I was wondering, given two fermionic canonical operators: $$\{\hat{\chi}_{\alpha a}(x),\hat{\bar{\chi}}_{\beta b}(y)\} = \delta_{\alpha\beta}\delta_{ab}\delta^{(3)}(x-y)$$ they should have as eigenstates the Grassmann coherent states which give rise to the usual completness relation that maps a Fock space into a Grassmann algebra: $$\mathbb{I}_{Fock} = \int \mathcal{D}[\chi]\mathcal{D}[\bar{\chi}]\ e^{-\bar{\chi}\cdot \chi}\ \rvert \bar{\chi},\chi \rangle\langle \bar{\chi},\chi \rvert$$ I was wondering, is it possible to define in the same Hilbert space a scalar product like: $$\langle \Phi\rvert \Psi \rangle = \int \mathcal{D}[\chi]\mathcal{D}[\bar{\chi}]\ e^{-\bar{\chi}A\cdot \chi}\ \Phi^*(\bar{\chi},\chi) \Psi(\bar{\chi},\chi)$$ where $A$ is a positive definite matrix? Or should I need to redefine the $\bar{\chi},\chi$ operators and their coherent states?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.