Continue evolution while preserving its analytic meaning

Start with a regulated self-adjoint Hamiltonian bounded below. Real-time evolution is exp(−iHt/ℏ). Along the continuation t=−iτ with τ positive, it becomes exp(−Hτ/ℏ). Subtracting a fixed reference energy may be convenient, but changes the overall normalization and should be declared. A finite time-sliced representation carries this continuation before a formal continuum limit is attempted; exchanging continuation, integration, and regulator removal requires an analytic justification.

For a particle with real-time Lagrangian m qdot²/2−V, the continued velocity is i times the Euclidean derivative and dt=−i dτ. Consequently the Minkowski action evaluated on that contour is i times the Euclidean action with positive kinetic term plus V. Multiplication by the original i produces minus the Euclidean action in the exponent. Rotating only the time measure while leaving the velocity unchanged would give the wrong kinetic sign.

Expand an initial state in energy eigenstates. Euclidean evolution suppresses an excitation relative to the ground state by an amplitude factor determined by the energy gap and τ. Normalizing the evolved vector therefore suppresses relative probabilities by twice that exponent. Projection requires nonzero initial overlap with the desired ground sector; a conserved symmetry can prevent that overlap. This is an imaginary-time preparation device, not unitary physical evolution preserving the original vector norm.

A thermal partition function instead traces exp(−βH), with β inverse energy and τ=βℏ. Bosonic paths are periodic around the resulting Euclidean time circle; fermionic thermal fields use antiperiodic boundary conditions. Thermal level probabilities carry exp(−βE), not the squared amplitude of an imaginary-time-evolved pure state. Confusing these constructions introduces a factor of two and can also replace a mixed thermal ensemble by a pure projected state.

A positive quadratic Euclidean kinetic operator gives convergent finite bosonic integrals, but interactions, unbounded potentials, chemical potentials, or fermion determinants may spoil a positive sampling interpretation. Gauge constraints still require gauge fixing or a quotient construction. Continuum reconstruction of a unitary real-time theory requires structural conditions such as reflection positivity and appropriate regularity, not just a small Monte Carlo error. Singularities and the vacuum iε prescription determine how time-ordered real-time boundary values are recovered.

The heat lab demonstrates spectral damping in a separate equation. Reset and set mode number to 1 and diffusivity to 1, then Step twice to time 1; predict mode amplitude e⁻¹. Reset and set mode to 2 and diffusivity to 1, then Step twice; predict e⁻⁴. Its readout also gives the spatial L² norm as √π times that amplitude. This is one periodic heat-equation mode, not a Wick-rotation engine, quantum projection algorithm, or general field-theory solver.