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Work in the coordinate representation, $\hat O(\hat{\mathbf{r}}) |x\rangle=O(\mathbf{r})|x\rangle$, so, then, [\hat{\mathbf{p}}, \hat{U}_0] \mapsto -i\hbar {U}_0( {U}_0^\dagger \nabla_{\mathbf{r}} ... • 65.2k 0 votes ### Schrodinger basis representation of modes in 2D CFT (Polchinski section 2.8) \psi(x) are represented by a= (ix+p)/\sqrt{2} and a^\dagger = (ix-p)/\sqrt{2}. Here p= i\partial_x when acting on wave functions. In the Schr"odinger picture we thus have \begin{align} \... • 3,136 0 votes ### Schrodinger basis representation of modes in 2D CFT (Polchinski section 2.8) Yes, the second term in eq. (2.8.25a) is needed in order for the commutation relations\begin{align}[X_n,X_m]~=~&0, \cr [\frac{\partial}{\partial X_n},X_m]~=~&\delta^n_m, \cr [\frac{\partial}...
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Yes, they commute. However, it's not correct to talk about the commutator $[A,B]$ because these are operators defined on different Hilbert spaces. When you have a system of two particles, the state of ...