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This comes from the microscopic origin of the model. For example, in the case of the hydrogen atom, the dipole operator is given by (up to some signs) $\hat d=e \hat z E$ where I have assumed that the electric field is in the direction $z$, and $\hat z$ is the position operator of the electron (of charge $-e$). Let's now have a look at the effects of the ...

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The statement you cited does not imply that a complex representation of a gauge group implies a chiral gauge theory in general. This only holds true if the gauge group corresponds to a chiral symmetry in the first place. A chirally symmetric theory contains massless fermions. Regarding your counterexample: it is true that QCD contains fermions in the ...

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The process could in general take place. A sample diagram is: $\hspace{4cm}$ With regards to Parity: Assuming the electron-positron pair don't have any angular momentum, the initial Parity is $-1$. Assuming the $\eta_C\eta_C$ pair don't have any angular momentum, their Parity is $+1$. Thus in this case the reaction cannot occur. If we assume the ...

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Now when we operate parity operator, does that mean we are taking any physical entity at x to −x. Or we are just reverting axes of the co-ordinate system? Well, either operation should adhere to the same rules, and you mention the correct term: it depends on whether we see the operation as active or passive. Either view has the same end result: we move ...

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I slightly deviate from your notation and use $\phi$ to denote the scalar field as its more standard. Also I should point out that quantum fields are operators and thus under a transformation they get acted on from both the left and the right. The complex scalar field is given by, \phi (x) = \int \frac{ \,d^3p }{ (2\pi)^3 } \frac{1}{ ...

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I believe I'm ready to answer my own question. The pin group can alternately be defined as the set of all invertible elements $S_{\Lambda} \in \mathrm{Cl}(p,q)$ satisfying $S_{\Lambda} S_{\Lambda}^{\tau} = \pm 1$ and $$\alpha(S_{\Lambda}) \gamma^a S_{\Lambda}^{-1} = {\Lambda^a}_b \gamma^b$$ for some element ${\Lambda^a}_b \in \mathrm{O}(p,q)$. The map ...

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