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klein_gordon_kernel

deep_causality_physics

Function klein_gordon_kernel 

Source 
pub fn klein_gordon_kernel(
    psi_manifold: &Manifold<f64, f64>,
    mass: f64,
) -> Result<CausalTensor<f64>, PhysicsError>
Expand description

Calculates the Klein-Gordon operator action: $(\Delta + m^2)\psi$.

Computes the action of the Klein-Gordon operator on a scalar field $\psi$. The result is the “source” or “force” required to maintain that field configuration, or zero if the field satisfies the free equation.

§Sign Convention

This implementation uses the Euclidean/Riemannian convention where the Laplacian $\Delta$ is positive-definite (akin to $k^2$). The equation is $(\Delta + m^2)\psi = 0$.

If Manifold contains a Lorentzian metric, the laplacian method (defined via Hodge stars) naturally becomes the d’Alembertian $\square$ (Laplace-Beltrami). In that case, the equation form $(\square + m^2)\psi = 0$ is preserved if the signature (+—) is handled by the metric.

§Arguments

  • psi_manifold - Manifold containing the scalar field $\psi$ (0-form).
  • mass - Mass $m$.

§Returns

  • Result<CausalTensor<f64>, PhysicsError> - The result of applying the operator $O(\psi)$.