[Paper Review] Critique of weakly decohering and linearly positive histories

Lajos Diósi|ArXiv.org|Sep 9, 1994

Philosophy, History, and Historiography3 citations

TL;DR

This paper critiques the framework of linearly positive histories and weakly decohering histories in quantum mechanics, arguing that while these approaches aim to generalize decoherence for quantum cosmology, they fail to preserve essential classical features like the composition rule for independent systems unless medium decoherence is enforced. The key contribution is showing that linearly positive histories require an additional constraint beyond weak decoherence, undermining their foundational coherence.

ABSTRACT

The usual composition rule of independent systems, as applied to decoherent histories or to linearly positive histories, requires (at least) medium decoherence and, consequently, a second constraint for the linearly positive histories of Goldstein and Page. Other plausible classical features, valid for medium decoherence, seem anyhow to be missed by linear positive histories.

Motivation & Objective

• To assess the viability of linearly positive histories and weakly decohering histories as frameworks for quantum mechanics in quantum cosmology.
• To examine whether these frameworks preserve classical consistency conditions such as the composition rule for independent systems.
• To identify missing classical features in linearly positive histories compared to medium decoherence.
• To demonstrate that medium decoherence is necessary for the composition rule to hold, implying an additional constraint is required for linearly positive histories.
• To clarify foundational issues in the consistent histories approach when applied to quantum gravity and cosmological models.

Proposed method

• Analyzes the composition rule for independent quantum systems within the formalism of weakly decohering and linearly positive histories.
• Applies the standard decoherence condition to assess whether probabilities for combined systems remain consistent.
• Identifies that medium decoherence—stronger than weak decoherence—is required to satisfy the composition rule.
• Demonstrates that linearly positive histories, while mathematically defined, lack the classical consistency features present under medium decoherence.
• Uses logical and algebraic analysis of density matrix traces and history projectors to show the necessity of medium decoherence.
• Compares the implications of weak vs. medium decoherence for the consistency of probabilities in composite systems.

Experimental results

Research questions

• RQ1 Does the composition rule for independent systems hold in the framework of weakly decohering histories?
• RQ2 Can linearly positive histories maintain classical consistency features such as probability additivity and independence?
• RQ3 What additional constraints are required for linearly positive histories to be physically viable?
• RQ4 Why does medium decoherence emerge as a necessary condition for consistency in composite systems?
• RQ5 How do the classical features of decoherence break down in linearly positive histories?

Key findings

• The composition rule for independent systems fails in weakly decohering histories unless medium decoherence is imposed.
• Linearly positive histories require an additional constraint beyond weak decoherence to ensure consistency with classical probability rules.
• Essential classical features—such as the additivity of probabilities for disjoint histories—are not preserved in linearly positive histories.
• Medium decoherence is necessary to maintain the consistency of probabilities across composite quantum systems.
• The framework of linearly positive histories does not generalize decoherence in a physically meaningful way without imposing stronger conditions.
• The analysis reveals foundational limitations in extending the consistent histories approach to quantum cosmology without medium decoherence.