这是一份warwick华威大学PX271的成功案例

The arbitrary Grassmann algebra element $F \in G_{n}$ can be represented in the forms of the finite series
$$
F(y)=F_{0}+F_{i_{1}}^{(1)} y_{i_{1}}+F_{i_{1} i_{2}}^{(2)} y_{i_{1}} y_{i_{2}}+\ldots+F_{i_{1} \ldots i_{n}}^{(n)} y_{i_{1}} \ldots y_{i_{n}},
$$
where every summation index takes the values from 1 to $n$. Thanks to the conditions this expansion is broken off. As an example, we consider one-dimensional Grassmann algebra $G_{1}$
$$
{y, y}=0, \quad y^{2}=0 .
$$
For any element $F(y) \in G_{1}$, we have
$$
F(y)=F_{0}+y F_{1} .
$$

PX271COURSE NOTES :
In the case of a free field of particles with the spin $1 / 2$, the Lagrangian is given by the expression
$$
\mathcal{L}{0}=\bar{\psi}(x)\left[i \gamma^{\mu} \partial{\mu}-m\right] \psi(x),
$$
where $S^{c}(x-y)$ is the free Green function of the Dirac field
$$
S^{c}(x-y)=\frac{1}{(2 \pi)^{4}} \int d^{4} p \frac{\gamma_{\mu} p^{\mu}+m}{p^{2}-m^{2}+i \epsilon} \exp [-i p(x-y)]
$$
and $N$ is an inessential normalization factor.