# Fermion zero modes extra conditions?

A fermion zero mode is a zero eigenfunction, $$i\gamma^\mu(\partial_\mu-iA_\mu)\psi=0$$ The number of zero modes is apparently related to the instantons of the gauge field.

But now my question is about 'ordinary' solutions to the Dirac equation. Even if there is no gauge field and even in Euclidean space with a mass term the Dirac equation has solutions $$(i\gamma^\mu\partial_\mu + m)\psi=0$$ For instance, a possible basis choice in 2D is, $$\gamma^0=\sigma^1, \gamma^1=\sigma^2$$. Then there is the simple solution with no $$x^1$$ dependence, $$(\psi_L,\psi_R)=(e^{i m x^0},e^{i m x^0}).$$ Why are these ordinary solutions not considered when zero modes are considered? In Luboš Motl's answer here, he goes so far as to say solutions with non-zero mass don't exist in Euclidean space, but I don't see why not, I just explicitly found an obvious one. Is there some extra condition that goes into the definition of the zero mode that I am missing?

• zero modes are normalisable. Jan 21 '20 at 20:31
• @AccidentalFourierTransform, why is that important? Plane wave solutions are important in bosonic path integrals Jan 21 '20 at 20:34
• @AccidentalFourierTransform, Also if we work on the torus, this is normalizable Jan 21 '20 at 20:37

The eigenvalues of the Euclidean Dirac opertor are of the form $$i\lambda+m$$, $$\lambda,m,\in {\mathbb R}$$. Instanton backgrounds can allow solutions with $$\lambda=0$$, but if $$m\ne 0$$ you cannot get $$i\lambda+m=0$$. Your confusion comes from the fact that the Euclidean Dirac operator with hermitian $$\gamma^\mu$$ obeying $$\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu= 2\delta^{\mu\nu}$$ is $$\gamma^\mu \partial_\mu+m.$$ There is no "$$i$$" before the gammas. This absence of "$$i$$" is essential precisely because there must be no zero modes in the Euclidean theory so that the Euclidean propagator $$(\gamma^\mu \partial_\mu+m)^{-1}$$ always exists. It's only when we go back to Minkowski signature that we can go "on shell."
Your zero mode solution has $$p_0=m$$ which is a Minkowski energy momentum relation.
• There are different sign and gamma conventions so I can't say what Lubos had in mind. Even in Minkowski space people write Dirac differently between the mostly plus metric and the mostly minus metric. One has "$i$" before the gamma and the other does not as changing the sign of the metric multiplies the gammas by "$i$." What is true that the plane wave Euclidean eigenvalues $\Lambda$ obey $|\Lambda|^2=p_0^2+{\bf p}^2+m^2$ and cannot be zero, while in Minkowski they are real and obey $\Lambda^2=-p_0^2+{\bf p}^2+m^2$ and can can be zero when $p_0^2={\bf p}^2 +m^2$ Jan 22 '20 at 12:26
• The Euclidean Dirac opertor $D\!\! /=\gamma^\mu D_\mu$, with $D_\mu$ the covariant derivative on spinors, is skew-self-adjoint on a compact closed Riemanian manifold, so has a well defined eigenvlaue problem with complete sets of $L^2(M)$ eigenfunctions. In this it is little different from the laplacian on $M$ . Jan 23 '20 at 15:06