When first introducing the gravitational interaction for a spin-3/2 Rarita-Schwinger field, Majorana fields are usually used (see for example here at chapter 4, or in Ramond, (6.4.112) ).

Why is this? What are the advantaged of imposing a Majorana condition in this context?


In this context, the word "Majorana" doesn't mean that it is the "real one-half" of the ordinary simple Dirac spinor. It means that it is any half-integer field with a reality (Majorana) condition!

For example, in the Chapter 4 of the DAMTP lectures, the field introduced is a standard spin-3/2 field with one spinor index and one vector index, as required for a gravitino. The gravitino's spin has to be 3/2 because it differs by 1/2 from $j=2$ of the gravitons, and $j=5/2$ is already too much and would require too large gauge invariance.

The Majorana reality condition is imposed on the gravitinos simply because the metric tensor is naturally a real (or, quantum mechanically, Hermitian) field which is why it must be possible to naturally impose the reality condition on the superpartner field, the gravitino field, too.

The minimum $N=1$ supersymmetry is generated by 4 real supercharges which may be imagined to combine to one Weyl (complex chiral) spinor or one Majorana (real non-chiral) spinor. Using the latter approach, the gravitino superpartners of the real graviton by the real supercharges are real, too.

Analogously, we usually describe the gauginos, superpartners of (naturally real) gauge bosons, as Majorana fermions while the superpartners of (complex but not only complex) scalar fields are naturally Weyl fermions – even though the information in one Weyl fermion and one Majorana fermion is really "the same".

  • $\begingroup$ Thanks for the answer! Could you clarify in the first paragraph what you mean by "real one-half" of the ordinary simple Dirac spinor and any half-integer field with a Majorana condition? The second one is what I would normally call a Majorana field, what do you mean with the first one? (some math would be helpful) $\endgroup$ – glS Jan 19 '15 at 16:01
  • $\begingroup$ By the "real one-half of the Dirac spinor", I mean one-half of the degrees of freedom that are present in the Dirac spinor, a spinor with 4 complex components transforming as a spin-1/2 representation of the Lorentz group, and the one-half is obtained by demanding a reality condition so that the 4 complex components become 4 real components. If you understand that the word "Majorana field" in that explanation may mean a field with any $j\in Z+1/2$ and a real condition, then it's just fine, but then I don't understand too well why you don't understand the original text. $\endgroup$ – Luboš Motl Jan 19 '15 at 17:33

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