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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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Unfortunately I haven't studied GR yet. Could you explain what do you mean by the currents being the sources of the fields?Well, they are the sources of gravity in GR but not of the electromagnetic 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.You need a current of a gauge field.

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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. I guess the reason is this: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].

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

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

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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