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

Abstract

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.

Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology

“…Such an exact factorisation was considered in [32,54,[57][58][59]67,68] and it avoids the complication of having to consider the dynamics of each of the ψ k states. Evidently, this factorisation is ambiguous, as one can redefine each factor as follows:…”

“…In the literature [9,10,24,25,28,29,32,54], equation (10) is often interpreted as the Schrödinger equation for the 'light' system alone, in which the 'heavy' variables provide the clock which parametrises the evolution of the 'light' degrees of freedom. The (real part of the) source term J in (10) can be removed by a phase transformation of ψ [32,60].…”

“…Such an interpretation was also adopted by Arce [54] in the context of the problem of time in (non-relativistic) quantum mechanics. In [54], Arce referred to the partial averages defined in (11) as conditional expectation values. Such an interpretation is only possible when the 'light'-sector unitarity is enforced [58,59].…”

“…Thus, the TISE would be promoted to the fundamental equation of quantum mechanics. This approach was further pursued by Arce [54].…”

Potential energy surfaces in atomic structure: The role of Coulomb correlation in the ground state of helium

“…Equations (18) and (19) are arranged in a form analogous to Hunter's nonadiabatic electronic-nuclear separation [20,22,23,26], which, in turn, is a sort of exact version of the BO approximation. Thus, r 12 and r 1 , r 2 are the analogs of the electronic and the nuclear coordinates, respectively, andΩ,T 0 and U are the analogs of the clamped-nuclei Hamiltonian (with the electronnucleus attractions included), the nuclear kinetic energy operator, and the molecular NAPES.…”