Timeline for Simultaneous transformations on states and observables which leave predictions invariant
Current License: CC BY-SA 4.0
16 events
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| Nov 25, 2022 at 16:53 | comment | added | Norbert Schuch | After some extra thought, I would say that showing that $\varphi$ is an algebra isomorphism is the only real problem: If it is an isomorphism of the positive cone, then by linearity it is an isomorphism on the full matrix algebra (which is finite dimensional). However, isomorphisms of the full matrix algebra are always of the form $\varphi(X)=AXA^{-1}$ by the Skolem-Noether theorem, and that $A$ is unitary follows since $\varphi$ maps hermitian operators to hermitian operators. | |
| Nov 24, 2022 at 8:50 | comment | added | Tobias Fünke | To make sure I understand your edit: If $\varphi \circ \pi$ is a representation, then there is a unitary $U$ (since now $\varphi\circ \pi$ and $\pi$ are quasi-equivalent (and assuming an irreducible rep) s.t. $(\varphi \circ\pi)(A) = U\pi(A)U^\dagger$ and if $(^\dagger \circ \varphi\circ\pi)(A)$ is a representation, then there exists a $U$ such that $(\varphi\circ\pi)(A) =U \pi(A)^\dagger U^\dagger$. This exhausts all possibilities, assuming $\varphi$ is surjective. And is it true that if $\pi$ is irep., then so is $\varphi\circ \pi$? | |
| Nov 24, 2022 at 8:50 | comment | added | ACuriousMind♦ | @RyderRude I mean the algebra to which the SvN theorem applies, which consists of the exponentiated form of the Heisenberg algebra, i.e. the algebra generated by $\{1,\mathrm{e}^{\mathrm{i}xs},\mathrm{e}^{\mathrm{i}pt}\}$ where $s,t\in\mathbb{R}$. | |
| Nov 24, 2022 at 6:19 | vote | accept | Tobias Fünke | ||
| Nov 24, 2022 at 5:52 | comment | added | Ryder Rude | Also, would you say that the Clifford Algebra of Pauli Matrices is a special case of this $C^*$ algebra? | |
| Nov 24, 2022 at 5:28 | comment | added | Ryder Rude | By "algebra of canonical commutation relations", do you mean the Lie bracket of the Heisenberg algebra? Is it that, to abstractly define quantum theories, we postulate that there is a group $C^*$ with an additional operation defined : the Lie bracket of the Heisenberg alegebra? | |
| Nov 24, 2022 at 0:17 | comment | added | Norbert Schuch | ... I was already doubting my mind.) | |
| Nov 24, 2022 at 0:15 | comment | added | Norbert Schuch | Thanks, that indeed helps. Showing that $\varphi\circ\pi$ is a representation is indeed the key point. Yet (sorry!), you are assuming that $\varphi$ is surjective (on the density operators, or a suitable subspace). (It was a bit along the lines of what I had in mind, but I indeed did (and do) not see how to get rid of the trace, without knowing that $\varphi$ is surjective.) Furthermore, is it clear that a solution has to be either a homomorphism or antihomomorphism -- all you show (leaving the surjectivity aside) is that it is linear on the vector space. (P.S.: Happy you see the point, ... | |
| Nov 23, 2022 at 23:43 | comment | added | ACuriousMind♦ | @NorbertSchuch I've finally understood your point about the vagueness of $\varphi$ in the question and I've added some paragraphs discussing that. | |
| Nov 23, 2022 at 23:42 | history | edited | ACuriousMind♦ | CC BY-SA 4.0 |
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| S Nov 23, 2022 at 23:25 | history | mod moved comments to chat | |||
| S Nov 23, 2022 at 23:25 | comment | added | ACuriousMind♦ | Comments are not for extended discussion; this conversation has been moved to chat. | |
| Nov 23, 2022 at 23:25 | history | edited | ACuriousMind♦ | CC BY-SA 4.0 |
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| Nov 23, 2022 at 22:24 | comment | added | Norbert Schuch | @ACuriousMind Ok, then I'm not sure I understand your answer. Do you claim it answers the question as it is stated, without using anything else not stated in the question? (And if yes, how does it rule out dual representations -- because these are included in the question, as is complex conjugation.) My feeling is that the representation structure of $(H,\pi)$ is not implied by the question as stated. | |
| Nov 23, 2022 at 19:13 | vote | accept | Tobias Fünke | ||
| Nov 23, 2022 at 20:57 | |||||
| Nov 23, 2022 at 17:30 | history | answered | ACuriousMind♦ | CC BY-SA 4.0 |