I Is Observable A Invariant Under Symmetry Generators G_i?

[Kara386]

How do I know if an observable is invariant, specifically under some set of transformations described via the generators $G_i$? Which conditions would this observable have to fulfil?

 

[dextercioby]

How is the observable defined? Can it be written in terms of G_i?

 

[Kara386]

How is the observable defined? Can it be written in terms of G_i?

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. :)

 

[Kara386]

Apparently the answer is that the observable must commute. With what exactly I don't know, but there we are!

 

[strangerep]

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).

 

[Kara386]

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).

Yeah, good plan. Quite a lot of textbooks needed, I think. Thanks for your help! :)