[Paper Review] Equivalent Sets of Histories and Multiple Quasiclassical Realms

Murray Gell‐Mann, James B. Hartle|ArXiv.org|Apr 8, 1994

Mathematics and ApplicationsMathematics20 citations

TL;DR

This paper establishes a formal criterion for physical equivalence of sets of histories in quantum mechanics of closed systems, showing that triples of initial conditions, Hamiltonians, and histories are physically equivalent under any fixed unitary transformation. It demonstrates that multiple quasiclassical realms—each with distinct coarse-grained histories governed by classical-like laws—can coexist in quantum cosmology, and that IGUSes (information-gathering and utilizing systems) could evolve in such distinct realms, with probabilistic inferences possible via hybrid realms combining alternatives from different realms.

ABSTRACT

We consider notions of physical equivalence of sets of histories in the quantum mechanics of a closed system. We show first how the same set of histories can be relabeled in various ways, including the use of the Heisenberg equations of motion and of passive transformations of field variables. In the the usual approximate quantum mechanics of a measured subsystem, two observables re- presented by different Hermitian operators are physically distinguished by the different apparatus used to measure them. In the quantum mechanics of a closed system, however, any apparatus is part of the system and the notion of physically distinct situations has a different character. We show that a triple consisting of an initial condition, a Hamiltonian, and a set of histories is physically equivalent to another triple if the operators representing these initial conditions, Hamiltonians, and histories are related by any fixed unitary transformation. We apply this result to the question of whether the universe might exhibit physically inequivalent quasiclassical realms (which we earlier called quasiclassical domains), not just the one that includes familiar experience. We describe how the probabilities of alternative forms, behaviors, and evolutionary histories of information gathering and utilizing systems (IGUSes) using the usual quasiclassical realm could in principle be calculated in quantum cosmology, although it is, of course, impractical to perform the computations. We discuss how, in principle, the probabilities of occurence of IGUSes could be calculated in realms distinct from the usual quasiclassical one. We discuss how IGUSes adapted mainly to two different realms could draw inferences about each other using a hybrid realm consisting of alternatives drawn from each.

Motivation & Objective

• To clarify the notion of physical equivalence between sets of histories in the quantum mechanics of a closed universe.
• To investigate whether the universe might support multiple physically inequivalent quasiclassical realms beyond the familiar one.
• To develop a framework for computing probabilities of IGUSes (information-gathering and utilizing systems) evolving in non-standard quasiclassical realms.
• To explore how IGUSes in different realms might infer the existence and properties of each other using hybrid decohering sets of histories.
• To establish that no single quasiclassical realm is privileged by quantum mechanics, and that all are equally valid in principle.

Proposed method

• Defines physical equivalence of a triple (initial condition, Hamiltonian, set of histories) under fixed unitary transformations of all three components.
• Applies the criterion to show that different quasiclassical realms—defined by distinct sets of decohering histories—can be physically equivalent if related by unitary transformations.
• Uses the consistent histories formalism to compute probabilities for coarse-grained histories in both standard and non-standard quasiclassical realms.
• Introduces the concept of hybrid realms by combining alternatives from two distinct quasiclassical realms into a single decohering set.
• Analyzes how IGUSes in one realm can make probabilistic inferences about IGUSes in another by exploiting nearly perfect correlations between non-quasiclassical and quasiclassical observables.
• Applies the framework to hypothetical scenarios such as the universe as a quantum computer, showing that IGUSes could in principle evolve in such non-standard realms.

Experimental results

Research questions

• RQ1 What conditions define physical equivalence between different sets of histories in the quantum mechanics of a closed system?
• RQ2 Can the universe support multiple quasiclassical realms that are physically inequivalent yet consistent with quantum mechanics?
• RQ3 How can probabilities for the emergence of IGUSes be computed in quasiclassical realms other than the standard one?
• RQ4 What mechanisms allow IGUSes in one quasiclassical realm to draw probabilistic inferences about IGUSes in another realm?
• RQ5 To what extent can non-quasiclassical operators (e.g., spin or quantum-computational variables) be correlated with quasiclassical ones to enable cross-realm inference?

Key findings

• A triple consisting of an initial condition, a Hamiltonian, and a set of histories is physically equivalent to another triple if all components are related by a fixed unitary transformation.
• Multiple quasiclassical realms—each exhibiting classical-like behavior through decoherence and approximately deterministic correlations—can coexist in quantum cosmology without contradiction.
• The probability of IGUSes evolving in a non-standard quasiclassical realm (e.g., one based on quantum-computational variables) can be computed in principle, though such calculations are intractable in practice.
• Hybrid realms combining alternatives from two distinct quasiclassical realms can decohere and allow probabilistic inferences about IGUSes in one realm based on observations in the other.
• Even in the absence of perfect correlations, probabilistic inferences about IGUSes in different realms are possible when the hybrid set of histories decoheres.
• The existence of IGUSes in multiple quasiclassical realms does not violate quantum mechanics, and no realm is fundamentally privileged over another.