Struct SU3 

pub struct SU3 {
    pub matrix: Array2<Complex64>,
}

SU(3) group element - 3×3 complex unitary matrix with determinant 1

Represents a color rotation in QCD.

Representation

We use Array2<Complex64> from ndarray to represent the 3×3 unitary matrix.

Constraints

• Unitarity: U† U = I
• Determinant: det(U) = 1

Examples

use lie_groups::su3::SU3;
use lie_groups::traits::LieGroup;

let id = SU3::identity();
assert!(id.verify_unitarity(1e-10));

Fields

matrix: Array2<Complex64>

3×3 complex unitary matrix

Implementations

impl SU3

pub fn identity() -> Self

Identity element

pub fn verify_unitarity(&self, tolerance: f64) -> bool

Verify unitarity: U† U = I

pub fn inverse(&self) -> Self

Matrix inverse (equals conjugate transpose for unitary matrices)

pub fn adjoint(&self) -> Self

Conjugate transpose (adjoint)

pub fn distance_to_identity(&self) -> f64

Distance from identity element

Trait Implementations

impl Clone for SU3

fn clone(&self) -> SU3

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for SU3

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Display for SU3

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl LieGroup for SU3

const DIM: usize = 3

Matrix dimension in the fundamental representation.

type Algebra = Su3Algebra

Associated Lie algebra type.

fn identity() -> Self

The identity element e ∈ G.

fn compose(&self, other: &Self) -> Self

Group composition (multiplication): g₁ · g₂

fn inverse(&self) -> Self

Group inverse: g⁻¹

fn adjoint(&self) -> Self

Adjoint representation element (for matrix groups: conjugate transpose).

fn adjoint_action(&self, algebra_element: &Su3Algebra) -> Su3Algebra

Adjoint representation: Ad_g: 𝔤 → 𝔤

fn distance_to_identity(&self) -> f64

Geodesic distance from identity: d(g, e)

fn exp(tangent: &Su3Algebra) -> Self

Exponential map: 𝔤 → G

fn log(&self) -> LogResult<Su3Algebra>

Logarithm map: G → 𝔤 (inverse of exponential)

fn dim() -> usize

Dimension of the fundamental representation.

fn trace(&self) -> Complex64

Trace in the fundamental representation.

fn distance(&self, other: &Self) -> f64

Distance between two group elements: d(g, h)

fn is_near_identity(&self, tolerance: f64) -> bool

Check if this element is approximately the identity.

fn trace_identity() -> f64

Trace of the identity element

fn reorthogonalize(&self) -> Self

Project element back onto the group manifold using Gram-Schmidt orthogonalization.

impl Mul<&SU3> for &SU3

Group multiplication: U₁ · U₂

type Output = SU3

The resulting type after applying the * operator.

fn mul(self, rhs: &SU3) -> SU3

Performs the * operation.

impl Mul<&SU3> for SU3

type Output = SU3

The resulting type after applying the * operator.

fn mul(self, rhs: &SU3) -> SU3

Performs the * operation.

impl MulAssign<&SU3> for SU3

fn mul_assign(&mut self, rhs: &SU3)

Performs the *= operation.

impl PartialEq for SU3

fn eq(&self, other: &SU3) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl Compact for SU3

SU(3) is compact

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

impl SemiSimple for SU3

SU(3) is semi-simple

impl Simple for SU3

SU(3) is simple

It has no non-trivial normal subgroups (except center ℤ₃).

impl StructuralPartialEq for SU3