Semi-Classical Limit and Minimum Decoherence in the Conditional Probability Interpretation of Quantum Mechanics

Corbin, Vincent; Cornish, N.

doi:10.1007/s10701-009-9298-5

Cited by 9 publications

(13 citation statements)

References 11 publications

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“…Note that, interpreting the reduction maps as the "quantum coordinate maps" taking one from the clock-neutral physical Hilbert space to a specific "clock perspective, " any such TFC map in Equation (68) takes the same compositional form as coordinate changes on a manifold. In particular, any such temporal frame change proceeds by mapping via the clockneutral physical Hilbert space in analogy to how coordinate changes always proceed via the manifold (see Figure 3).…”

Section: State Transformationsmentioning

confidence: 99%

Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings

2021

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We have previously shown that three approaches to relational quantum dynamics—relational Dirac observables, the Page-Wootters formalism and quantum deparametrizations—are equivalent. Here we show that this “trinity” of relational quantum dynamics holds in relativistic settings per frequency superselection sector. Time according to a clock subsystem is defined via a positive operator-valued measure (POVM) that is covariant with respect to the group generated by its (quadratic) Hamiltonian. This differs from the usual choice of a self-adjoint clock observable conjugate to the clock momentum. It also resolves Kuchař's criticism that the Page-Wootters formalism yields incorrect localization probabilities for the relativistic particle when conditioning on a Minkowski time operator. We show that conditioning instead on the covariant clock POVM results in a Newton-Wigner type localization probability commonly used in relativistic quantum mechanics. By establishing the equivalence mentioned above, we also assign a consistent conditional-probability interpretation to relational observables and deparametrizations. Finally, we expand a recent method of changing temporal reference frames, and show how to transform states and observables frequency-sector-wise. We use this method to discuss an indirect clock self-reference effect and explore the state and temporal frame-dependence of the task of comparing and synchronizing different quantum clocks.

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Section: State Transformationsmentioning

confidence: 99%

Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings

2021

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“…(21) by ρ mar (R)Ψ * (x|R) and then integrate over x and R, taking into account Eqs. (16) and (17). Comparison of both results indicates that E = dRλ(R) = dRρ mar (λ(R)/ρ mar (R)) , so that λ(R)/ρ mar can be interpreted as a local energy.…”

Section: Marginal-conditional Factorization Of the Joint Eigenfumentioning

confidence: 93%

“…(20) by X * (R) and then integrate over R , taking into account Eq. (16). With a little additional manipulation this yields ǫ = dR Φ(R)|Ĥ|Φ(R) = E .…”

Section: Marginal-conditional Factorization Of the Joint Eigenfumentioning

confidence: 98%

“…More radical programs on the problem of time pursue the construction of a conceptually and formally consistent quantum theory that includes the PI but avoids any a priori conception of time [6,7,9,[13][14][15][16]. Some of them seek an internal relational "time", which in a sense resembles the external classical time, and an associated law that provides an effective description of dynamical evolution [6,7,[14][15][16], in tune with the second sentence of Peres's statement quoted above. A strategy of this sort is the conditional probability interpretation (CPI) of Page and Wootters [6] whose basic tenet is to extract the dynamical evolution of the system's state from the conditional dependence of probabilities on the value of an internal clock variable.…”

Section: Introductionmentioning

confidence: 99%

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Unification of the conditional probability and semiclassical interpretations for the problem of time in quantum theory

Arce

2012

Phys. Rev. A

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We show that the time-dependent Schrödinger equation (TDSE) is the phenomenological dynamical law of evolution unraveled in the classical limit from a timeless formulation in terms of probability amplitudes conditioned by the values of suitably chosen internal clock variables, thereby unifying the conditional probability interpretation (CPI) and the semiclassical approach for the problem of time in quantum theory. Our formalism stems from an exact factorization of the Hamiltonian eigenfunction of the clock plus system composite, where the clock and system factors play the role of marginal and conditional probability amplitudes, respectively. Application of the Variation Principle leads to a pair of exact coupled pseudoeigenvalue equations for these amplitudes, whose solution requires an iterative self-consistent procedure. The equation for the conditional amplitude constitutes an effective "equation of motion" for the quantum state of the system with respect to the clock variables. These coupled equations also provide a convenient framework for treating the back-reaction of the system on the clock at various levels of approximation. At the lowest level, when the WKB approximation for the marginal amplitude is appropriate, in the classical limit of the clock variables the TDSE for the system emerges as a matter of course from the conditional equation. In this connection, we provide a discussion of the characteristics required by physical systems to serve as good clocks. This development is seen to be advantageous over the original CPI and semiclassical approach since it maintains the essence of the conventional formalism of quantum mechanics, admits a transparent interpretation, avoids the use of the Born-Oppenheimer approximation, and resolves various objections raised about them.

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“…Since the PW interpretation requires specifying a clock system, it is of interest to investigate what types of clocks are consistent with such a framework. Indeed, this was pursued by Cornish and Corbin (CC) in [10], where they built upon the formalism that was already developed by Dolby. Whereas Dolby's specific example was a toy model with the Hamiltonian for the clock given by H C = p , CC rather employed a free particle as the clock so that In order to further assess the applicability of the conditional probability framework, an investigation into realistic clocks would be helpful.…”

Section: Introductionmentioning

confidence: 99%

Realistic Clocks for a Universe Without Time

Bryan

Medved

2017

Found Phys

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There are a number of problematic features within the current treatment of time in physical theories, including the "timelessness" of the Universe as encapsulated by the Wheeler-DeWitt equation. This paper considers one particular investigation into resolving this issue; a conditional probability interpretation that was first proposed by Page and Wooters. Those authors addressed the apparent timelessness by subdividing a faux Universe into two entangled parts, "the clock" and "the remainder of the Universe", and then synchronizing the effective dynamics of the two subsystems by way of conditional probabilities.The current treatment focuses on the possibility of using a (somewhat) realistic clock system; namely, a coherent-state description of a damped harmonic oscillator. This clock proves to be consistent with the conditional probability interpretation; in particular, a standard evolution operator is identified with the position of the clock playing the role of time for the rest of the Universe. Restrictions on the damping factor are determined and, perhaps contrary to expectations, the optimal choice of clock is not necessarily one of minimal damping.

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Semi-Classical Limit and Minimum Decoherence in the Conditional Probability Interpretation of Quantum Mechanics

Cited by 9 publications

References 11 publications

Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings

Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings

Unification of the conditional probability and semiclassical interpretations for the problem of time in quantum theory

Realistic Clocks for a Universe Without Time

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