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(Disclaimer: The more rigourously inclined individual may be better suited by looking at the Stone-von Neumann theorem, as Qmechanic notes) One can deduce that the momentum operator takes the form $\hat p = -\mathrm{i}\hbar\partial_x$ in the position representation from the fact that the momentum operator generates the infinitesimal translations as ...

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I) Comment to the question (v1): The Schrödinger position representation $$\hat{p}_k ~=~ \frac{\hbar}{i} \frac{\partial }{\partial x^k}, \qquad \hat{x}^j ~=~x^j,$$ correctly reproduces the canonical commutation relations $$[\hat{x}^j,\hat{p}_k ]~=~i\hbar ~\delta^j_k ~{\bf 1},$$ while the proposal $$\hat{p}_k ~=~ \frac{\hbar}{i} \frac{1}{x^k}, ... 1 The answer to both questions is that D act on Hilbert space states. I'll answer them in reverse order. what exactly do we mean by the eigenvectors of D? Are they fields in space-time? No, in this context, eigenvectors of D are states living in the Hilbert space of the field theory. Because it is only in this sense that the commutation relations between ... 4 The commutation relations$$ [D,P_{\mu}] = +i P_{\mu} , \qquad [D,K_{\mu}] = -i K_{\mu} $$show that P_{\mu} and K_{\mu} raise and lower the conformal dimension of a state. In other words, if you have a state |\phi\rangle of dimensions \Delta, so that D\, |\phi\rangle = i\Delta |\phi\rangle, then$$ D \, P_{\mu} \, |\phi\rangle = [D,P_{\mu}]\, ...

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In addition to Qmechanic's interesting answer, you might want to explicitly see that $\mathrm{SU(2)}$ only has three independent variables. Let us start by writing the $U \in \mathrm{SU(2)}$ as: $$U=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ with $a,b,c,d \in \mathbb{C}$. Since we are looking at the special unitary ...

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I) Quantum mechanically, the Lie group associated with rotational symmetry is $$G~:=~Spin(3)~\cong~ SU(2),$$ which is a double cover of $SO(3)$, and has a 3-dimensional real Lie algebra $$L~:=~so(3)~\cong~ su(2)$$ with generators $J_i$ satisfying $$[J_i,J_j]~=~i\hbar\sum_{k=1}^3\epsilon_{ijk} J_k.$$ II) In quantum mechanics, we are interested in Lie ...

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I) Before we start let us briefly recall certain aspects of the formalism from Ref. 1. The Minkowski sign convention is $(+,-,-,-)$. The 'phase space' measure for a particle $A$ is $$d\tilde{k}~:=~ \frac{d^3k}{(2\pi)^3 2\omega_{k,A}} ~=~ \frac{d^4k}{(2\pi)^3} \delta(k^2-m^2_A)\theta(k^0),$$ $$\omega_{k,A}~:=~\sqrt{{\bf k}^2+m^2_A}~>0~. \tag{3-35}$$ ...

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You obtain this by Wick's Theorem, which can be stated as $$T\{\phi_1\phi_2...\phi_n\}=N\{\phi_1\phi_2...\phi_n+\sum\text{all possible contractions of }\phi_1\phi_2...\phi_n\}$$ where N is the normal ordering operator which puts all the daggered fields on the left ( for example ...

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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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