Renormalization

renormalization

• way of pulling out sensible answers from Feynman diagrams that explode
• there are two parts in the renormalization program
  
  • regularization - the divergences must be written down in some way so that they can be compared, added and subtracted
    
  • renormalization proper - the various divergences must be gathered together and extracted from the rest of the calculation
    
• set of techniques used to understand a given quantum field theory in a certain energy or length interval
  
  • http://en.wikipedia.org/wiki/Effective_field_theory
  • effective ~ restricted to some interval
• easiest to grasp using functional integrals

 

 

 

regularization

• the values of observable quantities cannot depend on the way we've chosen to take the cutoff (regularize)
• introducing momentum or distance cutoff so as to render finite the large momentum or short distance limits of correlation functions
• methods of regularization
  
  • momentum regularization (modify the propagator by introducing cutoff dependent mass couplings)
  • lattice regulatization (replace R^d by a lattice, uses a small space cutoff)
  • dimensional regularization
• root of the problem
  
  • probability of creating particles of colossal energies
    
  • in terms of Feynman diagrams, the momentum in a loop can run away to infinity
    
• dimensional regularization
  
  • regularization scheme (especially good in QCD)
    
  • regard the dimension as a continuous quantity
    
  • coupling constant changes accordingly as the dimension changes continuously
    
  • we get a new parameter called regularization scheme
    

 

 

QED renormalization

• QED renormalization
  

 

 

electroweak renormalization

• spontaneous local symmetry breaking or Higgs mechanism
  
• mass term for gauge field is zero in the Lagranaian, but these bosons (W,Z bosons) have mass term. To resolve this, we employ Higgs mechanism
  

 

 

books

• Renormalization: an introduction to renormalization, the renormalization John C. Collins
• Kevin Costello
• 찾아볼 수학책

 

 

articles

• Renormalization and quantum field theory R. E. Borcherds, 2010
  
• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://arxiv.org/
• http://www.pdf-search.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://dx.doi.org/