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Unformatted text preview: 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 ) = ( | T ( x 1 ) ... ( x N ) | ) (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 ) = ( | T ( x 1 ) ... ( x N ) | ) , (2) can be computed from the generating functional Z [ J ] Z [ J ] = ( | Te i integraldisplay d D xJ ( x ) ( x ) | ) (3) In D = d + 1-dimensional Minkowski space-time Z [ J ] is given by the path integral Z [ J ] = integraldisplay D e iS [ ] + i integraldisplay d D xJ ( x ) ( x ) (4) where the action S [ ] 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 ] N J ( x 1 ) ...J ( 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 ( | T ( x 1 ) ( x 2 ) | ) = i 1 Z [ J ] 2 J ( x 1 ) J ( 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 4 interaction, but the methods are very general. We 1 will work in Euclidean space-time ( i.e., in imaginary time) where the generating function takes the form Z [ J ] = integraldisplay D e S [ ] + integraldisplay d D xJ ( x ) ( x ) (7) where S [ ] now is S [ ] = integraldisplay d D x bracketleftbigg 1 2 ( ) 2 + m 2 2 2 + 4! 4 bracketrightbigg (8) In the Euclidean theory the N-point functions are G N ( x 1 ,...,x N ) = ( ( x 1 ) ... ( x N ) ) = 1 Z [ J ] N J ( x 1 ) ...J ( x N ) Z [ J ] vextendsingle vextendsingle vextendsingle J =0 (9) Let us denote by Z [ J ] the generating action for the free scalar field, with action S [ ]. Then Z [ J ] = integraldisplay D e S [ ] + integraldisplay d D xJ ( x ) ( x ) = bracketleftbig Det ( 2 + m 2 )bracketrightbig 1 / 2 e 1 2 integraldisplay d D x integraldisplay d D yJ ( x ) G ( x y ) J ( y ) (10) where 2 is the Laplacian operator in D-dimensional Euclidean space, and G ( x y ) is the free field Euclidean propagator (...
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- Fall '08