What is the meaning of relativistic invariance?

It is usually assumed that in order to be relativistically invariant a physical theory must be formulated in terms of manifestly covariant quantities, i.e., quantities that transform as scalars, vectors, tensors, etc. with respect to boost transformations. For example, relativistic quantum theories of single particles were initially formulated by using wave equations (Klein-Gordon, Dirac, etc.) in which the time and position variables were formally equivalent. We already discussed that the tensor transformation laws of observables and the formal equivalence of the time and position coordinates are approximations. Then what is the exact meaning of relativistic invariance? How one can say whether a theory is relativistically invariant or not?

The key to understanding the relativistic invariance is not in the formal equivalence of space and time coordinates. The key is in the (Poincare) group properties of the ten types of inertial transformations (1 time translation, 3 space translations, 3 rotations, and 3 boosts). A quantum theory if fully relativistic if there is an unitary representation of the Poincare group in the Hilbert space of the system, and if the generators H for time translations, P for space translations and J for spatial rotations of this representation are identified with the observables of the total energy, total momentum and total angular momentum, repsectively, of the system. A relativistic classical theory is defined similarly. The difference is that instead of the "Hilbert space" one should use the "phase space", and instead of "unitary transformations" one should use "contact transformations", i.e., those conserving the Poisson brackets.