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We assume that OP asks apart from the facts that: Dirac representations by definition are complex; It is much easier to work with an algebraically closed field; Any real representation can be extended to a (possibly reducible) complex representation, so one is not missing anything by going complex. In other words, OP is interested in why certain real ...


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Generically, given a representation $V$ of a group, the tensor representation $V\otimes V$ will decompose into the symmetric and antisymmetric parts $$ V\otimes V = \Lambda^2 V \oplus S^2 V$$ and in the case of the rotation group (or its universal cover), the symmetric 2-tensors have a certain invariant under rotations - their trace! So when $j_1 = j_2$, the ...


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The claim is that $|0(\theta)>$ lies outside the Hilbert space built on the original vacuum |0>. To check that this true, consider the overlap of the new vacuum $|0(\theta)>$ and the (unnormalized) basis states $(a_k^\dagger)^n|0>$ generated from |0>, taking into account that $a_k(\theta) = a_k + \theta_k$, $a_k = a_k(\theta) - \theta_k$, and ...


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The defining property of the fundamental representation of the Lorentz group $\mathrm{SO}(1,3)$ $$ M^T\eta M = \eta \quad \forall M\in\mathrm{SO}(1,3)$$ and hence the defining property of the Lorentz group itself does not make sense in representations other than the fundamental, because those are not naturally equipped with a metric "$\eta$" from a physics ...


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Taking the trace of an operator over all the states/particles effectively means taking a sum over all the eigenvalues of the operator (the charges) over these states/particles. So what matters is how many states you have and with what charges. The numbers that you are referring to are just the appropriate multiplicities of the states. For example the first ...


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This question inspired me to try to write a conceptual introduction at the wikipedia article. To save you the trouble of clicking, I copied it below. (It's slightly inspired by what @Kostia wrote here) Motivating example: Position operator matrix elements for 4d→2s transition Let's say we want to calculate transition dipole moments for an electron to ...


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I have worked recently on something related to this and I have understood that if you have a time evolution that respects a certain symmetry group, then the state space breaks down into its algebra multiplets. For instance, if you have 2 bosonic modes, you can recover the angular momentum algebra su(2) by labelling them by the total photon number and by ...


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Yes you would need a new representation. The reason is the Lorentz group is connected with the group of translations and as you point out the time-translation is different. If $K^i$ is the generator of boosts in the $i$ direction and $P^j$ the generator of space translations (i.e. the linear momentum) $$[K^i,P^j]=iH\delta_{ij}$$ So if the interacting ...



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