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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 ... 2 Consider $$M_{31} = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -i & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} \text{ and } P_0 = -i \begin{pmatrix} 0 & 0 & 0 & 0 & 1 ... 2 You may always promote "couplings constants" (charge, mass, etc...) to fields. Now, as a physicist, you need to make some contact with reality. So you have to tell why and which field you are using (for instance the Higgs field (up to a constant), which has a SU(2) charge, is used to replace a constant mass coupling in the interaction m (\bar e_R e_L + ... 1 The Einstein equivalence principle states : The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime. Emphasis added. Note that this principle has done well in explaining quite a few things about gravity. So there is no a priori reason why you ... 0 Every Lie group has a set of generators, and typically a group element is found by exponentiate (linear combinations) of these generators. Since the fundamental definition of say SU(N) [similarly SO(N)] is something like The group of unitary (orthogonal), n by n matrices with unit determinant then, the fundamental representation is given by ... 3 Group actions in classical field theory. Let a classical theory of fields \Phi:M\to V be given, where M is a base" manifold and V is some target space, which, for the sake of simplicity and pedagogical clarity, we take to be a vector space. Let \mathscr F denote the set of admissible field configurations, namely the set of admissible functions ... 4 First of all, we are dealing with unitary representations, so that the T^as are always self-adjoint and the representations have the form$$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 ... 2 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 ... 2 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) + ... -1 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$(since it lives ... 1 For the relation between the abstract position basis and the$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 ...