Feynman Diagrams

1. S-Matrix

   Annotations:

   • This is the limit of a sequence of unitary operators connecting the in and out states for time approaching infinity. These in and out states are defined as the states where the wavepackets represent some initial and final states set up in the remote past and future. The limit where the momenta of these wavepackets become concentrated around definite momenta is then taken to be the in state.
   • The S matrix has the form S=1+iT, where T is the part of the matrix that accounts for interactions of the fields.
   1. Interactions

      Annotations:

      • Interactions occur when particles collide and scatter off each other. The interaction is then represented by the T part of the S matrix.
      1. Cross sections
      2. Decay rates
   2. Feynman Amplitudes

      Annotations:

      • This is the invariant matrix element left over when a momentum conserving Dirac delta is extracted from the T-matrix element.
      1. Feynman Rules

Annotations:

• Since no interacting field theory is exactly solvable in more than two space-time dimensions, we use the approach of evaluating the two-point correlation function perturbatively.

1. Perturbation Expansion of Correlation Functions

   Annotations:

   • Wrt. new ground states \Omega of the interacting theory, and including Heisenberg fields. We thus need to express both the ground states and fields in terms of the free field operators and ground state, which we know how to manipulate.
   • The final expression depends on the interaction picture fields and the interaction picture interaction Hamiltonian, and is obtained by time evolving the free field ground state to infinity.
   • Rewriting the Heisenberg picture fields to interaction picture fields introduces a unitary operator which solves the Schrödinger equation. The form of this operator is an exponentiation of the time integral of the interaction picture interaction Hamiltonian. This exponential function is time ordered.
   1. Wick's Theorem

      Annotations:

      • This gives a neat relation between the time-ordering and the normal-ordering of a product in terms of contractions of fields.
      1. Propagators

         Annotations:

         • These arise as contractions of fields when applying Wick's theorem.
         1. Diagrams

            Annotations:

            • These can now be drawn as points in spacetime connected by propagators.
            • Diagrams represent the probability amplitude for a spacetime process, such as a scattering or propagation, to occur. The total amplitude is the norm squared of the sum of all possible processes.
            1. Correlation function numerator

               Annotations:

               • This is now represented by the sum of all possible diagrams with two external points.
            2. Correlation function denominator

               Annotations:

               • Since the exponential of all disconnected diagrams factor out in the numerator, and this exponential is exactly the denominator, the denominator cancels out from the expression.
            3. Final expression for n-point correlation functions

               Annotations:

               • This becomes the sum of all connected diagrams with n external points.
   2. Tree Level
      1. Loop Level