A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an `almost ideal clock'. Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of `time' or `evolution', the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, we derive the complete formalism of external time quantum mechanics. The explicit relation between the physical Hilbert space and the usual Hilbert space for a system with an external time is also derived and the limit of an ideal clock is found to be closely associated with the Klein - Gordon inner product and the Newton - Wigner formalism.
A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an almost ideal clock'. Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of time' or `evolution', the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, we derive the complete formalism of external time quantum mechanics. The explicit relation between the physical Hilbert space and the usual Hilbert space for a system with an external time is also derived and the limit of an ideal clock is found to be closely associated with the Klein - Gordon inner product and the Newton - Wigner formalism.
A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an `almost ideal clock.' Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of `time' or `evolution,' the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, the work below derives the complete formalism of external time quantum mechanics. The limit of an ideal clock is found to be closely associated with the Klein-Gordon inner product and the Newton-Wigner formalism and, in addition, this limit is shown to fail for a clock that measures metric-defined proper time near a singularity in Bianchi models.
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.
Nonrelativistic model quantum cosmologies are studied in which the basic time variable is the position of a clock indicator and the time parameter of the Schr\"odinger equation is an unobservable label. Familiar Schr\"odinger-Heisenberg quantum mechanics emerges if the clock is ideal---arbitrarily accurate for arbitrarily long times. More realistically, however, the usual formulation emerges only as an approximation appropriate to states of this model universe in which part of the system functions approximately as an ideal clock. It is suggested that the quantum kinematics of spacetime theories such as general relativity may be analogous to those of this model. In particular it is suggested that our familiar notion of time in quantum mechanics is not an inevitable property of a general quantum framework but an approximate feature of specific initial conditions.
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.
In quantum mechanics it is usual to represent physical reality as a vector in Hilbert space at a particular time, with evolution being governed by Schrodinger's equation which involves an externally imposed time parameter. This leads to difficulties if one wishes to regard the universe as a quantum mechanical system, because there should no time external to such a system. The approach in this paper is to represent the totality of reality as a vector in Hilbert space. The author shows how time evolution follows, where time now is defined in terms of the states of a quantum mechanical clock which is part of the system. Rather than the correlation between the clock states and the states of the rest of the system arising because both are governed by an imposed law involving an external time parameter, it is seen that this correlation is of the separation-independent Einstein-Podolsky-Rosen type. The total reality vector, which incorporates the whole history of the system, is shown to be a zero-energy eigenstate of the system Hamiltonian. He discusses systems of finite and infinite lifetime, and is able to answer the question: what was the state before the initial state? He concludes that the quantum mechanical system of this paper is a reasonable representation of the observed universe.
The `problem of time' remains an unresolved issue in all known physical descriptions of the Universe. One aspect of this problem is the conspicuous absence of time in the Wheeler-Dewitt equation, which is the analogue of the Schrodinger equation for the Universal wavefunction. Page and Wootters famously addressed this problem by providing a mechanism for effectively introducing time evolution into this timeless cosmological picture. Their method, which is sometimes called the conditional probability interpretation (CPI), requires the identification of an internal clock system that is meant to keep time for the remainder of the Universe. Most investigations into this idea employ the idealized limit of a non-interacting clock system, the so-called ideal clock. However, by allowing for interactions, we have found the counter-intuitive result that a non-interacting clock is not necessarily the optimal choice, even if it is `ideal'. In particular, the uncertainty that is associated with the physical measurement of an atomic clock is found to decrease monotonically as the interactions grow stronger. This observation, which is reinforced by a previous study using a semi-classical clock, paves the way to an independent argument that is based on the energy conservation of any isolated system. Our conclusion is that ideal clocks must be prohibited from the CPI when recovering cosmological time evolution. Interactions are necessary for describing time evolution as a strict matter of principle. Lastly, we also consider the implications of this result for the experience of time in the evolution of the Universe.