A Quantum Canonical Framework for Order Book Dynamics: Proper-Time Relaxation in Non-Equilibrium Markets

Traditional quantitative finance models predominantly rely on continuous Brownian motion or discrete Markov jump processes calibrated against synchronous clock-time intervals. These frameworks fail to capture the non-equilibrium topological phase transitions and memory effects inherent in ultra-high-frequency order books. In this paper, we formulate a non-equilibrium quantum canonical framework that maps discrete Limit Order Book (LOB) dynamics onto an open quantum system governed by an effective Hamiltonian and canonical relaxation operators. By quantizing price and volume via an invariant financial Planck constant (h_f = Delta p_min x Delta v_min), we eliminate subjective sampling artifacts and define the state transition manifold over an adaptive Hilbert space. We introduce the Proper-Time Dwell Relaxation operator, which explicitly formalizes the state collapse and decoherence induced by liquidity dissipation and passive quote cancellations. Furthermore, we derive the Canonical Partition Function for order book configurations, demonstrating that market instability corresponds to localized entropy reduction within bounded phase-space manifolds (termed Golden Windows). Empirical validation conducted on ultra-high-frequency micro-futures tick data confirms that our canonical relaxation formulation significantly outperforms classical Markovian state transition benchmarks in predicting post-shock probability density distributions while maintaining strict mathematical invariance against arbitrary temporal discretization.