Talk on 'introduction to conformal field theory(CFT)'

introduction

• scaling and power law
• scale invariance and conformal invariance
• critical phenomena
  
  • http://scienceon.hani.co.kr/archives/13339
  • http://scienceon.hani.co.kr/archives/13941
  • http://scienceon.hani.co.kr/archives/14664

 

 

scale invariacne and power law

Scale Invariance of power law functions
The function y=xp is "scale-invariant" in the following sense.  Consider an interval such as (x,2x), where y changes from xp to  2pxp.  Now scale x by a scale factor a.  We look at the interval (ax,2ax) where y changes from (ax)p to 2p(ax)p.  We see that y in this interval is the same (except for a scale change of ap) as y in the unscaled interval.
Most functions do not behave this way.  Consider y=exp(x), which goes [from exp(x) to exp(2x)] over the interval (x,2x), and [from exp(ax) to exp(2ax)] in the scaled interval (ax,2ax).  There is no scale factor that can be removed from y to make the second interval of y appear the same as the first.
The sum of two different powers is usually not scale invariant.  However, special cases may still be.  y=Ax2 + Bx can be written as A(x+B/2A)2 -B2/4A.  If the new variable X=x+B/2A is scaled to aX, then y(aX)=a2Y(X)+(a2-1)(B2/4A).  In other words, the function y(x) is scale invariant (with scaling exponent 2) after finding the right scaling variable X and allowing for a scale-dependent shift of y.

 

 

critical phenomena

• critical phenomena

 

 

correlation at large distance

• universality class and critical exponent
• appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature.
• the critical exponent describes the behavior of physical quantities around the critical temperature
  e.g. magnetization \(M\sim (T_C-T)^{1/8}\)
  
• magnetization and susceptibility can be written as correlation functions
• large distance behavior of spin at criticality \(\eta=1/4\)
  \(<\sigma_{i}\sigma_{i+n}>=\frac{1}{|n|^{\eta}}\)
  
• correlation length critivel exponent \(\nu=1\)
  \(<\epsilon_{i}\epsilon_{i+n}>=\frac{1}{|n|^{4-\frac{2}{\nu}}}\)
  

 

conformal transformations

• roughly, local dilations
  
  • this is also equivalent to local scale invariance
• correlation functions do not change under conformal transformations

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• 5 conformal field theory