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Kara386
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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?
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:How is the observable defined? Can it be written in terms of Gi?
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).
Invariance of an observable A refers to the property of the observable remaining unchanged under a given transformation. In other words, the value of the observable A does not depend on the choice of coordinates or reference frame.
The most common types of transformations that can affect the invariance of an observable A are rotations, translations, and boosts. These transformations can change the coordinates or reference frame, potentially altering the value of the observable A.
Invariance of an observable A is important because it allows for consistency and reliability in scientific measurements. If an observable is not invariant, it may vary depending on the chosen coordinates or reference frame, leading to inconsistent or inaccurate results.
Invariance of an observable A is mathematically represented by a transformation law, which describes how the observable changes under a given transformation. This allows for the calculation of the observable in different reference frames, ensuring its invariance.
No, not all observables are invariant. Some observables, such as position and momentum, are invariant, while others, such as energy, are not. It depends on the specific observable and the type of transformation being applied.