Is a Time Rotation Operator Analogous to Spatial Rotations Possible?

[phyahmad]

We studied about the time translation operator and that its generator is the hamiltonian the question is could there be a time rotation operator in analogy with rotations in space and what would be it's relation to relativity?

 

[anuttarasammyak]

Lorentz boost seems what you are looking for.

 

[PeterDonis]

could there be a time rotation operator in analogy with rotations in space

There can't be a "time rotation operator" in the sense of just "rotating time" and not affecting "space" at all, because (heuristically) "time" is only one dimension and it makes no sense to "rotate" in just one dimension; you need at least two. So you need to include at least one dimension of space in the rotation.

As @anuttarasammyak points out in post #2, Lorentz boosts are the "time-space rotation" analogues of rotations in ordinary space. Boosts require two dimensions, one of "time" and one of "space" (the "space" dimension is the spatial direction in which the boost is applied).