Multi-Component Fields¶

When fields carry component indices, propagators and couplings gain index structure, and the output includes summations over internal indices.

Four-Point Function with Component Indices¶

\(\langle\phi_a(x)\,\phi_b(y)\,\phi_c(z)\,\phi_d(w)\rangle_{S_0}\):

from sft_wick import Field, Action, compute_moment, reset_uid_counter

reset_uid_counter()
phi = Field('phi', 'physical', n_components=3)

obs = [phi('a', 'x'), phi('b', 'y'), phi('c', 'z'), phi('d', 'w')]
result = compute_moment(obs, Action(vertices=[]), order=0)
print(result.order(0).to_latex())

C_{ab}(x, y) C_{cd}(z, w) + C_{ac}(x, z) C_{bd}(y, w) + C_{ad}(x, w) C_{bc}(y, z)

The three pairings now carry component indices matching the external operators.

Mixed Physical and Response Fields¶

\(\langle\psi_a(x)\,\phi_b(x)\,\phi_c(x)\,\phi_d(x)\rangle_{S_0}\):

reset_uid_counter()
phi = Field('phi', 'physical', n_components=3)
psi = Field('psi', 'response', n_components=3)

obs = [psi('a', 'x'), phi('b', 'x'), phi('c', 'x'), phi('d', 'x')]
result = compute_moment(obs, Action(vertices=[]), order=0)
print(result.order(0).to_latex())

Each \(R\) propagator now has the convention that the physical-field index is on the left: \(R_{ba}(x,x)\) means \(\langle\phi_b(x)\,\psi_a(x)\rangle_{S_0}\).

First-Order with Coupling Tensor¶

With a vertex \(\int F_{ijk}\,\phi_i\,\phi_j\,\psi_k\,\mathrm{d}z\):

reset_uid_counter()
v = Vertex(fields=[phi, phi, psi], coupling='F')
action = Action(vertices=[v])

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

The output includes:

Summation \(\sum_{i_0}\sum_{i_1}\sum_{i_2}\) over internal component indices
The coupling tensor \(F_{i_0 i_1 i_2}\)
Integration \(\int\mathrm{d}y_0\) over the vertex position
Products of \(C\) and \(R\) propagators with the appropriate index combinations