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---------- (16) |

which translates S

It represents the process of the annihilation of a pair of nucleon and anti-nucleon and the creation of a pion.

In the next higher order term the normal operator will generate Green's function such as or S

The pairing is referred to as propagator or internal line and graphically represented by a line running from a vertex at x to another vertex at y as shown in the diagram below:

which represents the annihilation of a pair of nucleon and anti-nucleon through a virtual nucleon.

Figure 01h corresponds to the scattering of two nucleons by exchanging a pion. The internal line represents the probability amplitude for a virtual particle to travel from one place to another (x y) in a given time with greater than light speed, or to travel with off mass-shell 4-momentum k, which could have arbitrary value in violation of energy-momentum conservation but allowed by the uncertainty principle. Mathematically, it is expressed by the Green's function: | |

## Figure 01h Nucleon-Nucleon Scattering [view large image] |

where is a small positive real constant - a mathematical device taking advantage of the technique of contour integral; + 0 will be taken after the integration. The subscript F refers to the Feynman prescription for integrating the Green's functions. They are in a form such that positive energy solutions are carried forward in time, while negative energy solutions are carried backward in time. The latter solution can be interpreted as the anti-particle with positive energy moving forward in time.

Another graph such as the one below:

this loop diagram represents the process of virtual pair creation and annihilation - the vacuum fluctuation (time runs horizontally in this graph). It is this kind of graphs, which give rise to divergent results.

The Feynman rules are summarized in the tables below:

## Table 02 Feynman Rules [view large image] |
## Table 03 Momentum Representations [view large image] |

Then the Feynman rules (see Table 03) can be expressed in terms of the energy

- As an example to demonstrate the power of Feynman diagram, the nucleon-nucleon scattering in Figure 01h will be used to evaluate the corresponding S matrix from the Feynman diagram (in momentum space, see Table 03):
- Collect the 4 quantum fields for the external nucleon lines (
_{p1}, '_{p2}, *_{p3}, '*_{p4}), plus other factors to be shown later. - Multiply 2 coupling constants g
_{0}and a delta function either (k - p_{1}+ p_{3}), or (k + p_{2}- p_{4}) for the vertex x or x'. - Write down a propagator
_{F}(k) for the internal line. It represents the virtual pions for all values of the 4-momentum k. - Integrate over internal momenta. In explicit form :
- In this case of tree diagram, the delta functions enforce the rule for energy-momentum conservation such that the internal momenta k = p
_{1}- p_{3}= p_{4}- p_{2}. Thus, the integration can be carried out trivially. Such is not the case for a loop diagram, which diverges quadratically (see for example the self-energy diagram in Figure 03f). - The integration of the k-space yields: 1/[(p
_{1}- p_{3})^{2}- m^{2}] - The S-matrix becomes :

S = (M/V)^{2}(E_{1}E_{3}E'_{2}E'_{4})^{-1/2}(_{p1}'_{p2}*_{p3}'*_{p4}) {(g_{0}^{2})/[(p_{1}- p_{3})^{2}- m^{2}]} ----- (29),

where M and E_{p}s are the mass and energy of the nucleons, V is the normalization volume. - The scattering cross section is proportional to the squared S-matrix: |S|
^{2}.

| |

## Feynman and Friends [view large image] |

It is worthwhile to repeat once again that Feynman diagrams can be divided into two types, "trees" and "loops", on the basis of their topology. Tree diagrams only have branches. They describe process such as scattering, which yields finite result and reproduces the classical value. Loop diagrams, as their name suggests, have closed loops in them such as the one for vacuum fluctuation. The loop diagrams involve "off mass-shell" virtual particles and is usually divergent (becomes infinity). Such virtual particles can appear and disappear violating the rules of energy and momentum conservation as long as the uncertainty principle is satisfied. They are said to be "off mass-shell", because they do not satisfy the relationship E

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