# Minimal coupling of a Field with electromagnetism

1. Feb 24, 2015

### Jesus

I have some doubts about minimal coupling of a field of spin 2 for example, with the electromagnetism and I hope someone can help me to clarify them.

According to Pauli and Fierz one couples the field with electromagnetism introducing the covariant derivative at the level of the Lagrangian, but, does the field need to be complexified in the lagrangian in order to maintain the same degrees of freedom than before we turn on the electromagnetic interaction?

At least that is what I have seen in the literature but I don't have clear why complexify the field.

Thank you.

2. Feb 24, 2015

### dextercioby

Well, you have two different problems: 1. Coupling of gravity with electromagnetism in the absence of (presumably electrically charged) matter. 2. Coupling of electromagnetism to electrically charged matter in the presence of gravity.

2. Electromagnetism is a linear U(1) gauge theory. It's necessary for the matter field to be either complex-valued (thus at least 2 types of fields) or more generally with values in a Z2-graded algebra (= involuted algebra), else the coupling will not occur. So indeed there are particular requirements on the matter fields before trying to see how they couple to an e-m field. Gravity couples to both the e-m field and to matter. There's no requirement on the matter field coming from the gravity sector.
1. Electromagnetism couples to gravity automatically if you write down the former's Lagrangian action in a curved space-time. The metric enters in a nice way (through F2) and in a pesky way through √-g. For details, see the wonderful <80 p. brochure on General Relativity by Dirac.

3. Feb 24, 2015

### samalkhaiat

Recall the interaction Lagrangian $\mathcal{L} = A_{\mu} J^{\mu}$.
1) What is $J^{\mu}$?
2) If the matter field (of any spin) is real, does $J^{\mu}$ exist?

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