Effective Action for Scalar and Fermion Fields

[The black vegetable]

TL;DR Summary

Using effective action to derive the one loop contribution of aa scalar field and fermion field

I am reading Peskin and Schroeder Section 11.4. They derive a formula for the effective action p.372 Equation 11.63 using a scalar field interaction,
They use this formula to determine the effective potential. If I want to do the same for a Lagrangian with with a scalar field and fermion field, can I use the same Formula reasoning and technique to get a formula for a scalar field and fermion field?

$$\Gamma \left ( \phi _{cl} \right )=\int d^{4}L_{1}\left [ \phi _{cl} \right ]+\frac{i}{2}\log\text{Det}\left [ \frac{\partial ^{2}L_{1}}{\partial \phi \partial \phi } \right ] - i\: Connected\: Diagrams+\int d^{4}x\delta L\left [ \phi _{cl} \right ].\tag{11.63}$$

If so do you treat the Dirac Fermion $$\bar{\Psi } \Psi$$ as two different fields?

 

[vanhees71]

You can use the same formula also for fermionic fields, but you must be aware that these fields must be described as Grassmann-number valued and that the differentiation and integration with respect to Grassmann numbers/fields are modified. This is all necessary to get the anticommutation properties of the corresponding field operators correctly mapped to the functional formalism.

 

[The black vegetable]

Ok Thanks, very helpful I have some notes on Grassmann variables that I will revisit, but can I start as Peskin and Schroeder did with a new Lagrangian but this time containing a scaler field and a fermionic field expanding both
$\phi \rightarrow \phi _{cl}+\eta$
$\Psi \rightarrow \Psi _{cl}+\xi$

Then comparing the $\eta^{2}$ to get a value for $\frac{\delta ^{2}L_{1}}{\delta \phi \delta\phi }$

Or maybe I'm not following it very well

 

[vanhees71]

The general technique is the same for fermions and bosons. You have to be only careful with the signs for the fermionic case using Grassmann variables. Peskin&Schroeder also treats fermions in the path-integral formalism (Sect. 9.5).

Another very good book using the path-integral formalism from the very beginning is

D. Bailin and A. Love, Introduction to Gauge Field Theory,
Adam Hilger, Bristol and Boston (1986).

 

[The black vegetable]

The general technique is the same for fermions and bosons. You have to be only careful with the signs for the fermionic case using Grassmann variables. Peskin&Schroeder also treats fermions in the path-integral formalism (Sect. 9.5).

Another very good book using the path-integral formalism from the very beginning is

D. Bailin and A. Love, Introduction to Gauge Field Theory,
Adam Hilger, Bristol and Boston (1986).

Okay thanks for your help