I have some trouble understanding what exactly is parity transformation.
The definition of parity transformation is a flip in the sign of all three spatial coordinates, ie $$(x,y,z) \rightarrow (-x,-y,-z).$$
Consider a stationary particle at a position $(a,b,c)$ in space described by a coordinate system $(x,y,z)$. Does parity transformation mean that the particle is still at the exact point in space but its position is now described by $(-a,-b,-c)$?
But often parity is talked about as a mirror reflection and it seems to me that a mirror reflection means physically moving the particle from point $(a,b,c)$ to $(-a,-b,-c)$ in a coordinate system $(x,y,z)$.
Which of the above 2 cases is parity transformation really referring to? If it refers to both cases, why are the two cases the same? In one case a particle is fixed in space while in another case a particle is moved to another point in space.