Timeline for Does there exist finite dimensional irreducible rep. of Poincare group where translations act nontrivially?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| S Dec 3, 2020 at 15:53 | history | suggested | Nihar Karve | CC BY-SA 4.0 |
Minor typos
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| Dec 3, 2020 at 15:32 | review | Suggested edits | |||
| S Dec 3, 2020 at 15:53 | |||||
| Jan 31, 2020 at 13:51 | comment | added | Noix07 | $g\in\mathcal{P}$ can be written as $𝑔′ \cdot 𝑔′=𝑔$ so that the representation has to be unitary ($\rho(g)$ is a square $=\rho(g')\circ \rho(g')$ and cannot be antiunitary). I may have been a little too elliptical about Wigner's theorem: the action has to preserve transition probabilities. | |
| Jan 31, 2020 at 13:37 | comment | added | Noix07 | "Hence the field transforms in a finite dimensional representation $\sigma_{\text{fin}}$": what is transformed? the field. In what space? the space of fields, i.e. of functions. So, as it does not been explicitly mentionned, the reason one is interested in unitary representations of the Poincaré group has to do with the axiom of quantum mechanics that says that states are described by rays. Then by Wigner's theorem, projective representations on ray may be lifted to true representation, either unitary or antiunitary. And finally, for the connected subgroup of Poincaré containing the identity.. | |
| Jan 30, 2020 at 16:17 | comment | added | ACuriousMind♦ | @Noix07 $V$ is not the space of fields but the finite-dimensional target space of the fields. You're right that there is an infinite-dimensional representation on the space of functions but I don't talk about that in this answer | |
| Jan 30, 2020 at 16:11 | comment | added | Noix07 | @ACuriousMind Once again, interesting and does point to the crucial fact that requiring unitarity has to do with quantum theory. But again, the representation $\sigma_{\text{fin}}$ of translation on the field is obviously infinite dimensional!! (Unless I'm missing an argument saying the the space of fields is finite dimensional...) | |
| Apr 22, 2016 at 4:15 | vote | accept | 346699 | ||
| Apr 8, 2016 at 13:03 | comment | added | 346699 | Yes. we don't need to consider this case in physics. The question 2 is only a mathematical interest. | |
| Apr 8, 2016 at 11:52 | comment | added | ACuriousMind♦ | @user34669: I have no idea, except that they are probably physically irrelevant. We don't need any non-trivial finite-dimensional representations of translations since the fields always have to transform trivially. | |
| Apr 8, 2016 at 5:06 | comment | added | 346699 | Thanks. According to your answer, I have understood my question 1,3. Do you have any idea about question 2? | |
| Apr 7, 2016 at 18:33 | history | answered | ACuriousMind♦ | CC BY-SA 3.0 |