Timeline for Meaning of the term 'states carry an irreducible representation of a group $G$'
Current License: CC BY-SA 4.0
17 events
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| Aug 24, 2021 at 20:03 | comment | added | DIRAC1930 | So we have to explicitly specify which irreps the state space carries? I.e. for the (n,m) representation of the Lorentz group we have to specify that the one-particle Hilbert space carries an irrep of $(0,0)$, $(0,1/2)$, $(1/2,0)$ etc. however does not carry a known irrep for $(n,m)$ such that $n,m > 3/2$? | |
| Aug 24, 2021 at 16:40 | comment | added | ZeroTheHero | I don’t know of any particles with $j=72/2$. They may have angular momentum (spatial degree of freedom) that much, but not spin (internal degree of freedom). There is nothing that mathematically prevents particles with $s=36$, but we don't see them so any model with that contents is tossed aside. | |
| Aug 24, 2021 at 16:26 | comment | added | DIRAC1930 | I'm just trying to figure out how a physical theory is built from symmetries and at the moment I'm confused because the rules seem arbitrary. For example, a single particle can have $j=72/2$ but it can't have $s=73/2$. Yet colloquially, we talk about them in exactly the same way. | |
| Aug 24, 2021 at 16:19 | comment | added | ZeroTheHero | It seems you’re looking for a math answer to a physics problem… | |
| Aug 24, 2021 at 16:18 | comment | added | ZeroTheHero | but why would that contain all representations? There are no particles with spin $73/2$…. Moreover there are plenty of models with symmetries that don’t refer to particles: the harmonic oscillator states with fixed number $N$ of total excitations carry irreps of U(N), and irreps of Sp(2N,R) for variable N. | |
| Aug 24, 2021 at 16:14 | comment | added | DIRAC1930 | A state-space that describes every single particle that can exist in the universe. | |
| Aug 24, 2021 at 16:11 | comment | added | ZeroTheHero | what is the state space of the Universe? | |
| Aug 24, 2021 at 15:32 | comment | added | DIRAC1930 | But doesn't the state space of the universe carry all the irreducible representations? When we focus on the $s=1/2$ sector, i.e. the state space of the Pauli Hamiltonian, it only carries an irrep of the $s=1/2$ $SU(2)$ representation. | |
| Aug 24, 2021 at 15:29 | comment | added | ZeroTheHero | @DIRAC1930 We must be talking at cross purposes. Why do you insist in 2.? Of course not. The Hilbert space need not contain all the representations. The Hilbert space for a single spin-1/2 particle is 2-dimensional, not infinite-dimensional. The Hilbert space for a 1-D harmonic oscillator contains a single $sp(2,\mathbb{R})$ representations, not all of them... | |
| Aug 24, 2021 at 15:21 | comment | added | DIRAC1930 | Thanks for this. I've updated my question (see Edit). Could you clear up my confusions in the edit if possible? | |
| Aug 24, 2021 at 14:54 | comment | added | ZeroTheHero | see physics.stackexchange.com/a/594553/36194 and associated reference: Gatland, I.R., 2006. Integer versus half-integer angular momentum. American journal of physics, 74(3), pp.191-192. | |
| Aug 24, 2021 at 14:22 | comment | added | DIRAC1930 | Why isn't spin represented on $L^2$ functions like $SO(3)$ is? | |
| Aug 24, 2021 at 3:25 | history | edited | ZeroTheHero | CC BY-SA 4.0 |
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| Aug 24, 2021 at 3:09 | comment | added | ZeroTheHero | Yes but (2) must hold, and it’s nontrivial to find $T_{ij}(g)$ that will then make (1) and (2) hold. | |
| Aug 24, 2021 at 3:08 | history | edited | ZeroTheHero | CC BY-SA 4.0 |
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| Aug 24, 2021 at 1:33 | comment | added | DIRAC1930 | But the equation $T^\lambda(g_1) T^\lambda(g_2) | \psi_k \rangle = T^\lambda(g) | \psi_k \rangle$ is true for any vector $| \psi_k \rangle$ with the same dimension as $T^\lambda(g_i)$. | |
| Aug 23, 2021 at 23:19 | history | answered | ZeroTheHero | CC BY-SA 4.0 |