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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?