Four-momentum of a free real scalar field ($P^{\mu }=\int T^{0\mu }{d}^{3}x$)

For the free real scalar field with signature $(+,-,-,-)$,The canonical stress-energy tensor is $T^{\mu \nu }={\partial }^{\mu }\varphi {\partial }^{\nu }\varphi -{\eta }^{\mu \nu }\mathcal{L}$. Its divergence vanishes by the Klein-Gordon equation, giving constant integrated charges when integrals and boundary flux are well behaved. The minus sign in contravariant spatial momentum is from ${\partial }^i=-{\partial }_i$; lower-index spatial charges have the opposite sign. A plane wave $\cos (Et-\mathbf{k}\cdot \mathbf{x})$ has momentum density pointing along $\mathbf{k}$.

For a canonically quantized real scalar field, the quadratic Hamiltonian contains $\frac{1}{2}{\sum }_{\mathbf{p}}{E}_{\mathbf{p}}$ in a regulated box. Normal ordering subtracts this zero-point energy, leavingThe vacuum momentum is zero with an inversion-symmetric regulator. The canonical commutation relation then gives $[P^{\mu },a_{\mathbf{p}}^{†}]=p^{\mu }{a}_{\mathbf{p}}^{†}$, identifying the momentum and energy added by one particle. The un-subtracted energy differs by a constant and is not literally the same operator.

The canonical commutation relation gives $[P^{\mu },a_p^{†}]=p^{\mu }{a}_p^{†}$ for a free real scalar field. Differentiate the exponential conjugation: its derivative is a scalar multiple of the same operator, yielding the displayed phase. The annihilation operator receives the opposite phase, so at equal time $e^{-i\mathbf{P}\cdot \mathbf{y}}\varphi (\mathbf{x})e^{i\mathbf{P}\cdot \mathbf{y}}=\varphi (\mathbf{x}+\mathbf{y})$. The spatial translation operator translates a state's position wavefunction in the opposite argument direction.