Renormalization

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Renormalization

• Kesheng Yang

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1 The Divergence in EM
(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
EM, we just care about the interaction energy,
or the radiation energy, which is finite, so
infinity doesnt bother us.
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(2) Interaction Energy
Finite
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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.
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(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.
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? 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.
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  ------Adiabatic Approximation

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? 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
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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.
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CM
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Renormalized Coupling Constant

• The difference between the above two equations
  doesnt matter, because the difference is just a
  higher order small quantity, which goes beyond
  the experiment precision.

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? 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
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