Quartic Interaction - Feynman Integral Quantization

Feynman Integral Quantization

The Feynman diagram expansion may be obtained also from the Feynman path integral formulation. The time ordered vacuum expectation values of polynomials in φ, known as the n-particle Green's functions, are constructed by integrating over all possible fields, normalized by the vacuum expectation value with no external fields,

All of these Green's functions may be obtained by expanding the exponential in J(x)φ(x) in the generating function

A Wick rotation may be applied to make time imaginary. Changing the signature to (++++) then gives a φ4 statistical mechanics integral over a 4-dimensional Euclidean space,

Normally, this is applied to the scattering of particles with fixed momenta, in which case, a Fourier transform is useful, giving instead

The standard trick to evaluate this functional integral is to write it as a product of exponential factors, schematically,

The second two exponential factors can be expanded as power series, and the combinatorics of this expansion can be represented graphically. The integral with λ = 0 can be treated as a product of infinitely many elementary Gaussian integrals, and the result may be expressed as a sum of Feynman diagrams, calculated using the following Feynman rules:

• Each field in the n-point Euclidean Green's function is represented by an external line (half-edge) in the graph, and associated with momentum p.
• Each vertex is represented by a factor -λ.
• At a given order λk, all diagrams with n external lines and k vertices are constructed such that the momenta flowing into each vertex is zero. Each internal line is represented by a factor 1/(q2 + m2), where q is the momentum flowing through that line.
• Any unconstrained momenta are integrated over all values.
• The result is divided by a symmetry factor, which is the number of ways the lines and vertices of the graph can be rearranged without changing its connectivity.
• Do not include graphs containing "vacuum bubbles", connected subgraphs with no external lines.

The last rule takes into account the effect of dividing by . The Minkowski-space Feynman rules are similar, except that each vertex is represented by, while each internal line is represented by a factor i/(q2-m2 + i ε), where the ε term represents the small Wick rotation needed to make the Minkowski-space Gaussian integral converge.