I'm referring here to invariance of the Lagrangian under Lorentz transformations.
There are two possibilities:
- Physics does not depend on the way we describe it (passive symmetry). We can choose whatever inertial frame of reference we like to describe a physical system. For example, we can choose the starting time to be $t_0=0$ or $t_0=4$ (connected by a translation in time $t \rightarrow t' = t + a_0$). Equivalently it does not matter where we put the origin of our coordinate system (connected by a translation in space $x_i \rightarrow x_i' = x_i + a_i$)) or if we use a left-handed or a right-handed coordinate system (connected by a parity transformation). Physics must be independent of such choices and therefore we demand the Lagrangian to be invariant under the corresponding transformations.
- Physics is the same everywhere, at any time (active symmetry). Another perspective would be that translation invariance in time and space means that physics is the same in the whole universe at any time. If our equations are invariant under time translations, the laws of physics were the same $50$ years ago and will be tomorrow. Equations invariant under spatial translations hold at any location. Furthermore, if a given Lagrangian is invariant under parity transformations, any experiment whose outcome depends on this Lagrangian finds the same results as an equivalent, mirrored experiment. A basic assumption of special relativity is that our universe is homogeneous and isotropic and I think this might be where the justification for these active symmetries comes from.
The first possibility is really easy to accept and for quite some time I thought this is why we demand physics to be translation invariant etc.. Nevertheless, we have violatíon of parity. This must be a real thing, i.e. can not mean that physics is different if we observe it in a mirror. Therefore, when we check if a given Lagrangian is invariant under parity, we must transform it by an active transformation and do not only change our way of describing things.
What do we really mean by symmetries of the Lagrangian? Which possibility is correct and why? Any reference to a good discussion of these matters in a book or likewise would be aweseome!