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Module sun

lie_groups

Module sun 

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Generic SU(N) - Special unitary N×N matrices

This module provides a compile-time generic implementation of SU(N) for arbitrary N. It elegantly generalizes SU(2) and SU(3) while maintaining type safety and efficiency.

§Mathematical Structure

SU(N) = { U ∈ ℂᴺˣᴺ | U† U = I, det(U) = 1 }

§Lie Algebra

The Lie algebra su(N) consists of N×N traceless anti-Hermitian matrices:

su(N) = { X ∈ ℂᴺˣᴺ | X† = -X, Tr(X) = 0 }
dim(su(N)) = N² - 1

§Design Philosophy

  • Type Safety: Const generics ensure dimension errors are caught at compile time
  • Efficiency: Lazy matrix construction, SIMD-friendly operations
  • Elegance: Unified interface for all N (including N=2,3)
  • Generality: Works for arbitrary N ≥ 2

§Examples

ⓘ
use lie_groups::sun::SunAlgebra;
use lie_groups::LieAlgebra;

// SU(4) for grand unified theories
type Su4Algebra = SunAlgebra<4>;
let x = Su4Algebra::zero();
assert_eq!(Su4Algebra::DIM, 15);  // 4² - 1 = 15

// Type safety: dimensions checked at compile time
let su2 = SunAlgebra::<2>::basis_element(0);  // dim = 3
let su3 = SunAlgebra::<3>::basis_element(0);  // dim = 8
// su2.add(&su3);  // Compile error! Incompatible types

§Physics Applications

  • SU(2): Weak force, isospin
  • SU(3): Strong force (QCD), color charge
  • SU(4): Pati-Salam model, flavor symmetry
  • SU(5): Georgi-Glashow GUT
  • SU(6): Flavor SU(3) × color SU(2)

§Performance

  • Algebra operations: O(N²) [optimal]
  • Matrix construction: O(N²) [lazy, only when needed]
  • Exponential map: O(N³) via scaling-and-squaring
  • Memory: (N²-1)·sizeof(f64) bytes for algebra

Structs§

SUN
SU(N) group element - N×N unitary matrix with det = 1
SunAlgebra
Lie algebra su(N) - (N²-1)-dimensional space of traceless anti-Hermitian matrices

Type Aliases§

SU4
Type alias for SU(4) - Pati-Salam model
SU5
Type alias for SU(5) - Georgi-Glashow GUT
SU2Generic
Type alias for SU(2) via generic implementation
SU3Generic
Type alias for SU(3) via generic implementation