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The classical expression for the total momentum operator is $$P^{i} = -\int d^3x \, \pi(x) \, \partial_{i} \phi(x),$$

which, after second quantisation, using $$\hat{\phi}(x) = \int \frac{d^3k}{(2 \pi)^3} \, \frac{1}{ \sqrt{2 E_{k}}} \left( \hat{a}_{k} + \hat{a}^{\dagger}_{-k} \right) e^{i k \cdot x}$$ and $$\hat{\pi}(x) = \int \frac{d^3k}{(2 \pi)^3} \, \sqrt{\frac{E_{k}}{2}} \left( \hat{a}_{k} - \hat{a}^{\dagger}_{-k} \right) e^{i k \cdot x},$$

it becomes: $$\int \frac{d^3p}{(2\pi)^3} \, p^i \left( \hat{a}_{p}^{\dagger} \hat{a}_{p} + \frac12 (2\pi)^3 \delta^{(3)}(0) \right).$$

What is the significance of the diverging second term?

I know we get a similar expression for the ground state energy, which we just ignore by arguing that absolute energy is not an observable since we can only measure energy differences. But momentum is surely an observable? We can measure absolute momentum yes?

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    That second term is not a divergence. You have an odd integral so it vanishes. – FenderLesPaul Dec 30 '14 at 18:43
  • You are integrating the momentum $\vec{p}$ over all directions so it adds up to zero. More explicitly you can send $\vec{p} \rightarrow -\vec{p}$ which gives $\int d^3p \vec{p}\delta^3(0) = -\int d^3p \vec{p}\delta^3(0) \Rightarrow \int d^3p \vec{p} \delta^3(0) = 0$. – FenderLesPaul Dec 30 '14 at 19:03
  • I am not sure about this argument. It may arise confusion because $\delta(\textbf{0})$ being $\infty$ and the momentum integral being $0$ lead to $\infty \cdot 0$. – rainman Oct 30 '21 at 22:23

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Well, quantization of a classical system may not be a unique procedure. Classically, all variables commute, and, say, $\pi ~\partial_i \phi = \partial_i\phi~\pi.$

When we quantize, why should we choose $\hat{\pi}\hat{\phi}$ ordering, as OP does in his example? Why not instead the opposite $\hat{\phi}\hat{\pi}$ ordering? Or perhaps symmetrize? Or use normal ordering? Or something else?

As FenderLesPaul points out in a comment, the operator ordering ambiguity happens to disappear in OP's case. But this is not always the case.

More generally, during quantization, we often replace classical expressions with normal ordered expression plus intercept parameters that parametrize our ignorance on how to order operators. Consistency and other physical principles might later fix these ambiguities for us.

See also my Phys.SE answer here.

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