# Coupling a spinor field to a preexisting scalar field?

So I'm not a physicist, but I'm thinking about a mathematical problem where I think physical insight might be useful.

We're working on a Riemannian manifold $(M,g)$ (positive definite metric) with a distinguished smooth function $f$. The metric and the smooth function are related by a tensor equation $$Rc(g)+\nabla^2f=\displaystyle\frac{1}{2}g$$ (The second summand is the Hessian wrt the Levi-Civita connection.)

I want to study spinor fields $\psi$ which solve some kind of Dirac equation but somehow involve this function $f$. (Is it appropriate to say I want to "couple" $\psi$ and $f$?) I thought perhaps physicists had thought about such things and may have insights.

So, my question: Are there natural equations (from a physics POV) to write down for $\psi$ that involve $f$?

I'm aware of the Yukawa interaction (thanks Google), which is a Lagrangian you can write down for undetermined scalar and matter fields, but in this case the scalar field is fixed ahead of time, so I don't know how that figures in.

Any thoughts at all are appreciated.

• You can write down the Yukawa interaction even if the scalar field is a fixed function. – Robin Ekman May 26 '16 at 19:24
• There is no sense in which the Yukawa interaction need a "fixed" scalar field. (Robin posted his comment 3 seconds before mine. This sentence was edited and added into my comment later.) – Luboš Motl May 26 '16 at 19:24
• Of course you're correct. Perhaps I should've told more of the story. What got me interested in the Yukawa interaction in the first place was the Euler-Lagrange equation coming from varying $\phi$ (see wiki article for notation). It looks like $\Delta\phi +\phi=g\bar{\psi}\psi=g|\psi|^2$, but $\phi=f-2$ from above is known to satisfy $\Delta\phi +\phi=|\nabla\phi|^2$. So I thought I could possibly link $|\psi|^2$ to $|\nabla\phi|^2$, which would potentially be interesting to my research. But this comes from varying $\phi$ and $f$ is fixed above, so I'm trying to figure out if there's... – Brian Klatt May 26 '16 at 19:41
• ...something to be made of this (possibly by some modification) or if this line of thought is doomed and I'm in over my head with this physics stuff. – Brian Klatt May 26 '16 at 19:42
• It would be very useful if you explained your goal. Why introduce a spinor? What are you hoping to accomplish? It's easy to write down a PDE involving the spinor and scalar -- but what properties do you expect the spinor to have? – Bob Knighton Jul 11 at 12:32

## 1 Answer

Your first equation looks a bit like GR with a dilaton. IIRC, the analogous equations in supergravity will naturally have spinors coupling to the dilaton.

• Excuse my ignorance, but what is IIRC? – Brian Klatt May 26 '16 at 20:16
• If I Recall Correctly, in this case. – user1504 May 26 '16 at 20:20
• Any solid reference for me to check out about this? – Brian Klatt May 26 '16 at 20:22
• I can't offer much: No specific page or equation. I think I'd start by looking in one of the canonical string theory textbooks: Green, Schwarz, and Witten or Polchinski. (Or possibly Weinberg, Vol III?) See if you can find a Lagrangian, and then work out its Euler-Lagrange equation, and you might see something like what you want. – user1504 May 26 '16 at 22:24