Why is the spacelike conserved charge due to spacetime translations the momentum?

Question

Whilst reading several books on QFT, I have come across the derivation of the conserved charges due to the symmetry under spacetime-translations. I can follow the derivations, and have that the conserved charges are

$$P^{\alpha}=\int \mathrm{d}^3xT^{0\alpha}~,$$

where $T^{\mu\nu}$ is the energy-momentum tensor. The time-like component gives straightforwardly that

$$P^0 = \int\mathrm{d}^3x~T^{00} = \int\mathrm{d}^3x~\mathcal{H}~,$$

and the space-like components give

$$P^k = \int\mathrm{d}^3x~T^{0k} = \int\mathrm{d}^3x~\pi(x)\partial^k\phi(x)~.$$

Which corresponds to the momentum. However, I do not immediately see this. Naively, I would have expected that the momentum would correspond to $\int\mathrm{d}^3x~\pi(x)$, but I would also expect physically that the symmetry due to space-translations would result in conservation of momentum.

In short, I can't seem to connect the above equation for $P^k$, what I would expect the momentum to look like naively. Is there a better way than seeing this than that the momentum is defined as the conserved space-like charge due to spacetime translations?

Answer 1

The key conceptual insight lies in distinguishing between the concepts of "physical space" - the Minkowski space $\mathbb{R}^4$ - and "field space" - the space of possible values that the field can attain, which is $\mathbb{R}$ for a real scalar field, $\mathbb{C}$ for a complex scalar field, etc. Intuitively, "momentum" means "how fast something is moving through space." But which space?

$\pi(x)$ refers to "field momentum" - i.e. the speed with which the value of the scalar field is changing, or "moving through field space." Note that this has nothing to do with spatial translations - it's defined entirely at a single point, and it doesn't matter what's happening at even an infinitesimal distance away. This quantity doesn't have or need a coordinate index, because there's only one "direction" in field space for a scalar field to "move."

On the other hand, $\pi(x) \partial^k \phi(x)$ refers to physical momentum, or the speed with which the value of the field changes as you move in physical space. It needs a coordinate index $k$, because you can move through physical space in any of three different directions. This is naturally related to infinitesimal translations, because you're related the value of the field at one point in physical space to its value at a different point in physical space (even though the second point is only an infinitesimal distance away).

As a very rough analogy, if you think of a field in two spatial dimensions as a rubber sheet, then $\pi(x) \partial^k \phi(x)$ is kind of like the in-plane ("longitudinal") momenta of the sheet's tiny area elements at each point $x$ that compose the sheet. $\pi(x)$ is kind of like the sheet's transverse momenta at each point $x$ - it moves through an entirely different "space" (the $z$-direction) from the 2D space on which the sheet itself is defined.