The discussion revolves around the conditions under which a quantum mechanical observable is invariant under a set of transformations described by symmetry generators. Participants explore the definitions and requirements for invariance, particularly focusing on the mathematical relationships involved.
Participants generally agree that the observable must commute with the symmetry generators for invariance, but there is uncertainty regarding the specifics of the observable's definition and the implications of this condition.
Some participants express confusion about the nature of the observable and the symmetry generators, indicating that additional context or definitions may be necessary for a complete understanding.
This discussion may be useful for those studying quantum mechanics, particularly in the context of symmetries and observables, as well as for individuals seeking clarification on the mathematical framework involved.
It's just a quantum mechanical observable, A. I have no more information than that about it. I'm not sure what ##G_i## is, so I'm not sure if it can be written like that. Strange question really, can't seem to find the answer on google. :)dextercioby said:
Yeah, good plan. Quite a lot of textbooks needed, I think. Thanks for your help! :)strangerep said:The observable is represented as an operator "##A##" (say) on Hilbert space. The symmetry generators ##G_i## are represented as operators on the same Hilbert space. The notion that the observable is invariant under those symmetries is implemented by requiring ##[A, G_i] = 0##. I.e., the observable operator must commute with the symmetry generators.
My only other suggestion is: "get thee to a copy of Ballentine" (quickly).