Struct HopfState 

pub struct HopfState(/* private fields */);

A point on the 3-Sphere ($S^3$), representing a unit spinor or rotor in 3D Euclidean space.

The Hopf Fibration maps this state to a point on the 2-Sphere ($S^2$) via the projection $h(R) = R \sigma_3 \tilde{R}$.

This structure captures both the Direction (the point on $S^2$) and the Phase/Twist (the position on the $S^1$ fiber).

Applications

• Quantum Mechanics: Represents a Qubit state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$. The projection is the Bloch Vector.
• Robotics: Represents a rotation without Gimbal Lock (Quaternion). The fiber is the “twist” redundancy.
• Electromagnetism: Represents Hopfion field configurations.

Implementations

impl HopfState

pub fn new(data: Vec<f64>) -> Result<Self, CausalMultiVectorError>

Creates a new HopfState from raw coefficients. Enforces Euclidean(3) metric and Normalization.

pub fn from_spinor(alpha: Complex<f64>, beta: Complex<f64>) -> Self

Constructs a HopfState from two Complex numbers (Spinor formalism).

Maps $(\alpha, \beta) \in \mathbb{C}^2$ to the 3-Sphere. $|\alpha|^2 + |\beta|^2 = 1$.

This connects Standard QM notation to Geometric Algebra.

pub fn project(&self) -> CausalMultiVector<f64>

The Projection Map $h: S^3 \to S^2$.

Returns the vector on the 2-Sphere (The “Shadow” or “Bloch Vector”). $v = R \sigma_3 \tilde{R}$.

pub fn fiber_shift(&self, angle_rad: f64) -> Self

Traverses the Fiber ($S^1$).

Rotates the state by phase radians without changing the projection on $S^2$. This corresponds to the global phase $e^{i\theta}$ in QM or the “Twist” in robotics.

$R’ = R e^{-\frac{\theta}{2} \mathbf{I}}$ (Where I is the generator of rotation around the pole, typically e12 for Z-axis).

pub fn as_inner(&self) -> &CausalMultiVector<f64>

Access underlying algebra

Trait Implementations

impl Clone for HopfState

fn clone(&self) -> HopfState

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for HopfState

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

Formats the value using the given formatter.

impl Display for HopfState

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

Formats the value using the given formatter.

impl PartialEq for HopfState

fn eq(&self, other: &HopfState) -> 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 TryFrom<&HilbertState> for HopfState

Conversion: Quantum State (Spinor) -> Topological Rotor (Hopf).

Maps a 2-level Quantum System (Qubit) into the geometry of the 3-Sphere. $\psi = \alpha|0\rangle + \beta|1\rangle \to R \in S^3$.

This allows you to calculate the “Hopf Invariant” or “Berry Phase” of a quantum state.

type Error = CausalMultiVectorError

The type returned in the event of a conversion error.

fn try_from(quantum_state: &HilbertState) -> Result<Self, Self::Error>

Performs the conversion.

impl TryFrom<HopfState> for HilbertState

Conversion: Topological Rotor (Hopf) -> Quantum State (Spinor).

Maps a geometric orientation back into Quantum Hilbert Space. Useful for initializing a Qubit based on a geometric rotation.

type Error = CausalMultiVectorError

The type returned in the event of a conversion error.

fn try_from(hopf: HopfState) -> Result<Self, Self::Error>

Performs the conversion.

impl StructuralPartialEq for HopfState