Schwinger-Keldysh Contour in Quantum Dynamics

• Schwinger-Keldysh Contour is a closed real and imaginary time path designed to compute contour-ordered Green’s functions in both equilibrium and non-equilibrium settings.
• It unifies Matsubara, retarded, and advanced Green’s functions, enabling systematic treatment of both low- and high-frequency spectral features.
• The contour is pivotal in numerical analytic continuation via the Maximum Entropy Method, balancing stability at low frequencies with high-frequency resolution.

The Schwinger–Keldysh (SK) contour, also known as the closed time path (CTP) contour, is a piecewise trajectory in the complex time plane devised to formulate real-time quantum dynamics and thermodynamics, particularly suited for studying non-equilibrium processes, response functions, and analytic continuation problems in many-body systems. Its structure naturally enables consistent computation of contour-ordered correlation functions, incorporating both forward and backward time evolution, and in equilibrium, it provides a unified framework encompassing Matsubara (imaginary-time), real-time, and mixed Green’s functions. The SK contour is also central to maximum-entropy approaches for analytic continuation and is fundamental in the path-integral formalism for both classical and quantum statistical mechanics. Its segments encode complementary information about spectral and dynamical properties, and the contour's generalizations admit systematic treatment of both equilibrium and far-from-equilibrium phenomena.

1. Formal Structure of the Schwinger–Keldysh Contour

The canonical Schwinger–Keldysh contour $\mathcal{C}$ is composed of three connected segments in the complex time plane:

• An imaginary-time (Matsubara) leg from $t=0$ to $t=-i\beta$, encoding thermal preparation.
• A forward real-time branch from $t=0$ to $t=+t_{\text{max}}$.
• A backward real-time branch from $t=+t_{\text{max}}$ back to $t=0$.

This may be parameterized by a single variable $t\in[0,2 t_{\text{max}}+\beta]$ with piecewise mapping

$$z(t) = \begin{cases} -i t & 0 \leq t \leq \beta\ t - i\beta & \beta \leq t \leq \beta + t_{\text{max}} \ (2 t_{\text{max}} + \beta - t) - i\beta & \beta + t_{\text{max}} \leq t \leq 2 t_{\text{max}} + \beta\ \end{cases}$$

The segments traverse down the imaginary axis, run forward along the real axis (displaced by $-i\beta$), and return backward, forming a closed loop suitable for computing traces and expectation values in the real-time path-integral representation (Dirks et al., 2012).

2. Contour-Ordered Green’s Functions and Kernel Properties

On $\mathcal{C}$, the contour-ordered Green’s function is defined by

$$G(z, z') \equiv -i \langle T_\mathcal{C} \psi(z) \psi^\dagger(z')\rangle,$$

where $T_\mathcal{C}$ orders operators according to their position on the contour. For equilibrium states, such $G(z,z')$ admits a spectral representation

$$G(z, z') = \int_{-\infty}^\infty d\omega\, A(\omega)\, e^{-i\omega(z-z')}\, F_{z,z'}(\omega)$$

with the “contour-Fermi factor”

$$F_{z,z'}(\omega) = \begin{cases} f(-\omega) & z' \prec_\mathcal{C} z \ - f(\omega) & \text{otherwise} \end{cases}, \qquad f(\omega) = \frac{1}{e^{\beta\omega} + 1}.$$

Restricting $z, z'$ to segments of $\mathcal{C}$ yields canonical functions:

• Matsubara Green’s function: Imaginary-time data, sensitive to low-frequency (quasiparticle) structure.
• Greater/Lesser propagators: Real-time data, sensitive to high-frequency (satellite, band-edge) features.
• Retarded Green’s function: Causal response, obtained from the real-time branches.

Each kernel probes $A(\omega)$ differently. The Matsubara kernel $K_M(\tau, \omega)$ is smooth in $\omega$ and thus exponentially suppresses high-frequency features, making spectral reconstruction from imaginary time alone an ill-posed inversion for large $|\omega|$, but well-conditioned for low frequencies. Conversely, real-time kernels $e^{-i\omega t}$ afford greater sensitivity at high $|\omega|$, with truncation-induced oscillatory “ringing” in $\omega$ (Dirks et al., 2012).

3. Numerical Implementation and the Maximum Entropy Method

For practical analytic continuation, one discretizes both imaginary and real-time data:

• Imaginary time: $N_\tau$ points $\tau_j = j\beta/N_\tau$, yielding $D_{\text{imag}} = \{G_M(\tau_j)\}$.
• Real time: $N_t$ points $t_j = j t_{\text{max}}/(N_t-1)$; both $\text{Re}\, G^>(t_j,0)$, $\text{Im}\, G^>(t_j,0)$, etc., are evaluated, giving a composite data vector $D_{\text{real}}$ of length $4 N_t$.

Each data point admits a kernel representation $D_i = \int d\omega\, K_i(\omega)A(\omega)$. The discretized inverse problem $D = KA$ (with $K$ an $N_{\text{data}}\times M$ matrix) is ill-conditioned and addressed via the Maximum Entropy Method (MEM), wherein one minimizes a functional $Q[A] = \chi^2[A] - \alpha S[A]$, combining the goodness-of-fit to the data and an entropic regularizer relative to a default model. Singular-value decomposition (SVD) of $K$ identifies those spectral directions strongly determined by the data (large singular values) and suppresses noise-dominated components (Dirks et al., 2012).

4. Physical Interpretation: Information Content of Contour Segments

Contour Segment	Primary Spectral Content	Limiting Factors or Advantages
Matsubara (Imaginary)	Low-frequency, quasi-particle peaks	Suppresses high-frequency; robust at low $\omega$
Real-Time (Finite $t_{\text{max}}$)	High-frequency, satellites, band-edges	Sensitive to cutoff; enhances large $|\omega|$

The Matsubara branch imposes strong constraints on $A(\omega)$ for small $|\omega|$, anchoring features like quasiparticle peaks and gap edges. The real-time branches (finite $t_{\text{max}}$) furnish a singular-value plateau whose width is set by $t_{\text{max}}$, encoding significant information about high-$\omega$ behavior. As $t_{\text{max}}$ increases, the plateau widens, enhancing high-frequency resolution.

Combining both branches leverages the smoothing kernel in Matsubara time to stably determine low-frequency features, while using short real-time windows (typically $t_{\text{max}}\sim 2\!-\!10$ in bandwidth units, limited by Monte Carlo sign problems) to recover global high-energy structure—at moderate numerical cost (Dirks et al., 2012).

5. Practical Aspects and Limitations

Several factors must be optimized for robust recovery of spectral information:

• The number of imaginary and real-time points ($N_\tau, N_t$) must be large enough to resolve the singular-value plateau of $K$ and avoid machine-precision truncation.
• Imaginary-time data should be most densely sampled at small $\tau$ for optimal low-$\omega$ feature resolution.
• The overall numerical instability is controlled by the depth of the plateau and the degree of noise in both data channels.
• The main practical difficulty with real-time data is the dynamical sign problem, which fundamentally limits $t_{\text{max}}$, constraining the attainable high-frequency resolution.

The inclusion of mixed Matsubara–real components (off-diagonal elements) can further improve reconstructions but increases computational complexity (Dirks et al., 2012).

6. Unified Framework and Broader Significance

The Schwinger–Keldysh contour construction unifies the treatment of different Green’s functions (Matsubara, retarded, advanced, greater, lesser) as different projections of the same contour-ordered correlator. This approach situates both equilibrium and nonequilibrium spectral calculations within a single formal structure compatible with modern analytic and numerical techniques such as MEM. Contour-ordered Green’s functions $G(z,z')$ admit reduction onto familiar components for both spectral and time-dependent observables, providing a powerful conceptual and computational infrastructure for strongly correlated quantum systems.

A plausible implication is that the SK contour formulation not only informs equilibrium analytic continuation, but naturally extends to nonequilibrium time evolution and allied inverse problems, bridging stochastic, dynamical, and thermodynamical viewpoints within a single, operationally transparent path-integral or operator framework (Dirks et al., 2012).

7. Extensions and Future Directions

Possible extensions include:

• Systematic exploitation of mixed and off-diagonal contour data for improved reconstructions.
• Extension to more general contours (e.g., “multi-fold” contours or Kostantinov–Perel’ constructions that facilitate more general initial-state correlators).
• Integration with quantum Monte Carlo methods that directly sample the SK contour, thereby bypassing analytic continuation when possible but relying on the same contour infrastructure for observable reconstruction.

The Schwinger–Keldysh contour thereby remains central to ongoing theoretical and algorithmic developments in the quantum many-body and nonequilibrium statistical mechanics communities.