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I am reading Zee's Quantum Field Theory in a Nutshell. In the section about Grassmann numbers, there is an identity:$$\int dx\int dy\,e^{yAx}=\det A\tag{II.5.13}$$ where $x=(x_1,x_2,\dots,x_N),y=(y_1,y_2,\dots,y_N)$ are $N$ Grassmann numbers and $A$ is an antisymmetric $N$ by $N$ matrix. If $A$ is an arbitary matrix (not necessarily antisymmetric), does $(II.5.13)$ still hold? When I try to prove $(II.5.13)$ for an arbitrary $A$, I encounter a problem. To save writing, I only show the case for $N=2$.

For $N=2$ and arbitrary $A$, $$\int dx\int dy\,e^{yAx}=\int dx\int dy\,(y_1A_{11}x_1y_2A_{22}x_2+y_1A_{12}x_2y_2A_{21}x_1)=\int dx\int dy\,(-y_1A_{11}y_2x_1A_{22}x_2-y_1A_{12}y_2x_2A_{21}x_1)=-A_{11}A_{22}+A_{12}A_{21}=-\det A.\tag{2}$$ The equation $(2)$ seems wrong since there is an additonal minus sign even for an antisymmetric $A$. Where is the problem in my proof?

Qmechanic
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rioiong
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1 Answers1

1

Yes, eq. (II.5.13) holds for an arbitrary complex $n\times n$ matrix $A$ up to sign convention$^1$. Zee's assumption that $A$ is antisymmetric is not necessary; he does note that $A$ does not have to be invertible.

Sketched proof for invertible $A$: Make substitution $x^{\prime}=Ax$, and remember that Grassmann integration is the same as Grassmann differentiation, so that the Grassmann measure transforms with the inverse Jacobian. $\Box$

Sketched proof for non-invertible $A$: Consider a sequence of invertible matrices that converge to $A$. Now use a continuity argument. $\Box$

References:

  1. A. Zee, QFT in a nutshell, eq. (II.5.13).

  2. M. Srednicki, QFT, 2007; eq. (44.35). A prepublication draft PDF file is available here.

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$^1$ To get a consistent sign convention, one could e.g. define the Grassmann measure as $$ d^nx~d^ny~:=~ dx_n~dy_n\ldots dx_1~dy_1 ~=~(-1)^{[n/2]}dx_n\ldots dx_1~dy_n\ldots dy_1, \tag{44.35} $$

cf. Ref. 2. This e.g. explains OP's extra minus sign for $n=2$.

Qmechanic
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