Theorem. The odd Wick rule pairs inverse alternating entries [fdip-008K]
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  • November 1, 2026

Let \(A^T=-A\) be an invertible \(2m\times 2m\) matrix. Its nonzero Pfaffian defines the normalized linear functional \[W_A(f)=\frac {I_\xi (f e^{-\xi ^TA\xi/2})} {\operatorname {Pf}(A)}.\] For linear Grassmann elements \(\lambda _a=\sum _i\ell _{ai}\xi _i\), put \(C_{ab}=\ell _aA^{-1}\ell _b^T\). For every \(r\ge 0\), \[W_A(\lambda _1\cdots \lambda _{2r})=\operatorname {Pf}(C),\] where \(C\) is the \(2r\times 2r\) alternating matrix just defined. Every odd product has value zero. Repeated or dependent \(\lambda _a\) are allowed. The inverse formula requires invertible \(A\); it asserts no singular extension.

Proof. 

  • November 1, 2026

Define the left derivative on a monomial by \[\partial _i(\xi _{j_1}\cdots \xi _{j_s}) =\sum _{a=1}^s(-1)^{a-1}\delta _{i,j_a} \xi _{j_1}\cdots \widehat {\xi _{j_a}}\cdots \xi _{j_s}.\] This respects the exterior relations. Splitting the removed positions between two factors gives the graded product rule \[\partial _i(fg)=(\partial _i f)g+(-1)^s f(\partial _i g)\] when \(f\) is homogeneous of degree \(s\). It lowers degree, so \(I_\xi (\partial _i h)=0\) for every \(h\).

Write \(E=e^{-\xi ^TA\xi/2}\). Differentiating its finite series, using that its quadratic exponent is even, gives \[\partial _iE=-\sum _jA_{ij}\xi _jE.\] For odd homogeneous \(f\), the product rule and coefficient of the total derivative imply \[\begin {aligned} 0&=I_\xi ((\partial _i f)E+f\sum _jA_{ij}\xi _jE),\\ \sum _jA_{ij}W_A(\xi _jf)&=W_A(\partial _i f). \end {aligned}\] The second line uses \(f\xi _j=-\xi _jf\). Multiplying by \(A^{-1}\) gives \[W_A(\xi _kf)=\sum _i(A^{-1})_{ki}W_A(\partial _i f).\]

Apply this to \(f=\lambda _2\cdots \lambda _{2r}\) and expand \(\lambda _1\). The derivative removes its \(a\)-th listed factor with sign \((-1)^{a-2}\), yielding \[\begin {aligned} W_A(\lambda _1\cdots \lambda _{2r}) &=\sum _{a=2}^{2r}(-1)^a C_{1a}\\ &\quad \cdot W_A(\lambda _2\cdots \widehat {\lambda _a} \cdots \lambda _{2r}). \end {aligned}\] Starting from \(W_A(1)=1\), this is exactly the Pfaffian recursion. Odd products have odd degree, while \(E\) has only even degrees and the selected top degree \(2m\) is even; hence their integrals vanish.

The left derivative is contraction by the coordinate functional \(\xi _j\mapsto \delta _{ij}\). TauCeti's TauCeti.ExteriorAlgebra.contractLeft_ιMulti gives the alternating deletion formula. At zero quadratic form, CliffordAlgebra.contractLeft_mul gives the product rule: the grade involution acts by \((-1)^s\) in degree \(s\). These support the contraction identities; the Berezin normalization and Wick recurrence are the direct argument above.

For \(A=\left (\begin {smallmatrix}0&a\\-a&0\end {smallmatrix}\right )\) with \(a\ne 0\), the two-generator calculation gives \(W_A(\xi _1\xi _2)=-1/a=(A^{-1})_{12}\). This checks the inverse-pairing sign. In [etingof2024mathematical, v2, Theorem 9.11, p. 132], the increasing top order gives normalization \(\operatorname {Pf}(-A)\). Changing the volume multiplies both numerator and normalization by the same sign, leaving this normalized Wick rule unchanged.