Theorem. Accretive complex Gaussian integrals and cancellation [fdip-009K]
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  • November 1, 2026

Call \(K\in M_n(\mathbb C)\) strictly accretive when its Hermitian part \(H=(K+K^*)/2\) is positive definite. For \(z_j=x_j+iy_j\), use ordinary Lebesgue volume \(d^{2n}z=\prod _j dx_j\,dy_j\). Then \(K\) is invertible, the following integral converges absolutely, and \[I(K):=\int _{\mathbb C^n}e^{-z^*Kz}\,d^{2n}z =\frac {\pi ^n}{\det K}.\] Consequently, with \(Z_B(K)=\pi ^{-n}I(K)\) and the paired Grassmann value \(Z_F(K)=J_n(e^{-\bar \psi K\psi })\), \[Z_B(K)Z_F(K)=1.\] For \(n=0\) both factors are one.

Proof. 

  • November 1, 2026

For \(n>0\), choose \(h>0\) with \(H\ge hI\). Then \[|e^{-z^*Kz}|=e^{-z^*Hz}\le e^{-h\|z\|^2}.\] The right side is integrable by the product of scalar Gaussian integrals. If \(Kz=0\), its real quadratic value is \(z^*Hz=0\), so \(z=0\); finite dimension makes \(K\) invertible.

First suppose \(K=K^*>0\). A unitary eigenbasis gives positive eigenvalues \(\lambda _j\). The unitary change of variables preserves real volume, since its real Jacobian has absolute determinant \(|\det _{\mathbb C}U|^2=1\). In each complex coordinate the scalar Gaussian formula gives \[\int _{\mathbb R^2}e^{-\lambda _j(x^2+y^2)}dx\,dy =\frac \pi {\lambda _j}.\] Multiplication gives \(I(K)=\pi ^n/\det K\).

For general \(K\), write \(K=H+A\), where \(A^*=-A\), and set \(K(w)=H+wA\) for a complex parameter \(w\). Its Hermitian part is \(H+i(\operatorname {Im}w)A\). Choose \(\epsilon >0\) such that \(\epsilon \|A\|<h/2\), with any positive choice if \(A=0\). On the connected strip \(|\operatorname {Im}w|<\epsilon \), this part is at least \(hI/2\), so \(K(w)\) is invertible.

The integral \(I(K(w))\) is holomorphic on that strip. Differentiation of the integrand brings down powers of \(-z^*Az\); on every smaller neighborhood these are bounded by a polynomial in \(\|z\|\) times \(e^{-h\|z\|^2/2}\). This integrable bound justifies differentiation under the integral. The function \(\pi ^n/\det K(w)\) is holomorphic there too. For \(w=it\) with real \(|t|<\epsilon \), the matrix \(H+itA\) is Hermitian positive, so the preceding calculation identifies the two functions. These points accumulate at the interior point zero. The one-variable identity theorem therefore identifies the functions throughout the strip, in particular at \(w=1\). The Grassmann determinant then proves cancellation.

For Hermitian positive \(K\), the normalized density \((\det K/\pi ^n)e^{-z^*Kz}\) is a genuine Gaussian probability law on \(2n\) real coordinates. For general accretive \(K\), the weight can be complex: \(K=(1+i)\) gives \(Z_F=1+i\) and \(Z_B=(1+i)^{-1}\). Normalization and determinant cancellation still hold, but there is no positive Gaussian law with this density. A single real Gaussian has only a determinant power \(-1/2\); Grassmann algebra by itself supplies no convergence for a commuting integral.

Accretivity also survives the Schur operation. Its principal block \(D\) is accretive by testing \(K\) on \((0,u)\), hence invertible. For \(v\ne 0\), put \(x=(v,-D^{-1}Cv)\). Then \(Kx=(Sv,0)\), and \[\operatorname {Re}(v^*Sv) =\operatorname {Re}(x^*Kx)>0.\] Thus \(S\) is accretive. This extra conclusion uses accretivity of the full matrix, not just invertibility of \(D\).

[wingate2025lattice, Section 9.1, equations (9.3)-(9.5), p. 19] describes integrating a fermion matrix and re-expressing its determinant with commuting complex pseudofermions. Its equation (9.5) uses \(K^{-1}\) in the commuting quadratic form, yielding a determinant rather than its inverse when convergence holds. The cancellation here uses \(K\) in both forms and the specified factor \(\pi ^{-n}\); the proof above states its convergence hypotheses explicitly.