# 117 the φ x term already saturates the equality so

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Chapter 10 / Exercise 38
Discrete Mathematics With Applications
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(117), the φ ( x ) term already saturates the equality, so the remaining terms must add to zero. Thus integraldisplay d 4 z bracketleftBig ε ( z ) bracketrightBig integraldisplay D φe i integraltext d 4 y parenleftBig 1 2 φ square y φ parenrightBig bracketleftBig φ ( x ) square z φ ( z ) + 4 ( z x ) bracketrightBig = 0 (120) Since the path integral does not depend on z except through the field insertion, the square z can be pulled outside of integral. For the equality to hold for any ε ( z ) , we must have ( i ) 2 parenleftbigg square z 2 Z [ J ] ∂J ( z ) ∂J ( x ) parenrightbiggvextendsingle vextendsingle vextendsingle vextendsingle J =0 = 4 ( z x ) Z  (121) In terms of correlation functions, this is square z ( φ ˆ ( z ) φ ˆ ( x ) ) = 4 ( z x ) (122) which is of course nothing but the Green’s function equation for the Feynman propagator. It is also the Schwinger-Dyson equation for the two-point function in a free scalar field theory. For an interacting theory, let us add a potential so that L = 1 2 φ square φ + L int [ φ ] . Then the classical equations of motion are square φ = L int [ φ ] . In the path integral, the addition of the potential contributes a term i integraltext d 4 z ε ( z ) L int [ φ ( z )] to the {} in Eq.(119) and Eq.(120) gets modified to integraldisplay d 4 z bracketleftBig ε ( z ) bracketrightBig braceleftbigg square z integraldisplay D φ [ e iS φ ( z ) φ ( x )] integraldisplay D φe iS φ ( x ) L int [ φ ( z )] + 4 ( z x ) integraldisplay D φe iS bracerightbigg = 0 (123) This can be written as a statement about correlation functions in the canonical picture: square z ( φ ˆ ( z ) φ ˆ ( x ) ) = (L int bracketleftbig φ ˆ ( z ) bracketrightbig φ ˆ ( x ) ) − 4 ( z x ) (124) which is the Schwinger-Dyson equation for the two-point function in the presence of interac- tions. If we have more field insertions, the Schwinger-Dyson equations add contact interactions con- tracting the field on which the operator acts with all the other fields in the correlator. For example, with three fields square x ( φ ˆ ( x ) φ ˆ ( y ) φ ˆ ( z ) ) = (L int bracketleftbig φ ˆ ( x ) bracketrightbig φ ˆ ( y ) φ ˆ ( z ) ) − 4 ( x z ) ( φ ˆ ( y ) ) − 4 ( x y ) ( φ ˆ ( z ) ) (125) and so on. In this way, the complete set of Schwinger-Dyson equations can be derived. Similar equations hold for theories with spinors or gauge bosons. For example, write the QED Lagrangian as L = 1 2 A µ square µν A ν + ψ ¯ ( i∂ m ) ψ eA µ ψ ¯ γ µ ψ (126) with square µν = square g µν (1 1 ξ ) µ ν in covariant gauges. The classical equations of motion for A ν are square µν A ν = ej µ = ¯ γ µ ψ . By varying A µ ( x ) A µ ( x ) + ε µ ( x ) and considering the correlation func- tion ( A α ψ ¯ ψ ) we would find square µν x ( A ν ( x ) A α ( y ) ψ ¯ ( z 1 ) ψ ( z 2 ) ) = e ( j µ ( x ) A α ( y ) ψ ¯ ( z 1 ) ψ ( z 2 ) ) − 4 ( x y ) δ µ α ( ψ ¯ ( z 1 ) ψ ( z 2 ) ) (127) 18 Section 7
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Chapter 10 / Exercise 38
Discrete Mathematics With Applications
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Another Schwinger-Dyson equation, for QED, is ( κρ µ µ + κρ ) ( ψ ¯ κ ( x ) ψ α ( y ) ψ ¯ β ( z ) ψ γ ( w ) ) = e ( ψ ¯ κ ( x ) A κρ ψ α ( y ) ψ ¯ β ( z ) ψ γ ( w ) ) γρ δ 4 ( x w ) ( ψ α ( y ) ψ ¯ β ( z ) ) − αρ δ 4 ( x
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