Two-particle QFT in the real-frequency Keldysh formalism

Alternatively to the imaginary-time (Matsubara) formalism used on the other pages, two-particle quantum field theory can also be formulated in the real-frequency Keldysh formalism. This is particularly useful for studying non-equilibrium steady states, but can also be applied to equilibrium situations, avoiding the need for analytic continuation from imaginary to real frequencies.

Many-body action in the Keldysh formalism¶

We begin again with the many-body Hamiltonian in second quantization. In the Keldysh formalism, the partition function is given by the path integral

\begin{align} Z &= \int \mathcal{D}[\bar{c}, c] \,e^{i S[\bar{c}, c]} \\ S[\bar{c}, c] &= \int_\mathcal{C} dt \left\{ \sum_1 \bar{c}_1(t) i \partial_t c_1(t) - H[c_1^\dagger(t), c(t)] \right\} = S_0 + S_I \, , \end{align}

(1)

where the time integral runs along the Keldysh contour $\mathcal{C}$ , which consists of a forward and backward branch running from $t=-\infty$ to $t=+\infty$ and back. The fields $\bar{c}_1(t)$ and $c_1(t)$ are defined on the contour, with the index 1 again denoting all single-particle quantum numbers (e.g., momentum, spin, ...).

For an arbitrary quartic interaction, we can write the action as

\begin{align} S = S_0 + S_I = \overline{c}_1 (G_0^{-1})_{12} c_2 + \frac{1}{4} \overline{c}_1 \overline{c}_3 F_{0,1234} c_2 c_4 \, , \end{align}

(2)

where repeated indices imply summation/integration. Here, $G_0$ is the bare propagator and $F_0$ the bare four-point vertex, defined on the Keldysh contour.

Correlation functions¶

The two-point and four-point correlation functions are defined as in the Matsubara formalism, with an additional contour-ordering operator $\mathcal{T}_\mathcal{C}$ and one factor of $i$ ,

\begin{align} G_{21} &= -i \langle \mathcal{T}_\mathcal{C} c_2 \overline{c}_1 \rangle \, , \\ G^{(4)}_{4231} &= i \langle \mathcal{T}_\mathcal{C} c_4 c_2 \overline{c}_3 \overline{c}_1 \rangle \, . \end{align}

(3)

The factors of $i$ are motivated as follows. Correlation functions are most cleanly derived as functional derivatives of a generating functional with respect to source fields. The generating function in the Keldysh formalism reads

\begin{align} Z[\bar{J}, J] = \int \mathcal{D}[\bar{c}, c] \,e^{i S[\bar{c}, c] + i \bar{J} c + i \overline{c} J} \, . \end{align}

(4)

Now, due to the factor of $i$ in the exponent, each functional derivative with respect to a source field brings down a factor of $i$ ,

\begin{align} \frac{\delta}{\delta \bar{J}} &\leftrightarrow i c \, ; & \frac{\delta}{\delta J} &\leftrightarrow i \overline{c} \, . \end{align}

(5)

Hence, a contour-ordered $2n$ -point correlation function will contain a prefactor of $1/i^{(2n)}$ ,

\begin{align} \langle T_\mathcal{C} c_1 \ldots c_n \overline{c}_{n+1} \ldots \overline{c}_{2n} \rangle = \frac{1}{i^{2n}} \frac{1}{Z} \frac{\delta^{2n} Z}{\delta \bar{J}_1 \ldots \delta \bar{J}_n \delta J_{n+1} \ldots \delta J_{2n}} \Bigg|_{\bar{J}=J=0} \, . \end{align}

(6)

Now, for an interacting theory, such computations cannot be done exactly. But we can do them for a non-interacting theory, for which we can compute the Gaussian path integral. The generating functional for the non-interacting theory reads

\begin{align} Z_0[\bar{J}, J] &= \int \mathcal{D}[\bar{c}, c] \,e^{i \bar{c} (G_0^{-1}) c + i \bar{J} c + i \overline{c} J} \\ &= Z_0[0,0] \, e^{-i \bar{J} G_0 J} \, , \end{align}

(7)

where $Z_0[0,0]$ is the partition function of the non-interacting theory without sources. Taking functional derivatives, we find that each pair of fields contributes a factor of $-i G_0$ to the correlation function,

\begin{align} \frac{\delta^2 Z_0[\bar{J}, J]}{\delta \bar{J} \delta J} &= (-i) G_{0} Z_0 \, . \end{align}

(8)

We hence find that the two-point correlation function is given by

\begin{align} \langle T_\mathcal{C} c \overline{c} \rangle_0 = \frac{1}{i^2} (-i) G_0 = i G_0 \ \Leftrightarrow \ G_0 = -i \langle T_\mathcal{C} c \overline{c} \rangle_0 \, . \end{align}

(9)

By analogy, we demand the same normalization for the interacting theory.

For higher-point correlation functions, the situation is not so clear and indeed different conventions exist in the literature. Here, we follow the commonly used convention that an $n$ -point correlation function contains a prefactor of $(-i)^{(n-1)}$ .

As a consequence, the tree expansion of the four-point correlation function reads

\begin{align} iG^{(4)}_{4231} = G_{41} G_{23} + \zeta G_{43} G_{21} + i G^{(4)}_{c,4321}\, , \end{align}

(10)

with the connected part given by

\begin{align} G^{(4)}_{c,4321} = - G_{4\tilde{1}} G_{2\tilde{3}} F_{\tilde{1}\tilde{2}\tilde{3}\tilde{4}} G_{\tilde{2}3} G_{\tilde{4}1}\, . \end{align}

(11)

Keldysh index structure of two- and four-point functions¶

Since a general $n$ -point function is given via an expectation value of a product of $n$ fields, which can each be placed on either of the two branches of the Keldysh contour, it will have $2^n$ Keldysh components in total.

Propagator¶

Starting, with the two-point Green’s function, we can arrange its four Keldysh components in a $2\times 2$ matrix,

\begin{align} (G^{c_2 c_1}) = \begin{pmatrix} G^{--} & G^{-+} \\ G^{+-} & G^{++} \end{pmatrix} \, , \end{align}

(13)

where the superscripts $c_2, c_1 \in \{-, +\}$ denote whether the corresponding time argument is on the forward or backward branch of the contour, respectively. Explicitly, the components read

\begin{align} G^{-|+}(t_2, t_1) &= -i \langle \mathcal{T}_\mathcal{C} c^-(t_2) \overline{c}^+(t_1) \rangle = - \zeta i \langle \overline{c}(t_1) c(t_2) \rangle \equiv G^<(t_2, t_1) \\ G^{+|-}(t_2, t_1) &= -i \langle \mathcal{T}_\mathcal{C} c^+(t_2) \overline{c}^-(t_1) \rangle = -i \langle c(t_2) \overline{c}(t_1) \rangle \equiv G^>(t_2, t_1) \\ G^{--}(t_2, t_1) &= -i \langle \mathcal{T}_{\mathcal{C}} c^-(t_2) \overline{c}^-(t_1) \rangle = G^>(t_2, t_1) \theta(t_2 - t_1) + G^<(t_2, t_1) \theta(t_1 - t_2) \\ G^{++}(t_2, t_1) &= -i \langle \mathcal{T}_{\mathcal{C}} c^+(t_2) \overline{c}^+(t_1) \rangle = G^<(t_2, t_1) \theta(t_2 - t_1) + G^>(t_2, t_1) \theta(t_1 - t_2) \, , \end{align}

(14)

where we have suppressed all other dependencies for ease of notation. Here, the contour-time-ordering has been made explicit and the greater and lesser Green’s functions $G^>$ and $G^<$ have been introduced.

As can be confirmed by direct calculation, the four Keldysh components are not independent, but satisfy the relation

\begin{align} G^{-|+} + G^{+|-} = G^{--} + G^{++} \, . \end{align}

(15)

Making use of this redundancy motivates the Keldysh rotation, defined as

\begin{align} G^{k_2 k_1} = D^{k_2 c_2} G^{c_2 c_1} (D^{-1})^{c_1 k_1} \, , \end{align}

(16)

with the transformation matrix

\begin{align} D &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}\, ; & D^{-1} &= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} \, . \end{align}

(17)

The Keldysh-rotated Green’s function then takes the form

\begin{align} (G^{k_2 k_1}) = \begin{pmatrix} 0 & G^A \\ G^R & G^K \end{pmatrix} \, , \end{align}

(18)

where the component $G^{11}$ vanishes by construction. $G^R(t_2, t_1) = -i \langle [c(t_2), \overline{c}(t_1)]_\zeta \rangle \theta(t_2 - t_1)$ is the physically relevant retarded Green’s function, $G^A(t_2, t_1) = [G^R(t_1, t_2)]^*$ is the advanced component, and $G^K(t_2, t_1) = G^>(t_2, t_1) + G^<(t_2, t_1) = -[G^K(t_1, t_2)]^*$ is the Keldysh component.

Bubble¶

This Keldysh structure of the propagator carries over to the bubble, following directly from its definition as a product of two propagators. In Keldysh space, the bubble reads

\begin{align} \chi_0^{k_4, k_3, k_2, k_1} = G^{k_4 k_1} G^{k_2 k_3} \, . \end{align}

(19)

Formally organizing the 16 Keldysh components in a $4\times 4$ matrix, we find the explicit structure

\begin{align} (\chi_0^{k_4 k_3 k_2 k_1}) &= \begin{pmatrix} 1111 & 1112 & 1121 & 1122 \\ 1211 & 1212 & 1221 & 1222 \\ 2111 & 2112 & 2121 & 2122 \\ 2211 & 2212 & 2221 & 2222 \end{pmatrix} \\ &= \begin{pmatrix} G^{11} G^{11} & G^{12} G^{11} & G^{11} G^{21} & G^{12} G^{21} \\ G^{11} G^{12} & G^{12} G^{12} & G^{11} G^{22} & G^{12} G^{22} \\ G^{21} G^{11} & G^{22} G^{11} & G^{21} G^{21} & G^{22} G^{21} \\ G^{21} G^{12} & G^{22} G^{12} & G^{21} G^{22} & G^{22} G^{22} \end{pmatrix} \\ &= \begin{pmatrix} 0 & 0 & 0 & G^A G^R \\ 0 & G^A G^A & 0 & G^A G^K \\ 0 & 0 & G^R G^R & G^K G^R \\ G^R G^A & G^K G^A & G^R G^K & G^K G^K \end{pmatrix}\, . \end{align}

(20)

Self-energy¶

The Keldysh rotation can be applied to the self-energy, $\Sigma^{k_1 k_2} = D^{k_1 c_1} \Sigma^{c_1 c_2} (D^{-1})^{c_2 k_1}$ , leading to the following Keldysh structure,

\begin{align} (\Sigma^{k_1 k_2}) = \begin{pmatrix} \Sigma^{K} & \Sigma^{R} \\ \Sigma^{A} & 0 \end{pmatrix} \, . \end{align}

(21)

Due to causality, the component $\Sigma^{22}$ vanishes identically. Note that, compared to $G$ , the Keldysh indices $k_1$ and $k_2$ of the self-energy are interchanged.

Dyson equation¶

Speaking of the Dyson equation, which reads in its most general form

\begin{align} G = G_0 + G_0 \circ \Sigma \circ G \, , \end{align}

(23)

where the connector ∘ denotes contractions in time and single-particle quantum numbers. Making the Keldysh structure explicit, separate Dyson equations for the individual Keldysh components can be derived. They read

\begin{align} G^{R} &= G_0^{R} + G_0^{R} \bullet \Sigma^{R} \bullet G^{R} \, , \\ G^{A} &= G_0^{A} + G_0^{A} \bullet \Sigma^{A} \bullet G^{A} \, , \\ G^{K} &= (1 + G^{R} \bullet \Sigma^{R}) \bullet G_0^{K} \bullet (1 + \Sigma^{A} \bullet G^{A}) + G^{R} \bullet \Sigma^{K} \bullet G^{A} \, , \end{align}

(24)

where the bullet $\bullet$ denotes contractions over all indices and variables except the Keldysh indices.

To derive these equations, we start from the general Dyson equation written explicitly in Keldysh space,

\begin{align} \begin{pmatrix} 0 & G^{A} \\ G^{R} & G^{K} \end{pmatrix} &= \begin{pmatrix} 0 & G_0^{A} \\ G_0^{R} & G_0^{K} \end{pmatrix} + \begin{pmatrix} 0 & G_0^{A} \\ G_0^{R} & G_0^{K} \end{pmatrix} \bullet \begin{pmatrix} \Sigma^{K} & \Sigma^{R} \\ \Sigma^{A} & 0 \end{pmatrix} \bullet \begin{pmatrix} 0 & G^{A} \\ G^{R} & G^{K} \end{pmatrix} \\ &= \begin{pmatrix} 0 & G_0^{A} \\ G_0^{R} & G_0^{K} \end{pmatrix} + \begin{pmatrix} 0 & G_0^{A} \bullet \Sigma^{A} \bullet G^{A} \\ G_0^{R} \bullet \Sigma^{R} \bullet G^{R} & (G_0^{R} \bullet \Sigma^{K} + G_0^{K} \bullet \Sigma^{A}) \bullet G^{A} + G_0^{R} \bullet \Sigma^{R} \bullet G^{K} \end{pmatrix} \, . \end{align}

(25)

We directly obtain the two independent Dyson equations for the retarded and advanced components,

\begin{align} G^{R} &= G_0^{R} + G_0^{R} \bullet \Sigma^{R} \bullet G^{R} \, , \\ G^{A} &= G_0^{A} + G_0^{A} \bullet \Sigma^{A} \bullet G^{A} \, , \end{align}

(26)

as well as the equation for the Keldysh component,

\begin{align} G^{K} &= G_0^{K} + G_0^{R} \bullet \Sigma^{K} \bullet G^{A} + G_0^{K} \bullet \Sigma^{A} \bullet G^{A} + G_0^{R} \bullet \Sigma^{R} \bullet G^{K} \, . \end{align}

(27)

To solve the last equation for $G^K$ , we first rewrite it as

\begin{align} (1 - G_0^{R} \bullet \Sigma^{R}) \bullet G^{K} &= G_0^{K} \bullet (1 + \Sigma^{A} \bullet G^{A}) + G_0^{R} \bullet \Sigma^{K} \bullet G^{A} \, . \end{align}

(28)

Next, we observe that we can utilize the Dyson equation for $G^R$ (twice) to write

\begin{align} (1 - G_0^{R} \bullet \Sigma^{R})^{-1} &= G^R \bullet (G^{R})^{-1} (1 - G_0^{R} \bullet \Sigma^{R})^{-1} \\ &= G^R \bullet (G^R - G_0^R \bullet \Sigma^R \bullet G^R)^{-1} \\ &= \boxed{G^R \bullet (G_0^R)^{-1}} \\ &= G^R \bullet [(G^R)^{-1} + \Sigma^R] = \boxed{1 + G^R \bullet \Sigma^R} \, . \end{align}

(29)

Hence, we obtain for the Keldysh component

\begin{align} G^{K} &= \underbrace{(1 - G_0^{R} \bullet \Sigma^{R})^{-1}}_{= 1 + G^R \bullet \Sigma^R} \bullet G_0^{K} \bullet (1 + \Sigma^{A} \bullet G^{A}) + \underbrace{(1 - G_0^{R} \bullet \Sigma^{R})^{-1}}_{= G^R \bullet (G_0^R)^{-1}} \bullet G_0^{R} \bullet \Sigma^{K} \bullet G^{A} \\ &= (1 + G^{R} \bullet \Sigma^{R}) \bullet G_0^{K} \bullet (1 + \Sigma^{A} \bullet G^{A}) + G^{R} \bullet \Sigma^{K} \bullet G^{A} \, . \ \checkmark \end{align}

(30)

In practice, one will calculate all Keldysh components of $\Sigma$ , evaluate the Dyson equations for the retarded and advanced components first (usually in the more practical form $(G^{R/A})^{-1} = (G_0^{R/A})^{-1} - \Sigma^{R/A}$ , and finally evaluate the third equation to obtain $G^K$ .

Four-point vertex¶

The Keldysh rotation for the four-point vertex is defined as

\begin{align} F^{k_1 k_2 k_3 k_4} = D^{k_1 c_1} D^{k_3 c_3} F^{c_1 c_2 c_3 c_4} (D^{-1})^{c_2 k_2} (D^{-1})^{c_4 k_4} \, . \end{align}

(32)

This leads to a $2^4=16$ -component object in Keldysh space. As for the self-energy, the component that vanishes identically due to causality is the one where all Keldysh indices are 2,

\begin{align} F^{2222} = 0 \, . \end{align}

(33)

For an instantaneous bare interaction (i.e., local in time), such as, e.g., the Hubbard interaction, the bare interaction strongly simplifies in Keldysh space, taking the form

\begin{align} F_0^{k_1 k_2 k_3 k_4} = \begin{cases} F_0 / 2 & \text{if } k_1 + k_2 + k_3 + k_4 \text{ odd} \\ 0 & \text{otherwise}\, . \end{cases} \end{align}

(34)