Non-Local Interactions

Non-local vertices allow the coupling to depend on multiple spatial arguments. This is useful for modelling spatially extended interactions, kernels in Fourier space, or non-Markovian dynamics.

A Non-Local Kernel

Consider \(S_{\mathrm{int}} = \iint K_{ij}(x,x')\,\psi_i(x)\,\psi_j(x')\,\mathrm{d}x\,\mathrm{d}x'\):

from sft_wick import Field, Vertex, Action, compute_moment, reset_uid_counter

reset_uid_counter()
phi = Field('phi', 'physical', n_components=2)
psi = Field('psi', 'response', n_components=2)

v_nl = Vertex(fields=[psi, psi], coupling='K', local=False)
action = Action(vertices=[v_nl])

obs = [phi('a', 'x'), phi('b', 'y')]
result = compute_moment(obs, action, order=1)
print(result.order(1).to_latex())

Because the vertex is non-local:

• Each \(\psi\) in the vertex gets its own spatial variable (\(y_0\) and \(y_1\))

• The coupling symbol carries spatial arguments: \(K_{i_0 i_1}(y_0, y_1)\)

• Two spatial integrations appear: \(\int\mathrm{d}y_0\int\mathrm{d}y_1\)

Mixing Local and Non-Local Vertices

Actions can contain both local and non-local vertices:

reset_uid_counter()
v_local = Vertex(fields=[phi, phi, psi], coupling='F')
v_nonlocal = Vertex(fields=[psi, psi], coupling='K', local=False)

action = Action(vertices=[v_local, v_nonlocal])

obs = [phi('a', 'x'), phi('b', 'y')]
result = compute_moment(obs, action, order=1)

At first order, each vertex contributes separately (one diagram set from the local vertex, another from the non-local vertex). At second order and beyond, cross terms appear where different vertex types mix.

Comparing Local vs. Non-Local Output

The key differences in the symbolic output:

Feature	Difference
Spatial variables	Local: all fields share one variable. Non-local: each field gets its own.
Coupling symbol	Local: F_{ijk} (no spatial args). Non-local: K_{ij}(y_0, y_1).
Integrations	Local: single \(\int\mathrm{d}y_0\). Non-local: one per spatial variable.