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Compact

lie_groups::traits

Trait Compact 

Source 
pub trait Compact: LieGroup + SealedCompact { }
Expand description

Marker trait for compact Lie groups.

A Lie group is compact if it is compact as a topological space.

§Examples

  • U(1), SU(2), SU(3), SO(3) - all compact

§Significance

  • Representations are completely reducible
  • Yang-Mills theory well-defined
  • Haar measure exists

§Sealed Trait

This trait is sealed - only verified compact groups can implement it.

Dyn Compatibility§

This trait is not dyn compatible.

In older versions of Rust, dyn compatibility was called "object safety", so this trait is not object safe.

Implementors§

Source§

impl Compact for SO3

SO(3) is compact

The rotation group is diffeomorphic to ℝP³ (real projective 3-space). All rotations are bounded: ||R|| = 1.

Source§

impl Compact for SU2

SU(2) is compact.

All elements are bounded: ||U|| = 1 for all U ∈ SU(2). The group is diffeomorphic to the 3-sphere S³.

Source§

impl Compact for SU3

SU(3) is compact

All elements are bounded: ||U|| = 1 for all U ∈ SU(3).

Source§

impl Compact for U1

U(1) is compact.

The circle group {e^{iθ} : θ ∈ ℝ} is diffeomorphic to S¹. All elements have unit modulus: |e^{iθ}| = 1.

§Topological Structure

U(1) ≅ SO(2) ≅ S¹ (the unit circle)

  • Closed and bounded in ℂ
  • Every sequence has a convergent subsequence (Bolzano-Weierstrass)
  • Admits a finite Haar measure

§Physical Significance

Compactness of U(1) ensures:

  • Well-defined Yang-Mills action
  • Completely reducible representations
  • Quantization of electric charge
Source§

impl<const N: usize> Compact for SUN<N>

SU(N) is compact for all N ≥ 2.