Theorem. Gauge covariance and skew-adjointness of the Dirac operator [fdip-009P]
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  • November 1, 2026

For the declared unitary internal connection, \[\begin {aligned} (T_j^U)^*&=(T_j^U)^{-1},\\ D_U^*&=-D_U,\\ D_{U^g}\mathcal G&=\mathcal G D_U. \end {aligned}\] Consequently the spectrum and determinant of \(D_U\), and of \(mI+D_U\) for a fixed scalar \(m\), are gauge invariant.

Proof. 

  • November 1, 2026

The operator \(T_j^U\) is the product of pointwise unitary multiplication by \(I_S\otimes U_j(x)^{-1}\) and the periodic forward shift. Reindexing the counting inner product shows that the shift is unitary, as in Definition [fdip-000R]. Their product is therefore unitary, with the inverse already displayed in the definition. Thus \((\nabla _j^U)^*=-\nabla _j^U\). Since \(\Gamma _j^*=\Gamma _j\) and \(\Gamma _j\) commutes with \(\nabla _j^U\), the adjoint of their product is its negative. Summing proves the second identity.

The transformed inverse link is \[U_j^g(x)^{-1}=g_xU_j(x)^{-1}g_{x+e_j}^{-1}.\] Therefore \[\begin {aligned} (T_j^{U^g}\mathcal G\psi )(x) &=(I_S\otimes g_xU_j(x)^{-1})\psi (x+e_j)\\ &=(\mathcal G T_j^U\psi )(x). \end {aligned}\] Both shifts are invertible, so the same intertwining identity holds for their inverses and hence their centered differences. Commutation of \(\mathcal G\) with each \(\Gamma _j\) then gives \(D_{U^g}\mathcal G=\mathcal G D_U\). The operator \(\mathcal G\) is unitary and invertible. The two Dirac matrices are conjugate, as are their scalar translates, proving the spectral and determinant statements.

For a general connection, the coordinate shifts need not commute. Accordingly the free identity \(D^2=\sum _j\nabla _j^2\) cannot be imported without its commutation hypothesis. In the present notation direct expansion gives \[\begin {aligned} D_U^2&=\sum _j(\nabla _j^U)^2\\ &\quad +\sum _{i<j}\Gamma _i\Gamma _j [\nabla _i^U,\nabla _j^U], \end {aligned}\] where \([A,B]=AB-BA\). The first sum includes the identity on \(S\); the second sum need not vanish. Skew-adjointness still implies \(D_U^*D_U=-D_U^2\ge 0\). This does not identify that square with the nearest-neighbor connection Laplacian.

The formal adjoint identities linked in the free square theorem support the algebraic ingredients; the shifted-link reindexing and the internal-factor commutation are the explicit calculations here. In any chosen orthonormal basis of \(\mathcal H\), the paired Grassmann formula applies to \(mI+D_U\) as a finite complex matrix. This observation requires neither positivity nor invertibility and is not an ordinary probability interpretation.