Quantum Field Theory: Stationary Point of the Effective Action

[latentcorpse]

We have the effective action which obeys $\frac{\delta \Gamma[\varphi]}{\delta \varphi(x)}=J(x)$ where and we are told the stationary point, $\varphi_0$, of this action, $\frac{\delta \Gamma[\varphi_0]}{\delta \varphi(x)}=0$, corresponds to the vacuum expectation value.
(This is out of my notes - there is a discussion around p380 of Peskin & Schroeder on this although it goes a bit more in depth than my notes...)

Anyway, on the next page, he just says that we can write

$\Gamma[\varphi]=i \displaystyle\sum_{n=0}^\infty \frac{1}{n!} \int d^dx_1 \dots \int d^dx_n \varphi(x_1) \dots \varphi(x_n) \Gamma_n (x_1, \dots , x_n)$

Can anybody explain to me where this formula has come from? And what is $\Gamma_n$? He hasn't defined that either.

Thanks.

 

[fzero]

Up to the factor of $$i$$ this is just a power-series expansion for a functional. $$\Gamma_n$$ can be expressed in terms of functional derivatives of $$\Gamma$$ evaluated at $$\phi = 0$$ as written.

 

[latentcorpse]

Up to the factor of $$i$$ this is just a power-series expansion for a functional. $$\Gamma_n$$ can be expressed in terms of functional derivatives of $$\Gamma$$ evaluated at $$\phi = 0$$ as written.

Ok. So I found that $\Gamma_n = - i \frac{\delta^n \Gamma [ \varphi ] }{ \delta \varphi(x_1) \dots \delta \varphi(x_n)}|_{\varphi=0}$

But shouldn't this be evaluated at $\varhpi = \varphi_0$ i.e. the minimum of the effective potential?

 

[fzero]

Ok. So I found that $\Gamma_n = - i \frac{\delta^n \Gamma [ \varphi ] }{ \delta \varphi(x_1) \dots \delta \varphi(x_n)}|_{\varphi=0}$

But shouldn't this be evaluated at $\varhpi = \varphi_0$ i.e. the minimum of the effective potential?

The whole expansion should be made around $$\varphi=\varphi_0$$. Your reference is either being sloppy or you've left out information.

 

[paranormal]

Quantum Field Theory (QFT) is a mathematical framework used to describe the behavior of particles and fields at a quantum level. In QFT, the effective action is a mathematical object that encodes the dynamics of the fields in a given system. It is a functional of the fields, denoted by \Gamma[\varphi], and is defined as the Legendre transform of the generator of connected Green's functions.

The effective action satisfies the equation \frac{\delta \Gamma[\varphi]}{\delta \varphi(x)}=J(x), where J(x) is the external source term. This equation is known as the Schwinger-Dyson equation and it is a fundamental equation in QFT. It relates the behavior of the fields with the external sources.

The stationary point of the effective action, \varphi_0, is a special field configuration that satisfies the equation \frac{\delta \Gamma[\varphi_0]}{\delta \varphi(x)}=0. This means that the effective action is minimized at this point, and it corresponds to the vacuum expectation value (VEV) of the fields.

In order to calculate the effective action, we can use a perturbative expansion in terms of the fields. This is where the formula given in your notes comes from. The term \Gamma_n (x_1, \dots , x_n) represents the n-point connected Green's function, which is a function of the field variables at different points in space and time. The summation over n in the formula indicates that we are considering all possible combinations of n fields. The factor of \frac{1}{n!} ensures that we do not double count the same terms in the expansion.

In summary, the formula given in your notes is a perturbative expansion of the effective action in terms of the fields, and \Gamma_n represents the n-point connected Green's function.