Proving a Grassmann integral identity

Question

How to prove the following identity

$$ \begin{align} \int {\rm d} \eta_{1} {\rm d} \bar{\eta}_{1} \exp\left(a \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right)\left(\eta_{1}-\eta_{0}\right) + b \left(\bar{\eta}_{2}-\bar{\eta}_{1}\right)\left(\eta_{2}-\eta_{1}\right)\right) = (a+b) \exp\left(\frac{\left(\bar{\eta}_{2}-\bar{\eta}_{0}\right)\left(\eta_{2}-\eta_{0}\right)}{\frac{1}{a}+\frac{1}{b}}\right). \end{align} $$

Here $\eta_{0},\eta_{1},\eta_{2}$ are Grassmann variables and $a,b$ are complex numbers.

Answer 1

Hints:

• Method 1: Use that Berezin integration is the same as differentiation, and that the Taylor series for the exponentials truncate.

• Method 2: Use the WKB/saddle point/stationary phase approximation, which is exact since the action is quadratic. The saddle point is $$ (\eta_1,\bar{\eta}_1)~=~\left(\frac{a\eta_0+b\eta_2}{a+b},\frac{a\bar{\eta}_0+b\bar{\eta}_2}{a+b}\right).$$

• Method 3: Complete the square in the Gaussian Grassmann integral.

Also note that $\eta_1$ and $\bar{\eta}_1$ can be viewed as independent (not necessarily complex conjugate) complex Grassmann numbers.

Answer 2

Using the hint in @Qmechanic's answer I am able to prove the identity

Method 1:

$$ \begin{align} & \frac{\partial}{\partial \eta_{1}}\frac{\partial}{\partial \bar\eta_{1}} \left(1+a \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right)\left(\eta_{1}-\eta_{0}\right) + b \left(\bar{\eta}_{2}-\bar{\eta}_{1}\right)\left(\eta_{2}-\eta_{1}\right) + ab \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right)\left(\eta_{1}-\eta_{0}\right) \left(\bar{\eta}_{2}-\bar{\eta}_{1}\right)\left(\eta_{2}-\eta_{1}\right) \right) \\ & = \frac{\partial}{\partial \eta_{1}} \left(a \left(\eta_{1}-\eta_{0}\right) - b \left(\eta_{2}-\eta_{1}\right) + ab \left(\eta_{1}-\eta_{0}\right) \left(\bar{\eta}_{2}-\bar{\eta}_{1}\right)\left(\eta_{2}-\eta_{1}\right) - ab \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right)\left(\eta_{1}-\eta_{0}\right) \left(\eta_{2}-\eta_{1}\right) \right) \\ & = a + b + ab \left(\bar{\eta}_{2}-\bar{\eta}_{1}\right)\left(\eta_{2}-\eta_{1}\right) - ab \left(\eta_{1}-\eta_{0}\right) \left(\bar{\eta}_{2}-\bar{\eta}_{1}\right) + ab \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right) \left(\eta_{2}-\eta_{1}\right) + ab \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right)\left(\eta_{1}-\eta_{0}\right) \\ & = (a+b) \left(1+\frac{ab}{a+b} \left(\left(\bar\eta_2-\bar\eta_0\right)\left(\eta_2-\eta_1\right)+\left(\bar\eta_2-\bar\eta_0\right)\left(\eta_1-\eta_0\right)\right)\right) \\ & = (a+b) \left(1+\frac{ab}{a+b} \left(\bar\eta_2-\bar\eta_0\right)\left(\eta_2-\eta_0\right)\right) \\ & = (a+b) \exp\left(\frac{ab}{a+b} \left(\bar\eta_2-\bar\eta_0\right)\left(\eta_2-\eta_0\right)\right) \end{align} $$