We challenge the common belief that if there is no absolute time parameter in physics, a quantum system described without reference to an external clock can be assumed to be in a stationary state of its Hamiltonian. We present a time-reparameterisation invariant quantum evolution law, which for a given initial condition predicts the same trajectory in state space as the Schr\"odinger equation, except that for nontrivial trajectories it does not predict the speed at which the trajectory is traversed. The solutions of this evolution law are all time-reparameterised solutions of the Schr\"odinger equation. We show how the predictions of the Schr\"odinger equation are recovered relative to an internal clock in this framework. In contrast to the Page-Wootters formalism or Dirac's quantisation of the Hamiltonian constraint, here the global sate is not stationary. We discuss the assumptions leading to the common conclusion that the state can be taken stationary and suggest that they need to be revisited.
Quantum mechanics is widely recognised as being incomplete. It is not consistent with the second law of thermodynamics and does not provide a scientifically credible physical account of the measurement process, the means by which coherence is broken and classically observable states are recorded. This has led to many ad hoc assumptions being used to account for various properties of quantum systems, among which is the coherence time of quantum devices that determines their ability to perform computations. Here, we show that all these properties can be accommodated naturally and consistently in the context of mixing quantum systems which exhibit continuous spectra, as arises in the thermodynamic limit in quantum statistical mechanics and quantum gravity. In particular, for isolated systems we show that the time-reversal symmetry associated with unitary time evolution of the quantum state gives rise to time-symmetry breaking and a semi-group evolution which attains thermodynamic equilibrium at long times. Moreover, the emergence of this non-unitary time-asymmetry leads to microcanonical equilibrium states in which all quantum coherence is lost and is accompanied by the transformation of pure states into mixtures, leading in turn to an increase in entropy. Inclusion of a macroscopic measurement apparatus shows how the outcome of a measurement corresponds to the von Neumann projection postulate, arising with probabilities in conformance with the Born rule. The mathematical structure of the theory which applies to quantum systems with continuous spectra is closely analogous to the classical ergodic theory of dynamical systems and the conditions under which they attain equilibrium states.
We investigate the unitarity of quantum evolution relative to an internal time defined by a local observer's clock. The observer is modeled as a relativistic particle carrying both a clock and a matter-field detector, analyzed first on a fixed curved background and subsequently within a fully diffeomorphism-invariant theory of dynamical gravity. In the former case, we find that evolution with respect to the internal clock time is generally nonunitary, implying a violation of the Einstein equivalence principle at the quantum level. In contrast, in the latter case, diffeomorphism invariance allows us to adopt observer-centric coordinates without loss of generality. On the resulting partially-reduced phase space, one of the diffeomorphism generators becomes linear in the clock Hamiltonian, generating a relational evolution that is consistent with the remaining diffeomorphism constraints. Assuming that an effective quantum field theory exists to be consistent with the diffeomorphism invariance, these features ensure unitary evolution relative to the internal clock, thereby preserving the equivalence principle even in the quantum regime. Our results highlight the fundamental role of diffeomorphism invariance in shaping relational unitary evolution from the perspective of a local observer.
We introduce a quantum stochastic resetting protocol with uniform memory, in which each resetting event returns the system to a state visited at a time chosen uniformly from its entire history. The resulting dynamics is nonunitary, non-Markovian and a direct quantum generalization of the classical preferential relocation model. Working in the energy eigenbasis, we derive the exact evolution of every density-matrix element for an arbitrary time-independent Hamiltonian and show that the Hamiltonian enters the dynamics only through the corresponding Bohr frequencies. This leads to a natural distinction between two classes of quantum systems: gapped and gapless. In \emph{gapped systems} (systems with a discrete energy spectrum), while the diagonal elements remain unchanged, the off-diagonal elements of the density matrix in the energy eigenbasis decay algebraically with a continuously varying exponent and with an amplitude that oscillates periodically in $\log t$. The system therefore approaches a stationary state that is independent of the resetting rate and retains a strong memory of the initial state. In \emph{gapless systems} (systems with a continuous energy spectrum), arbitrarily small Bohr frequencies prevent stationarity. Instead, the position distribution spreads on the universal (ultra-slow) scale $\log(rt)/r$, independently of the initial state and of the details of the Hamiltonian. We illustrate these results with a two-level system, a harmonic oscillator, and a free quantum particle, and contrast them with their classical counterparts.
Gabriele de Mauro, Manas Kulkarni, S. Majumdar· 0 citations
Reverse Physics is a methodology that breaks physical theories into separate mathematical and physical conditions to establish their logical relationships. To showcase the power of the methodology, we present several results for quantum mechanics and their related insights. The standard Hilbert-space formulation conflicts with basic physical requirements, while a minimal topological modification can solve these problems. The ensemble space, rather than the pure-state space, distinguishes classical from quantum systems. The Born rule is an additional assumption linking orthogonality, mutual exclusivity and information entropy. Under explicit background conditions, unitary evolution is equivalent to deterministic and reversible evolution. Nonselective projective measurements can be characterized as Lindblad equilibration processes, while unitary evolution can be characterized as a limit of infinitesimal projective processes. Classical mechanics is recovered as the high-entropy limit of quantum mechanics, and every quantum state, pure or mixed, is a dynamical, spectral and thermodynamic equilibrium. These results are self-contained, use the standard vector-space representation and can thus be used as common tools and constraints for teaching, interpretations, reconstructions and future theories.
G. Carcassi, Tobias Thrien, C. Aidala· 0 citations
Quantum algorithms for simulating linear differential equations have attracted growing interest, driven by applications ranging from Hamiltonian dynamics to general non-unitary dynamics. While time-independent cases are well studied, time-dependent non-unitary dynamics remains considerably less explored, and it is unclear how to systematically adapt existing solvers for time-independent systems to such problems. In this work, we address this gap by introducing an autonomization framework based on the clock-variable formulation, a technique originally developed for time-dependent Hamiltonian systems in~\cite{CJL23TimeSchr}. By lifting the original non-autonomous system to an autonomous transport-type equation on an extended space and applying the Fourier spectral discretization in the clock variable, we obtain an explicit time-independent linear system, together with a suitable initial state and a recovery map for the target solution. Crucially, this formulation decouples the treatment of time dependence from the choice of the quantum ODE solver, thereby enabling the direct application of existing solvers designed for time-independent systems to the resulting autonomous problem. We combine this framework with Schr\"odingerization and a Taylor-expansion-based quantum ODE solver. In the Schr\"odingerization-based combination, our complexity analysis shows that the precision dependence can scale as $\log^{5/4}(1/\varepsilon)$, improving upon the $\log^2(1/\varepsilon)$ scaling found in existing approaches. Numerical experiments validate the autonomization formulation and confirm the successful recovery of the target solution.
In timeless formulations of quantum theory, the Page-Wootters proposal (PW) recovers time and dynamics as relative clock-and-world states. By interpreting the total timeless system as a clock entangled with the world, the states having a definite clock reading appear to recover the temporal states of the world as relative states, and dynamics emerges. But an entangled state admits infinitely many decompositions as a superposition of product states, each decomposition corresponding to a different but equally valid time operator of the same clock system. Different valid choices of the time operator for the same clock system lead to different physical histories, where the world states in a history are superpositions of the world states from another history. Fixing one operator as"the"clock time would import the temporal meaning that the Page-Wootters proposal is meant to explain in the first place. Therefore, the interpretation of PW that the world state is passively conditioned on the clock state cannot hold. The timeless state is resolved into time-dependent states by the intrinsic pointer observables of the world, without having to outsource the role of the time operator to a separate clock. Fortunately, the formalism of PW remains valid, provided that it is interpreted in terms of intrinsic pointer observables of the world itself.
O. C. Stoica· 0 citations
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