### 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

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.

*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.

## 3 Comments:

Are there cases, where no such representations of the PG exist? Doesn't at least the trivial representation exist always?

You wrote that, so far, there are no experiments detecting the corrections you have found for the boost transformation of quantum system. May be, such ones are possible for the corresponding corrections for classical systems? There are discussions about the interpretation of certain effects, like the perihelion rotation. Could your corrections of special-relativistic results be within the order of magnitude of such effects?

Peter:

Are there cases, where no such representations of the PG exist?No, there can be no such cases. The existence of a unitary representation of the Poincare group in the Hilbert space of the system is necessary for any relativistic quantum theory. This was demonstrated by Wigner as early as in 1939.

Doesn't at least the trivial representation exist always?The trivial representation of the Poincare group corresponds to the most trivial physical system - the vacuum.

Regarding your question about the perihelion rotation: When we consider the perihelion precession we are talking about the time evolution of Mercury's position. The time evolution is related to the interaction present in the Hamiltonian (in the system Sun-Mercury). These kinds of effects are possible to describe within my approach (though I hesitate to make any claims about gravitational interactions at this moment), and they are relatively easy to observe. On the other hand, the effects related to interactions in the boost operator are very difficult to observe. Their observation requires comparison of measurements in two inertial systems moving with respect to each other. We don't have many experiments like that.

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