Keldysh-Schwinger Field Theory Conventions

This document outlines the conventions used in Keldysh-Schwinger field theory, including the system's action, Green's functions, and the self-consistent Born approximation. These conventions are essential for understanding the mathematical framework and methods employed in the package.

System

The system is described by the action $S$, which governs the dynamics of the bosonic fields $\psi_+$ and $\psi_-$ on the forward $(+)$ and backward $(-)$ contours, respectively. The action is given by:

\[\begin{aligned} S & \left[\psi_{+},\left(\psi_{+}\right)^*, \psi_{-},\left(\psi_{-}\right)^*\right]\\ & =\int \mathrm{d}^d x \mathrm{~d} t^{\prime} \left(\psi_{+}\right)^*\left[i \partial_t+D \nabla^2-V(x)\right] \psi_{+} -\left(\psi_{-}\right)^*\left[i \partial_t+D \nabla^2-V(x)\right] \psi_{-} \\ & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+i\left[L_{+}\left(L_{-}\right)^*-\frac{1}{2}\left(\left(L_{+}\right)^* L_{+}-\left(L_{-}\right)^* L_{-}\right)\right] \end{aligned}\]

Here, $\psi_+$ and $\psi_-$ are the bosonic fields on the forward $(+)$ and backward $(-)$ contours, respectively.

We can rewrite this in the Retarded-Advanced-Keldysh (RAK) basis by defining:

• \[\psi_c = (\psi_+ + \psi_-)/\sqrt{2}\]

  (classical field)

• \[\psi_q = (\psi_+ - \psi_-)/\sqrt{2}\]

  (quantum field)

In this basis, the diffusion part of the action becomes:

\[S_{\mathrm{diff}}^{R A K}=\int d t d^d x\left\{\bar{\psi}_q\left[i \partial_t+D \nabla^2-V(\vec{x})\right] \psi_c+\bar{\psi}_c\left[i \partial_t+D \nabla^2-V(\vec{x})\right] \psi_q\right\}.\]

Green's Functions

Green's functions describe the propagation of fields and encode information about the system's response to perturbations. In the RAK basis, the Green's function is represented as a matrix:

\[\begin{aligned} \hat{G}^{R A K}\left(x_1, x_2\right) &=\left(\begin{array}{cc}G^K\left(x_1, x_2\right) & G^R\left(x_1, x_2\right) \\ G^A\left(x_1, x_2\right) & 0\end{array}\right) \\ &=-i\left(\begin{array}{cc}\left\langle\phi_c\left(x_1\right) \bar{\phi}_c\left(x_2\right)\right\rangle & \left\langle\phi_c\left(x_1\right) \bar{\phi}_q\left(x_2\right)\right\rangle \\ \left\langle\phi_a cornerstone\]

Here, $G^R$, $G^A$, and $G^K$ are the retarded, advanced, and Keldysh Green's functions, respectively.

The two-point dressed Green's function $G^{\mu \nu}$, labeled by indices $\mu=c,q$ and $\nu=c,q$, is defined as:

\[i G^{\mu \nu}\left(x_1, x_2\right)=\int \mathcal{D}\left[\phi_c, \bar{\phi}_c, \phi_q, \bar{\phi}_q\right] \phi_\mu\left(x_1\right) \bar{\phi}_\nu\left(x_2\right) e^{i S_0+i S_{\mathrm{int}}}.\]

Here, $S_\mathrm{int} = \int d^{d}x d t \mathcal{L}_\mathrm{int}$ is the interaction part of the action.

Self-Consistent Born Approximation

The self-consistent Born approximation is a perturbative method used to compute the Green's function by expanding it in terms of the interaction part of the action. The expansion is given by:

\[\begin{aligned} & i G^{\mu \nu}\left(x_1, x_2\right)=\int \mathcal{D}\left[\phi_c, \bar{\phi}_c, \phi_q, \bar{\phi}_q\right] \phi_\mu\left(x_1\right) \bar{\phi}_\nu\left(x_2\right) \sum_{k=0}^{\infty} \frac{i^k S_{\mathrm{int}}^k}{k!} e^{i S_0}= \\ & \quad=i G_0^{\mu \nu}\left(x_1, x_2\right)+i \int d^d y d t_y\left\langle\phi_\mu\left(x_1\right) \bar{\phi}_\nu\left(x_2\right) \mathcal{L}_{\mathrm{int}}(y)\right\rangle_0+\sum_{k=2}^{\infty}\left\langle\phi_\mu\left(x_1\right) \bar{\phi}_\nu\left(x_2\right) \frac{i^k S_{\mathrm{int}}^k}{k!}\right\rangle_0 . \end{aligned}\]

The perturbative expansion of the Green's function has the structure:

\[\hat{G}=\hat{G}_0+\hat{G}_0 \circ \hat{\Sigma} \circ \hat{G}_0+\hat{G}_0 \circ \hat{\Sigma} \circ \hat{G}_0 \circ \hat{\Sigma} \circ \hat{G}_0+\ldots=\hat{G}_0+\hat{G}_0 \circ \hat{\Sigma} \circ \hat{G}.\]

Here, $\circ$ denotes space-time convolution and Keldysh-matrix multiplication. The self-energy $\hat{\Sigma}$ is defined as:

\[\hat{\Sigma}\left(y_1, y_2\right)=\left(\begin{array}{cc}0 & \Sigma^A\left(y_1, y_2\right) \\ \Sigma^R\left(y_1, y_2\right) & \Sigma^K\left(y_1, y_2\right)\end{array}\right).\]

This leads to the Dyson equation, which relates the full Green's function $\hat{G}$ to the non-interacting Green's function $\hat{G}_0$ and the self-energy $\hat{\Sigma}$:

\[[\hat{G}_0^{-1} -\hat{\Sigma} ]\hat{G}=\mathbb{1}.\]

The Dyson equation is central of many-body physics, providing a framework for systematically incorporating interactions.