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

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

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

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

In the original formulation, y lables the source point while x is the field point. In the parlance of Quantum Field Theory the Green's function is better known as propagator denoted by , which is visualized as the propagation of virtual particles from x to x' with an infinite range of momentum via the Fourier transform (Figure 01e, and see formula below). The particle at space-time point x has x

The 4-momentum k is associated with k^{} = (iE, k), k_{} = (-iE, k). The two particles interact via the virtual particles with different values of k (see Feynman diagram in Figure 01f). For (x - y)^{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-y) = <0|T(x)(x')|0>, where T is called the time-order operator. It is used to arrange the later time on 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 propagator for the ferminon is

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

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

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