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Explicit form of the translation operator generators in the poincarePoincare group?

Let $P_0$ be the generator for temporal translation and $P_1, P_2, P_3$ be for spatial translations. Let $p_μ$ be the momentum operator in the $x_μ=x^μ$ direction. I watched a lecture where the guy said that $P_μ=p_μ=i\eta^{μ \nu}\partial_\nu$ (this is all in planck units for simplicity and $\eta^{μ\nu}$ is the (+,-,-,-)$(+,-,-,-)$ Minkowski metric) and for μ=0, P=H$μ=0, P=H$ where H$H$ is the Hamiltonian operator (which makes sense to me because of the schrodingerSchrodinger equation). But, with a little wikipedia research, I found that the spatial translation operator is $T=exp(-ix^μp_μ)$$T=\exp(-ix^μp_μ)$ and the temporal translation operator is $T=exp(iHt)$$T=\exp(iHt)$. So shouldntshouldn't the translation generators of the Poincare group then be $$P_μ=i\eta^{μ\nu}x_μp_{\nu}$$$$P_μ=i\eta^{μ\nu}x_μp_{\nu}~?$$ What am I missing here?

Explicit form of the translation operator generators in the poincare group?

Let $P_0$ be the generator for temporal translation and $P_1, P_2, P_3$ be for spatial translations. Let $p_μ$ be the momentum operator in the $x_μ=x^μ$ direction. I watched a lecture where the guy said that $P_μ=p_μ=i\eta^{μ \nu}\partial_\nu$ (this is all in planck units for simplicity and $\eta^{μ\nu}$ is the (+,-,-,-) Minkowski metric) and for μ=0, P=H where H is the Hamiltonian operator (which makes sense to me because of the schrodinger equation). But, with a little wikipedia research, I found that the spatial translation operator is $T=exp(-ix^μp_μ)$ and the temporal translation operator is $T=exp(iHt)$. So shouldnt the translation generators of the Poincare group then be $$P_μ=i\eta^{μ\nu}x_μp_{\nu}$$ What am I missing here?

Explicit form of the translation operator generators in the Poincare group?

Let $P_0$ be the generator for temporal translation and $P_1, P_2, P_3$ be for spatial translations. Let $p_μ$ be the momentum operator in the $x_μ=x^μ$ direction. I watched a lecture where the guy said that $P_μ=p_μ=i\eta^{μ \nu}\partial_\nu$ (this is all in planck units for simplicity and $\eta^{μ\nu}$ is the $(+,-,-,-)$ Minkowski metric) and for $μ=0, P=H$ where $H$ is the Hamiltonian operator (which makes sense to me because of the Schrodinger equation). But, with a little wikipedia research, I found that the spatial translation operator is $T=\exp(-ix^μp_μ)$ and the temporal translation operator is $T=\exp(iHt)$. So shouldn't the translation generators of the Poincare group then be $$P_μ=i\eta^{μ\nu}x_μp_{\nu}~?$$ What am I missing here?

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Explicit form of the translation operator generators in the poincare group?

Let $P_0$ be the generator for temporal translation and $P_1, P_2, P_3$ be for spatial translations. Let $p_μ$ be the momentum operator in the $x_μ=x^μ$ direction. I watched a lecture where the guy said that $P_μ=p_μ=i\eta^{μ \nu}\partial_\nu$ (this is all in planck units for simplicity and $\eta^{μ\nu}$ is the (+,-,-,-) Minkowski metric) and for μ=0, P=H where H is the Hamiltonian operator (which makes sense to me because of the schrodinger equation). But, with a little wikipedia research, I found that the spatial translation operator is $T=exp(-ix^μp_μ)$ and the temporal translation operator is $T=exp(iHt)$. So shouldnt the translation generators of the Poincare group then be $$P_μ=i\eta^{μ\nu}x_μp_{\nu}$$ What am I missing here?