Example. The Hermitian convention can retain extra Wilson zeros [fdip-009W]
Example. The Hermitian convention can retain extra Wilson zeros [fdip-009W]
Set \(d=1\), \(N=4\), \(a=r=1\), trivial links and \(F=\mathbb C\). The chosen spin space is \(S=\mathbb C^2\), with gamma matrix \[X=\begin {pmatrix}0&1\\1&0\end {pmatrix}.\] The free skew Dirac block is \(D_1(p)=iX\sin p\). Replacing it by the Hermitian operator \(iD_1\) and then adding the same Wilson term gives the block \[H_W(p)=-X\sin p+(1-\cos p)I.\] This is a different operator from Theorem [fdip-009V].
The normalized vectors \((1,1)/\sqrt 2\) and \((1,-1)/\sqrt 2\) are \(X\)-eigenvectors of eigenvalues \(+1\) and \(-1\). Thus the two eigenvalues of \(H_W(p)\) are \(1-\cos p\mp \sin p\). On the 4 allowed momenta: \[\begin {array}{c|cc|c} p&X=+1&X=-1&\dim \ker H_W(p)\\\hline 0&0&0&2\\ \pi /2&0&2&1\\ \pi &2&2&0\\ 3\pi /2&2&0&1 \end {array}\] The total kernel dimension is 4. The skew Wilson operator \(D_1+L_1/2\) instead has kernel dimension two, consisting of constant fields. Tensoring this example with an internal space of dimension \(q\) multiplies both counts by \(q\).
The difference is elementary: a real scalar can cancel a real eigenvalue of the Hermitian first difference. For the skew first difference the squared norm is a sum of nonnegative squares, as in the preceding proof. Adjoint conventions therefore cannot be changed silently in a claim that the Wilson term removes all nonzero-momentum zeros.