In Peskin and Schroeder, after having derived a conserved tensor $T^{\mu \nu}$ associated with translations in space-time (the stress-energy tensor), it is said that the charges $\int d^3 x T^{0i}$: $$P^{i} = -\int d^3 \mathbf{x} \hphantom{ii} \pi(\mathbf{x}) \partial_i \phi(\mathbf{x}) $$ are to be "interpreted" as the physical momentum, as opposed to the canonical momentum, whose density is

$$\pi(\mathbf{x})~=~\frac{\partial{\cal L}}{\partial \dot{\phi}(\mathbf{x})}.$$

Is there a way of showing that the above equation does indeed correspond to the physical momentum?

  • 3
    $\begingroup$ Physical momentum is that which is conserved under spatial translations. $\endgroup$ – ACuriousMind Dec 29 '14 at 19:45
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    $\begingroup$ That is the physical interpretation of the stress-energy tensor, as opposed to the canonical momentum, which is just a generalised (co)coordinate $\endgroup$ – Phoenix87 Dec 29 '14 at 19:47

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