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 Ψj with zero eigenvalue for some operator O. Say,
OΨj=λjΨj,
with some
λa=0 for some
a.
This can be Dirac operator of some fermion fields, such as (iγμDμ(A,ϕ)−m)Ψj=λjΨj
here there may be nontrivial gauge profile
A and soliton profile
ϕ in spacetime. If zero mode exists then with
λa=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
∫[DΨ][DˉΨ]eiS[Ψ]=∫[DΨ][DˉΨ]eiˉΨ(iγμDμ(A,ϕ)−m)Ψ=det(iγμDμ(A,ϕ)−m)=∏jλj
In this case, if there exists
λa=0, then we need to be very careful about the possible long range correlation of
Ψa, 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 λj become energy eigenvalues, then the zero modes becomes zero energy modes: HΨj=λjΨj
if there exists some
λa=0.
(3) Zero modes ϕ0 and conjugate momentum winding modes Pϕ
In the chiral boson theory or heterotic string theory, the bosonic field Φ(x) Φ(x)=ϕ0+Pϕ2πLx+i∑n≠01nαne−inx2πL
contains zero mode
ϕ0.
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