We study the quantum dynamics of a time reparametrization invariant system with a vanishing Hamiltonian. The evolution of the physical degrees of freedom of the system is described, both at the classical and at the quantum level, in relational terms by the construction of an internal time parameter. We use the Pegg-Barnett phase operator formalism in finite dimensional Hilbert space as an essential ingredient.
We study the quantum dynamics of a time reparametrization invariant system with a vanishing Hamiltonian. The evolution of the physical degrees of freedom of the system is described, both at the classical and at the quantum level, in relational terms by the construction of an internal time parameter. We use the Pegg-Barnett phase operator formalism in finite dimensional Hilbert space as an essential ingredient.
The classical and quantum dynamics of simple time-reparametrization-invariant models containing two degrees of freedom are studied in detail. Elimination of one ``clock'' variable through the Hamiltonian constraint leads to a description of time evolution for the remaining variable which is essentially equivalent to the standard quantum mechanics of an unconstrained system. In contrast with a similar proposal of Rovelli, evolution is with respect to a geometrical proper time, and the Heisenberg equation of motion is exact. The physical phase space contains no coordinate associated with the eliminated ``clock'' variable. Therefore, time evolution is not with respect to the observable readings of a physical clock. Rather, the eliminated variable can be construed as intrinsic to a particular observer, and, hence, as unobservable, in principle, by this observer. The possibility of a ``test clock,'' which would reveal time evolution while contributing negligibly to the Hamiltonian constraint, is examined, and found to be viable in the semiclassical limit of large quantum numbers. \textcopyright{} 1996 The American Physical Society.
We study classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the physical time, which is assumed to be discrete. In this way, a physical clock with discrete states is introduced, which presently is still treated as decoupled from the system. This is motivated by the recent discussion of ``timeless'' reparametrization invariant models, where discrete physical time has been constructed based on quasi-local observables. Employing the path-integral formulation of classical mechanics developed by Gozzi et al., we show that these deterministic classical systems can be naturally described as unitary quantum mechanical models. We derive the emergent quantum Hamiltonian in terms of the underlying classical one. Such Hamiltonians typically need a regularization - here performed by discretization - in order to arrive at models with a stable groundstate in the continuum limit. This is demonstrated in several examples, recovering and generalizing a model advanced by 't Hooft.
This paper establishes the Quantum Temporal Non-Constructibility Theorem (QTNT): the external parameter t appearing in the Schrödinger equation carries no operational meaning for an observer embedded within a closed quantum system. Working strictly within the standard Hilbert-space formalism, using only Stone’s theorem, POVM measurement theory, and unitary evolution, the paper proves that all observable quantum data (POVM probabilities, conditional statistics, and multi-time correlators) are invariant under smooth, strictly monotonic reparametrizations t' = f(t), provided the Hamiltonian transforms covariantly as H' = H (dt/dt'). The theorem shows that internal observers, who have access only to reduced states and subsystem POVMs, cannot detect or reconstruct the external Schrödinger-time parameter. Observable histories depend solely on the geometric orbit traced by the state in projective Hilbert space, not on the particular parameter used to traverse it. As a result, Schrödinger-time functions as a gauge label rather than a measurable physical quantity. The analysis provides an operational reinterpretation of Pauli’s theorem, clarifies the correlational nature of quantum clocks, and completes the temporal non-constructibility program alongside its classical and relativistic counterparts. Appendices supply supporting derivations from Stone’s theorem, Pauli’s theorem, quantum clocks, explicit examples, and the path-integral formulation. Note. The corresponding classical and relativistic results are developed in the companion papers The Temporal Non-Constructibility Theorem in Classical Dynamical Systems (Zenodo, DOI: 10.5281/zenodo.17753466) and Temporal Non-Constructibility in General Relativity (Zenodo, DOI: 10.5281/zenodo.17765323) respectively.
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.
This work addresses the long-standing frozen formalism problem in the quantization of reparametrization-invariant systems. In Dirac quantization, physical states are defined by a quantum constraint of the form Ĉ Ψ = 0, a condition often interpreted as implying the absence of dynamics. This paper demonstrates that such an interpretation is structural rather than physical. For finite-dimensional constrained Hamiltonian systems, the paper introduces the concept of a renewal ordering: an intrinsic, locally monotonic ordering of distinguishable internal configurations constructed from relational observables. When the quantum constraint is expressed in variables adapted to this ordering, the condition Ĉ Ψ = 0 is shown to imply, under explicit and minimal assumptions, a Schrödinger-type evolution equation with respect to the renewal parameter. The analysis is purely operator-theoretic and local in character. It covers constraint operators that are either first order or quadratic with hyperbolic principal part in the renewal-direction derivative. No external time variable, physical Hamiltonian, reference fluid, or auxiliary clock system is introduced. No physical Hilbert space, inner product, or probabilistic interpretation is assumed or constructed. The central result establishes that the frozen formalism does not represent physical stasis. Instead, whenever a reparametrization-invariant quantum system admits a locally monotonic relational ordering, internal relational evolution is already encoded in the constraint equation itself and becomes explicit when expressed in renewal variables. The scope of the paper is deliberately restricted. It applies only to finite-dimensional constrained systems and makes no claims regarding global unitarity, factor-ordering uniqueness, gravitational field theories, cosmology, or the Wheeler–DeWitt operator. Within this domain, the results are exact and structural.
We analyze the relational quantum evolution of generally covariant systems in terms of Rovelli's evolving constants of motion and the generalized Heisenberg picture. In order to have a well-defined evolution, and a consistent quantum theory, evolving constants must be self-adjoint operators. We show that this condition imposes strong restrictions to the choices of the clock variables. We analyze four cases. The first one is nonrelativistic quantum mechanics in parametrized form; we show that, for the free particle case, the standard choice of time is the only one leading to self-adjoint evolving constants. Second, we study the relativistic case. We show that the resulting quantum theory is the free particle representation of the Klein-Gordon equation in which the position is a perfectly well defined quantum observable. The admissible choices of clock variables are the ones leading to spacelike simultaneity surfaces. In order to mimic the structure of general relativity we study the SL(2R) model with two Hamiltonian constraints. The evolving constants depend in this case on three independent variables. We show that it is possible to find clock variables and inner products leading to a consistent quantum theory. Finally, we discuss the quantization of a constrained model having a compact constraint surface. All the models considered may be consistently quantized, although some of them do not admit any time choice such that the equal time surfaces are transversal to the orbits.