pycauset.field.ScalarField

class ScalarField(Field)

Represents a massive scalar field defined on a Causal Set.

This class implements the generalized Retarded Propagator $K_R$ for a scalar field $\phi$ satisfying the Klein-Gordon equation on the discrete causal set structure.

Constructor

ScalarField(causet: CausalSet, mass: float = 0.0)

causet (CausalSet): The causal set on which the field lives. Must have density information available.
mass (float): The mass of the field ($m$). Defaults to 0.0 (massless).

Methods

propagator

def propagator(self, a: float = None, b: float = None) -> TriangularFloatMatrix

Computes the Retarded Propagator $K_R$.

The propagator is defined as: $$ K_R = \Phi(I - b\Phi)^{-1} $$ where $\Phi = a C$.

Automatic Coefficient Derivation: If a and b are not provided, they are automatically calculated based on the spacetime dimension ($d$), the sprinkling density ($\rho$), and the field mass ($m$).

For $d=2$ (Minkowski): $$ a = 1/2, \quad b = -m^2/\rho $$
For $d=4$ (Minkowski): $$ a = \frac{\sqrt{\rho}}{2\pi\sqrt{6}}, \quad b = -m^2/\rho $$

Parameters:

a (float, optional): Manual override for coefficient $a$.
b (float, optional): Manual override for coefficient $b$.

Returns:

TriangularFloatMatrix: The computed propagator matrix.

Raises:

ValueError: If the causal set density is unknown and coefficients are not manually provided.
NotImplementedError: If the spacetime dimension/type is not supported for automatic derivation.

pauli_jordan

def pauli_jordan(self) -> AntiSymmetricFloat64Matrix

Computes the Pauli-Jordan function $i\Delta$, where $\Delta = K - K^T$.

This function returns an AntiSymmetricFloat64Matrix representing the operator $i\Delta$. The matrix stores the values of $\Delta$, but its scalar property is set to 1j (the imaginary unit), so that accessing elements or performing arithmetic operations treats it as $i\Delta$.

Returns:

AntiSymmetricFloat64Matrix: The matrix $i\Delta$.