Quantum Domain: k=0 Optimal
For quantum state geometry problems without explicit singularities, k=0(classical calculus) is optimal. CASCADE shows no advantage when problems are naturally smooth.
0/2
Quantum Wins
k=0
Classical Optimal
TIE
No Singularity
CASCADE advantage appears only near singularities. Smooth problems favor classical methods.
View full CASCADE singularity analysis →Numerical Methods Application - Requires Validation
This page demonstrates how the numerical methods toolkit might apply to quantum state geometry. This is an exploratory application for educational purposes, not validated physics.
For verified mathematical results, see Spectral Gap Verification.
Quantum State Geometry
Different ways to measure distance in quantum state space.
Quantum states live in a complex Hilbert space with multiple natural distance metrics. Each metric represents a C-scheme choice. Scheme-invariant features are those that all metrics agree upon - these are the physical properties of quantum systems.
1. Quantum State Metrics
Overview
This tool compares different ways to measure distance between quantum states. Just like you can measure distance on Earth using straight lines or along the curved surface, quantum states have multiple valid distance definitions.
Different metrics answer different questions: Fubini-Study measures the angle between states, Bures measures fidelity (how well one state can mimic another), and trace distance measures distinguishability in experiments. Understanding when they agree tells us what properties are truly physical vs. mathematical artifacts.
High correlation (near 100%) means metrics agree on distance ordering - the geometry is scheme-invariant. For pure states, Fubini-Study and Bures should correlate perfectly. Trace distance may differ due to its operational interpretation.
2. Chiral Anomaly Detection
Overview
This tool demonstrates one of the most important discoveries in quantum field theory: the chiral anomaly. A symmetry that exists in the classical theory FAILS to survive quantization. The path integral measure is not invariant under chiral rotations!
The chiral anomaly explains why neutral pions decay into two photons (pi0 to 2*gamma), why protons are stable (baryon number conservation), and constrains the number of fermion generations in the Standard Model. It is one of the deepest results in physics.
With Wilson fermions, the Jacobian acquires a non-trivial phase proportional to the gauge field (the anomaly). The phase plot should show a linear increase matching the theoretical Fujikawa prediction. The Ward identity is VIOLATED by exactly this amount.
The ABJ Anomaly Story
Discovery (1969)
Adler, Bell, and Jackiw independently discovered that the axial-vector current conservation fails at the quantum level. The divergence picks up an extra term proportional to the electromagnetic field strength: d_mu j^5_mu = (alpha/pi) * E.B
Fujikawa Method (1979)
Fujikawa showed that the anomaly arises from the non-invariance of the path integral measure under chiral transformations. The Jacobian of the transformation has a non-trivial phase - exactly what our visualizer demonstrates!
The Unifying Principle
Anomalies are obstructions to extending classical symmetries to quantum theory. They are the ultimate example of scheme-dependence: physics that exists in one formalism but not another.
The meta-calculus framework provides systematic tools for detecting such obstructions and understanding when they represent genuine physics vs. mathematical artifacts.
3. Quantum Phase Parameter Scan
MOO Results: k=0 Optimal
A systematic parameter scan over (k, w, p) space reveals that classical calculus (k=0) is optimal for quantum state geometry. This is expected: smooth problems without singularities do not benefit from NNC regularization.
Data from quantum_phase_corrected.json (524KB, 1000+ parameter combinations).
Parameter Space Exploration
MOO Objectives
How much does state norm change?
How long before instability?
Stability under perturbations
Key Finding
All Pareto-optimal solutions cluster near k=0 (classical calculus). Non-zero k values cause norm drift and instability because quantum evolution is naturally smooth - there are no singularities for NNC to regularize.
Why This Matters (Honest Assessment)
CASCADE works by regularizing singularities. Quantum state geometry has no singularities to regularize. This is a negative result that shows the boundaries of the approach:
- + NNC/CASCADE excels for: 1/r singularities, crack tips, shock waves, power laws
- - NNC/CASCADE offers no benefit for: smooth evolution, quantum mechanics, diffusion
- = Classical calculus (k=0) is optimal when problems are inherently smooth
Quantum Geometry Summary
3
Distance Metrics
(Fubini-Study, Bures, Trace)
>99%
Metric Correlation
(Pure states are scheme-invariant)
1
Anomaly Detected
(Chiral ABJ anomaly)
k=0
Optimal Parameter
(Classical is best here)