Axiomatic Framework for a Discrete Quantum Reality

Abstract

This document outlines a candidate theory for fundamental physics based on a discrete, informational ontology. The theory posits that the continuous formalisms of Quantum Field Theory (QFT) and General Relativity (GR) are emergent, statistical approximations of an underlying discrete reality. This framework successfully resolves the vacuum energy problem and the black hole information paradox. Crucially, it re-evaluates the process of decoherence, predicting a long-term residual quantum coherence orders of magnitude larger than predicted by standard models. This residual coherence provides a potential physical mechanism for single-system, historical anomalies, such as the Mandela Effect (M.E.), without contradicting existing experimental data on decoherence rates.

1. Core Axioms

The theory is founded upon the following four axioms:

Axiom 1: Discreteness.

The universe is a discrete informational structure. All physical quantities, including space and time, are fundamentally quantized.

Axiom 2: The Universal State Vector.

The complete physical state of our single universe at a discrete time step $t$ is described by a state vector $| Ψ(t) \rangle$. This vector exists in a vast, finite-dimensional Hilbert space and is a superposition of orthogonal basis states $|S_i\rangle$, where each $|S_i\rangle$ represents a complete, discrete configuration of the universe.

$$ | Ψ(t) \rangle = ∑i=1N c_i |S_i\rangle $$

Here, $c_i$ are complex amplitudes, and $N$ is the total (finite) number of possible universal configurations.

Axiom 3: Discrete Unitary Evolution.

The evolution of the state vector is governed by a fundamental, discrete, unitary update matrix $\mathbf{M}$.

$$ | Ψ(t+1) \rangle = \mathbf{M} | Ψ(t) \rangle $$

The unitarity of $\mathbf{M}$ ensures the conservation of the total squared length of the state vector, $L^2 = \langle Ψ | Ψ \rangle = ∑ |c_i|^2$. This length $L$ is a conserved constant of the universe, not assumed to be 1.

Axiom 4: The Born Rule as an Interaction Law.

The probability $P(S_k)$ of an observer (a subsystem) experiencing the outcome corresponding to a specific configuration $|S_k\rangle$ upon a measurement-like interaction is given by the Born Rule. This rule is postulated as the law governing how subsystems interact with the universal state vector.

$$ P(S_k) = \frac{|c_k|^2}{∑i=1N |c_i|^2} = \frac{|c_k|^2}{L^2} $$

This axiom ensures that experienced probabilities are normalized, explaining the empirical success of the $∑|c_i|^2 = 1$ formalism in standard quantum mechanics.

2. Emergence of Classicality and Decoherence

2.1. Observation and Collapse

“Collapse” is not a fundamental process. It is the emergent phenomenon of decoherence-driven partitioning of the universal state vector $| Ψ(t) \rangle$. An “observation” occurs when a simple quantum subsystem becomes entangled with a complex, macroscopic subsystem (an observer, a measuring device). This entanglement partitions the state vector into a superposition of macroscopically distinct, orthogonal, and non-interfering branches. A conscious observer, being part of one such branch, experiences a single, definite outcome.

2.2. A Re-evaluation of Decoherence

The standard model of decoherence assumes an infinite, continuous environment, leading to a perfectly exponential decay of off-diagonal coherence terms in the density matrix $ρ$. This is described by the Lindblad Master Equation:

$$ \frac{dρ}{dt} = -i[H, ρ] + \mathcal{L}(ρ) $$

Our discrete theory, based on a finite environment of $N$ states, makes a crucial modification. While the initial rate of decoherence predicted by our model is identical to the Lindblad equation (thus agreeing with all current experiments), the long-term behavior is different. The off-diagonal terms do not decay to zero. They decay to a residual “floor” of coherence.

The magnitude of this residual coherence is approximately:

$$ | ρij |\text{residual} \propto \frac{1}{\sqrt{N\text{eff}}} $$

where $N_\text{eff}$ is the effective number of environmental states the system is entangled with. This value, while extremely small, is many orders of magnitude larger than the near-zero coherence predicted by the standard continuous model.

3. Key Problems Resolved

1. The Vacuum Energy Problem: The infinite energy predicted by QFT is an artifact of integrating over a continuous spacetime. In our discrete model, the sum is finite. The observed vacuum energy is the small, baseline ground state energy of the discrete network $S$ evolving via $\mathbf{M}$.
2. The Black Hole Information Paradox: Information is not destroyed. It is scrambled and stored within the finite, discrete degrees of freedom that constitute the black hole’s event horizon. The evaporation process releases this information back into the universe, preserving the unitarity of the evolution matrix $\mathbf{M}$.

4. Central Falsifiable Prediction

The theory’s central prediction is that the standard model of decoherence is an over-aggressive approximation.

Prediction: An experiment capable of tracking the coherence of a complex, isolated quantum system over a long duration will find that the coherence does not decay to zero. Instead, it will decay to a non-zero asymptotic floor. The magnitude of this floor will be orders of magnitude larger than the residual coherence allowed by the standard continuous model.

5. A Potential Mechanism for Anomalous Subjective Experience (e.g., The Mandela Effect)

The predicted non-zero “fuzziness” between decohered branches provides a potential physical mechanism for rare, single-system anomalies.

• Hypothesis: The Mandela Effect is not a memory error, but a subjective experience of informational crosstalk between nearly-separate branches of reality, made possible by the larger-than-expected residual coherence. A complex information-processing system, like the human brain, could be sensitive enough to be affected by this residual coherence, leading to the formation of a physical memory record (informational inertia) of a non-dominant reality branch.

This explanation is a direct consequence of the theory’s core tenet that decoherence is not absolute. Its rarity is a qualitative measure of the universe’s incredibly fine-grained, but not perfectly continuous, nature.