# Fractional derivatives in a QFT Lagrangian

There are is at least one question asking about fractional powers of fields in QFT (and why they're not expected to occur), and several others asking about the physical relevance of fractional derivatives. Regarding the former, fractional powers of fields seem to always renormalize to integer powers. Regarding the latter, the main objection seems to be that once you introduce fractional powers to any Lagrangian, it immediately becomes non-local. Let's try to excuse that for a moment.

If we want to use fractional derivatives in a Lagrangian, there's the question of how to preserve Lorentz invariance: with integer derivatives, we can have expressions like $$(\partial_\mu \varphi)(\partial^\mu \psi)$$. It's hard to find valid forms with fractional derivatives. One option I see is using roots of the $$\partial^2$$ operator, $$(\partial^2)^{r}$$. So a possible-seeming Lagrangian would be something like:

$$\mathcal{L} = (\partial\varphi)^2-m_\varphi\varphi^2 + (\partial\psi)^2-m_\psi\psi^2 + (|\partial^2|^{1/2}\varphi)(|\partial^2|^{1/2}\psi)$$

Indeed, I can't see any way to make this invariant except for $$(\partial^2)^r$$. This is also what's done in https://en.wikipedia.org/wiki/Fractional_quantum_mechanics, but that is usually non-relativistic and not subject to such serious restrictions. It also seems to focus on describing existing models in a new approximation, as opposed to discussing possible new interactions or particles.

On a vaguely related note, I did find this paper discussing how $$(\partial^2)^{1-\epsilon}$$ can be used to formulate a renormalization scheme.

So,

Question 1: To what degree are fractional derivatives like the above admissable under constraints (1) Lorentz invariance, (2) renormalizability, and/or (3) locality?

and

Question 2: Assuming a sufficiently positive answer to question 1, what kinds of new particles or interactions could be possible? Are any of these discussed seriously as a way to resolve known problems?

Consider a quantum field theory with degrees of freedom $$\phi$$ and $$\psi$$, described by a local Lagrangian. If you integrate the $$\psi$$ field exactly, you get a theory described by an action of $$\phi$$ alone, in general non-local. If the field $$\psi$$ is massive, then for excitations of energies small compared to this scale, the dynamics will be approximately local, but nothing like this holds for massless fields. One example for this is Schwinger QED, with Lagrangian

$$\mathscr{L} = \overline{\psi} \gamma^\mu \mathcal{D}_\mu \psi - \frac{1}{4} F_{\mu \nu} F^{\mu \nu} \ .$$

If the fermions are integrated out, the resulting action for $$A$$ alone is

$$S = \int_x \int_y A_\mu(x)\left(g^{\mu \nu} - \frac{\partial^\mu \partial^\nu}{\partial^2}\right)A_\nu(y) + \text{Maxwell Term} \ .$$

This appears non-local, but since it originated from a local theory, it is in fact local as well.

This gives an important subclass of non-local dynamics which are Lorentz-invariant, renormalizable, local: namely those that can be made manifestly local by introducing additional (massless) degrees of freedom.

• This is a cool example of "non-local" formulations behaving nicely, I'll definitely upvote. It doesn't address the fractional derivatives though, so I won't accept it as an answer. – Alex Meiburg Feb 14 at 11:58