
Question: What is concretely an "unphysical" state or field from viewpoint of gauge theory?
TL;DR: Ghost states do not belong to the physical Hilbert space.
This begs OP's next question: What exactly is the physical Hilbert space in a gauge theory? If we transcribe the gauge symmetry into a BRST symmetry, we can give a precise technical definition: The physical Hilbert space is given by the BRST-cohomology of zero ghost number.
References:
There are different ways of defining what "unphysical" state is. Often states with negative norm $\langle \Psi | \Psi \rangle \lt 0$ are considered to be unphysical.
A third view on this. Ghosts are fields that are introduced when you remove the gauge redundancy in the path integral.
In a path integral of a gauge theory you integrate over equivalent fields due to the gauge redundancy. You can fix the gauge and isolate the infinite volume of the gauge group but you are left with a determinant that you can then rewrite as the exponential of an action with Grassmann fields. These Grassmann fields are the ghosts. They appear in your theory and interact with the original matter fields of your theory, but resulting type of interaction ensures that they can never appear as external fields. They are not physical fields, i.e. fields that correspond to an observable degree of freedom.
I am aware that the use of the two words path integral may scare off mathematicians, but this is imo the most physical explanation of what ghosts are.
An unphysical state or field in a gauge theory is a ghost, ghost field, ghost particle, or gauge ghost. Ghosts are necessary to keep gauge invariance in theories where the local fields exceed a number of physical degrees of freedom. There are different types of ghosts, such as Faddeev–Popov ghosts, Goldstone bosons, and Pauli–Villars ghosts. Some ghosts are "good", meaning they are introduced for regularization and do not violate unitarity. Some ghosts are "bad", meaning they have negative norm or negative kinetic energy and allow negative probabilities.
