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We have a number m that shows up in a purely mathematical context and then we interpret it as having a physical meaning. The question is: What is the intuition that connects the two? The intuition connecting the two is essentially identical with the reason why Wigner connects the purely physical notion of an elementary particle with the purely ...

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Here we will for simplicity just consider an arbitrary finite-dimensional complex$^1$ semisimple Lie algebra $\mathfrak{g}$. I) One may show that the CSAs are precisely the maximal toral Lie subalgebras of $\mathfrak{g}$. In particular CSAs are abelian. Also the Killing form $\kappa:\mathfrak{g}\times \mathfrak{g}\to \mathbb{C}$ (which is ...

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A representation of the Lorentz group implies a set of matrices $(S_{\mu\nu})_a{}^b$ (one for each $\mu,\nu$) that satisfy the Lorentz algebra, i.e. satisfies a relation of the form (I am not keeping track of signs) $$[S_{\mu\nu} , S_{\rho\sigma} ] = i ( \eta_{\mu\rho} S_{\nu\sigma} + \cdots )$$ What this means is that there is a vector space, with vectors ...

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Notation: I will write a Poincaré transformation as ${x'}^\mu = {\Lambda^\mu}_\nu x^\nu + a^\mu$, the operator representing this transformation on the Hilbert space is $U(\Lambda, a)$. An infinitesimal transformation with ${\Lambda^\mu}_\nu = \delta^\mu_\nu + {\omega^\mu}_\nu$ and $a^\mu = \epsilon^\mu$ can be expanded as  U(\delta + \omega, \epsilon) = 1 ...

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There is a subtle difference between saying $(2,2)$ and $2\otimes 2$. In the latter case we are thinking of both reps as transforming under the same element of the group $SU(2)$. In the former case we are thinking of $(2,2)$ as transforming under the Lorentz group, which contains two distinct copies of $SU(2)$. Call one copy the $L$ copy and the other the ...

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