- #1

Plantation

- 14

- 1

- Homework Statement
- $$\int d\bar{\vec{\theta}}d \vec{\theta} e^{-(\bar{\vec{\theta}} - \bar{\vec{\eta}} A^{-1})A( \vec{\theta}-A^{-1}\vec{\eta})} = \operatorname{det}(A)$$

- Relevant Equations
- $$ \int d\bar{\theta}_1d\theta_1 \cdots d\bar{\theta}_n d\theta_n e^{-\bar{\theta}_i A_{ij} \theta_{j} } = \operatorname{det}(A) \tag{14.98}$$

I am reading the Schwartz's Quantum field theory, p.269~p.272 ( 14.6 Fermionic path integral ) and some question arises.

In section 14.6, Fermionic path integral, p.272, (14.100), he states that

$$ \int d\bar{\theta}_1d\theta_1 \cdots d\bar{\theta}_n d\theta_n e^{-\bar{\theta}_i A_{ij} \theta_{j} + \bar{\eta}_i \theta_{i}+ \bar{\theta}_i \eta_i} = e^{\bar{\vec{\eta}} A^{-1} \vec{\eta}} \int d\bar{\vec{\theta}}d \vec{\theta} e^{-(\bar{\vec{\theta}} - \bar{\vec{\eta}} A^{-1})A( \vec{\theta}-A^{-1}\vec{\eta})}= \operatorname{det}(A) e^{\bar{\vec{\eta}} A^{-1}\vec{\eta}} \tag{14.100}$$

where ##\theta_i## are grassmann numbers ( C.f. His book p.269 ) and ##\bar{\theta}_i## are defined in p.271. And ##\eta_i## and ##\bar{\eta}_i## are external currents.

Q. Why ##\int d\bar{\vec{\theta}}d \vec{\theta} e^{-(\bar{\vec{\theta}} - \bar{\vec{\eta}} A^{-1})A( \vec{\theta}-A^{-1}\vec{\eta})} = \operatorname{det}(A)## ?

In his book, p.271, (14.98), he deduced that

$$ \int d\bar{\theta}_1d\theta_1 \cdots d\bar{\theta}_n d\theta_n e^{-\bar{\theta}_i A_{ij} \theta_{j} } = \operatorname{det}(A) \tag{14.98}$$

Can we use this? How? Or by similar argument for deduction of the (14.98)?

In section 14.6, Fermionic path integral, p.272, (14.100), he states that

$$ \int d\bar{\theta}_1d\theta_1 \cdots d\bar{\theta}_n d\theta_n e^{-\bar{\theta}_i A_{ij} \theta_{j} + \bar{\eta}_i \theta_{i}+ \bar{\theta}_i \eta_i} = e^{\bar{\vec{\eta}} A^{-1} \vec{\eta}} \int d\bar{\vec{\theta}}d \vec{\theta} e^{-(\bar{\vec{\theta}} - \bar{\vec{\eta}} A^{-1})A( \vec{\theta}-A^{-1}\vec{\eta})}= \operatorname{det}(A) e^{\bar{\vec{\eta}} A^{-1}\vec{\eta}} \tag{14.100}$$

where ##\theta_i## are grassmann numbers ( C.f. His book p.269 ) and ##\bar{\theta}_i## are defined in p.271. And ##\eta_i## and ##\bar{\eta}_i## are external currents.

Q. Why ##\int d\bar{\vec{\theta}}d \vec{\theta} e^{-(\bar{\vec{\theta}} - \bar{\vec{\eta}} A^{-1})A( \vec{\theta}-A^{-1}\vec{\eta})} = \operatorname{det}(A)## ?

In his book, p.271, (14.98), he deduced that

$$ \int d\bar{\theta}_1d\theta_1 \cdots d\bar{\theta}_n d\theta_n e^{-\bar{\theta}_i A_{ij} \theta_{j} } = \operatorname{det}(A) \tag{14.98}$$

Can we use this? How? Or by similar argument for deduction of the (14.98)?