Schwinger–Keldysh and open quantum fields

Begin with a normalized initial density operator $\rho_0$, rather than silently selecting a vacuum. Introduce sources on forward and backward time evolution and take the trace after both branches meet. The branches carry opposite action signs, and the initial density matrix supplies their initial boundary condition. Doubling the fields is bookkeeping for the expectation value; it does not introduce a second physical copy that an experiment can independently observe.

When the two sources agree, unitary forward evolution cancels backward evolution inside the trace, giving a generating functional equal to one. Derivatives with respect to differently arranged sources then generate contour-ordered correlations and retarded responses. This normalization constrains an approximate real-time action. It is stronger than demanding a plausible-looking damping curve, because an inconsistent truncation can generate correlations while violating probability conservation or the causal structure needed for a physical response.

Integrating environmental degrees of freedom produces an influence functional coupling the two histories. Its real and imaginary parts encode response and fluctuations in a convention-dependent organization. In a general environment the kernels have memory, so the resulting equation need not be local in time. Thermal equilibrium imposes a fluctuation–dissipation relation with a specified temperature and frequency convention; a driven nonequilibrium state does not automatically satisfy the same relation.

A time-local Lindblad equation is a useful special case under appropriate assumptions, not a consequence of merely drawing the contour. Consider pure dephasing with nonnegative constant rate $\gamma$ and zero Hamiltonian. The diagonal populations stay fixed, while an off-diagonal density-matrix element decays as $e^{-\gamma t}$. This map preserves trace and positivity. It captures a simple Markovian channel, but cannot represent arbitrary strong system-environment coupling, initial correlations, or long-lived environmental memory.

For an initial equal superposition, dephasing leaves both measurement populations at one-half. Nevertheless, the state changes: its off-diagonal coherence shrinks and its purity falls from one toward one-half. A coherent Hamiltonian that only rotates relative phase also leaves these populations unchanged, but preserves purity. Measuring basis populations alone therefore cannot diagnose decoherence; interference measurements or state reconstruction are needed to access the missing information.

In a field theory, the same distinction appears among occupation, response, and noise correlations. An effective dissipative action should be tested against normalization, causality, positivity constraints, and any equilibrium relations actually assumed. Deriving and maintaining those properties through an approximation can be difficult, especially beyond weak coupling or a Markov limit. The contour supplies a systematic framework for asking those questions; it does not certify an arbitrary noise model or justify replacing all quantum correlations by classical random numbers.