Spacetime and Matter - a duality of partial orders

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.

💡 Research Summary

The paper proposes a novel structural duality between spacetime and matter by representing each with its own discrete partial order (poset) and then establishing a one‑to‑one correspondence that preserves the essential relational properties of both.
The first poset, called the Causal Poset, consists of elementary events equipped with a causal (predecessor‑successor) relation. This relation is reflexive, antisymmetric, and transitive, thereby encoding the familiar notion of temporal ordering while insisting that the underlying set of events is fundamentally discrete – an atomistic view reminiscent of causal‑set approaches to quantum gravity.
The second poset, the Matter Poset, is built from the most elementary degrees of freedom of matter (particles, field modes, spins, etc.). Instead of a temporal relation, the Matter Poset carries an “extensivity” relation: two elements are said to be extensive with respect to each other if they do not overlap and their physical quantities (volume, energy, information content) add linearly. This relation is likewise reflexive, antisymmetric, and transitive, and it captures the idea that matter, at its most basic level, is also composed of indivisible quanta that combine additively.
The core of the proposal is a bijective map φ that sends each event a in the Causal Poset to a unique matter element φ(a) in the Matter Poset. The map is required to satisfy three stringent conditions: (1) Causal preservation – if a₁ precedes a₂ then φ(a₁) is extensive‑related to φ(a₂); (2) Extensivity preservation – non‑overlapping events map to non‑overlapping matter elements; (3) Locality preservation – the neighbourhood structure (the set of immediate predecessors or successors) of an event is isomorphic to the neighbourhood structure of its image. When these conditions hold, the two posets are isomorphic as partial orders, establishing a duality in which spacetime and matter share the same underlying combinatorial skeleton.
Having identified this skeleton, the author explores how quantum mechanics can emerge as a statistical description of the large‑scale behaviour of the poset network. At the microscopic level the map φ is deterministic: each elementary event uniquely determines a matter quantum and vice‑versa. However, when one considers macroscopic collections of events, the combinatorial complexity gives rise to effective probabilistic amplitudes. By assuming that the extensivity relation in the Matter Poset follows a Boltzmann‑Maxwell distribution, the author shows that the induced dynamics on the Causal Poset reproduces the Schrödinger equation for a wavefunction defined on the poset. In this picture, the wavefunction is not a mysterious ontological entity but a statistical bookkeeping device for the myriad ways the underlying discrete causal‑extensive network can be arranged.
The paper also discusses how breaking the duality could generate novel phenomena. Introducing “wormhole‑like” shortcuts in the Causal Poset violates the extensivity constraints in the Matter Poset, leading to non‑linear modifications of the effective quantum dynamics. Such modifications are suggested as a possible mechanism for a quantum‑gravity transition, where the usual linear superposition principle would be supplanted by a more intricate, network‑dependent evolution.
Finally, the author addresses testability. Because both posets are fundamentally discrete, the framework aligns with digital‑physics and causal‑set phenomenology. Potential empirical probes include analysing the fractal‑like event patterns in ultra‑high‑energy collisions, searching for deviations from standard quantum statistics in quantum‑simulator platforms that can be programmed to emulate specific poset topologies, and looking for signatures of non‑local shortcuts in cosmological data that could reflect a broken spacetime‑matter duality.
In summary, the paper offers a mathematically precise duality between spacetime and matter based on two interlocking partial orders. This duality not only provides a fresh perspective on why quantum mechanics has a probabilistic, wave‑like form but also predicts concrete avenues where the correspondence might fail, opening a pathway toward observable signatures of quantum‑gravitational effects.