Is a Time Rotation Operator Analogous to Spatial Rotations Possible?

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The discussion centers on the concept of a time rotation operator and its analogy to spatial rotations. It is established that a time rotation operator cannot exist independently of spatial dimensions, as time is a single dimension. Instead, Lorentz boosts serve as the appropriate analogy, combining time and space dimensions to illustrate how transformations occur in relativistic physics. This conclusion emphasizes the interdependence of time and space in the context of rotations.

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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?
 
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Lorentz boost seems what you are looking for.
 
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phyahmad said:
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).
 
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