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.

Status: exploratory numerical-methods application that requires validation. It shows how the toolkit might apply to quantum state geometry, for education, and is not validated physics. For the toolkit's tested mathematical results, see the Spectral-Gap Boundary audit.

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.

For pure states all three are functions of the same overlap |<psi|phi>|, so they order pairs identically by construction: d_FS = arccos|<psi|phi>|, d_B = sqrt(2 - 2|<psi|phi>|) and d_T = sqrt(1 - |<psi|phi>|^2). Their agreement is a mathematical identity, not evidence of scheme invariance.

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 the rate of the neutral pion decay into two photons (pi0 to 2*gamma), and anomaly cancellation constrains the fermion content of each Standard Model generation. It is one of the deepest results in quantum field theory.

The panel below is a schematic, not a lattice computation: the Jacobian phase is set to the continuum Fujikawa value Q = Phi/(2 pi), so the "measured" and predicted curves agree by construction.

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 (one Dirac fermion of charge e): d_mu j^5_mu = (2 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 - the schematic above draws that phase; it does not compute it.

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.

This page only illustrates the textbook story; it does not provide a meta-calculus method that detects anomalies.

3. Quantum Phase Parameter Scan

Stored Scan: No Unitary Configuration

A grid of 1000 (k, w, p) points in public/quantum_phase_corrected.json records 0 of 1000 configurations as unitary. The smallest norm drift is 2.56e-4 at k = 0.111 (w = -0.5, p = 1). The 10 k values run from -1.000 to 1.000 and the grid contains no k = 0 point, so the scan cannot say anything about classical calculus; there is also no Pareto front in the file.

The separate robustness test in the same file perturbs the k = w = p = 0 baseline and stays unitary, which is the expected behaviour of the classical integrator, not a result of the scan.

Honest Assessment

  • - Every non-zero k configuration in the scan breaks unitarity
  • - The scan has no k = 0 point and no multi-objective optimisation output
  • - No claim about an optimal k is supported by this artifact

Quantum Geometry Summary

3

Distance Metrics

(Fubini-Study, Bures, Trace)

Schematic

Chiral Anomaly Panel

(continuum formula, not a lattice computation)

0/1000

Unitary Configurations

(stored k, w, p scan)