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I'm trying to get a better sense of how the various group theory applications in physics fit together and I have some outstanding issues in my understanding:

The gauge group of the standard model is $SU(3) \times SU(2) \times U(1)$. The symmetries of spacetime are described by the proper Lorentz group $SO^+(3,1)$. 3 of the generators of the defining representation of the Lorentz group share an algebra with $SU(2)$, which describes spin. Does that mean spin comes from the Lorentz group and not from the Standard Model? I know the $SU(2)$ in the standard model describes the weak interactions through isospin, but how does regular spin fit in? Aside from grand unified theories, am I missing any other groups that contribute?

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  • $\begingroup$ en.wikipedia.org/wiki/Poincaré_group $\endgroup$
    – G. Smith
    Aug 26 '20 at 22:40
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    $\begingroup$ Spin comes from Poincaré symmetry (or pedantically from representations of the Poincaré group) and how this is basically unrelated to the gauge group of the SM is the content of the famous no-go theorem of Coleman and Mandula. $\endgroup$
    – DanielC
    Aug 27 '20 at 0:52
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The Lorentz group is a subgroup of the poincare group which is a global space time symmetry. Every fundamental particle lives in an irreducible representation of the Lorentz group, this determines if the particle is a scalar, spin 1/2 fermion, vector, etc..

The standard model gauge groups are internal, local symmetries under which each SM particle transforms. These are completely separate from the Lorentz group and it is up to experiment to determine how each particle transforms (gluons are vectors (space time) and are charged under SU(3) but not SU(2) or U(1))

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I know the SU(2) in the standard model describes the weak interactions through isospin, but how does regular spin fit in?

In curved spacetime, spin (or more precisely the spin current) is coupled to spin connection $\omega^{ab}_\mu$, which is the gauge field of the (double cover of) local Lorentz gauge symmetery $spin(1,3)$. Given that the spacial rotation portion $spin(0,3)$ of local Lorentz symmetry group is isomorphic to $SU(2)$, you can make an analogy between iso-spin/weak interaction $W^a_\mu$ with spin/spin connection interaction $\omega^{ab}_\mu$.

The weak $SU(2)$ covariant derivative of an iso-spin doublet $\psi$ is $$ D_\mu \psi = (\partial_\mu + W^a_\mu T_a)\psi, $$ while the $spin(1,3)$ covariant derivative of a Dirac spinor $\psi$ (a spin doublet) is $$ D_\mu \psi = (\partial_\mu + \omega^{ab}_\mu\gamma_a\gamma_b)\psi, $$ where $\gamma_a$ are Gamma matrices and $\gamma_a\gamma_b (a\neq b)$ are the generators of the $spin(1,3)$ local Lorentz gauge group.

That said, the spin-spin interaction via $\omega^{ab}_\mu$ is so weak that it is not detectable experimentally. Only at extremely high densities, it could be significant in fermionic matter. For example, such an interaction could potentially avert the Big Bang singularity. See here.

In flat/Minkowski spacetime, which is the usual context the Standard Model is discussed, the Lorentz symmetry is global, characterized by zero spin connection $\omega^{ab}_\mu=0$, hence there is no weak-like-gauge-interaction between regular spins. That is the reason spin connection $\omega^{ab}_\mu$ is not normally mentioned in regular quantum field theory books.

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