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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?

Qmechanic
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SuperCiocia
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1 Answers1

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Conservation of charge does not mean conservation of particle number. It means conservation of total charge during interactions. Conservation of particle number is automatic if there are no other fields that your complex scalar could interact with, in which case conservation of energy-momentum leads to conservation of particle number. Similarly, in the real scalar field case with no other fields, particle number is conserved because energy-momentum is conserved.

By the way, there is a shift-symmetry in your Lagrangian of the real scalar. Although you can break it by adding a mass term.

Kosm
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