Foundational Structures
Dualities, symmetries, and quantization ambiguities.
The deepest structures in physics often involve dualities (equivalent descriptions of the same physics) and ordering ambiguities (choices that arise when quantizing classical systems). This page illustrates both with exact formulas and stored toolkit runs.
Status: exploratory educational demonstration, not validated physics. It shows how the numerical methods toolkit might apply to foundational physics questions. For the toolkit's tested mathematical results, see the Spectral-Gap Boundary audit.
1. Kramers-Wannier Duality
Overview
This visualizer demonstrates one of the most beautiful results in statistical mechanics: the Kramers-Wannier duality of the 2D Ising model. The partition function at high temperature is tied exactly to the partition function at low temperature through the duality transformation K* = -0.5 * ln(tanh(K)), computed here from Onsager's exact solution.
Dualities are profound because they reveal hidden symmetries in physics. The same mathematical structure describes both the ordered (low-T) and disordered (high-T) phases. This duality also predicts the exact critical temperature where phase transitions occur (the self-dual point K = K*). Similar dualities appear throughout physics, from string theory to quantum field theory.
2. Wheeler-DeWitt Operator Ordering
Overview
This visualizer explores the Wheeler-DeWitt equation - the central equation of quantum cosmology that describes the quantum state of the universe. The key issue: when promoting classical variables (like the scale factor a) to quantum operators, different orderings give different equations. This tool shows a toy of the ordering prefactor (which is not ordering-invariant) and the stored pymoo search over the ordering parameter.
The operator ordering problem is one of the deepest unsolved issues in quantum gravity. In ordinary quantum mechanics, ordering affects subleading corrections (hbar^2 terms). But in quantum cosmology, the universe IS the quantum system - there is no external classical background. Different orderings can predict different probabilities for the universe to nucleate, tunnel, or evolve. Which ordering is correct?
Domain Restrictions & Validity
Kretschmann Formula: K ~ r^(-6), D_BG[K] = e^(-6)
Valid only for r > 0 (outside singularity). The bigeometric derivative captures the power-law scaling but does NOT resolve the r=0 singularity itself.
FRW Radiation Era: n=0.5 parameter
Ricci scalar can be zero in radiation-dominated cosmology. This is a physical feature, not a singularity - the universe was smooth and homogeneous.
The Unifying Principle
Both dualities and ordering ambiguities represent choices in how we describe physics. The panels above illustrate the two cases: an exact duality, and an ordering choice that changes the answer.
Treating both as "scheme choices" is an organising heuristic of this project, not a demonstrated result; neither panel tests a meta-calculus prediction.
Foundational Structures Summary
Exact
Kramers-Wannier Duality
(Onsager free energy, textbook identity)
Not invariant
Toy WDW Ordering
(<a> depends on p by construction)