Question

Coupling a spinor field to a preexisting scalar field?

Answer 1

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)(M,g)(positive definite metric) with a distinguished smooth function ff. The metric and the smooth function are related by a tensor equation

Rc(g)+∇2f=12gRc(g)+∇2f=12g

(The second summand is the Hessian wrt the Levi-Civita connection.)

I want to study spinor fields ψψ which solve some kind of Dirac equation but somehow involve this function ff. (Is it appropriate to say I want to "couple" ψψ and ff?) 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 ψψ that involve ff?

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.



plzz mark as brainliest answer