Quantum Pixel Model as a Physical Realization of the Amplituhedron
Description
Quantum Pixel Model as a Physical Realization of the Amplituhedron
We establish a direct correspondence between the Quantum Pixel Model (QPM)—a hexagonal
Planck-scale lattice of spacetime—and the amplituhedron, a geometric object encoding particle
interactions. By mapping QPM’s topological defects to the amplituhedron’s boundary strata, we show
that:
- (1) The amplituhedron’s ”volume” represents the quantum amplitude of defect configurations
in QPM, - (2) Scattering processes in the Standard Model emerge from hexagonal lattice dynamics,
and - (3) Unitarity and locality are emergent properties of the pixelated spacetime structure. This
work bridges the amplituhedron’s abstract formalism with a concrete quantum gravity model
The amplituhedron, a positive geometry in the Grassmannian 𝐺(𝑘, 𝑘 + 4), revolutionizes scattering
amplitude calculations by eliminating spacetime and unitarity as fundamental concepts.
However, its physical interpretation remains opaque. We demonstrate that the Quantum
Pixel Model (QPM) provides the missing link: the amplituhedron is the configuration space
of QPM’s topologicaldefects.
Table Decoded
| QPM Defect | Amplituhedron Component | Geometric Condition | Meaning |
|---|---|---|---|
| Electron | Vertex | 〈1234〉 > 0 | The electron defect maps to a vertex in the amplituhedron, characterized by a positive 4-bracket (Grassmannian positivity condition). |
| Muon | 14D Face | det(C14×18) = 0 | The muon's 14.5-winding defect corresponds to a 14-dimensional boundary of the amplituhedron, defined by the rank condition of a 14 × 18 matrix. |
| Gluon | Edge | 〈12〉 ∼ sin(π/6) | The gluon defect maps to an edge, with its kinematic variable constrained by the hexagonal lattice symmetry (π/6 = 30°) |
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Consistency Check: The geometric conditions ensure QPM defects satisfy the amplituhedron's positivity constraints.
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Predictive Power: The sin(π/6) relation for gluons directly stems from QPMs hexagonal lattice.
-
Validation: The muon's det = 0 condition matches its role as a "composite" defect in QPM (14.5 windings → higher-dim boundary).
QPM:
https://doi.org/10.5281/zenodo.15748400
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Quantum_Pixel_Model_as_a_Physical_Realization_of_the_Amplituhedron_Preprint.pdf
Additional details
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- Is part of
- Preprint: https://doi.org/10.5281/zenodo.15748400 (URL)