There have been several Phys.SE questions on the topic of zero modes. Such as, e.g.,

Here I would like to understand further **whether "Zero Modes" may have physically different interpretations** and **what their consequences are**, or **how these issues really are the same, related or different.** There at least 3 relevant issues I can come up with:

## (1) **Zero eigenvalue modes**

By definition, **Zero Modes** means zero eigenvalue modes, which are modes $\Psi_j$ with zero eigenvalue for some operator $O$. Say,

$$O \Psi_j = \lambda_j \Psi_j,$$ with some $\lambda_a=0$ for some $a$.

This can be Dirac operator of some fermion fields, such as $$(i\gamma^\mu D^\mu(A,\phi)-m)\Psi_j = \lambda_j \Psi_j$$ here there may be nontrivial gauge profile $A$ and soliton profile $\phi$ in spacetime. If zero mode exists then with $\lambda_a=0$ for some $a$. **In this case, however, as far as I understand, the energy of the zero modes may not be zero.** This zero mode contributes nontrivially to the path integral as $$\int [D\Psi][D\bar{\Psi}] e^{iS[\Psi]}=\int [D\Psi][D\bar{\Psi}] e^{i\bar{\Psi}(i\gamma^\mu D^\mu(A,\phi)-m)\Psi } =\det(i\gamma^\mu D^\mu(A,\phi)-m)=\prod_j \lambda_j$$ In this case, if there exists $\lambda_a=0$, then we need to be very careful about the possible long range correlation of $\Psi_a$, seen from the path integral partition function (**any comments at this point?**).

## (2) **Zero energy modes**

If said the operator $O$ is precisely the hamiltonian $H$, i.e. the $\lambda_j$ become energy eigenvalues, then the zero modes becomes zero energy modes: $$ H \Psi_j= \lambda_j \Psi_j $$ if there exists some $\lambda_a=0$.

## (3) **Zero modes $\phi_0$ and conjugate momentum winding modes $P_{\phi}$**

In the chiral boson theory or heterotic string theory, the bosonic field $\Phi(x)$ $$ \Phi(x) ={\phi_{0}}+ P_{\phi} \frac{2\pi}{L}x+i \sum_{n\neq 0} \frac{1}{n} \alpha_{n} e^{-in x \frac{2\pi}{L}} $$ contains zero mode $\phi_0$.

Thus: **Are the issues (1),(2) and (3) the same, related or different physical issues?** If they are the same, why there are the same? If they're different, how they are different?

I also like to know when people consider various context, which issues they are really dealing with: such as the **Jackiw-Rebbi** model, the **Jackiw-Rossi** model and **Goldstone-Wilczek** current computing induced quantum number under soliton profile, **Majorana zero energy modes**, such as the Fu-Kane model (arXiv:0707.1692), Ivanov half-quantum vortices in p-wave superconductors (arXiv:cond-mat/0005069), or the issue with **fermion zero modes under QCD instanton** as discussed in Sidney Coleman's book ``Aspects of symmetry''.

ps. since this question may be a bit too broad, it is totally welcomed that anyone attempts to firstly answer the question partly and add more thoughts later.

This post imported from StackExchange Physics at 2014-03-12 15:56 (UCT), posted by SE-user Idear