The pseudo-quantum representation of finite reversible Markov chains
The pseudo-quantum representation re-encodes a finite, irreducible, reversible continuous-time Markov chain with $M$ states as a complex-orthogonal flow on a doubled space of dimension $2M$. After uniformisation, a square-root gauge, a doubling of the state space and a diagonal unitary twist, the chain generates an entire one-parameter group $W_z = e^{zK}$ with $z$ complex, and this single group is the object of the paper. Its real slice, $z$ real, reproduces the stochastic semigroup exactly through an explicit decoding whose probabilistic meaning is the parity of the number of ticks of the uniformised chain. Its imaginary slice, $z = iθ$, is a finite-dimensional unitary quantum system. The chain and the quantum system are therefore not two analogous models but two restrictions of one entire representation. The setup also produces, with no further input, a pseudo-Schroedinger equation, a bilinear von Neumann equation for a complex-symmetric pseudo-density, and a pseudo-Bloch vector equation on a non-compact quadric. We develop the construction from first principles and work out one model completely, the usual symmetric Ehrenfest urn. Its gauged generator equals $(2/n) J_x$, the spin-$n/2$ operator, its relaxation modes are Lorentz boosts, and its imaginary slice is a depth-one quantum circuit whose Born distribution is the classical urn law under the clock $θ(t) = \arccos(e^{-2t})$. The same clock identifies the Fisher-Rao lift of the urn trajectory with a rigid spin-coherent-state orbit. This is a preliminary simplified version of a longer paper. It treats only finite state spaces and only the symmetric Ehrenfest urn, and it previews without proofs the asymmetric urn, two queues, the symmetric simple exclusion process and the stochastic Ising model. Appendix primers make the quantum, geometric and information-theoretic language self-contained.