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The question is to show that the unitary transformation corresponding to spacetime translation [itex]\delta_\alpha[/itex] of a scalar field is [itex]U = e^{-(\mathrm{i}/\hbar) \delta_\alpha P^\alpha }[/itex] where [itex]P^\alpha[/itex] is the energy-momentum 4-vector of the field.

[itex]\varphi (x) \mapsto \varphi'(x') = \varphi(x_\alpha - \delta_\alpha) = U\varphi(x)U^\dag[/itex].

Essentially this boils down to showing that

[itex]\varphi(x_\alpha-\delta_\alpha) = U \varphi(x_\alpha)U^\dag[/itex].

I'm sure I need to use the identity

[itex][P^\alpha, U] = -\mathrm{i}\hbar\frac{\partial U}{\partial x_\alpha}[/itex],

but I'm not sure how to contort it into a form that will give me what I want.

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# Mandl and Shaw 2.5

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