### Approximating the effective action

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This is both a basic and an advanced problem, related to anomalous mean-fields, field theory and the many-body problem in general. The question is:

\begin{align}\label{eq:partFunc}

Z[J^{*}_{},J^{}_{}]&=\int \mathcal{D}[\phi^{*}_{},\phi^{}_{}]e^{\mathrm{i} S[\phi^{*}_{},\phi^{}_{}]} e^{\int \mathrm{i}\text{d}t(J^{}_{}\phi^{*}_{} + J^{*}_{}\phi^{}_{})},

\end{align}

where \( S \) is supposed to be a non-quadratic (i.e.

\begin{align}

W[J^{*}_{},J^{}_{}]=-\mathrm{i}\ln{Z[J^{*}_{},J^{}_{}]}.

\end{align}

Averages can then be obtained by functional differentiation with respect to the sources, according to

\begin{align}

\Phi_{}=\langle\phi_{}\rangle=\frac{\delta W}{\delta J_{}^{*}}.

\end{align}

The

\begin{align}\label{eq:effAction}

\Gamma[\Phi^{*}_{},\Phi^{}_{}]=W[J^{*}_{},J^{}_{}]-\int \mathrm{d}t(J^{}_{}\Phi^{*}_{} + J^{*}_{}\Phi^{}_{}).

\end{align}

If one decomposes the field into average and fluctuation,

\begin{align}\label{eq:flucs_decomp}

\phi_{}=\Phi_{}+\delta\phi_{},

\end{align}

after re-exponentiating the effective action may be written

\begin{align}

& e^{\mathrm{i}\Gamma[\Phi^{*}_{},\Phi^{}_{}]}=\int \mathcal{D}[\delta\phi^{*}_{}, \delta\phi^{}_{}] \exp{\left\{\mathrm{i} S_{}[\Phi^{*}+\delta\phi^{*}, \Phi_{}+\delta\phi_{}]-\mathrm{i}\int\text{d}t\left(\frac{\delta\Gamma}{\delta\Phi^{*}_{}}\delta\phi^{*}_{}

+\frac{\delta\Gamma}{\delta\Phi^{}_{}}\delta\phi^{}_{}\right)\right\}},

\end{align}

where we have replaced the sources with the inverse relation from the Legendre transform. Note that the source terms for the average from the Legendre transform have cancelled those from the definition of \( Z \). I believe to understand that the essence of the many-body problem is contained in this formula: for the trouble is that we now have \( \Gamma \) on both sides of the equation! That might sound banal, but since one obviously cannot simply solve for \( \Gamma \), we have a conundrum. The farthest I have gotten so far is the lowest level of approximation, which amounts to exploiting the fact that to lowest order the effective action must equal the action evaluated at the average, \( \Gamma_0[\Phi^{*}_{},\Phi^{}_{}]= S[\Phi^{*}_{},\Phi^{}_{}]\). If one uses this on the right-hand side, and also drops all fluctuation terms of higher order than quadratic, one seemingly comes out with the Hartree-Fock approximation. To next order, however, things are already more involved. As I gather from standard QFT books, approximating \( \Gamma \) on the right-hand side is also related to renormalization (in the sense of generating counter-terms).

Some references:

- Diehl, S. et al. "Keldysh Field Theory for Driven Open Quantum Systems" [arXiv:1512.00637].

- Chp. 11.4 of Peskin, M.E. and Schroeder, D.V. (1995). “An Introduction to Quantum Field Theory”.

- Chp. 2 here is a very good starting point: Berges, J. (2004). “Introduction to nonequilibrium quantum field theory”. AIP Conf. Proc. 739:1, pp. 3-62 [arXiv:hep-ph/0409233].
Have been reading this, but am partially dissatisfied with the offered degree of explanation... He never says why he approximates \(\Gamma\) the way he does!

**How to rigorously approximate the effective action for an interacting field theory?**

First let me outline the construction of the effective action. Starting point is the partition function\begin{align}\label{eq:partFunc}

Z[J^{*}_{},J^{}_{}]&=\int \mathcal{D}[\phi^{*}_{},\phi^{}_{}]e^{\mathrm{i} S[\phi^{*}_{},\phi^{}_{}]} e^{\int \mathrm{i}\text{d}t(J^{}_{}\phi^{*}_{} + J^{*}_{}\phi^{}_{})},

\end{align}

where \( S \) is supposed to be a non-quadratic (i.e.

*interacting*) action, and \( Z \) is a function of the external sources \( J \). The generating functional for the connected diagrams is given by the logarithm of the partition function,\begin{align}

W[J^{*}_{},J^{}_{}]=-\mathrm{i}\ln{Z[J^{*}_{},J^{}_{}]}.

\end{align}

Averages can then be obtained by functional differentiation with respect to the sources, according to

\begin{align}

\Phi_{}=\langle\phi_{}\rangle=\frac{\delta W}{\delta J_{}^{*}}.

\end{align}

The

**effective action**is finally defined as the*Legendre**transform*of \( W \),\begin{align}\label{eq:effAction}

\Gamma[\Phi^{*}_{},\Phi^{}_{}]=W[J^{*}_{},J^{}_{}]-\int \mathrm{d}t(J^{}_{}\Phi^{*}_{} + J^{*}_{}\Phi^{}_{}).

\end{align}

If one decomposes the field into average and fluctuation,

\begin{align}\label{eq:flucs_decomp}

\phi_{}=\Phi_{}+\delta\phi_{},

\end{align}

after re-exponentiating the effective action may be written

\begin{align}

& e^{\mathrm{i}\Gamma[\Phi^{*}_{},\Phi^{}_{}]}=\int \mathcal{D}[\delta\phi^{*}_{}, \delta\phi^{}_{}] \exp{\left\{\mathrm{i} S_{}[\Phi^{*}+\delta\phi^{*}, \Phi_{}+\delta\phi_{}]-\mathrm{i}\int\text{d}t\left(\frac{\delta\Gamma}{\delta\Phi^{*}_{}}\delta\phi^{*}_{}

+\frac{\delta\Gamma}{\delta\Phi^{}_{}}\delta\phi^{}_{}\right)\right\}},

\end{align}

where we have replaced the sources with the inverse relation from the Legendre transform. Note that the source terms for the average from the Legendre transform have cancelled those from the definition of \( Z \). I believe to understand that the essence of the many-body problem is contained in this formula: for the trouble is that we now have \( \Gamma \) on both sides of the equation! That might sound banal, but since one obviously cannot simply solve for \( \Gamma \), we have a conundrum. The farthest I have gotten so far is the lowest level of approximation, which amounts to exploiting the fact that to lowest order the effective action must equal the action evaluated at the average, \( \Gamma_0[\Phi^{*}_{},\Phi^{}_{}]= S[\Phi^{*}_{},\Phi^{}_{}]\). If one uses this on the right-hand side, and also drops all fluctuation terms of higher order than quadratic, one seemingly comes out with the Hartree-Fock approximation. To next order, however, things are already more involved. As I gather from standard QFT books, approximating \( \Gamma \) on the right-hand side is also related to renormalization (in the sense of generating counter-terms).

Some references:

- Diehl, S. et al. "Keldysh Field Theory for Driven Open Quantum Systems" [arXiv:1512.00637].

- Chp. 11.4 of Peskin, M.E. and Schroeder, D.V. (1995). “An Introduction to Quantum Field Theory”.

- Chp. 2 here is a very good starting point: Berges, J. (2004). “Introduction to nonequilibrium quantum field theory”. AIP Conf. Proc. 739:1, pp. 3-62 [arXiv:hep-ph/0409233].

written
6 months ago by
Tim

### 1 Answer

1

related post:

“When can we handle a quantum field like a classical field?”

https://physics.stackexchange.com/q/385470

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