We complete the quantum formulation of History-Dependent Gravity (HDG) by demonstrating that its phenomenological memory kernel admits a rigorous quantum environment realization. Starting from a microscopic system-environment decomposition with spectral weight J(μ) = ρ(μ)², we derive the exact non-Markovian master equation for the reduced dynamics of matter. Crucially, the HDG memory kernel K(t) = ∫ J(μ) e^{-iμt} dμ is shown to coincide identically with the environment correlation function, closing the theory at the microscopic level. Our formalism yields three fundamental results. First, the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation is derived as the rigorous Davies limit of HDG, rather than being postulated ad hoc. Second, we prove that the positivity of the spectral weight J(μ) ≥ 0 guarantees a completely positive and trace-preserving (CPTP) unitary dilation of the reduced dynamics, linking the geometric no-ghost condition directly to information-theoretic consistency. Third, decoherence rates are not free parameters but are computed explicitly as γ_i(ω_i) = 2π J_i(ω_i), fixed by the renormalization group flow of the spectral density. The theory predicts a universal short-time survival probability P(t) = 1 - αt² + O(t³), with α = ∫ J(μ) dμ, and a super-linear mass scaling γ_eff ∝ M² (under linear mass-coupling assumptions), providing sharp falsifiable criteria for matter-wave interferometry and superconducting qubit experiments. Furthermore, we establish a spectral tomography protocol allowing direct reconstruction of the effective memory spectrum J(μ) from decoherence data. This work elevates HDG from an effective model to a testable quantum-gravitational decoherence framework, bridging quantum information, non-Markovian dynamics, and the infrared structure of gravity.
We complete the quantum formulation of History-Dependent Gravity (HDG) by demonstrating that its phenomenological memory kernel admits a rigorous quantum environment realization. Starting from a microscopic system-environment decomposition with spectral weight J(μ) = ρ(μ)², we derive the exact non-Markovian master equation for the reduced dynamics of matter. Crucially, the HDG memory kernel K(t) = ∫ J(μ) e^{-iμt} dμ is shown to coincide identically with the environment correlation function, closing the theory at the microscopic level. Our formalism yields three fundamental results. First, the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation is derived as the rigorous Davies limit of HDG, rather than being postulated ad hoc. Second, we prove that the positivity of the spectral weight J(μ) ≥ 0 guarantees a completely positive and trace-preserving (CPTP) unitary dilation of the reduced dynamics, linking the geometric no-ghost condition directly to information-theoretic consistency. Third, decoherence rates are not free parameters but are computed explicitly as γ_i(ω_i) = 2π J_i(ω_i), fixed by the renormalization group flow of the spectral density. The theory predicts a universal short-time survival probability P(t) = 1 - αt² + O(t³), with α = ∫ J(μ) dμ, and a super-linear mass scaling γ_eff ∝ M² (under linear mass-coupling assumptions), providing sharp falsifiable criteria for matter-wave interferometry and superconducting qubit experiments. Furthermore, we establish a spectral tomography protocol allowing direct reconstruction of the effective memory spectrum J(μ) from decoherence data. This work elevates HDG from an effective model to a testable quantum-gravitational decoherence framework, bridging quantum information, non-Markovian dynamics, and the infrared structure of gravity.
This preprint presents a self-contained and mathematically explicit reconstruction of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation starting from exact microscopic unitary dynamics through the Nakajima-Zwanzig projection operator formalism. Beginning with the exact non-local generalized master equation, we systematically analyze the hierarchy of approximations required for the emergence of a Markovian quantum dynamical semigroup. The Born (weak-coupling), Markov (time-localization), and secular approximations are derived and interpreted as successive information-reduction steps, each associated with clearly identifiable mathematical assumptions and physical consequences. Particular attention is devoted to the role of secularization in transforming the Redfield generator into a completely positive GKSL structure. To provide a concrete worked example, the formalism is applied to a damped two-level system coupled to a bosonic thermal reservoir, where the Redfield and GKSL equations are derived explicitly and compared at the microscopic level. Beyond the derivation itself, the work develops a systematic map of validity limits and failure regimes. The breakdown of the Born, Markov, secular, factorization, and stationarity assumptions is analyzed in terms of environmental correlation times, spectral-density structure, non-Markovian memory effects, information backflow, and loss of semigroup behavior. The resulting framework clarifies the precise operational domain in which GKSL dynamics can be regarded as a physically reliable effective description and provides a rigorous foundation for applications involving entropy-production theorems, quantum thermodynamics, and open-system information theory.
Abstract We show that general relativistic time dilation can act as a controllable source of non-Markovianity in the dynamics of delocalized quantum clocks coupled to structured environments. As a concrete setting, we model a delocalized quantum clock, a two-level Unruh-DeWitt detector interacting with a thermal scalar field with finite correlation time in a weak, static gravitational field. The gravitational redshift between spatially sepa- rated branches leads to a path-dependent sampling of the bath correlation functions and to an effective spectral mismatch between the arms of the interferometer. Within a second-order time-convolutionless expansion we derive a time-local master equation whose time-dependent rates depend explicitly on the redshift parameter and reduce to the usual Markovian Gorini-Kossakowski-Sudarshan-Lindblad generator when gravity is switched off or the bath becomes memoryless. Numerical evaluation for an Ohmic environment reveals gravitationally in- duced coherence revivals and information backflow, providing a genuine non-Markovian signature that can be quantified by standard trace-distance measures. We identify a resonance-like regime in which the gravitational detuning becomes comparable to the dominant bath frequencies, leading to a suppression of decoherence and an enhancement of entanglement lifetimes relative to the flat-space Markovian case. Finally, by comparing the size of the gravity-induced deviations to quantum projection noise in state-of-the-art optical clocks, we derive an analytic signal-to-noise ratio and show that, within our model, engineered low-frequency reservoirs could al- low gravity-controlled memory effects to be resolved above the shot-noise limit on few-hour integration times, whereas for fast thermal baths the effect remains QPN-limited and effectively unobservable.
Markovian open quantum theory assumes evolution with respect to an external classical time parameter, yet no preferred notion of time exists fundamentally in relativistic physics. We resolve this tension by formulating relativistic open quantum dynamics relationally through finite-resolution quantum clocks. Using the Schwinger-Tomonaga formalism, we derive a covariant master equation directly on spacetime hypersurfaces and show that the resulting reduced dynamics is generically non-Markovian even for local interactions. Environmental correlations and clock fluctuations jointly generate the memory kernel, while the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) structure emerges only after relational coarse-graining, implying that exact Markovian evolution is effective rather than fundamental. In the sharp-clock limit, the formalism reduces to the Anastopoulos-Hu gravitational decoherence equation.
The scaling of fault-tolerant quantum computers is fundamentally constrained by temporal nonlocality. In this work, we apply the History-Dependent Quantum Dynamics (HDQD) framework to analyze the Quantum Zeno Effect under frequent quantum error correction (QEC) measurements. We demonstrate that frequent syndrome measurements truncate the non-Markovian memory kernel, trapping the system in a short-time quadratic regime and effectively renormalizing the rate of coherent gate evolution. This leads to a fundamental memory–measurement tradeoff. We derive a universal scaling law for the maximum achievable circuit depth: Dmaxopt∝T21/3Bmem−2/3Dmaxopt∝T21/3Bmem−2/3, where T2T2 is the decoherence time and BmemBmem is the spectral memory bandwidth of the physical substrate. This result establishes that temporal spectral memory is not merely a source of decoherence, but a structural resource that dictates the scalability horizon of quantum hardware. Furthermore, we propose a falsifiable experimental protocol for reconstructing the spectral memory density ρ(μ)ρ(μ) via two-quadrature Ramsey interferometry. This work extends the universal spectral memory formalism, previously applied to nuclear scattering and mass defects, into the domain of quantum information science.
We define non-Markovian quantum dynamics as evolution in which the current state depends on all past states, and completely characterize its structure under the assumptions of complete positivity and non-signalling. The resulting continuous-time dynamics is an integro-differential equation that augments the Gorini-Kossakowski-Sudarshan-Lindblad equation with a memory integral, and is capable of describing the quantum state of systems exposed to noise with any integrable power spectral density with no further approximations. We then establish a formalism to evaluate multi-time correlations of measurement outcomes in this general setting, obviating the need for a regression theorem. As an application, we derive the emission spectrum of a driven two-level system coupled to a non-Markovian bath: the familiar Mollow triplet acquires a frequency-dependent linewidth that encodes the memory of the bath. Our work provides a rigorous yet transparent description of the quantum state of non-Markovian systems, opening the door for state estimation and state-based quantum control beyond the Markovian regime.
Contact Jonathan.duchesne@hotmail.com 418-953-0596 This thesis presents Time–Decoherence Gravity (TDG), a unified theoretical framework in which spacetime geometry, gravitation, and the physical flow of time arise from fundamental quantum decoherence. Rather than treating decoherence as an environmental or effective process, TDG builds on a covariant extension of the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation, where four directional Lindblad operators encode intrinsic, irreversible information loss. These operators define a “decoherence tensor,” which acts both as a memory of quantum irreversibility and as the generator of emergent spacetime geometry through a nonlinear constitutive relation. The theory provides a coherent gravitational collapse mechanism based on the incompatibility of decoherence patterns across quantum branches. This leads to an intrinsic “decoherence energy,” whose magnitude determines the collapse timescale, extending and generalizing the well-known Penrose–Diósi models. TDG predicts a smooth transition from persistent microscopic coherence to rapid macroscopic collapse. At cosmological scales, a homogeneous decoherence tensor naturally generates modified Friedmann dynamics, early-universe stiff-fluid behavior, and late-time acceleration without requiring a fine-tuned cosmological constant. Black-hole entropy emerges as a saturation of decoherence memory at the horizon, while Hawking radiation appears as the dissipative relaxation of this saturation. The theory also predicts short-range deviations from Newtonian gravity and a universal spectrum of fundamental gravitational noise. Finally, TDG offers several experimentally testable signatures involving optical clocks, mesoscopic interferometry, short-range force measurements, and correlated phase noise in global timing networks. The framework is internally consistent, falsifiable, and provides a novel and unified approach to the relationship between quantum mechanics, gravity, and the arrow of time.