Is there some fundamental reason observables should be…
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Is there some fundamental reason observables should be equivalent to continuous transformations?
In (continuum) classical mechanics, observables are functions on phase space. By adding in the poisson bracket, these observables turn into a Lie algebra which generates continuous transformations of your physical system.
Similarly, in quantum mechanics, observables are hermitian operators. By treating the commutator as a Lie bracket, we get a Lie algebra that generates continuous transformations of the physical system.
Based on those examples, it seems like there's a kind of duality between observables and continuous transformations. I understand the math behind this, but I'm curious if anyone has any physical justification for *why* this should be the case.
If I were living in a cave with no knowledge of our universe's physics, trying to dream up some alternate world's physics, is there some physical postulate that would force me to introduce the observable/transformation duality into my theory to get a consistent set of physical laws?
#science
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In (continuum) classical mechanics, observables are functions on phase space. By adding in the poisson bracket, these observables turn into a Lie algebra which generates continuous transformations of your physical system.
Similarly, in quantum mechanics, observables are hermitian operators. By treating the commutator as a Lie bracket, we get a Lie algebra that generates continuous transformations of the physical system.
Based on those examples, it seems like there's a kind of duality between observables and continuous transformations. I understand the math behind this, but I'm curious if anyone has any physical justification for *why* this should be the case.
If I were living in a cave with no knowledge of our universe's physics, trying to dream up some alternate world's physics, is there some physical postulate that would force me to introduce the observable/transformation duality into my theory to get a consistent set of physical laws?
#science
source
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