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

introduction

• scaling and power law
• scale invariance and conformal invariance
•  

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.  In this sense, the thermodynamic functions seem to display scale invariance around the critical point (for systems that have a critical point!), with the scaling variable t=(1-T/Tc).  Note that the logarithm y=log x obeys y(ax)=y(x) + log a.  It is scale invariant with exponent 0 (and a scale-dependent shift.)  This is related to the famous formula
    limp-->0 (xp-1)/p = log x
which shows that logs are a special case of power law functions with power 0.

 

 

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related items

• 5 conformal field theory

 

 

encyclopedia

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• http://www.proofwiki.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

 

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

 

 

 

articles

 

• 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/

 

 

question and answers(Math Overflow)

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experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/