Skip to content

README.md

Latest commit

 

History

History
62 lines (39 loc) · 1.8 KB

README.md

File metadata and controls

62 lines (39 loc) · 1.8 KB

Algebraic Scanner: Automated Theory Search

This example demonstrates an automated search through Clifford Algebras to find dimensions that admit a complex structure (I² = -1).

How to Run

cargo run -p physics_examples --example algebraic_scanner

Engineering Value

In theoretical physics, finding algebras with complex structure is crucial for:

  • Quantum Mechanics: Complex numbers are fundamental (wavefunctions)
  • Supersymmetry: Requires specific Clifford algebra properties
  • Signal Processing: Complex structure enables Fourier analysis

This example automates what would otherwise be tedious manual calculation.

What It Does

Scans Clifford Algebras Cl(p,q) from dimension 1 to 9 and checks if the pseudoscalar I satisfies:

I² = -1  (Complex Structure)

Pseudoscalar

The pseudoscalar I is the highest-grade element of the algebra (product of all basis vectors). Its square depends on the metric signature:

  • Euclidean Cl(n,0): I² = ±1 depending on dimension
  • Minkowski Cl(p,q): I² depends on signature

Output Interpretation

[MATCH] Dimension 3: Euclidean signature Cl(3, 0) admits Complex Structure. I² = -1.0000

This means Cl(3,0) - the algebra of 3D Euclidean space - has a complex structure, which is why quaternions (related to Cl(3,0)) are so useful in 3D rotations.

Adapting This Example

  1. Change dimension range: Modify max_dim (note: memory grows as 2^n)
  2. Add custom signatures: Use Metric::Custom for arbitrary (p,q) signatures
  3. Check other properties: Extend to check for division algebra structure, etc.

Key APIs Used

  • CausalMultiVector::new() - Create multivector with metric
  • geometric_product() - Compute I²
  • Metric::Euclidean(n), Metric::Minkowski(n) - Signature selection