Questions: Functional Methods and Generating Functionals

Question 1

The generating functional Z[J] = integral D[phi] e^{i(S[phi] + integral J phi)} is the path integral with an external source J. The n-point correlation function <0|T{phi(x1)...phi(xn)}|0> is obtained from Z[J] by taking functional derivatives with respect to J and setting J = 0. Why is this a useful formulation?

• A. Because the path integral is easier to evaluate than operator products
• B. Because Z[J] contains ALL correlation functions simultaneously — once Z[J] is known (even approximately), every Green's function, scattering amplitude, and physical observable can be extracted by differentiation, making it the master object of the theory
• C. Because Z[J] is always exactly solvable
• D. Because external sources J correspond to measurable physical fields

Answer: (B)

Z[J] is a generating function in the same sense as in probability theory: all moments of the field (all n-point functions) are encoded as coefficients of the Taylor expansion of Z[J] in powers of J. The n-th functional derivative (delta/delta J(x))^n Z[J] evaluated at J = 0 gives the n-point function. This is computationally powerful because general properties of Z[J] (symmetries, Ward identities, saddle-point approximations) translate into statements about all correlation functions simultaneously.

Question 2

W[J] = -i ln Z[J] generates connected Green's functions only — diagrams where all external points are linked by propagators. Why is it useful to separate connected from disconnected diagrams?

• A. Because disconnected diagrams are always zero
• B. Because disconnected diagrams factorize into products of lower-point connected functions — they contain no new information beyond what is already in the connected functions, and the connected functions are what enter into the S-matrix via the LSZ formula
• C. Because disconnected diagrams violate momentum conservation
• D. Because only connected diagrams are Lorentz invariant

Answer: (B)

A disconnected four-point function <phi phi phi phi>_disconnected is just a product of two-point functions <phi phi><phi phi>, which you already know. The connected part contains the genuinely new four-point interaction information. The LSZ reduction formula extracts S-matrix elements from connected, amputated Green's functions. The logarithm in W = -i ln Z is the functional analog of the cumulant expansion in probability theory: it extracts the connected (irreducible) part. This is also related to the linked cluster theorem: the S-matrix exponent is a sum of connected diagrams.

Question 3

The effective action Gamma[phi_cl] is the Legendre transform of W[J]. Its significance is that Gamma[phi_cl] is the quantum generalization of the classical action — the tree-level approximation of Gamma gives the full quantum result.

• T. True
• F. False

Answer: True

This is the remarkable property of the effective action. If you compute Gamma[phi_cl] exactly and use it at tree level (no loops), you reproduce the full quantum theory including all loop corrections. This is because Gamma generates one-particle-irreducible (1PI) vertices, and the full Green's functions are obtained by connecting these 1PI vertices with exact propagators — a tree-level exercise. In practice, Gamma is computed perturbatively (the loop expansion of Gamma), but the conceptual point is that Gamma packages all quantum effects into an effective classical action.

Question 4

Explain what a one-particle-irreducible (1PI) diagram is and why the effective action generates exactly these objects.

Model answer: A 1PI diagram is a connected diagram that cannot be separated into two disconnected pieces by cutting a single internal line. Examples: the one-loop self-energy (a circle with two external legs) is 1PI; a diagram that is two self-energies connected by a single propagator is NOT 1PI (cutting the connecting propagator disconnects it). The effective action Gamma[phi_cl] generates 1PI diagrams because the Legendre transform from W[J] to Gamma[phi_cl] algebraically removes all 'tree-level gluings' of subdiagrams. What remains are the irreducible building blocks — the 1PI vertices — from which all Green's functions can be reconstructed by tree-level Feynman rules using the exact (dressed) propagator.

The decomposition into 1PI building blocks is practically important because the 1PI vertices are what get renormalized. Each counterterm in the Lagrangian corresponds to a specific 1PI function (self-energy, vertex correction, etc.). The effective action Gamma is the natural object for studying renormalization and symmetry breaking.