KeldyshQFT

Conventions for real-frequency quantum field theory in the Keldysh formalism, as used in ^{1 2 3 4}.

Basic building blocks

Variable	Definition	Diagram	Comment
Partition function	\[ \mathcal{Z} = \int \mathcal{D}[\overline{c},c]\, e^{iS[\overline{c},c]} \]	\(-\)	\(-\)
Action	\[ \begin{aligned} S[\overline{c},c] &= S_0[\overline{c},c] + S_{\mathrm{int}} [\overline{c},c] \\ &= \overline{c}_{1'} [G_0^{-1}]_{1'|1} c_{1} + \tfrac{1}{4} \overline{c}_{1'} \overline{c}_{2'} [\Gamma_0]_{1'2'|12} c_{2} c_{1} \end{aligned} \]	\(-\)	Multi-indices: time, Keldysh contour index, spin, etc. Einstein summation convention is used. Contracting time and Keldysh indices means an integration along the Keldysh contour.
Bare interaction	\[ \begin{aligned} &[\Gamma_0]^{j_{1'}j_{2'}|j_1 j_2}_{\sigma_{1'}\sigma_{2'}|\sigma_1 \sigma_2}(t_{1'},t_{2'}|t_1,t_2 ; q_{1'},q_{2'}|q_1 q_2) \\ &= -j_1 \delta_{j_{1'}=j_{2'}=j_{1}=j_{2}} \delta(t_{1'}=t_{2'}=t_{1}=t_{2}) \\ &\quad \times (\delta_{\sigma_{1'},\sigma_2}\delta_{\sigma_{2'},\sigma_1} - \delta_{\sigma_{1'},\sigma_1}\delta_{\sigma_{2'},\sigma_2}) [\Gamma_0](q_{1'},q_{2'}|q_1 q_2) \end{aligned} \]		$j$: Keldysh contour index $\sigma$: Spin index $t$: real time variable $q$: everything else
Bare inv. propagator	$$[G_0^{-1}]_{1'|1} = \delta_{1'|1} i\partial_{t_1} - h_{1'|1}$$	\(-\)	$h_{1'\mid 1}$ is the single-particle Hamiltonian, $H_{0}[\overline{c},c] =\overline{c}_{1'} h_{1'\mid 1} c_1$
Bare propagator	\[ \begin{aligned} {[G_0]}_{1|1'} &= -i \langle \mathcal{T_\mathcal{C}}\, c_1 \overline{c}_{1'}\rangle_0 \\ &= \frac{-i}{\mathcal{Z}_0} \int \mathcal{D}[\overline{c},c]\, c_1 \overline{c}_{1'} e^{iS_0[\overline{c},c]} \end{aligned} \]		with the non-interacting partition function $\mathcal{Z}_0 = \int \mathcal{D}[\overline{c},c]\, e^{iS_0[\overline{c},c]}$
Full propagator	\[ \begin{aligned} G_{1|1'} &= -i\langle \mathcal{T_\mathcal{C}}\, c_1 \overline{c}_{1'}\rangle \\ &= \frac{-i}{\mathcal{Z}} \int \mathcal{D}[\overline{c},c]\, c_1 \overline{c}_{1'} e^{iS[\overline{c},c]} \end{aligned} \]		\(-\)
Dyson equation	\[ \begin{aligned} G_{1|1'} &= [G_0]_{1|1'} + [G_0]_{1|2'} \Sigma_{2'|2} G_{2|1'} \\ \Leftrightarrow \quad G^{-1}_{1'|1} &= [G_0^{-1}]_{1'|1} - \Sigma_{1'|1} \end{aligned} \]		\(-\)
Four-point (4p) function	$G^{(4)}_{12|1'2'} = i\langle \mathcal{T_\mathcal{C}}\, c_1 c_2 \overline{c}_{2'} \overline{c}_{1'} \rangle$		\(-\)
Tree expansion	$i G^{(4)}_{12|1'2'} = G_{1|1'} G_{2|2'} - G_{1|2'} G_{2|1'} + i G^{(4)}_{c;\,12|1'2'}$		\(-\)
Connected 4p function	$G^{(4)}_{c;\,12|1'2'} = - G_{1|3'} G_{2|4'} \Gamma_{3'4'|34} G_{3|1'} G_{4|2'}$		\(-\)

Still to do

Two-particle channels

• parquet decomposition
• frequency parametrization

Spin components

Keldysh contour, Keldysh rotation, FDT

Symmetries

• crossing
• parity
• complex conjugation
• …

References