I'm trying to learn about Grassmann numbers from the book "Condensed Matter Field Theory" by Altland and Simons, but I am currently encountering some difficulties. I have several smaller questions which are connected.

• Does a Grassmann number $\eta_i$ and a creation operator $a^\dagger_j$ commute or anti-commute? Altland and Simons only defined $[\eta_i, a_j]_+ =0$. I would suggest to take the conjugate of this and arrive at an equation for the creation operator, but the authors refused to define a conjugation for Grassmann numbers as it can lead to some difficulties.

• The authors defined functions of Grassmann numbers by its Taylor expansion: $f(\eta) = f(0)+f'(0)\eta$. But how do I get the equation $e^{-\eta a^\dagger} = 1-\eta a^\dagger$ which I encountered in several notes? The derivative of $e^{-x a^\dagger}$ is $-a^\dagger e^{-x a^\dagger}$. So I would guess that $e^{-\eta a^\dagger} = 1 - a^\dagger \eta$. One solution would be that a creation operator commutes with a Grassmann number, but I am not sure if this holds.

• Can anyone recommend a good introduction to Grassmann numbers and their use in field theory which does not delve too deep into the mathematics but concentrates on performing calculations?

I'm trying to learn about Grassmann numbers from the book "Condensed Matter Field Theory" by Altland and Simons, but I am currently encountering some difficulties. I have several smaller questions which are connected.

• Does a Grassmann number $\eta_i$ and a creation operator $a^\dagger_j$ commute or anti-commute? Altland and Simons only defined $[\eta_i, a_j]_+ =0$. I would suggest to take the conjugate of this and arrive at an equation for the creation operator, but the authors refused to define a conjugation for Grassmann numbers as it can lead to some difficulties.

• The authors defined functions of Grassmann numbers by its Taylor expansion: $f(\eta) = f(0)+f'(0)\eta$. But how do I get the equation $e^{-\eta a^\dagger} = 1-\eta a^\dagger$ which I encountered in several notes? The derivative of $e^{-x a^\dagger}$ is $-a^\dagger e^{-x a^\dagger}$. So I would guess that $e^{-\eta a^\dagger} = 1 - a^\dagger \eta$. One solution would be that a creation operator commutes with a Grassmann number, but I am not sure if this holds.

• Can anyone recommend a good introduction to Grassmann numbers and their use in field theory which does not delve too deep into the mathematics but concentrates on performing calculations?

I'm trying to learn about Grassmann numbers from the book "Condensed Matter Field Theory" by Altland and Simons, but I am currently encountering some difficulties. I have several smaller questions which are connected.

• Does a Grassmann number $\eta_i$ and a creation operator $a^\dagger_j$ commute or anti-commute? Altland and Simons only defined $[\eta_i, a_j] =0$. I would suggest to take the conjugate of this and arrive at an equation for the creation operator, but the authors refused to define a conjugation for Grassmann numbers as it can lead to some difficulties.

• The authors defined functions of Grassmann numbers by its Taylor expansion: $f(\eta) = f(0)+f'(0)\eta$. But how do I get the equation $e^{-\eta a^\dagger} = 1-\eta a^\dagger$ which I encountered in several notes? The derivative of $e^{-x a^\dagger}$ is $-a^\dagger e^{-x a^\dagger}$. So I would guess that $e^{-\eta a^\dagger} = 1 - a^\dagger \eta$. One solution would be that a creation operator commutes with a Grassmann number, but I am not sure if this holds.

• Can anyone recommend a good introduction to Grassmann numbers and their use in field theory which does not delve too deep into the mathematics but concentrates on performing calculations?

I'm trying to learn about Grassmann numbers from the book "Condensed Matter Field Theory" by Altland and Simons, but I am currently encountering some difficulties. I have several smaller questions which are connected.

• Does a Grassmann number $\eta_i$ and a creation operator $a^\dagger_j$ commute or anti-commute? Altland and Simons only defined $[\eta_i, a_j]_+ =0$. I would suggest to take the conjugate of this and arrive at an equation for the creation operator, but the authors refused to define a conjugation for Grassmann numbers as it can lead to some difficulties.

• The authors defined functions of Grassmann numbers by its Taylor expansion: $f(\eta) = f(0)+f'(0)\eta$. But how do I get the equation $e^{-\eta a^\dagger} = 1-\eta a^\dagger$ which I encountered in several notes? The derivative of $e^{-x a^\dagger}$ is $-a^\dagger e^{-x a^\dagger}$. So I would guess that $e^{-\eta a^\dagger} = 1 - a^\dagger \eta$. One solution would be that a creation operator commutes with a Grassmann number, but I am not sure if this holds.

• Can anyone recommend a good introduction to Grassmann numbers and their use in field theory which does not delve too deep into the mathematics but concentrates on performing calculations?