Struct HopfState 

pub struct HopfState<R: RealField>(/* 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<R: RealField> HopfState<R>

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

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

pub fn from_spinor(alpha: Complex<R>, beta: Complex<R>) -> 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 as_inner(&self) -> &CausalMultiVector<R>

Access underlying algebra

impl<R: RealField + FromPrimitive> HopfState<R>

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

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: R) -> 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).

Trait Implementations

impl<R: Clone + RealField> Clone for HopfState<R>

fn clone(&self) -> HopfState<R>

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl<R: Debug + RealField> Debug for HopfState<R>

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

Formats the value using the given formatter.

impl<R: RealField + Debug> Display for HopfState<R>

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

Formats the value using the given formatter.

impl<R: PartialEq + RealField> PartialEq for HopfState<R>

fn eq(&self, other: &HopfState<R>) -> 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<R: RealField> StructuralPartialEq for HopfState<R>

impl<R: RealField> TryFrom<&HilbertState<R>> for HopfState<R>

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

type Error = CausalMultiVectorError

The type returned in the event of a conversion error.

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

Performs the conversion.

impl<R: RealField> TryFrom<HopfState<R>> for HilbertState<R>

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

type Error = CausalMultiVectorError

The type returned in the event of a conversion error.

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

Performs the conversion.