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---------- (7) |

Invoking the Green's function technique, the solution of this equation is given by:

---------- (8) |

---------- (9a) |

In the parlance of Quantum Field Theory the Green's function is better known as propagator denoted by (the source point is often labeled by x' instead of y). The is visualized as virtual particles with an infinite range of momentum via the Fourier transform, which happens to be also its solution (Figure 01e, and see formula below). The particle at space-time point x has x

The 4-momentum k is associated with k^{} = (E, k), k_{} = (E, -k). The two particles interact via the virtual particles with different values of k (see Feynman diagram in Figure 01f). For (x - x')^{2} o the separation is said to be time-like or light-like meaning that the two particles can connect, otherwise they are space-like and cannot communicate with each other. It is forward in time for t > t' ; the reversed is called backward, which is interpreted as forward for anti-particle. Other often used notations are kx = k^{}x_{} = Et - kx, and d^{4}k = dE d^{3}k (all in natural unit with c = = 1).
| ||

## Figure 01e Fourier Transfrom |
## Figure 01f Feynman Diagram |
Negecting the distinction on the bare m_{o} and renormalized mass m for the moment, the solution for Eq.(9a) is (see "contour integral" for an important step in the derivation) : |

## Figure 01g Feynman Propagator [view large image] |
The Feynman propagator has the important feature that includes the anti-particle as going backward in time with negative energy (in the second term), while the first term is for particle going forward in time. |

- Here's some other properties of the Feynman propagator :
- The plane wave
_{k}(x) in the Feynman propagator can be replaced by any quantum field (x) in the vacuum expectation value of its product, i.e., i_{F}(x-x') = <0|T(x)(x')|0>, where T is called the time-order operator. It is used to arrange the latest time to the left, e.g., t > t' in the above example. This operation is to insure that a particle has to be created before it gets destroyed. - By repeating the same step as above, the commutative relation can be written as :

[(x), (x')] = i_{F}(x-x') for time-like separation, and = 0 for space-like.

This property makes sure that the theory obeys the laws of causality. - The
_{F}(x-x') is divergent as**k**. However for simple tree diagram such as the one in Figure 01f, the conservation of 4-momenta with the real particles (the**p**'s) presents a factor of (_{i}**k - p**), and (_{4}+ p_{2}**k - p**) to avoid the divergence in the integration. There would be no such luck in the loop-diagrams._{1}+ p_{3}

Similarly, the Green's function for the ferminon is defined by the equation (denoted by S

---------- (9b) |

where g

The chiral representation is in the form:

.

It splits the Dirac equation into 2 self-contained pieces (for the left-handed and right-handed leptons respectively) more suitable for formulating the Standard Model.

Now back to renormalization, it can be shown that the "bare field" can be expressed in terms of c

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