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arXiv:1202.4638 · Physical Review A

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

J. C. Arce

Published Apr 9, 2012• 13 citations
arXivHTMLPDF

Abstract

We show that the time-dependent Schr\"odinger 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.

DOI
10.1103/physreva.85.042108

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arXiv:1712.00081•Quantum•Jul 8, 2019•98 citations
We show that the time-dependent Schr"odinger 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.
Beginning with an entangled state of a time-independent (TI) quantum system coupled to its TI quantum environment, we show that a time-dependent Schrödinger equation (TDSE) for the quantum system alone can be derived in the limit that one of the environment variables becomes a classical variable. In the same limit the TI amplitude of the environment wavefunction becomes the TD amplitude of an eigenfunction expansion of the system TD wavefunction. Similarly, the phase of the TI environment wavefunction goes over into the TD phase of the system amplitude. By considering that more and more environment variables become successively classical, each provides a classical clock to give a multiple-time TDSE for the quantum system. Two examples of two-time TDSE given in the literature are derived. When all clocks are synchronized the multiple-time TDSE reduces to the usual one-time TDSE.
The local conservation of a physical quantity whose distribution changes with time is mathematically described by the continuity equation. The corresponding time parameter, however, is defined with respect to an idealized classical clock. We consider what happens when this classical time is replaced by a nonrelativistic quantum-mechanical description of the clock. From the clock-dependent Schr\"odinger equation (as an analog of the time-dependent Schr\"odinger equation) we derive a continuity equation, where, instead of a time derivative, an operator occurs that depends on the flux (probability current) density of the clock. This clock-dependent continuity equation can be used to analyze the dynamics of a quantum system and to study degrees of freedom that may be used as internal clocks for an approximate description of the dynamics of the remaining degrees of freedom. As an illustration, we study a simple model for coupled electron-nuclear dynamics and interpret the nuclei as quantum clock for the electronic motion. We find that whenever the Born-Oppenheimer approximation is valid, the continuity equation shows that the nuclei are the only relevant clock for the electrons.
The problem of describing the dynamics of quantum systems relative to quantum clocks with which they interact is relevant to the physics of quantum reference frames and quantum gravity alike. There have been different approaches that derive a relational time-dependent Schrödinger equation for the system relative to the clock from a time-independent Hamiltonian constraint. We show that some of these approaches, due to Smith and Ahmadi, Gemsheim and Rost, as well as the so-called Born-Oppenheimer approach, can be unified in a single framework for describing conditional quantum states. In particular, the Born-Oppenheimer approach to quantum cosmology is seen to correspond to a type of interacting Page-Wootters formalism. We discuss the issue of unitarity and linearity, or lack thereof, in the relational dynamics, and we conclude with a brief outlook to further developments.
We argue that (1) our perception of time through change and (2) the gap between reality and our observation of it are at the heart of both quantum mechanics and the dynamical mechanism of physical systems. We suggest that the origin of quantum uncertainty lies with the absence of infinities or infinitesimals in observational data and that our concept of time derives from observing changing data (events). We argue that the fundamentally important content of the Superposition Principle is not the "probability amplitude" of posterior state observation but future state availability conditional only on prior information. Since event detection also implies posterior conditions (e.g. a specific type of detectable event occurred) as well as prior conditions, the probabilities of detected outcomes are also conditional on properties of the posterior properties of the observation. Such posterior conditions cannot affect the prior state availabilities and this implies violation of counter-factual definiteness. A component of a quantum system may be chosen to represent a clock and changes in other components can then be expected to be correlated with clocks with which they are entangled. Instead of traditional time-dependent equations of motion we provide a specific mechanism whereby evolution of data is instead quasi-causally related to the relative \availability\ of states and equations of motion are expressed in terms of quantized clock variables. We also suggest that time-reversal symmetry-breaking in weak interactions is an artifice of a conventional choice of co-ordinate time-function. Analysis of a "free" particle suggests that conventional co-ordinate space-time emerges from how we measure the separation of objects and events.
We show that the emergence of time evolution in an otherwise timeless nonrelativistic closed quantum system -- viewed as a poor man's model of generally covariant quantum theory -- can be understood from the perspective of the path integral representation. As often happens in the functional integral approach, this viewpoint offers a more intuitive account of features that become cumbersome in the operator/Hilbert-space formulation. We show how Schrödinger evolution emerges once a clock degree of freedom is identified and placed in a suitable semiclassical `good-clock state'. Our analysis has a consequence that extends to path integral formulations of generally covariant systems with action SSS (including gravity). In such theories certain transition amplitudes take the form exp⁡(iS/ℏ)+exp⁡(−iS/ℏ)\exp(iS/\hbar)+\exp(-iS/\hbar)exp(iS/ℏ)+exp(−iS/ℏ) rather than the expected `forward propagating' exp⁡(iS/ℏ)\exp(iS/\hbar)exp(iS/ℏ). This feature, known as the {\em cosine problem}, appears in concrete regularizations of the path integral, for example in the spin foam representation defining the physical inner product between spin network states in loop quantum gravity. Both formally and in explicit regularizations, this apparent difficulty has led some authors to seek modifications of the basic amplitudes to eliminate backward propagation. Our model shows that the cosine problem is instead a natural consequence of time-reversal invariance of the fundamental dynamics together with the time-neutral boundary states commonly used in transition amplitudes. When a suitable clock system is identified and placed in a semiclassical `good-clock state', it introduces a time arrow selecting the `forward propagating' exp⁡(iS/ℏ)\exp(iS/\hbar)exp(iS/ℏ), without modifying the fundamental dynamics. The analysis clarifies how time emerges under suitable conditions and emphasizes that, in the canonical formulation, quantum gravity is fundamentally timeless.
This article generalizes the conditional probability interpretation of time in which time evolution is realized through entanglement between a clock and a system of interest. This formalism is based upon conditioning a solution to the Wheeler-DeWitt equation on a subsystem of the Universe, serving as a clock, being in a state corresponding to a time t . Doing so assigns a conditional state to the rest of the Universe | ψ S ( t ) ⟩ , referred to as the system. We demonstrate that when the total Hamiltonian appearing in the Wheeler-DeWitt equation contains an interaction term coupling the clock and system, the conditional state | ψ S ( t ) ⟩ satisfies a time-nonlocal Schrödinger equation in which the system Hamiltonian is replaced with a self-adjoint integral operator. This time-nonlocal Schrödinger equation is solved perturbatively and three examples of clock-system interactions are examined. One example considered supposes that the clock and system interact via Newtonian gravity, which leads to the system's Hamiltonian developing corrections on the order of G / c 4 and inversely proportional to the distance between the clock and system.