Quantum clocks observe classical and quantum time dilation

Abstract

At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times. We consider quantum clocks constructed from the internal degrees of relativistic particles that move through curved spacetime. The probability that one clock reads a given proper time conditioned on another clock reading a different proper time is derived. From this conditional probability distribution, it is shown that when the center-of-mass of these clocks move in localized momentum wave packets they observe classical time dilation. We then illustrate a quantum correction to the time dilation observed by a clock moving in a superposition of localized momentum wave packets that has the potential to be observed in experiment. The Helstrom-Holevo lower bound is used to derive a proper time-energy/mass uncertainty relation.

Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings

“…While a Newton-Wigner type localization is approximate and not fully Lorentz covariant, due to the relativistic localization nogo theorems of Perez-Wilde [92] and Malament [93] (see also [94,95]), it is generally accepted as the best possible localization in relativistic quantum mechanics (In quantum field theory localization is a different matter [90,94]). This demonstrates the advantage of using covariant clock POVMs in relational quantum dynamics [7,44,45,96,97]. The trinity also extends the probabilistic interpretation of relational observables: a Dirac observable describing the relation between a position operator and the covariant clock POVM corresponds to a Newton-Wigner type localization in relativistic settings.…”

“…In the second line we made use of Equation (45), in the third of Theorem 1, and in the fourth of Equation ( 71) and the fact that θ (−σ 1p1 ) commutes with the reduction map of the C 2 clock and withF O C 2 S|C 1 ,T 1 (τ 1 ) (see Lemma 5).…”

“…(i) a Dirac quantization scheme, wherein relational observables are constructed that encode correlations between evolving and clock degrees of freedom [1,2,4,, (ii) the Page-Wootters formalism, which defines a relational dynamics in terms of conditional probabilities for clock and evolving degrees of freedom [7,25,[39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57], and (iii) classical or quantum deparametrizations, which result in a reduced quantum theory that only treats the evolving degrees of freedom as quantum [1,2,7,10,30,31,[58][59][60][61][62][63][64][65].…”

“…This degeneracy is not covered by our previous construction. While quadratic clock Hamiltonians are standard in the literature on relational observables [approach (i)] and deparametrizations [approach (iii)], see e.g., [4,10,11,19,29,31], relativistic particle models have only recently been studied in the Page-Wootters formalism [approach (ii)] [45,[49][50][51]. However, Kuchař's criticism (a) that the Page-Wootters approach yields incorrect localization probabilities in relativistic settings has yet to be addressed.…”

Time Observables in a Timeless Universe

“…Perhaps a PaW formulation of spacetime may represent a first step towards removing this asymmetry. This might help to develop relativistic generalizations of the PaW formalism, see also [31,32,33,34].…”