We know that the laws of physics are invariant under proper orthochronous Lorentz transformations.
How did we come to this knowledge? Is it an experimental fact that has not been violated, or can it be shown mathematically that this is true for all the known forces?
Is it an experimental fact that has not been violated, or can it be shown mathematically that this is true for all the known forces?
This question sets up a false dichotomy: If you believe that "The laws of physics are Lorentz invariant" is a statement about reality, then you can never show it purely "mathematically" - mathematics alone can never derive statements about physical reality. If you believe that "The laws of physics are Lorentz invariant" is a statement about the mathematical structure of physical theories, then this cannot be an experimental fact, since experiments investigate reality, not the mathematical structure of our models of it.
The problem, ultimately, is that while this statement sounds nice, it doesn't actually mean anything without a lot more context:
What is a law of physics? I would suggest that it is a "law of physics" that electrons and positrons annihilate when they meet each other. There is no evident sense in which this statement "is Lorentz invariant" or "is not Lorentz invariant" - it is just a statement about what can happen when an electron and a positron meet each other. The usual choice here is to clarify that one is talking about the equations of motion when one says "laws of physics", possibly adding further modifiers about the theory having to be "fundamental" to avoid running into the problem that Newtonian mechanics a) is a physical theory with laws of physics and b) is not Lorentz invariant.
Invariance vs. covariance. The equations of motion are not, in fact, Lorentz invariant, as is obvious from the covariant form of the Lorentz force law. The Lorentz force law is covariant, not invariant, under Lorentz transformations, in much the same way as any equation between vectors is covariant, not invariant, under rotations in non-relativistic 3d physics. The focus on the phrase "Lorentz invariance" is just a weird linguistic quirk - few people go around hailing "rotation invariance" as the essential feature of Newtonian mechanics.
That last analogy makes clear what we're actually trying to say: The point is that the Lorentz transformations are a symmetry of our theory, just like ordinary 3d rotations are often a symmetry of ordinary 3d physics, and the thing that is invariant under symmetries are not some vague "laws of physics", but the action of our theory.
"The action is invariant under Lorentz transformations" is clearly a statement about the mathematics of our theory - it is perfectly possible to suggest theories in which this is not true. Given an action, you can purely mathematically show whether or not an action is Lorentz invariant, and this doesn't really have anything to do with experimental physics. Lorentz invariance becomes a statement about experimental reality when we claim that the Lorentz invariant action we've written down correctly models reality.
The sense in which "Lorentz invariance" is experimentally testable is therefore this: We start with a Lorentz invariant theory in the above mathematical sense and a modified version of it that is not Lorentz invariant, schematically by taking our usual relativistic theories with the Minkowski metric $\eta^{\mu\nu}$ and perturbing this to a small $\eta^{\mu\nu} + \epsilon h^{\mu\nu}$ where $h$ is such that the Lorentz transformations are not isometries of it. Then we do an experiment for a situation where the two theories predict different results and fit $\epsilon$ to our data - if $\epsilon = 0$ is consistent with our experimental results, then this is a successful "test of Lorentz invariance".
