Spiral-Time with Memory: A Structural Extension of Non-Markovian Quantum Dynamics

Marcel Krüger

Marcel Krüger

Partha Nandi, Giuseppe Vitiello

Can a closed quantum system generate time-crystal-like nonequilibrium behavior, self-similar scaling structures, and non-Markovian memory without external driving or coupling to a macroscopic environment? We address this question within the quantum Bateman oscillator formulated in a nonrelativistic (2 + 1)-dimensional phase-space noncommutative framework generated by spin-induced spatial deformation. The resulting doubled quantum dynamics is governed by a time-independent Hermitian Hamiltonian and exhibits an underlying SU(1, 1) structure with amplified and damped collective modes. We show that these modes satisfy an exact discrete scaling covariance, leading to self-similar temporal evolution without external driving. Upon tracing over one oscillator sector, the reduced dynamics becomes intrinsically non-Markovian and is governed by a history-dependent memory kernel. The same scaling structure admits a geometric representation in terms of logarithmic-spiral trajectories associated with the amplified and damped branches of the Bateman system. Because the mechanism relies on nonequilibrium reduced dynamics rather than equilibrium expectation values of local observables, it lies outside the assumptions underlying conventional no-go theorems for equilibrium time crystals. Our results identify spin as the common physical origin of the amplified and damped Bateman dynamics, self-similar scaling periodicity, logarithmic-spiral structures, and non-Markovian memory, also suggesting a natural extension of the mechanism to relativistic anyonic systems.

Gianpaolo Torre

Dariusz Chruściński, Andrzej Kossakowski

Marcel Krüger

We propose that time carries intrinsic memory, encoded in a triadic spiral-time structure Ψ(t) = (t,ϕ(t),χ(t)). In contrast to standard quantum mechanics, where time is treated as a memoryless parameter, spiral-time introduces a non-Markovian temporal substrate from which space, matter, and measurement emerge as effective phenomena. We argue that the projection postulate can be replaced by a dynamical stabilization of time-memory, outline how Born-type statistics may arise as a resonance-time measure, and provide explicit falsification criteria based on process-tensor non-Markovianity and CP-divisibility tests. The triadic phase operator ∆Φ can be formulated as a deterministic observer within a minimal control-theoretic framework [1].