sft-wick: Feynman-Diagram Expansion for Stochastic Field Theories

sft-wick automates perturbative calculations for stochastic (partial) differential equations in the Martin–Siggia–Rose (MSR) response-field formalism. Starting from a Langevin-type field equation — a deterministic drift plus noise that may be non-Gaussian and spatially correlated — it builds the interaction action, applies Wick’s theorem to expand arbitrary field moments order by order, and writes every term using just two two-point propagators: the correlation function \(C = \langle\phi\phi\rangle\) and the response (Green’s) function \(R = \langle\phi\psi\rangle\).

\[\langle \mathcal{O} \rangle_S = \sum_{n=0}^{N} \frac{(-1)^n}{n!}\, \langle \mathcal{O}\, S_{\mathrm{int}}^{\,n} \rangle_{S_0}\]

Key Features

  • Three-layer API — pick your abstraction: L0 symbolic / L1 workflow (System, Expansion, Propagators, SweepResult) / L2 YAML + CLI (sft-wick run config.yaml).

  • No SymPy dependency — custom lightweight expression tree with exact rational arithmetic via fractions.Fraction.

  • MSR-optimised contraction — exploits the constraint \(\langle\psi\,\psi\rangle = 0\) to skip vanishing pairings entirely.

  • Feynman diagram generation — networkx-based graph representation with matplotlib rendering (correlation \(C\) as solid blue lines, response \(R\) as dashed red arrows).

  • LaTeX output — every expression renders to publication-ready LaTeX, with configurable propagator names.

  • Numerical evaluation — QMC with causal simplex mapping and O(1/N) convergence; separable / rotational / general spatial homogeneity modes; optional closed-form C bypass.

  • Time-dependent linear operator and spacetime-dependent non-local couplings both supported end-to-end through the L1 API.

Quick Start (L2 — config file)

The recommended entry point for any new analysis. Write a YAML config once, run it from the shell, iterate with --override or edits. The CLI registers automatically on install.

# demo1_config.yaml
system:
  field: {name: phi, n_components: 2}
  linear: {type: diagonal, gamma: [1.0, 1.0]}
  vertices:
    - name: F
      coupling:                              # bare F; MSR factor
        - [[0.0, 0.0], [0.0, 1.0]]           # applied automatically
        - [[0.0, 0.5], [0.5, 0.0]]
  noise:
    kappa2:
      type: separable_translation
      temporal: {type: exponential, lam: 0.05, sigma_t: 0.3}
      spatial:  {type: exponential, sigma_x: 1.0}

expand:
  observable: ["phi_a(x)", "phi_b(y)"]
  orders: [0, 2, 4]

propagators: {t_max: 15.0, n_grid_t: 60}

sweep:
  positions_grid: {x: [0.0], y: [0.0, 0.5, 1.0, 2.5]}
  t_final_grid: [1.0, 15.0]
  component_pairs: [[0, 0], [1, 1]]
  n_samples: 8192
  seed: 42

sft-wick run demo1_config.yaml
sft-wick run demo1_config.yaml --override sweep.seed=7
sft-wick run demo1_config.yaml --dry-run            # validate only

See Workflow API (L1 + L2) for the complete YAML schema and examples/demo1_config.yaml / examples/demo2_config.yaml for non-local vertices, closed-form \(C\) hooks, and dynamic couplings.

Quick Start (L1 — Python, for programmatic use)

The same workflow expressed directly in Python — use this when embedding the pipeline inside a larger script:

import numpy as np
import sft_wick as sw

F = np.zeros((2, 2, 2))
F[0, 1, 1] = 1.0
F[1, 0, 1] = F[1, 1, 0] = 0.5

system = sw.System(
    field=sw.FieldSpec("phi", n_components=2),
    linear=sw.DiagonalA(gamma=[1.0, 1.0]),
    vertices=[sw.LocalVertex("F", coupling=F)],   # bare F
    noise=sw.GaussianNoise(kappa2=sw.SeparableTranslation(
        temporal=sw.ExponentialTemporal(lam=0.05, sigma_t=0.3),
        spatial=sw.ExponentialSpatial(sigma_x=1.0),
    )),
)

expansion = system.expand(("phi_a(x)", "phi_b(y)"),
                          orders=[0, 2, 4])
props = system.propagators(t_max=15.0, n_grid_t=60)

sweep = expansion.sweep(
    props,
    positions_grid={"x": [0.0], "y": [0.0, 0.5, 1.0, 2.5]},
    t_final_grid=[1.0, 15.0],
    component_pairs=[(0, 0), (1, 1)],
)

print(sweep.totals())

For the L0 symbolic API (fine-grained control over individual pairings, canonical forms, custom simplification), see User Guide.

Contents

Indices and Tables