# Conjugate momentum in the vacuum functional for the fermionic oscillator

The vacuum functional for the fermionic oscillator is given by

$$Z = N\int\mathcal{D}\overline{\psi}\mathcal{D}\psi \exp\left(i\int_0^Tdt\left(i\overline{\psi}\psi-w\overline{\psi}\psi \right)\right). \tag{5.80}$$

Using Weyl ordering and discretizing the time integral, it is claimed that this could be written as

$$Z = \lim_{\epsilon\rightarrow0}\lim_{N\rightarrow\infty}N\int d\overline{\psi}_1\dots d\overline{\psi}_{N-1}d\psi_1\dots d\psi_{N-1}$$ $$\times \exp\left(i\epsilon\sum_{n=1}^N\left(i\overline{\psi}_n\frac{\psi_n-\psi_{n-1}}{\epsilon}-w\overline{\psi}_n\frac{\psi_n+\psi_{n-1}}{2} \right)\right),\tag{5.81}$$

where the mid-point prescription of the Weyl ordering was used.

My question is pretty simple and naive. To write this expression, the author says on the bottom of p.92 that

$$\overline{\psi}$$ represents the momentum conjugate to $$\psi$$.

But if I make the computation I get (using left derivatives)

$$\Pi_\psi = \frac{\partial L}{\partial \dot{\psi}}=-i\overline{\psi}.\tag{5.43}$$

Why is it valid to ignore the $$-i$$ factor and just to consider $$\overline{\psi}$$ as the conjugate momentum?

• ${}$ Which page? – Qmechanic Jun 25 at 4:47
• p92, second edition. Thanks for looking it up! – user2820579 Jun 25 at 17:29
• Does it make you equation for $Z$ invalid? – Oбжорoв Jun 25 at 18:06

Ashok Das is not ignoring any factor of $$-i$$ in formulas. The word "represent" in the sentence $$\overline{\psi}$$ represents the conjugate to $$\psi$$ is here used semantically in a weaker sense than "is equal", e.g. "is equal up to a multiplicative constant".