Definition. A centered Dirac operator on a finite periodic grid [fdip-000R]
AGENTDRAFTED
To couple Clifford multiplication to spatial differences, fix an
integer \(d\ge 1\) and periods \(N_j\ge 3\) for \(1\le j\le d\).
The periodic grid and its scalar fields are
\[\begin {aligned}
\Lambda &=\prod _{j=1}^d\mathbb Z/N_j\mathbb Z,\\
M&=\prod _{j=1}^d N_j,\qquad H=\mathbb C^\Lambda .
\end {aligned}\]
Write \(e_j\in \Lambda \) for the unit step in coordinate \(j\).
Addition is periodic and the lattice spacing is one. Extend the
counting pairing to the Hermitian inner product
\[\langle f,g\rangle _H
=\sum _{x\in \Lambda }\overline {f(x)}g(x),\]
conjugate-linear in the first slot.
The spatial quadratic form is
\[q_{\mathrm {sp}}(z)=\sum _{j=1}^d z_j^2
\qquad (z\in \mathbb C^d).\]
It is distinct from the Hermitian norm squared \(\sum _j|z_j|^2\)
and from the real four-field configuration form.
For example, at \(i\varepsilon _1\), where \(\varepsilon _j\) is the
standard basis of \(\mathbb C^d\), the quadratic value is \(-1\)
and the Hermitian norm squared is \(1\). The grid steps \(e_j\),
these complex coordinate vectors, and the 4 configuration
coordinates have different carriers.
Choose \(S=(\mathbb C^2)^{\otimes d}\) with its standard tensor
basis and product Hermitian inner product. In the
Pauli convention, take
\[X=\begin {pmatrix}0&1\\1&0\end {pmatrix},\qquad
Z=\begin {pmatrix}1&0\\0&-1\end {pmatrix},\]
and put \(I=I_2\) and
\[\gamma _j=Z^{\otimes (j-1)}\otimes X
\otimes I^{\otimes (d-j)}.\]
An empty tensor product contributes the scalar identity.
Direct multiplication gives \(X^*=X\), \(Z^*=Z\),
\(X^2=Z^2=I\) and \(XZ=-ZX\). Ordinary tensor products obey
\[\begin {aligned}
(A\otimes B)^*&=A^*\otimes B^*,\\
(A\otimes B)(C\otimes E)&=AC\otimes BE.
\end {aligned}\]
The product formula follows on pure tensors, which span; the
adjoint formula follows from the product inner product on such
tensors. Thus \(\gamma _j^*=\gamma _j\) and
\(\gamma _j^2=I_S\). For \(i<j\), the only anticommuting slot is
\(i\), where \(X\) meets \(Z\). Before it both factors are \(Z\);
after it at least one factor is \(I\). Hence
\[\gamma _i\gamma _j+\gamma _j\gamma _i
=2\delta _{ij}I_S.\]
Mathlib's Matrix.kronecker,
Matrix.mul_kronecker_mul,
Matrix.conjTranspose_kronecker and
Matrix.one_kronecker_one give the corresponding
matrix construction and identities. Iterating them requires
identifying the product-index bases.
The complex-linear map \(c:\mathbb C^d\to \operatorname {End}_{\mathbb C}(S)\)
defined by \(c(z)=\sum _j z_j\gamma _j\) satisfies
\[\begin {aligned}
c(z)^2&=\sum _j z_j^2 I_S\\
&\quad +\sum _{i<j}z_i z_j
(\gamma _i\gamma _j+\gamma _j\gamma _i)\\
&=q_{\mathrm {sp}}(z)I_S.
\end {aligned}\]
The Clifford universal property therefore gives a
unique unital complex-algebra map
\[\begin {aligned}
\rho :\operatorname {Cl}(q_{\mathrm {sp}})&\longrightarrow
\operatorname {End}_{\mathbb C}(S),\\
\rho (\iota (\varepsilon _j))&=\gamma _j.
\end {aligned}\]
This makes \(S\) a Clifford module, in the
module-map convention.
The ordinary-algebra universal property is stated in
[meinrenken2013clifford, Proposition 2.2 and Remark 2.2, p. 29].
Its bilinear form is \(b(z,w)=\sum _j z_jw_j\), with relation
\(uv+vu=2b(u,v)1\); see
[meinrenken2013clifford, Definition 2.1, p. 27].
The displayed square identity is precisely the input required
by CliffordAlgebra.lift over \(\mathbb C\).
This chosen module is reducible. The involution
\[J=I^{\otimes (d-1)}\otimes X\]
commutes with every \(\gamma _j\): its last factor meets either
\(I\) or \(X\), and its other factors are identities.
Its two eigenspaces are
\[S_\pm =(\mathbb C^2)^{\otimes (d-1)}
\otimes \mathbb C(1,\pm 1).\]
Both are nonzero and have dimension \(2^{d-1}\). Commutation
with \(J\) makes them invariant under every generator and hence
under the generated Clifford action. We use \(\dim S=2^d\);
no minimality or faithfulness of this representation is needed.
Extend the cyclic shift coordinatewise:
\[(T_j f)(x)=f(x+e_j).\]
Its inverse shifts by \(-e_j\). Reindexing \(y=x+e_j\) gives
\[\begin {aligned}
\langle T_jf,g\rangle _H
&=\sum _y\overline {f(y)}g(y-e_j)\\
&=\langle f,T_j^{-1}g\rangle _H.
\end {aligned}\]
Thus \(T_j^*=T_j^{-1}\), so \(T_j\) is unitary. The shifts and
their inverses commute, since addition in \(\Lambda \) commutes.
The centered differences are the endomorphisms
\[\nabla _j=(T_j-T_j^{-1})/2.\]
They commute and satisfy \(\nabla _j^*=-\nabla _j\).
Their source and target are both \(H\), unlike a general
vertex-to-edge incidence map.
Set \(K=S\otimes _{\mathbb C}H\), with
\(\langle s\otimes f,t\otimes g\rangle _K
=\langle s,t\rangle _S\langle f,g\rangle _H\).
Identify \(s\otimes f\) with the field \(x\mapsto f(x)s\).
The centered Dirac operator is
\[D=\sum _{j=1}^d\gamma _j\otimes \nabla _j.\]
Equivalently,
\[\begin {aligned}
(D\psi )(x)
&=\frac 12\sum _{j=1}^d\gamma _j\psi (x+e_j)\\
&\quad -\frac 12\sum _{j=1}^d\gamma _j\psi (x-e_j).
\end {aligned}\]
For \(d=1\), this is \(X\) applied to the ordinary centered
difference of a two-component field. Wingate's gauge-covariant
centered difference reduces to this difference when all links
are one and the spacing is one
[wingate2025lattice, Section 9.1, Equation (9.2), p. 19].
The finite grid and tensor module here are specified directly;
this construction introduces no Grassmann integration or gauge
field. The Lorentz-sign gamma algebra is a
different choice of form, scalar field and representation.