The river of time may not be a smooth flow but a sequence of minuscule steps, like a stop-motion film shot with an unimaginably fast camera. In 1964, Yu. A. Golfand suggested that time moves not continuously but in tiny jumps, and derived an equation that predicted an inevitable increase in disorder (entropy). Now this old formalism has found new life: the authors of a recent paper have translated it into the language of modern open quantum systems.
They rigorously showed that Golfand's evolution is a special quantum channel that preserves the sum of probabilities but not the purity of the state. In the continuous limit, the dynamics reduce to the Lindblad equation with a fundamental time scale τ (tau). This parameter acts like an invisible metronome: each 'tick' destroys quantum coherences and forces the system to lose its quantum individuality. This is how, without any external environment, the arrow of time emerges — from the very granularity of the temporal fabric.
Such intrinsic 'noise' imposes strict limits on future technologies. Calculations show how the accuracy of quantum computations degrades due to the inevitable decay of quantum fidelity. Moreover, even exotic phases like discrete time crystals (structures that pulse with a period that is a multiple of the time step) are doomed to a finite lifetime. Golfand's dynamics sets a fundamental limit on their existence.
The figure of 10⁻²⁶ seconds is not just an abstract limit. It is a window into the physics where quantum mechanics meets gravity. If the granularity of time exists at this scale, it could be sought in next-generation experiments. The stop-motion filming of reality may already be underway, the camera is just hidden from view for now.
as written by the authors arXiv:2606.22187
In 1964, Yu. A. Gol'fand proposed an extension of quantum mechanics to discrete time, predicting intrinsic non-unitarity and entropy increase. While historically significant, this formalism predates the modern theory of open quantum systems. In this work, we rigorously recast Gol'fand's discrete evolution equation as a Completely Positive Trace-Preserving (CPTP) quantum channel and derive its continuous-time coarse-grained limit. We demonstrate that the dynamics converge to a specific Lindblad master equation characterized by a fundamental time scale $$\tau$$, which induces decoherence in both the energy basis and a fundamental operator basis $$W$$. We analyze the thermodynamic implications using Spohn's entropy production formalism, proving that the discrete time step induces a strictly positive entropy production rate driven by the decay of quantum coherences, thereby providing a microscopic foundation for the arrow of time independent of environmental coupling. Furthermore, we quantify the loss of quantum coherence via fidelity decay and purity loss, establishing exact constraints for fault-tolerant quantum computing. We further investigate the impact of this intrinsic decoherence on Discrete Time Crystals (DTCs), showing that Gol'fand dynamics impose a fundamental lifetime limit on time-translation symmetry breaking phases. Finally, we utilize precision data from optical lattice clocks, matter-wave interferometry, and neutrino oscillations to place stringent upper bounds on $$\tau$$. Our results constrain the fundamental time discretization to $$\tau \lesssim 10^{-26}$$ s, significantly tightening previous limits and offering a testable framework for quantum gravity phenomenology.
A time step of 10⁻²⁶ seconds is so small that in one second, there are more such steps than seconds have passed since the Big Bang.
In Greg Egan's novel 'The City of Permutations', people upload their consciousness into a simulation with discrete time, but encounter unpredictable quantum effects, as if reality itself resists granularity.