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The rotational and spatial symmetries both give currents, namely the components of the energy-momentum and angular momentum tensors. Is there something wrong with them?

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Well, they are the sources of gravity in GR but not of the electromagnetic field. .

Unfortunately I haven't studied GR yet. Could you explain what do you mean by the currents being the sources of the fields?

You need a current of a gauge field.

Buy why is that required? In Tong's notes http://www.damtp.cam.ac.uk/user/tong/qft/six.pdf a condition is derived for the interaction term of the coupling of the EM field to matter, and it just says we have to use a conserved current.

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samalkhaiat

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Not

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Notanyconserved current. The electromagnetic field couples toelectrically chargedfields. The real scalar field iselectrically neutral, because the (real scalar field) Lagrangianis notinvariant under theglobal phase transformation[itex]\exp (i \alpha)[/itex], i.e., there is no conserved Noether current, [itex]J^{\mu}[/itex], associated with the transformation [itex]\varphi (x) \to \exp (i \alpha) \varphi (x)[/itex]. So, in the case of real scalar field, there is no coupling of the form [itex]e A_{\mu} J^{\mu}[/itex].

While that explains the physics of it what I am trying to understand now is a mathematical reason why we need the matter field to be invariant under a phase transformation. I guess the reason is this:

but I don't understand most of it, so I'll wait until I have studied GR and come back here.

Just a final question: from your answers I gather that the reason the real scalar field can't be coupled to the Dirac field is not that there are no conserved currents, as I thought, but that there is no phase symmetry. Is that right?

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samalkhaiat

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While that explains the physics of it what I am trying to understand now is a mathematical reason why we need the matter field to be invariant under a phase transformation.

The electromagnetic field, [itex]A_{\mu}[/itex], is a gauge field. Gauge field can

When you study gauge field theories, you will realise that the gauge fields

Real scalar fieldJust a final question: from your answers I gather that the reason the real scalar field can't be coupled to the Dirac field is not that there are no conserved currents, as I thought, but that there is no phase symmetry. Is that right?

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Real scalar fieldcan(and actuallydoes) couple to Dirac’s fields: [itex]g \ \bar{\psi} (x) \varphi (x) \psi (x)[/itex] is such coupling. The pion [itex]\pi^{0}[/itex], which is described by real scalar field, interacts strongly with the protons and neutrons which are Dirac fields: [itex]g \ \pi^{0} \ (\bar{p}p - \bar{n}n)[/itex]. In this case, coupling to a conserved symmetry current isnot an issuebecause you are not dealing with a gauge theory.

Oh, sorry, I meant the electromagnetic field, not the Dirac.

Wait, so pions are described by the scalar field? I thought the Higgs boson was the only particle that was described by it. Do you know any place where I can read which particles are described by what fields?

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Donoghue, J. F., Golowich, E., Holstein, B. R.: Dynamics of the Standard Model, Cambridge University press, 1992

For a nice introduction to chiral perturbation theory, see

https://arxiv.org/abs/nucl-th/9706075

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