Prelocality No-Go Theorems and Exactly Solvable Relational History Hamiltonians

Imagine a universe built not from fixed points in space, but from a vast web of relationships between particles. In this view, space itself is not a pre-existing stage where events happen; rather, space emerges from the connections between things. If two particles interact, they define a distance; if they do not, they are far apart. This idea, known as emergent geometry, is a central dream in modern physics: to show how the smooth, familiar world of space and time can arise from a chaotic, microscopic soup of interacting quantum bits. However, for this dream to work, the underlying rules must allow for a specific kind of order. They must allow the system to pick out a sparse, local structure—where each particle only talks to a few neighbors—from a sea of possibilities where every particle could potentially talk to every other one.

The challenge lies in the fundamental symmetry of the microscopic world. At the smallest scales, particles of the same type are indistinguishable. You cannot label them "particle one" or "particle two" without breaking the laws of physics. Because of this, the rules governing their interactions must treat every possible group of particles exactly the same. If you have a rule that makes a connection between particle A and particle B, the same rule must apply to every other pair, and every group of three, and so on. This creates a massive, uniform web of potential connections. The question physicists have been asking is whether this perfect symmetry can naturally lead to a sparse, local world, or if the symmetry itself forces the universe to remain a tangled, non-local mess.

A new study by independent researcher Maxim V. Churilov tackles this problem by building precise mathematical models of how such a universe could evolve. The work does not claim to have discovered a new theory of gravity or a hidden dimension. Instead, it acts as a rigorous stress test, constructing specific scenarios to see if the dream of emergent space can survive the strict demands of symmetry. The findings are sobering but clarifying: the paper proves that perfect symmetry alone cannot create a sparse, local universe. It shows that without adding extra, specific ingredients, the system remains a chaotic tangle where distance and locality are impossible to define. However, the study also succeeds in building two distinct, solvable models that demonstrate exactly what is required to make progress, revealing the precise trade-offs between the resources needed to build a clock and the complexity of the resulting history.

The first major result of the paper is a set of "no-go" theorems, which are mathematical proofs that certain outcomes are impossible under specific conditions. The researchers focused on a scenario where the universe is defined by a Hamiltonian, a mathematical object that dictates how energy is distributed and how the system changes over time. They asked if a system that treats every particle and every group of particles identically could ever settle into a state where only a few connections exist, creating a sparse network like a map of roads rather than a mesh of every point connected to every other point. The answer is a definitive no. The paper proves that if the rules are perfectly symmetric, the energy of the system will be the same for every possible arrangement of connections. This means the system cannot choose one specific, sparse map as its lowest energy state. Instead, it remains stuck in a superposition of all possible maps, or a chaotic mixture, unable to distinguish a local neighborhood from a global mess.

Furthermore, the study shows that even if you try to measure the system locally, you cannot determine if the whole network is connected. Imagine looking at a small patch of a forest and seeing only the trees within a few steps. You might see a dense cluster of trees, but you cannot tell if that cluster is part of a vast, continuous forest or just an isolated island of trees surrounded by a void. The paper proves that no amount of local observation can reveal the global structure of the network. This implies that to create a universe with a clear sense of "here" and "there," the system needs a global witness or a mechanism that looks at the whole picture, not just the local details. Finally, the researchers showed that the laws of quantum mechanics, which preserve the distinctness of different states, prevent a system from naturally collapsing into a single, stable configuration without some external influence or a specific mechanism to break the symmetry.

Having established these limitations, the paper moves to a constructive phase, building two exact models to see how one might bypass these obstacles. The first model uses a "cyclic clock." In this setup, the universe evolves in a loop, like a clock hand moving around a face. The researchers designed a system where a single clock register drives a sequence of updates that flip the connections between particles. Because the clock is cyclic and the updates are reversible, the system settles into a unique, stable state. The beauty of this model is that it can be solved completely, meaning the researchers can calculate the exact energy gap—the difference between the lowest energy state and the next possible state. They found that this gap depends only on the length of the clock cycle, not on the number of particles in the system. As the clock gets longer, the gap shrinks in a predictable way, scaling with the square of the clock's length. This provides a precise, mathematical certification that the system is stable and that its ground state is unique, but it also highlights a cost: the system relies on a single, global clock to coordinate the entire history.

The second model takes a different approach, abandoning the idea of a single, ordered sequence of events. Instead, it uses a "Boolean clock," where every possible pair of particles has its own independent clock bit. This allows the system to record which connections have been made without forcing a specific order on when they happened. In this architecture, the updates for different pairs commute, meaning they can happen in any order without changing the final result. This model is also exactly solvable. The researchers found that it has a constant energy gap that does not shrink as the system grows larger, which is a desirable feature for stability. However, this comes with a different cost. Instead of a single unique ground state, this model has a vast number of possible ground states, specifically a number that grows exponentially with the number of particles. The system does not automatically select one specific history; it leaves the choice open, requiring an additional mechanism to pick out a single, symmetric state from the many possibilities.

The paper concludes by comparing these two architectures to highlight the fundamental trade-offs in building a universe from relational data. The cyclic model offers a unique, well-defined history but requires a global clock and has an energy gap that gets smaller as the system grows. The Boolean model offers a constant energy gap and no need for a global order, but it sacrifices uniqueness, leaving the system in a state of massive degeneracy where many different histories are equally valid. The study makes it clear that neither model spontaneously generates a sparse, local geometry on its own. They are finite, pre-local models that demonstrate the resources required to manage history and symmetry, but they do not solve the final step of creating an emergent geometric phase. The work serves as a precise map of the landscape, showing exactly where the obstacles lie and what tools are needed to navigate them, leaving the ultimate construction of a local universe as a challenge for future theories to solve.