Wick rotation and analytic continuation: Shared structures and physical interpretation

Wick rotation relates oscillatory quantum amplitudes to damped Euclidean evolution when both arise from a common analytic construction. The correspondence depends on more than a formal substitution of imaginary time: domains, convergence, self-adjointness, positivity, and the interpretation of states and measurements determine what can be recovered. For a Hamiltonian bounded below, a complex-time operator family connects Euclidean propagation, thermal weighting, and unitary boundary evolution. Transfer operators retain information that a partition function alone discards, while recovery from noisy Euclidean data can remain unstable. A separate continuation of the reversal-rate parameter connects the classical Kac process to a one-dimensional Dirac equation, with different probability-transfer laws before and after continuation. Near oscillatory onset, neural mass models provide another kind of correspondence: a common local amplitude equation with phase symmetry and a reconstruction map. Complex coordinates, phase symmetry, and holomorphy impose different constraints. These examples distinguish exact reformulation, controlled reduction, and shared symmetry, and specify the additional observation and interaction maps required for physical applications.