# 2PI-effective action and functional derivatives

I'm trying to work out the 2PI-effective action for complex scalar fields.

Introducing a multi field index $$(a,b,c...)$$ the complex conjugation and all other degrees of freedoms are suppressed, and field vectors are used $$\phi=(\phi_a,\phi_b,...)^T=(\phi_1,\phi_1^*,....)^T .$$

The generating functional $$Z[J,R]$$ and $$W[J,R]$$ is then given by;

$$Z[J,R]= \int \mathcal{D} \phi \, \exp (i \left[ S[\phi] + J_a \phi_a + \frac{1}{2} \phi_a R_{a,b} \phi_b \right] ) = \exp(i W[J,R]).$$

We can now define the connected two points function $$G_{a,b}$$ and the macroscopic field $$\Phi_a$$ and find for the derivative of $$W$$:

$$\qquad \Phi_a = \frac{\delta W}{\delta J_a} \qquad G_{a,b} = \frac{\delta^2 W}{\delta J_a \delta J_b} \qquad \Rightarrow \frac{\delta W}{\delta R_{a,b}} = \frac{1}{2} (\Phi_a \Phi_b + i G_{a,b}).$$

We can now look for the Legendre transformation of W w.r.t. to the two fields.

$$\Gamma[\Phi,G] = W[J,R] - \Phi_a J_a - \frac{1}{2} (\Phi_a \Phi_b + i G_{a,b}) R_{b,a}.$$

From this we find for the sources;

$$\frac{\delta \Gamma}{\delta \Phi_a} = - J_a - \frac{1}{2} \Phi_a (R_{a,b}+R_{b,a})= - J_a - \Phi_a R_{a,b}\qquad \frac{\delta \Gamma}{\delta G_{a,b}} =- \frac{i}{2} R_{b,a} = - \frac{i}{2} R_{a,b}.$$

And here is my first question. I can see that $$R$$ must be symmetric by definition, therefore the expression does not depend on taking the derivative with $$G_{a,b}$$ or $$G_{b,a}$$. But how to take the derivative in general?

$$\qquad \qquad \frac{\delta G_{d,c}}{\delta G_{a,b}} = \delta_{a,d} \delta_{c,b} \qquad or \qquad \frac{\delta G_{d,c} }{\delta G_{a,b}} =\frac{1}{2} ( \delta_{a,d} \delta_{c,b} +\delta_{c,d} \delta_{a,b} ).$$

The second question is if that statement also implies that $$G_{a,b}$$ must by symmetric?

Following further the development the effective action is then found as;

$$\Gamma[\Phi,G]= S[\Phi] +\frac{i}{2} \text{tr} \log G^{-1} + \frac{i}{2} \text{tr} G_{0}^{-1}G + \Gamma^2$$

Where $$\Gamma^2$$ contains all 2-PI diagrams and will contribute the self energy. The effective action has boundary conditions at $$J=0, R=0$$, from where the Greens function can be found. Therefore we just have to take the functional derivative w.r.t. to $$G$$ of the upper equation.

$$\frac{\delta \Gamma}{\delta G_{a,b}}=0 = \frac{-i}{2} G^{-1}_{b,a} + \frac{i}{2} (G_{b,a}^0)^{-1} + \frac{\delta \Gamma}{\delta G_{a,b}}$$

Does this also implies that the self-energy $$\Sigma_{b,a} = 2i \frac{\delta \Gamma}{\delta G_{a,b}}$$ is symmetric in $$a,b$$?

1. The connected two-point function $$G_{ab}$$ and the self-energy $$\Sigma_{ab}$$ are in the 2PI formalism defined as manifestly symmetric tensors wrt. the indices $$a\leftrightarrow b$$.
2. So, Yes, the functional derivatives wrt. $$G_{ab}$$ needs to be symmetrized, cf. e.g. this and this Phys.SE posts.