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# 582-chapter11 - 11 Perturbation Theory and Feynam Diagrams...

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11 Perturbation Theory and Feynam Diagrams We now turn our attention to the dynamics of a quantum field theory. All of the results that we will derive in this section apply equally to both relativistic and non-relativistic theories with only minor changes. Here we will use the path integrals approach we developed in previous chapters. The properties of any field theory can be understood if the N -point Green functions are known G N ( x 1 , . . . , x N ) = ( 0 | T φ ( x 1 ) . . . φ ( x N ) | 0 ) (1) Much of what we will do below can be adapted to any field theory of interest. We will discuss in detail the simplest case, the relativistic self-interacting scalar field theory. It is straightforward to generalize this to other theories of interest. we will only give a summary of results for the other cases. 11.1 The Generating Functional in Perturbation Theory The N -point function of a scalar field theory, G N ( x 1 , . . . , x N ) = ( 0 | T φ ( x 1 ) . . . φ ( x N ) | 0 ) , (2) can be computed from the generating functional Z [ J ] Z [ J ] = ( 0 | T e i integraldisplay d D xJ ( x ) φ ( x ) | 0 ) (3) In D = d + 1-dimensional Minkowski space-time Z [ J ] is given by the path integral Z [ J ] = integraldisplay D φ e iS [ φ ] + i integraldisplay d D xJ ( x ) φ ( x ) (4) where the action S [ φ ] is the action for a relativistic scalar field. The N -point function, Eq.(1), is obtained by functional differentiation, i.e., G N ( x 1 , . . . , x N ) = ( i ) N 1 Z [ J ] δ N δJ ( x 1 ) . . . δJ ( x N ) Z [ J ] vextendsingle vextendsingle vextendsingle J =0 (5) Similarly, the Feynman propagator G F ( x 1 x 2 ), which is essentially the 2-point function, is given by G F ( x 1 x 2 ) = i ( 0 | T φ ( x 1 ) φ ( x 2 ) | 0 ) = i 1 Z [ J ] δ 2 δJ ( x 1 ) δJ ( x 2 ) Z [ J ] vextendsingle vextendsingle vextendsingle J =0 (6) Thus, all we need to find is to compute Z [ J ]. We will derive an expression for Z [ J ] in the simplest theory, the relativistic real scalar field with a φ 4 interaction, but the methods are very general. We 1

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will work in Euclidean space-time ( i.e., in imaginary time) where the generating function takes the form Z [ J ] = integraldisplay D φ e S [ φ ] + integraldisplay d D xJ ( x ) φ ( x ) (7) where S [ φ ] now is S [ φ ] = integraldisplay d D x bracketleftbigg 1 2 ( ∂φ ) 2 + m 2 2 φ 2 + λ 4! φ 4 bracketrightbigg (8) In the Euclidean theory the N -point functions are G N ( x 1 , . . . , x N ) = ( φ ( x 1 ) . . . φ ( x N ) ) = 1 Z [ J ] δ N δJ ( x 1 ) . . . δJ ( x N ) Z [ J ] vextendsingle vextendsingle vextendsingle J =0 (9) Let us denote by Z 0 [ J ] the generating action for the free scalar field, with action S 0 [ φ ]. Then Z 0 [ J ] = integraldisplay D φ e S 0 [ φ ] + integraldisplay d D xJ ( x ) φ ( x ) = bracketleftbig Det ( 2 + m 2 )bracketrightbig 1 / 2 e 1 2 integraldisplay d D x integraldisplay d D yJ ( x ) G 0 ( x y ) J ( y ) (10) where 2 is the Laplacian operator in D -dimensional Euclidean space, and G 0 ( x y ) is the free field Euclidean propagator ( i.e., the Green function) G 0 ( x y ) = ( φ ( x ) φ ( y ) ) 0 = ( x | 1 2 + m 2 | y ) (11) where the sub-index label 0 denotes a free field expectation value.
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