Editing Feynman diagram (section)

==== Sources ====
Correlation functions are the sum of the connected Feynman diagrams, but the formalism treats the connected and disconnected diagrams differently. Internal lines end on vertices, while external lines go off to insertions. Introducing ''sources'' unifies the formalism, by making new vertices where one line can end.

Sources are external fields, fields that contribute to the action, but are not dynamical variables. A scalar field source is another scalar field {{mvar|h}} that contributes a term to the (Lorentz) Lagrangian:

:<math> \int h(x) \phi(x)\, d^dx = \int h(k) \phi(k)\, d^dk \,</math>

In the Feynman expansion, this contributes H terms with one half-line ending on a vertex. Lines in a Feynman diagram can now end either on an {{mvar|X}} vertex, or on an {{mvar|H}} vertex, and only one line enters an {{mvar|H}} vertex. The Feynman rule for an {{mvar|H}} vertex is that a line from an {{mvar|H}} with momentum {{mvar|k}} gets a factor of {{math|''h''(''k'')}}.

The sum of the connected diagrams in the presence of sources includes a term for each connected diagram in the absence of sources, except now the diagrams can end on the source. Traditionally, a source is represented by a little "×" with one line extending out, exactly as an insertion.

:<math> \log\big(Z[h]\big) = \sum_{n,C} h(k_1) h(k_2) \cdots h(k_n) C(k_1,\cdots,k_n)\,</math>

where {{math|''C''(''k''<sub>1</sub>,...,''k<sub>n</sub>'')}} is the connected diagram with {{mvar|n}} external lines carrying momentum as indicated. The sum is over all connected diagrams, as before.

The field {{mvar|h}} is not dynamical, which means that there is no path integral over {{mvar|h}}: {{mvar|h}} is just a parameter in the Lagrangian, which varies from point to point. The path integral for the field is:

:<math> Z[h] = \int e^{iS + i\int h\phi}\, D\phi \,</math>

and it is a function of the values of {{mvar|h}} at every point. One way to interpret this expression is that it is taking the Fourier transform in field space. If there is a probability density on {{math|'''R'''<sup>''n''</sup>}}, the Fourier transform of the probability density is:

:<math> \int \rho(y) e^{i k y}\, d^n y = \left\langle e^{i k y} \right\rangle = \left\langle \prod_{i=1}^{n} e^{ih_i y_i}\right\rangle \,</math>

The Fourier transform is the expectation of an oscillatory exponential. The path integral in the presence of a source {{mvar|''h''(''x'')}} is:

:<math> Z[h] = \int e^{iS} e^{i\int_x h(x)\phi(x)}\, D\phi = \left\langle e^{i h \phi }\right\rangle</math>

which, on a lattice, is the product of an oscillatory exponential for each field value:

:<math> \left\langle \prod_x e^{i h_x \phi_x}\right\rangle </math>

The Fourier transform of a delta-function is a constant, which gives a formal expression for a delta function:

:<math> \delta(x-y) = \int e^{ik(x-y)}\, dk </math>

This tells you what a field delta function looks like in a path-integral. For two scalar fields {{mvar|φ}} and {{mvar|η}},

:<math> \delta(\phi - \eta) = \int e^{ i h(x)\big(\phi(x) -\eta(x)\big)\,d^dx}\, Dh\,, </math>

which integrates over the Fourier transform coordinate, over {{mvar|h}}. This expression is useful for formally changing field coordinates in the path integral, much as a delta function is used to change coordinates in an ordinary multi-dimensional integral.

The partition function is now a function of the field {{mvar|h}}, and the physical partition function is the value when {{mvar|h}} is the zero function:

The correlation functions are derivatives of the path integral with respect to the source:

:<math> \left\langle\phi(x)\right\rangle = \frac{1}{Z} \frac{\partial}{\partial h(x)} Z[h] = \frac{\partial}{\partial h(x)} \log\big(Z[h]\big)\,.</math>

In Euclidean space, source contributions to the action can still appear with a factor of {{mvar|i}}, so that they still do a Fourier transform.