Renormalization

1. Renormalization Kesheng Yang

2. 1 The Divergence in E&M (1) Infinite Self Energy  In modern Physics, we think electron is point-like charge, as well as other fermions. But, who has ever found bare electron? Bare electron means there is no electric field around the electron. Since we can NOT separate electric field from its source, infinity is always there because of the point charge model. However, in E&M, we just care about the interaction energy, or the radiation energy, which is finite, so infinity doesn’t bother us.

3. (2) Interaction Energy Finite

4. 2 Divergence in QED (1) Free Particles  All these expressions are just valid for free fermions, but all of our actual experiments are devised to measure the interaction between real particles.

5. (2) Interacting Particles  This set of equations are nonlinear and high coupling, so it is impossible to give a simple analytical solution. To attack this equation, we have to turn to Perturbation Theory, just as in QM.  Following the Perturbation theory, we first have to assume that the interaction part is very weak, that is, the parameter e is very small. So we can express the solution as the Taylor Series of this small parameter. Fortunately, this assumption turns out to be the real world.

6. : initial particle state :final particle state Far past Far future  Just as we deal with the time-dependent perturbation, we interpret the solution as the Transition Amplitude between the initial state and the final state, both of which are just the solutions of first two parts, free terms. (3) S-Matrix • Here, we take the initial and final states are the solution of free terms, which means the initial and final states both denote the free particles, the so-called bare particles. ------Adiabatic Approximation

7.  This approximation is a bit contradictory to the real situation. Even in the far past and the far future, no charge is free from the interaction with photons. (4) Divergence in QED

9.  To attack the infinity, another way is the so-called renormalization. During the above calculation, we find that the polarization tensor is given as the function of the coupling constant between bare particles. However, any interaction happens between real particles. Polarization Tensor  This integral is logarithmically divergent----infinity. (5) Renormalization  Physicists think that the divergence comes from the point model of particles, or say the involved particles interact with each other only at the same space-time point. Based on this idea, They came up with the string theory in which no infinity shows up.

10. And nobody can measures the constant . So we should redefine the constant, or say we should get the relation between and , the real coupling constant, then express our theoretical result completely in terms of measurable quantities. In this case, we only need renormalize the coupling constant; in other cases, we have to renormalize the masses of free particles, or wave functions.  It is clear that the polarization tensor transforms like a covariant tensor under Lorentz transformation. And, after integration, there are only and , which are of the characteristic of covariant tensor, and may show up in the expression of polarization tensor. So the general form of this tensor can be given as Logarithmically divergent

11.  Transferred Moment satisfies:

12. CM: ( )

13. Renormalized Coupling Constant • The difference between the above two equations doesn’t matter, because the difference is just a higher order small quantity, which goes beyond the experiment precision.

14. Because increases monotonously, the running coupling constant increases as the transferred momentum.  Now we can express our theoretical calculation in terms of the parameters which can be measured experimentally.  Considering the finite correction term, we have another definition, Running Coupling Constant. The End