Struct CausalMultiField 

pub struct CausalMultiField<T> { /* private fields */ }

A field of multivectors stored as tensors.

CausalMultiField stores multivector data in Matrix Isomorphism representation using CausalTensor for efficient computation.

Type Parameters

• T - The scalar type (typically f32 or f64).

Memory Layout

Shape: [Nx, Ny, Nz, Matrix_Dim, Matrix_Dim]

Implementations

impl<T> CausalMultiField<T>

where T: Field + Copy + Default + PartialOrd,

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

Scales the field by a scalar: result = scalar * self.

Arguments

• scalar - The scalar value to multiply by

impl<T> CausalMultiField<T>

where T: Field + RealField + Copy + Default + PartialOrd,

pub fn normalize(&self) -> Self

Normalizes the field: result = self / ||self||.

Returns the field scaled to unit magnitude.

pub fn squared_magnitude(&self) -> T

Computes the squared magnitude of the field.

Uses the L2 norm of the matrix representation.

impl<T> CausalMultiField<T>

where T: Field + Copy + Default + PartialOrd + Neg<Output = T> + 'static,

pub fn reversion(&self) -> Self

Computes the reversion (reversal) of the field.

The reversion is computed by extracting multivectors, applying the reversion operation, and reconstructing.

pub fn inverse(&self) -> Self

where T: RealField,

Computes the multiplicative inverse of the field.

Uses matrix inverse for each cell.

impl<T> CausalMultiField<T>

where T: Field + Ring + Copy + Default + PartialOrd, CausalTensor<T>: BatchedMatMul<T>,

pub fn commutator_lie(&self, rhs: &Self) -> Self

Computes the Lie commutator: [A, B] = AB - BA.

The Lie bracket measures the non-commutativity of the geometric product.

pub fn commutator_geometric(&self, rhs: &Self) -> Self

Computes the geometric commutator: (AB - BA) / 2.

Equivalent to the Lie commutator scaled by 1/2.

impl<T> CausalMultiField<T>

pub fn metric(&self) -> Metric

Returns the metric signature of the algebra.

pub fn dx(&self) -> &[T; 3]

Returns the grid spacing.

pub fn shape(&self) -> &[usize; 3]

Returns the grid shape [Nx, Ny, Nz].

pub fn num_cells(&self) -> usize

Returns the total number of grid cells.

pub fn matrix_dim(&self) -> usize

Returns the matrix dimension for the algebra.

For Cl(p,q,r), the matrix dimension is 2^⌈N/2⌉ where N = p + q + r.

pub fn data(&self) -> &CausalTensor<T>

Returns a reference to the underlying tensor.

impl<T> CausalMultiField<T>

where T: Field + Copy + Default + PartialOrd,

pub fn zeros(shape: [usize; 3], metric: Metric, dx: [T; 3]) -> Self

Creates a field filled with zero multivectors.

Arguments

• shape - Grid dimensions [Nx, Ny, Nz]
• metric - The metric signature of the algebra
• dx - Grid spacing [dx, dy, dz]

pub fn ones(shape: [usize; 3], metric: Metric, dx: [T; 3]) -> Self

Creates a field filled with identity matrices (scalar 1).

Each cell contains the identity element of the algebra.

pub fn from_coefficients( mvs: &[CausalMultiVector<T>], shape: [usize; 3], dx: [T; 3], ) -> Self

where T: Neg<Output = T>,

Creates a field from a collection of CausalMultiVectors.

Arguments

• mvs - Flat array of multivectors in row-major order
• shape - Grid dimensions [Nx, Ny, Nz]
• dx - Grid spacing [dx, dy, dz]

pub fn to_coefficients(&self) -> Vec<CausalMultiVector<T>>

where T: Neg<Output = T>,

Downloads the field to a collection of CausalMultiVectors.

Converts from Matrix Representation back to coefficient form using trace projection.

pub fn compute_matrix_dim(n: usize) -> usize

Computes the matrix dimension for a given algebra dimension.

For Cl(p,q,r), the matrix dimension is 2^⌈N/2⌉ where N = p + q + r.

impl<T> CausalMultiField<T>

where T: Field + Copy + Default + PartialOrd,

pub fn curl(&self) -> Self

where T: Ring + Neg<Output = T>,

Computes the curl: ∇ × F.

Returns the grade-2 component of the gradient ∇F.

pub fn divergence(&self) -> Self

where T: Ring + Neg<Output = T>,

Computes the divergence: ∇ · F.

Returns the scalar part (Grade 0) of the gradient ∇F.

pub fn gradient(&self) -> Self

where T: Ring + Neg<Output = T>,

Computes the gradient: ∇F.

Returns the full geometric derivative field (all grades).

pub fn partial_derivative(&self, axis: Axis) -> CausalTensor<T>

where T: Ring,

Computes the partial derivative along a given axis.

Uses central difference: ∂F/∂x ≈ (F[x+1] - F[x-1]) / (2dx)

This implementation works on the flat data vector to avoid needing range-based slice operations.

impl<T> CausalMultiField<T>

where T: Field + Copy + Default + PartialOrd,

pub fn grade_project(&self, k: usize) -> Self

where T: Neg<Output = T>,

Projects the field onto grade k: ⟨F⟩_k.

Arguments

• k - The grade to project onto (0=scalar, 1=vector, 2=bivector, etc.)

pub fn scalar_part(&self) -> Self

where T: Neg<Output = T>,

Extracts the scalar part (grade 0): ⟨F⟩₀.

pub fn vector_part(&self) -> Self

where T: Neg<Output = T>,

Extracts the vector part (grade 1): ⟨F⟩₁.

pub fn bivector_part(&self) -> Self

where T: Neg<Output = T>,

Extracts the bivector part (grade 2): ⟨F⟩₂.

pub fn trivector_part(&self) -> Self

where T: Neg<Output = T>,

Extracts the trivector part (grade 3): ⟨F⟩₃.

pub fn pseudoscalar_part(&self) -> Self

where T: Neg<Output = T>,

Extracts the pseudoscalar part (highest grade).

impl<T> CausalMultiField<T>

where T: Field + Ring + Copy + Default + PartialOrd, CausalTensor<T>: BatchedMatMul<T>,

pub fn inner_product(&self, rhs: &Self) -> Self

where T: Neg<Output = T>,

Computes the inner product (grade-0 projection of geometric product).

pub fn outer_product(&self, rhs: &Self) -> Self

where T: Neg<Output = T> + AddAssign + SubAssign,

Computes the outer product (wedge product).

Implemented via coefficient extraction and basis blade logic to ensure correctness for mixed-grade multivectors. (AB - BA)/2 is only valid for vectors.

pub fn cross(&self, rhs: &Self) -> Self

where T: AddAssign + SubAssign + Neg<Output = T>,

Computes the cross product via Hodge dual of wedge.

A × B = -I(A ∧ B) where I is the pseudoscalar.

pub fn hodge_dual(&self) -> Self

where T: AddAssign + SubAssign + Neg<Output = T>,

Applies the Hodge dual operation.

A* = A · I⁻¹ where I is the pseudoscalar.

Trait Implementations

impl<T> Add for &CausalMultiField<T>

where T: Field + Copy + Default + PartialOrd,

type Output = CausalMultiField<T>

The resulting type after applying the + operator.

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

Performs the + operation.

impl<T> Add for CausalMultiField<T>

where T: Field + Copy + Default + PartialOrd,

type Output = CausalMultiField<T>

The resulting type after applying the + operator.

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

Performs the + operation.

impl<T: Clone> Clone for CausalMultiField<T>

fn clone(&self) -> CausalMultiField<T>

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl<T: Debug> Debug for CausalMultiField<T>

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

Formats the value using the given formatter.

impl<T> Mul<&CausalMultiField<T>> for CausalMultiField<T>

where T: Field + Ring + Copy + Default + PartialOrd, CausalTensor<T>: BatchedMatMul<T>,

type Output = CausalMultiField<T>

The resulting type after applying the * operator.

fn mul(self, rhs: &CausalMultiField<T>) -> Self::Output

Performs the * operation.

impl<T> Mul for &CausalMultiField<T>

where T: Field + Ring + Copy + Default + PartialOrd, CausalTensor<T>: BatchedMatMul<T>,

type Output = CausalMultiField<T>

The resulting type after applying the * operator.

fn mul(self, rhs: Self) -> Self::Output

Performs the * operation.

impl<T> Mul for CausalMultiField<T>

where T: Field + Ring + Copy + Default + PartialOrd, CausalTensor<T>: BatchedMatMul<T>,

type Output = CausalMultiField<T>

The resulting type after applying the * operator.

fn mul(self, rhs: Self) -> Self::Output

Performs the * operation.

impl<T> Neg for CausalMultiField<T>

where T: Field + Copy + Default + PartialOrd + Neg<Output = T>,

type Output = CausalMultiField<T>

The resulting type after applying the - operator.

fn neg(self) -> Self::Output

Performs the unary - operation.

impl<T> Sub for &CausalMultiField<T>

where T: Field + Copy + Default + PartialOrd,

type Output = CausalMultiField<T>

The resulting type after applying the - operator.

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

Performs the - operation.

impl<T> Sub for CausalMultiField<T>

where T: Field + Copy + Default + PartialOrd,

type Output = CausalMultiField<T>

The resulting type after applying the - operator.

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

Performs the - operation.

impl<T> Zero for CausalMultiField<T>

where T: Field + Copy + Default + PartialOrd + Zero + Neg<Output = T>,

fn zero() -> Self

Returns the additive identity element of Self, 0.

fn is_zero(&self) -> bool

Returns true if self is equal to the additive identity.

fn set_zero(&mut self)

Sets self to the additive identity element of Self, 0.