This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nonrelativistic 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 Npoint 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 selfinteracting 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 Npoint 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 + 1dimensional Minkowski spacetime 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 Npoint 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 2point 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 spacetime ( 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 Npoint 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 Ddimensional Euclidean space, and G ( x y ) is the free field Euclidean propagator (...
View
Full
Document
 Fall '08
 Leigh

Click to edit the document details