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 violatÃon 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!
Best Answer
I might be wrong in this, but despite the similiarities, these two things you describe are sternly different I think.
Your first point is related to the so-called "general covariance". It is something that is in effect all the time. You don't see any kind of coordinate grid anywhere when you look outside the window, and you don't see any spatial or temporal point designated as a "starting point" (ignoring stuff like the big bang now etc.), therefore, it is only logical that such constructs exist only to help describing stuff mathematically, so physics should be independent of coordinates.
The second thing you say doesn't always happen. Example, if you have an explicitly time-dependent Lagrangian, then time displacements will NOT leave the Lagrangian invariant, and energy won't be conserved (with that said, in reality, energy IS conserved, but for example, if you have friction then you generally say it is not conserved, since it gets removed from the sum of kinetic and potential energies).
Likewise, if you have a spherically symmetric potential field, then rotations of the physical system will leave the Lagrangian invariant, since the potential field is equal for all rotations around the fixed origin. BUT if you have a cylindrically symmetric potential field, whose axis is the $z$ axis, then rotations around $z$ will leave the Lagrangian invariant, but rotations around the $x$ or $y$ axes will NOT leave the Lagrangian invariant.