Lorentzian, Euclidean, and In-In Formulations

A transition amplitude, a Euclidean correlation function, and a real-time expectation value answer different questions. This chapter connects their boundary conditions, analytic continuations, and contours, explaining why a formal change of time coordinate cannot replace the state and causality assumptions.

Enter this chapter

Begin with Lorentzian Boundary Conditions and the iε Prescription to understand pole placement. For a specified initial state, enter through the in–out versus in–in comparison before studying the closed time path.

Topics

1. Lorentzian Boundary Conditions and the iε Prescription — Interprets pole prescriptions as distributional boundary values and state information.
2. Wick Rotation and Analytic Continuation — Tests the analytic and contour conditions needed to continue between signatures.
3. Euclidean Correlators and Schwinger Functions — Defines Euclidean correlation data and its source normalization.
4. Reflection Positivity within Osterwalder–Schrader Reconstruction — Separates reflected positivity from the full reconstruction hypotheses.
5. Retarded, Advanced, and Spectral Correlators — Relates commutators, causal support, and frequency-space spectral data.
6. In–Out versus In–In Expectation Values — Distinguishes transition matrix elements from initial-state expectations.
7. Closed-Time-Path Grammar — Builds the forward and backward branches of the initial-state generating functional.

How the ideas fit together

A positive Euclidean probability measure does not by itself guarantee a physical Lorentzian Hilbert space. Reflection positivity tests a different quadratic form involving time reflection; reconstruction also needs the other hypotheses of an appropriate Osterwalder–Schrader theorem. The corrected sufficiency conditions matter; see Osterwalder and Schrader 1975, introduction and § IV.1.

In real time, evolving an initial density matrix forward and backward gives an in–in expectation value. Specifying separate initial and final states gives an in–out matrix element. Consequently, varying an in–out effective action does not automatically produce real causal evolution, and a closed time path is not intrinsically thermal.

Review the chapter

Why must a normalized closed-time-path generating functional equal one when the sources on both branches coincide?

Solution

The same source produces a unitary evolution and its inverse on the two branches. They cancel inside the trace, leaving the trace of the initial density matrix, which is one. Independent branch sources are retained during differentiation to generate nontrivial correlations.

Where to go next

Continue to Retarded, Advanced, and Spectral Correlators for causal response, or to Structural Principles and Axiom Maps for the hypotheses behind reconstruction.

References

• Konrad Osterwalder and Robert Schrader, “Axioms for Euclidean Green’s Functions II (with an Appendix by Stephen Summers),” Communications in Mathematical Physics 42 (1975), 281–305. DOI.