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12. 191) where F[ϕ] is an arbitrary functional, g is some new field variable and ϕ is a solution of the equation f (ϕ(x)) − g(x) = 0. 191) contains the δ-functional λ(x) exp i d 4 x λ(x)( f (ϕ(x)) − g(x)) = δ( f (ϕ(x)) − g(x)). 193) Hint. 194) 48 Quantum field theory: the path-integral approach where d 4 y c1 (x, y)ϕ(y) + f (ϕ(x)) = d 4 y d 4 y c2 (x, y, y )ϕ(y)ϕ(y ) + · · · . 194) equal to unity; the consideration is trivially generalized for an arbitrary value of the coefficient. The determinant in the path integral can now be understood as the power expansion det δ f (ϕ) δϕ = det 1 + = exp δf δϕ d4x = exp Tr ln 1 + δ f (x) δϕ(y) + x=y 1 2 δf δϕ d4x d4 y δ f (x) δ f (y) + ··· .

5 (we have dropped all J -independent terms because they vanish under the action of the functional derivatives and hence do not contribute to the Green functions). For systematization and further use, let us recall some nomenclature from graph (diagram) theory: • • • Diagrams containing pieces not connected by lines are called disconnected diagrams. If any vertex of a diagram can be reached from any other vertex by moving along the lines of the graph, the diagram is said to be connected. 5. Connected diagrams contributing to the second-order contributions to the Green-function generating functional in the ϕ 4 -model.

The generating functional [η, ¯ η, J ] contains several fields and sources and therefore we need different graphical elements for them. 2. 163) gives a generating functional in the first (1) [η, second (2) [η, ¯ η, J ] order of perturbation theory. 2. Correspondence rules for the Yukawa model. Physical quantity Mathematical expression Diagram element Propagators Dc (x − y) Sc (x − y) Interaction vertex External sources g J (x) η(x) ¯ η(x) •∼ •=⇐ •⇒= + − ∼Ö Ö Ö Ö Ö  (2)  [η, ¯ η, J ] =  ¹ − =⇐Ö + 2 Ö=⇐ Ö +2 =⇐Ö −2 =⇐Ö Ö ∼ Ö=⇐ Ö Ö=⇐  Ö    +  ∼Ö   ¹ − =⇐Ö Ö⇐ = 2          ¯ η, 0 [η, J ].