Close the time contour without doubling the physics

An initial normalized state $\varrho_0$ evolves as $U\varrho_0U^\dagger$. Computing an expectation at a later time inserts an operator into the trace of that evolved state. The same structure can be drawn as evolution from the initial time to a final contour time and then back again. The final time is chosen later than all physical insertions. The initial density matrix supplies the boundary information and can include correlations; assuming only a few Gaussian initial correlators is an approximation to that input.

Introduce distinct sources on the forward and backward branches to generate the required orderings. Setting those sources equal restores unitary cancellation between the two evolutions, so the generating functional is one for every common source when the trace is normalized. This condition expresses conservation of probability. It does not say that an operator expectation cannot change under that common physical evolution. Observables arise through appropriately placed source derivatives before the equal-source identification, rather than by interpreting the value one as a complete state description.

A bosonic contour two-point function has four branch assignments. Two insertions on the forward branch give time ordering, two on the backward branch give reverse time ordering, and insertions on different branches give greater or lesser products. The contour ordering, not a new physical interaction, determines their relationship. At separated times they satisfy an identity expressing that the two possible ordinary time orderings together cover the two unordered products. Fermionic versions need their matching statistical signs and definitions.

A useful change of variables uses the average field $\phi_r=(\phi_++\phi_-)/2$ and the difference field $\phi_a=\phi_+-\phi_-$. It organizes causal response and fluctuations efficiently, but the letters are labels of a linear basis change. In particular, the difference field is not a second particle species, and setting its physical source to zero does not discard all fluctuations. The contour action contains a forward action minus a backward action, with matching and initial-state conditions required to reconstruct the trace.

The closed contour can start from a thermal density matrix, but it does not require one. Equilibrium permits KMS relations that reduce the number of independent functions, while general initial states need separate statistical information. An imaginary-time segment can encode a thermal initial ensemble in a contour construction; it is not evidence that every real-time problem must be periodic in time. A quench or driving protocol changes the actual Hamiltonian history and should not be represented by switching a parameter in a stationary equilibrium answer.

Approximations to contour dynamics should retain the normalization and causal identities relevant to the calculation. Violating them can create spurious responses or probability loss. Conservation-law checks are also valuable, but preserving one identity alone does not establish full accuracy: an approximation can conserve energy while missing scattering or long-time thermalization. Compare the intended observable, time interval, and scale hierarchy with the retained terms. The contour is an exact organizational framework; the practical closure or diagram truncation still needs a controlled justification.