Remark. Chirality and the scope of other doubling remedies [fdip-009X]
Remark. Chirality and the scope of other doubling remedies [fdip-009X]
In even dimension \(d=2m\), the specified Hermitian gamma matrices define a chirality operator on \(S\otimes F\) by \[C=i^m\Gamma _1\cdots \Gamma _{2m}.\] Write \(P=\Gamma _1\cdots \Gamma _{2m}\). Reversing the order uses \(m(2m-1)\) interchanges, hence \(P^*=(-1)^mP\) and \(P^2=(-1)^mI\). It follows that \(C^*=C\) and \(C^2=I\). Moving one gamma matrix across the other \(2m-1\) factors gives \(C\Gamma _j=-\Gamma _j C\). Since the links act only on \(F\), \(C\) commutes with each shift and with \(L_{a,U}\). Therefore \[\begin {aligned} CD_{a,U}+D_{a,U}C&=0,\\ CD_{W,a,U}+D_{W,a,U}C&=ra\,C L_{a,U}. \end {aligned}\] The second expression is generally nonzero: already at trivial links the nearest-neighbor Laplacian is nonzero on a nonconstant Fourier mode. Thus the Wilson term changes this exact finite anticommutation relation, the algebraic chiral cost discussed in [wingate2025lattice, Section 9.3, p. 21].
Other approaches change different data. The following are source comparisons, not additional equivalence theorems for these finite operators.
- Staggered fields. Spin diagonalization uses a site-dependent product of gamma matrices and then a one-component field with alternating difference signs [wingate2025lattice, Section 9.4, Equations (9.19)–(9.25), pp. 22–23]. The conventional four-dimensional count is 4 tastes relative to minimal four-component Dirac spinors. It is not the number of exact zero modes on a finite grid. In dimension \(4\) the present \(S\) has 16 complex components; no taste identification for that reducible carrier is asserted. On a periodic torus the site-parity transformation also requires compatible even periods or correspondingly changed boundary conditions. No global spin diagonalization on arbitrary odd periods, rooting assertion or continuum equivalence is proved here.
- Dirac–Kähler fields. A geometric route places components on cells of different degrees. Becher and Joos map forms to cochains by integration over cells and use the boundary and coboundary correspondences to write a lattice equation [becher1982dirac, Section 3.1, Equations (3.1)–(3.4), pp. 352–353]. This is taken up in Dirac–Kähler operators. A dimension count does not identify that construction with the present spinor/internal operator, and no such identification or optional cochain construction is made here.
- Ginsparg–Wilson operators. The modified relation \[C\mathcal D+\mathcal D C=a\mathcal D C\mathcal D\] replaces exact anticommutation [wingate2025lattice, Section 9.6, Equation (9.27), p. 24]. If \(\mathcal D\) is invertible, multiplication on both sides by its inverse gives \[C\mathcal D^{-1}+\mathcal D^{-1}C=aC,\] matching the source's contact-term identity (9.28). Neither relation constructs an overlap operator here or follows from the Wilson kernel theorem.
These finite statements do not derive a positive fermion transfer operator or an interacting continuum theory. The reflection-positive transfer construction has its own hypotheses; the choice \(r=1\) does not extend it to a coupled Wilson-fermion action.