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Using an elementary example based on two simple harmonic oscillators, we show how a relational time may be defined that leads to an approximate Schrodinger dynamics for subsystems, with corrections leading to an intrinsic decoherence in the energy eigenstates of the subsystem.
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In a seminal paper (Page and Wootters 1983) Page and Wootters suggest time evolution could be described solely in terms of correlations between systems and clocks, as a means of dealing with the problem of time stemming from vanishing Hamiltonian dynamics in many theories of quantum gravity. Their approach to relational time centres around the existence of a Hamiltonian and the subsequent constraint on physical states. In this paper we present a state-centric reformulation of the Page and Wootters model better suited to theories which intrinsically lack Hamiltonian dynamics, such as Chern--Simons theories. We describe relational time by encoding logical clock qubits into anyons---the topologically protected degrees of freedom in Chern--Simons theories. The timing resolution of such anyonic clocks is determined by the universality of the anyonic braid group, with non-universal models naturally exhibiting discrete time. We exemplify this approach using SU(2)$_2$ anyons and discuss generalizations to other states and models.
In the context of the de Broglie-Bohm pilot wave theory, numerical simulations for simple systems have shown that states that are initially out of quantum equilibrium - thus violating the Born rule - usually relax over time to the expected $|psi|^2$ distribution on a coarse-grained level. We analyze the relaxation of nonequilibrium initial distributions for a system of coupled one-dimensional harmonic oscillators in which the coupling depends explicitly on time through numerical simulations, focusing in the influence of different parameters such as the number of modes, the coarse-graining length and the coupling constant. We show that in general the system studied here tends to equilibrium, but the relaxation can be retarded depending on the values of the parameters, particularly to the one related to the strength of the interaction. Possible implications on the detection of relic nonequilibrium systems are discussed.
The problem of time in quantum gravity calls for a relational solution. Using quantum reduction maps, we establish a previously unknown equivalence between three approaches to relational quantum dynamics: 1) relational observables in the clock-neutral picture of Dirac quantization, 2) Page and Wootters (PW) Schrodinger picture formalism, and 3) the relational Heisenberg picture obtained via symmetry reduction. Constituting three faces of the same dynamics, we call this equivalence the trinity. We develop a quantization procedure for relational Dirac observables using covariant POVMs which encompass non-ideal clocks. The quantum reduction maps reveal this procedure as the quantum analog of gauge-invariantly extending gauge-fixed quantities. We establish algebraic properties of these relational observables. We extend a recent clock-neutral approach to changing temporal reference frames, transforming relational observables and states, and demonstrate a clock dependent temporal nonlocality effect. We show that Kuchav{r}s criticism, alleging that the conditional probabilities of the PW formalism violate the constraint, is incorrect. They are a quantum analog of a gauge-fixed description of a gauge-invariant quantity and equivalent to the manifestly gauge-invariant evaluation of relational observables in the physical inner product. The trinity furthermore resolves a previously reported normalization ambiguity and clarifies the role of entanglement in the PW formalism. The trinity finally permits us to resolve Kuchav{r}s criticism that the PW formalism yields wrong propagators by showing how conditional probabilities of relational observables give the correct transition probabilities. Unlike previous proposals, our resolution does not invoke approximations, ideal clocks or ancilla systems, is manifestly gauge-invariant, and easily extends to an arbitrary number of conditionings.
Recent work has shown that relativistic time dilation results in correlations between a particles internal and external degrees of freedom, leading to decoherence of the latter. In this note, we briefly summarize the results and address the comments and concerns that have been raised towards these findings. In addition to brief replies to the comments, we provide a pedagogical discussion of some of the underlying principles of the work. This note serves to clarify some of the counterintuitive aspects arising when the two theories are jointly considered.
We investigated the effects of the global monopole spacetime on the Dirac and Klein-Gordon relativistic quantum oscillators. In order to do this, we solve the Dirac and Klein-Gordon equations analytically and discuss the influence of this background which is characterized by the curvature of the spacetime on the energy profiles of these oscillators. In addition, we introduce a hard-wall potential and, for a particular case, determine the energy spectrum for relativistic quantum oscillators in this background.
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