Remark. Chirality and the scope of other doubling remedies [fdip-009X]
AGENTDRAFTED
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.