Definition. The covariant Wilson operator with lattice spacing [fdip-009U]

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Use the field space and unitary shifts \(\mathcal H=\ell ^2(\Lambda )\otimes S\otimes F\) and \(T_j^U\), with \(\dim _{\mathbb C}S=2^d\) and \(q=\dim _{\mathbb C}F>0\). A lattice spacing \(a>0\) rescales the differences, while \(x\in \Lambda \) remains an integer site label. Define \[\begin {aligned} D_{a,U}&=a^{-1}D_U,\\ L_{a,U}&=a^{-2}\sum _{j=1}^d \bigl (2I-T_j^U-(T_j^U)^{-1}\bigr ). \end {aligned}\] For \(r\in \mathbb R\setminus \{0\}\), the massless Wilson operator in the skew Dirac convention is \[D_{W,a,U}=D_{a,U}+\frac {ra}{2}L_{a,U}.\] The usual positive Wilson coefficient is \(r>0\). For \(a=1\) this is \(D_U+(r/2)L_{1,U}\). The Laplacian is the nearest-neighbor second difference, not the negative square of the centered first difference.

Each \(T_j^U\) is unitary, so \[2I-T_j^U-(T_j^U)^{-1} =(I-T_j^U)^*(I-T_j^U).\] Consequently \(L_{a,U}\) is self-adjoint and nonnegative, whereas \(D_{a,U}\) is skew-adjoint. For the internal gauge transformation \(\mathcal G\) of that proposition, conjugating every shift and its inverse gives \[\begin {aligned} L_{a,U^g}&=\mathcal G L_{a,U}\mathcal G^{-1},\\ D_{W,a,U^g}&=\mathcal G D_{W,a,U}\mathcal G^{-1}. \end {aligned}\] This defines a covariant operator for all the supplied unitary connections. The Fourier kernel theorem below assumes the trivial connection.

For \(r=1\), this is the second-derivative addition in [wingate2025lattice, Section 9.3, Equation (9.15), p. 21]: \(L_a=-\sum _j\nabla _j^+\nabla _j^-\) in the free difference notation. The coefficient \(ra/2\), rather than \(r/2\), is required when \(L_a\) contains \(a^{-2}\). A paired Grassmann quadratic form may use this finite matrix as its kernel, as in Theorem [fdip-008H]. Gauge covariance and the Wilson coefficient alone supply neither a positive probability density nor a fermionic reflection-positive transfer construction.