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Let me try to explain a vox populi secret. If you have a two sphere, $S^2$, and define a vector on a point $p$ (you can imagine a vector as an arrow), the vector does not lie on the sphere!. In fact the set of all possible vectors at that point generate (or live) in a flat and tangent space to the sphere at $p$, denoted as $T_p S^2$. First note: On $T_p ...


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Consider $$M_{31} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -i & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} \text{ and } P_0 = -i \begin{pmatrix} 0 & 0 & 0 & 0 & 1 ...


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You may always promote "couplings constants" (charge, mass, etc...) to fields. Now, as a physicist, you need to make some contact with reality. So you have to tell why and which field you are using (for instance the Higgs field (up to a constant), which has a $SU(2)$ charge, is used to replace a constant mass coupling in the interaction $m (\bar e_R e_L + ...


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The Einstein equivalence principle states : The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime. Emphasis added. Note that this principle has done well in explaining quite a few things about gravity. So there is no a priori reason why you ...


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Every Lie group has a set of generators, and typically a group element is found by exponentiate (linear combinations) of these generators. Since the fundamental definition of say $SU(N)$ [similarly $SO(N)$] is something like The group of unitary (orthogonal), $n$ by $n$ matrices with unit determinant then, the fundamental representation is given by ...


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Group actions in classical field theory. Let a classical theory of fields $\Phi:M\to V$ be given, where $M$ is a ``base" manifold and $V$ is some target space, which, for the sake of simplicity and pedagogical clarity, we take to be a vector space. Let $\mathscr F$ denote the set of admissible field configurations, namely the set of admissible functions ...


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First of all, we are dealing with unitary representations, so that the $T^a$s are always self-adjoint and the representations have the form $$U(v) = e^{i \sum_{a=1}^Nv^a T_a}$$ with $v \in \mathbb R^N$. When you say that $U$ is real you just mean that the representation is made of the very real, unitary, $n\times n$ matrices $U$. This way, the condition ...


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Just following up on Frederic's answer. I wouldn't get too hung up on thinking about a metric that includes $\eta_{-1-1}$. The metric really refers to spacetime and there is no new spacetime dimension we've introduced. It's just a way of labeling the generators - their indices don't necessarily refer to spacetime dimensions, though they do for most of ...


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General remarks on flows and their generators. Let an $\epsilon$-parameter flow $\Phi(\epsilon):\mathbb R^d\to \mathbb R^d$ be given. Let the flow be defined on some $\epsilon$-neighborhood containing $0$. Provided the flow is sufficiently smooth, we can expand the flow in the parameter $\epsilon$; \begin{align} \Phi(\epsilon) = \Phi(0) + ...


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Ok, my first attemp at answering this was a failure, let's try again: $$x'_\mu = x_\mu + a_\mu$$ is a finite translation, so you should not expect the $a_\mu$ to be the generator. You are totally right, since the infinitesimal translation is $x'_\mu = x_\mu + \mathrm{i}\partial_\mu$, the generator of translations is $\mathrm{i}\partial_\mu$ (since it lives ...


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For the relation between the abstract position basis and the $L^2$ spaces, I refer you to my answer here (read the other answers too, they're good ;) ) You are quite close with your understanding of the representations, but not quite there: First of all, for the 2-dim spin-$\frac{1}{2}$ Hilbert space $\mathcal{H}_{\uparrow\downarrow}$, the set of ...



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