# How could I derive the Noether charge for a real scalar field?

I know for a (free) complex scalar field $$\psi$$ the Lagrangian is: $$\mathcal{L} = \partial^\mu \psi^\ast\partial_\mu \psi$$ and that Noether's theorem from the $$U(1)$$ symmetry of the system gives a conserved current $$j_\mu \propto iq(\psi\partial_\mu\psi^\ast-\psi^\ast\partial_\mu\psi)$$, which can be interpreted as the difference of the number of particles and anti-particles and hence as the conservation of electrical charge.

For real scalar field, though, I would have: $$\mathcal{L} = \partial^\mu \phi\partial_\mu \phi$$ so there is not $$U(1)$$ symmtry... but I still expect particle number to be conserved? Shouldn't particle number be the conserved charge?

• Hint: note that $\mathcal{L} = \frac{1}{2} \partial^\mu \phi\partial_\mu \phi$ is invariant under spacetime symmetries i.e. the Poincaré symmetry-group; compute the noether's conserved current/charges for that group – user164843 Apr 14 '19 at 0:08
• yeah but that's just energy and momentum. I meant does particle number never come up as a conserved charge? – SuperCiocia Apr 14 '19 at 0:14
• $\mathcal{L} = \partial^\mu \phi\partial_\mu \phi$ defines an free classical field theory, and therefore the number of particles is arbitrary, however after the procedure of quantization, $\hat{N}=\hat{a}^{\dagger}\hat{a}$ (particle number operator) commutes with $\hat{H}$ and implies that the number of particles is conserved; also, i find an related question here physics.stackexchange.com/questions/332189/… – user164843 Apr 14 '19 at 1:07