Struct U1Algebra 

pub struct U1Algebra(pub f64);

Lie algebra u(1) ≅ ℝ

The Lie algebra of U(1) consists of purely imaginary numbers i·a where a ∈ ℝ. We represent this as a 1-dimensional real vector space.

Mathematical Structure

Elements of u(1) have the form:

X = i·a where a ∈ ℝ

The exponential map is simply:

exp(i·a) = e^{ia} ∈ U(1)

Examples

use lie_groups::u1::U1Algebra;
use lie_groups::traits::LieAlgebra;

// Create algebra element
let v = U1Algebra::from_components(&[1.5]);

// Scale
let w = v.scale(2.0);
assert_eq!(w.value(), 3.0);

Tuple Fields

0: f64

Implementations

impl U1Algebra

pub fn value(&self) -> f64

Get the real value a from the algebra element i·a

Trait Implementations

impl Add<&U1Algebra> for &U1Algebra

type Output = U1Algebra

The resulting type after applying the + operator.

fn add(self, rhs: &U1Algebra) -> U1Algebra

Performs the + operation.

impl Add<&U1Algebra> for U1Algebra

type Output = U1Algebra

The resulting type after applying the + operator.

fn add(self, rhs: &U1Algebra) -> U1Algebra

Performs the + operation.

impl Add<U1Algebra> for &U1Algebra

type Output = U1Algebra

The resulting type after applying the + operator.

fn add(self, rhs: U1Algebra) -> U1Algebra

Performs the + operation.

impl Add for U1Algebra

type Output = U1Algebra

The resulting type after applying the + operator.

fn add(self, rhs: Self) -> Self

Performs the + operation.

impl Clone for U1Algebra

fn clone(&self) -> U1Algebra

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for U1Algebra

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

Formats the value using the given formatter.

impl Display for U1Algebra

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

Formats the value using the given formatter.

impl LieAlgebra for U1Algebra

const DIM: usize = 1

Dimension of the Lie algebra as a compile-time constant.

fn zero() -> Self

Zero element (additive identity) 0 ∈ 𝔤.

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

Add two algebra elements: v + w

fn scale(&self, scalar: f64) -> Self

Scalar multiplication: α · v

fn norm(&self) -> f64

Euclidean norm: ||v||

fn basis_element(i: usize) -> Self

Get the i-th basis element of the Lie algebra.

fn from_components(components: &[f64]) -> Self

Construct algebra element from basis coordinates.

fn to_components(&self) -> Vec<f64>

Extract basis coordinates from algebra element.

fn bracket(&self, _other: &Self) -> Self

Lie bracket operation: [X, Y] ∈ 𝔤

fn dim() -> usize

Dimension of the Lie algebra (dim 𝔤).

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

Inner product: ⟨v, w⟩

impl Mul<U1Algebra> for f64

type Output = U1Algebra

The resulting type after applying the * operator.

fn mul(self, rhs: U1Algebra) -> U1Algebra

Performs the * operation.

impl Mul<f64> for U1Algebra

type Output = U1Algebra

The resulting type after applying the * operator.

fn mul(self, scalar: f64) -> Self

Performs the * operation.

impl Neg for U1Algebra

type Output = U1Algebra

The resulting type after applying the - operator.

fn neg(self) -> Self

Performs the unary - operation.

impl PartialEq for U1Algebra

fn eq(&self, other: &U1Algebra) -> 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 Sub for U1Algebra

type Output = U1Algebra

The resulting type after applying the - operator.

fn sub(self, rhs: Self) -> Self

Performs the - operation.

impl AntiHermitianByConstruction for U1Algebra

u(1) algebra elements are anti-Hermitian by construction.

The representation U1Algebra(f64) stores a real number a representing the purely imaginary element ia ∈ iℝ ⊂ ℂ. Since (ia)* = -ia, these are anti-Hermitian (as 1×1 complex matrices).

Note: NOT TracelessByConstruction

Unlike su(n), the u(1) algebra is NOT traceless. For a 1×1 matrix [ia], the trace is ia ≠ 0 (for a ≠ 0). This is why U(1) ≠ SU(1).

The theorem det(exp(X)) = exp(tr(X)) still holds, but gives:

• det(exp(ia)) = exp(ia) ≠ 1 in general
• This is correct: U(1) elements have |det| = 1, not det = 1

impl Copy for U1Algebra

impl StructuralPartialEq for U1Algebra