[Paper Review] Spacetime and Matter - a duality of partial orders

Hans-Thomas Elze|arXiv (Cornell University)|Jan 22, 2010

Quantum Mechanics and Applications11 references3 citations

TL;DR

This paper proposes a duality between spacetime and matter based on two partial orders—causality in spacetime and extensity in matter—showing that discrete, causal structures can give rise to quantum mechanics via a modified Liouville equation that reduces to the von Neumann equation under specific conditions. The key result is the emergence of quantum theory from a classical statistical framework with built-in decoherence due to spacetime discreteness and local potential fluctuations.

ABSTRACT

A new kind of duality between the deep structures of spacetime and matter is proposed here, considering two partial orders which incorporate causality, extensity, and discreteness. This may have surprising consequences for the emergence of quantum mechanics, which are discussed.

Motivation & Objective

• To resolve the foundational disparity between quantum theory (QT) and general relativity (GR) by rethinking the deep structures of spacetime and matter.
• To address the lack of a universal time in quantum mechanics by grounding dynamics in causal, discrete spacetime structures.
• To explore how quantum mechanics might emerge from a classical statistical framework when spacetime is discrete and matter is extensive.
• To investigate the role of spacetime discreteness and local potential fluctuations in generating quantum behavior, including decoherence.
• To propose a duality between causality (spacetime) and extensity (matter) as fundamental, with implications for quantum gravity.

Proposed method

• Formulates a classical statistical dynamics using a Liouville-type equation for a phase space probability distribution f(x,y;t), with Hamiltonians Hx and Hy acting on coordinates x and y.
• Introduces a superoperator interaction term 𝒪(x,y) that couples the bra and ket states, representing a non-standard interaction absent in standard quantum mechanics.
• Assumes the true potential v(x) is piecewise linear with a characteristic length δ ≫ lP, modeling spacetime discreteness and avoiding continuum infinities.
• Demonstrates that when 𝒪 ≡ 0 (true potential is constant, linear, or harmonic), the equation reduces to the von Neumann equation of quantum mechanics.
• Models local fluctuations δV as arising from both matter and spacetime discreteness, leading to a Lindblad-type decoherence term in the equation.
• Uses the framework to show that interactions become unresolvable at high energies due to Lorentz contraction, supporting the idea of a fundamental spacetime cutoff.

Experimental results

Research questions

• RQ1 Can quantum mechanics emerge from a classical statistical dynamics on a discrete, causally ordered spacetime structure?
• RQ2 How does spacetime discreteness at the Planck scale lead to the structure of quantum theory, including superposition and entanglement?
• RQ3 What role do local potential fluctuations play in generating natural decoherence and suppressing macroscopic superpositions?
• RQ4 How does the duality between causality (spacetime) and extensity (matter) constrain the form of the effective dynamics?
• RQ5 Can the absence of a universal time in quantum mechanics be resolved by grounding dynamics in a partial order of causal events?

Key findings

• The Liouville equation with the superoperator interaction 𝒪 reduces to the von Neumann equation of quantum mechanics when the true potential v(x) is constant, linear, or harmonic, implying quantum theory emerges from classical statistical dynamics under these conditions.
• The interaction term 𝒪 vanishes only for specific potentials, indicating that quantum mechanics arises as a special case of a more general classical statistical framework with nontrivial interactions.
• Local fluctuations δV in the true potential v(x), arising from spacetime discreteness and matter, induce a Lindblad-type decoherence term, providing a natural mechanism for wave function collapse.
• The model predicts that at high energies, interactions become unresolvable due to Lorentz contraction shrinking the interaction region below the Planck volume, supporting a fundamental cutoff in spacetime.
• The framework naturally accounts for the absence of macroscopic Schrödinger cat states through induced decoherence, offering a testable prediction.
• The duality between causality in spacetime and extensity in matter is proposed as a fundamental principle, with both structures reflecting discrete, partially ordered sets.