Stress tensor of complex scalar field

Question

The canonical stress tensor is defined by

$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi_a)}\partial^{\nu}\phi_a-\eta^{\mu\nu}\mathcal{L}.$$

where $\phi_a$ are the different fields in the theory. For the complex scalar field, the Lagrangian is

$$\mathcal{L}=\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi-m^2\phi^{\dagger}\phi$$

from which I calculate (treating $\phi$ and $\phi^{\dagger}$ as two different $\phi_a$)

$$T^{\mu\nu}=\partial^{\mu}\phi^{\dagger}\partial^{\nu}\phi+\partial^{\nu}\phi^{\dagger}\partial^{\mu}\phi-\eta^{\mu\nu}\mathcal{L}.$$

However, my book states the answer as

$$T^{\mu\nu}=2\partial^{\mu}\phi^{\dagger}\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}.$$

Is it obvious that this simplification is allowed (or is my calculation wrong)? To me it seems like $\partial^{\mu}\phi^{\dagger}\partial^{\nu}\phi$ could be complex, but I can't think of a simple example where it is.

Answer 1

Hints:

1. For a complex scalar field, the canonical stress-energy-momentum (SEM) tensor reads $$\begin{align}\mp T^{\mu}{}_{\nu}~=~&\frac{\partial {\cal L}}{\partial {\rm Re}\phi_{\mu}}{\rm Re}\phi_{\nu}+\frac{\partial {\cal L}}{\partial {\rm Im}\phi_{\mu}}{\rm Im}\phi_{\nu} -\delta^{\mu}_{\nu}{\cal L}\cr ~=~&\frac{\partial {\cal L}}{\partial \phi_{\mu}}\phi_{\nu}+\frac{\partial {\cal L}}{\partial \phi^{\ast}_{\mu}}\phi^{\ast}_{\nu} -\delta^{\mu}_{\nu}{\cal L}\end{align}$$ for Minkowski sign convention $(\mp,\pm,\pm,\pm)$, respectively.

2. It follows that the canonical SEM tensor $T^{\mu\nu}=T^{\nu\mu}$ is symmetric for a complex scalar field.