Real-Time Contours, Keldysh Bases, and Response

Real-time quantum field theory is reliable only when its contour, initial state, basis rotation, normalization identity, and source convention are fixed together. This chapter develops that complete structure. Its central check is simple but powerful: equal sources on the two contour branches must give $Z[J,J]=1$, and every transformed propagator and vertex must respect the causal consequences of that identity.

Enter the closed-time path

Begin with the normalized generating functional. The forward branch $+$ runs from $t_0$ to a return time $t_f$ and the backward branch $-$ returns from $t_f$ to $t_0$. Throughout the chapter,

$$\phi_r=\frac{\phi_++\phi_-}{2}, \qquad \phi_a=\phi_+-\phi_-,$$

and sources are rotated as $J_r=(J_++J_-)/2$, $J_a=J_+-J_-$, so that

$$J_+\phi_+-J_-\phi_-=J_a\phi_r+J_r\phi_a.$$

The physical source is $J_a=0$, $J_r=J$. Some references exchange the labels or use $1/\sqrt2$ normalizations; their formulas must be transformed rather than copied term by term.

The initial-state page encodes Gaussian covariances and non-Gaussian cumulants at $t_0$. Unitarity and largest-time identities derive the causal zeros. Retarded, advanced, and Keldysh bases then rotate the full contour matrix, and causal and statistical propagators isolate commutator response from state-dependent fluctuations.

The last three pages add dynamics and inference: Keldysh vertices organize causal diagrams, KMS and fluctuation–dissipation states the equilibrium condition that relates response to fluctuations, and nonlinear sources derive nested-commutator response.

Diagnosis by the missing ingredient in a real-time calculation

Symptom	Inspect	Decisive test
Wrong sign or acausal support	Branch orientation, source signs, and basis normalization	Recover $G^R(x,y)=0$ for $x^0
Unexplained early-time transient	Initial density matrix and switch-on protocol	Vary $t_0$ while preserving the physical preparation.
Fluctuation–dissipation violation	KMS, chemical potentials, and Fourier convention	Test the Wightman KMS relation before defining an effective temperature.
Forbidden diagram survives	$r/a$ vertex parity and latest-time index flow	Verify every closed retarded chain has incompatible strict time inequalities.

Scope boundary

The formalism computes in-in expectation values for a specified density matrix. Normalization and causality do not by themselves imply positivity of every truncated correlator, thermal equilibrium, Markovianity, or gauge consistency. Open-system trace preservation, KMS symmetry, and nonlinear fluctuation theorems require additional hypotheses stated on their respective pages.

References

• Keldysh, L. V. (1965). “Diagram Technique for Nonequilibrium Processes.” Soviet Physics JETP 20, 1018–1026. JETP PDF.
• Schwinger, J. (1961). “Brownian Motion of a Quantum Oscillator.” Journal of Mathematical Physics 2, 407–432. DOI.