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Real-Time Contours, Keldysh Bases, and Response

The closed-time path computes real-time expectation values from a prepared quantum state. Doubling the history exposes response and fluctuations while keeping unitarity and causality visible, even away from equilibrium.

Enter this chapter

Section titled “Enter this chapter”

Begin with Closed-Time-Path Generating Functionals in Practice to construct a normalized generating functional for forward and backward evolution.

Helpful background. Closed-Time-Path Generating Functionals in Practice and Closed-Time-Path Grammar develop the definitions used by the chapter’s opening pages.

Topics

Section titled “Topics”
  1. Closed-Time-Path Generating Functionals in Practice — Builds the generating functional that evolves and traces an initial density operator.
  2. Initial Density Matrices and Contour Boundary Conditions — Encodes Gaussian and higher initial correlations at the preparation surface.
  3. Unitarity, Normalization, and Largest-Time Identities — Derives the identities imposed by a normalized unitary evolution.
  4. Retarded, Advanced, and Keldysh Bases — Rotates contour components into response and fluctuation variables.
  5. Causal and Statistical Propagators — Separates state-dependent fluctuations from commutator response.
  6. Keldysh Vertices and Causal Diagrammatics — Turns causal identities into rules for interaction vertices and diagrams.
  7. KMS Relations and Fluctuation–Dissipation — Derives the equilibrium relation between response and fluctuations.
  8. Sources and Nonlinear Real-Time Response — Obtains higher response from causally ordered source derivatives.

How the ideas fit together

Section titled “How the ideas fit together”

When the two histories have equal sources, forward and backward evolution cancel under the trace. This normalization identity constrains both propagators and vertices. The Keldysh basis reorganizes those constraints so that causal response and statistical fluctuations are distinct components Kamenev and Levchenko 2009, §§ 2.2–2.4.

Away from equilibrium, the initial state supplies fluctuation information independently of the response. KMS adds the relation between them only in thermal equilibrium. Merely finding a stationary state does not justify inserting an equilibrium fluctuation–dissipation relation.

Where to go next

Section titled “Where to go next”

Continue to Nonequilibrium Green Functions and Kadanoff–Baym Evolution to identify the statistical and spectral functions needed for two-time evolution.

References

Section titled “References”
  • Kamenev, Alex, and Alex Levchenko. “Keldysh Technique and Non-Linear σ-Model: Basic Principles and Applications.” Advances in Physics 58, no. 3 (2009): 197–319. DOI.