Field Theory on Causal Sets

Causal Set Theory is not just about the discrete structure of spacetime itself; it is also a framework for doing Quantum Field Theory (QFT). To do this, we define Fields that live on the causal set.

In PyCauset, the geometry (the pycauset.CausalSet) and the matter (the pycauset.field.Field) are distinct objects. This separation allows you to study different fields (massless, massive, interacting) on the same underlying spacetime background.

The Scalar Field

The most fundamental field studied in Causal Set Theory is the Scalar Field.

The Retarded Propagator ($K_R$)

The Retarded Propagator $K_R$ is the inverse of the causal set d'Alembertian. In PyCauset, it is computed using the generalized formula:

\[ K_R = \Phi(I - b\Phi)^{-1} \]

where $\Phi = a C$ ($C$ is the Causal Matrix).

Parameters

The coefficients $a$ and $b$ are the bridge between the discrete matrix mathematics and the continuous physical parameters. They depend on:

Dimension ($d$): The dimension of the manifold (e.g., 2 or 4).
Density ($\rho$): The sprinkling density ($N/V$).
Mass ($m$): The mass of the field.

PyCauset automatically derives $a$ and $b$ for standard Minkowski spacetimes.

Dimension	$a$	$b$
2D	$1/2$	$-m^2/\rho$
4D	$\frac{\sqrt{\rho}}{2\pi\sqrt{6}}$	$-m^2/\rho$

Usage Guide

1. Define the Spacetime and Causal Set

First, generate your background geometry. Note that you must use density-based sprinkling (or provide the density manually) for the field coefficients to be calculated correctly.

import pycauset as pc

# 1. Define Spacetime (2D Minkowski)
st = pc.spacetime.MinkowskiDiamond(2)

# 2. Sprinkle Causal Set (Density is required!)
c = pc.CausalSet(density=1000, spacetime=st)

2. Define the Field

Create a pycauset.field.ScalarField instance attached to your causal set. This is where you specify the mass.

from pycauset.field import ScalarField

# Define a massive field (m=1.5)
field = ScalarField(c, mass=1.5)

3. Compute the Propagator

Call .propagator() to compute the $K_R$ matrix. This operation is computationally intensive ($O(N^2)$) and returns a pycauset.TriangularFloatMatrix.

# Compute K_R
K = field.propagator()

# K is a large matrix backed by disk storage
print(f"Propagator shape: {K.shape}")

Massless Limit

For a massless field ($m=0$), the parameter $b$ becomes 0. The formula simplifies to $K_R = aC$.

massless_field = ScalarField(c, mass=0.0)
K_0 = massless_field.propagator()

Advanced: Manual Coefficients

If you are working with a non-standard spacetime or want to experiment with different non-locality parameters, you can override $a$ and $b$ manually.

# Manually specifying coefficients (bypasses automatic derivation)
K_custom = field.propagator(a=0.5, b=-0.02)

The Pauli-Jordan Function ($i\Delta$)

The Pauli-Jordan function $\Delta$ is defined as the difference between the retarded and advanced propagators: $$ \Delta = K_R - K_A $$ Since $K_A = K_R^T$, this is equivalent to: $$ \Delta = K - K^T $$

In Quantum Field Theory, the operator of interest is often $i\Delta$. PyCauset provides a dedicated method to compute this efficiently.

# Compute i*Delta
Delta = field.pauli_jordan()

The result is an AntiSymmetricFloat64Matrix. To represent the factor of $i$ without storing complex numbers for every element (which would double the storage requirement), PyCauset stores the real values of $\Delta$ and sets the matrix's scalar property to 1j.

When you access elements or perform arithmetic with this matrix, the factor of $i$ is automatically applied.

Dimension	\(a\)	\(b\)
2D	\(1/2\)	\(-m^2/\rho\)
4D	\(\frac{\sqrt{\rho}}{2\pi\sqrt{6}}\)	\(-m^2/\rho\)