Struct ConnesDistance 

pub struct ConnesDistance {
    pub spectral_triple: String,
}

The Connes spectral distance on a noncommutative space.

Given a spectral triple (A, H, D), the Connes distance between two states φ, ψ on A is: d(φ, ψ) = sup { |φ(a) − ψ(a)| : a ∈ A, ‖[D, a]‖ ≤ 1 }. This recovers the geodesic distance on a Riemannian manifold.

Fields

spectral_triple: String

Description of the spectral triple used to define the distance.

Implementations

impl ConnesDistance

pub fn new(spectral_triple: impl Into<String>) -> Self

Constructs the Connes distance associated with the given spectral triple.

pub fn metric_dimension(&self) -> f64

Returns the metric dimension of the noncommutative space.

The metric dimension p is determined by the growth of the eigenvalues of the Dirac operator D: it is the infimum of {s : Tr(|D|^{-s}) < ∞}.

pub fn is_metric(&self) -> bool

Checks that the Connes distance is a genuine metric (positivity, symmetry, triangle inequality).

impl ConnesDistance

pub fn compute_lower_bound( &self, phi_values: &[f64], psi_values: &[f64], commutator_norms: &[f64], ) -> f64

Approximate the Connes distance between two states given as vectors of expectation values ⟨a_k⟩_φ and ⟨a_k⟩_ψ for generators {a_k}, and the corresponding commutator norms ‖[D, a_k]‖.

Uses the formula d(φ, ψ) = sup_k |φ(a_k) - ψ(a_k)| / ‖[D, a_k]‖.

pub fn satisfies_triangle_inequality( &self, d_phi_psi: f64, d_psi_chi: f64, d_phi_chi: f64, ) -> bool

Check the triangle inequality d(φ, χ) ≤ d(φ, ψ) + d(ψ, χ) for sampled distances.

Trait Implementations

impl Clone for ConnesDistance

fn clone(&self) -> ConnesDistance

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for ConnesDistance

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

Formats the value using the given formatter.