[Paper Review] Information and measurement in generally covariant quantum theory

S. Jay Olson, Jonathan P. Dowling|arXiv (Cornell University)|Jan 28, 2007

Quantum Mechanics and Applications2 references3 citations

TL;DR

This paper resolves the ambiguity in time ordering for multiple measurements in generally covariant quantum theory by introducing an information-theoretic framework where observer-based entropy relationships emerge as the basis for probabilistic predictions. By generalizing Cerf and Adami's observer-entropy formalism to a background-independent setting, the authors show that time ordering and collapse probabilities arise naturally from reduced density operators in extended configuration space, providing a self-contained, unitary probability interpretation without external time or classical observers.

ABSTRACT

Due to the absence of an external, classical time variable, the probabilistic predictions of covariant quantum theory are ambiguous when multiple measurements are considered. Here, we introduce an information theoretic framework to the covariant formalism, and use it to interpret the measurement process. We find that the time ordering of measurements emerges as an entropy relationship in the state of the observers, giving unique probabilities for multiple measurements. This approach suggests a new, fully self-contained probability interpretation for generally covariant quantum physics, which makes use of a quantum mechanical description of the observer, in contrast to standard quantum mechanics which assumes an external, classical observer.

Motivation & Objective

• To resolve the ambiguity in multiple-measurement probabilities in generally covariant quantum theories lacking a classical time variable.
• To develop a self-contained probability interpretation for quantum mechanics that does not rely on external observers or background causal structures.
• To generalize Cerf and Adami's information-theoretic measurement formalism to the covariant, background-free setting of quantum gravity and quantum cosmology.
• To demonstrate that time ordering and effective wavefunction collapse emerge from entropic relationships in observer subsystems rather than from external time parameters.

Proposed method

• Introduce a partial trace over the quantum system Q in extended configuration space to define reduced density operators for observers A and B.
• Define observer states in distinct regions of configuration space (S, S′, S′′) to model sequential measurements without assuming time ordering.
• Use von Neumann entropy of observer density matrices to derive classical probabilities, mirroring Cerf and Adami’s information-theoretic approach.
• Generalize the entropy-based time ordering to covariant settings by showing that increasing entropy in observer systems defines a causal sequence.
• Construct a formalism where probabilities for multiple measurements are uniquely determined by the entropy structure of observer subsystems, not by time-ordered projections.
• Demonstrate that in the small-configuration-space limit, the formalism reduces to standard Schrödinger picture quantum mechanics with the Born rule preserved.

Experimental results

Research questions

• RQ1 How can unique probabilities be assigned to multiple measurements in a generally covariant quantum theory without a classical time variable?
• RQ2 What mechanism can replace time-ordered projections to resolve the ambiguity in measurement sequence probabilities?
• RQ3 Can the Born rule and effective wavefunction collapse be recovered in a background-independent quantum formalism?
• RQ4 How do observer subsystems encode probabilistic information in the absence of an external observer or classical time?
• RQ5 What role does entropy play in defining a causal order for measurements in a generally covariant quantum framework?

Key findings

• The time ordering of measurements emerges not from a background time variable, but from increasing von Neumann entropy in the reduced density operators of observer systems.
• Probabilities for sequential measurements are uniquely determined by the entropy structure of observer subsystems, with p_B(j) = ∑_i |c_i|² |U_ij|² derived from observer B’s density matrix.
• The formalism reproduces the standard Born rule in the small-configuration-space limit, ensuring consistency with standard quantum mechanics.
• The approach eliminates the need for non-unitary collapse or external time by deriving effective probabilities from unitary evolution and partial traces.
• Observer-based entropy relationships provide a self-contained, background-independent probability interpretation for generally covariant quantum theories.
• The method resolves the measurement ambiguity in Reisenberger-Rovelli formalism by replacing time-ordered projections with entropy-based causal ordering.