"Talk on 'introduction to conformal field theory(CFT)'"의 두 판 사이의 차이

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(피타고라스님이 이 페이지의 이름을 talk on 'introduction to conformal field theory(CFT)'로 바꾸었습니다.)
1번째 줄: 1번째 줄:
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<h5>introduction</h5>
  
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* scaling and power law
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* scale invariance and conformal invariance
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*  
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Scale Invariance of power law functions<br> 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.<br> 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.<br> 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<br>     limp-->0 (xp-1)/p = log x<br> which shows that logs are a special case of power law functions with power 0.
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<h5>history</h5>
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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<h5>related items</h5>
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* [[5 conformal field theory(CFT)|5 conformal field theory]]
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
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* http://en.wikipedia.org/wiki/
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* http://www.scholarpedia.org/
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* http://www.proofwiki.org/wiki/
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
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<h5>books</h5>
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* [[2010년 books and articles]]<br>
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* http://gigapedia.info/1/
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* http://gigapedia.info/1/
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
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<h5>expositions</h5>
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
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* http://www.ams.org/mathscinet
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* http://www.zentralblatt-math.org/zmath/en/
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* http://arxiv.org/
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* http://www.pdf-search.org/
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* http://pythagoras0.springnote.com/
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
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* http://dx.doi.org/
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<h5>question and answers(Math Overflow)</h5>
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* http://mathoverflow.net/search?q=
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* http://mathoverflow.net/search?q=
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<h5>blogs</h5>
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*  구글 블로그 검색<br>
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**  http://blogsearch.google.com/blogsearch?q=<br>
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** http://blogsearch.google.com/blogsearch?q=
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* http://ncatlab.org/nlab/show/HomePage
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<h5>experts on the field</h5>
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* http://arxiv.org/
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<h5>links</h5>
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
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* http://functions.wolfram.com/

2010년 10월 14일 (목) 21:19 판

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.

 

 

history

 

 

related items

 

 

encyclopedia

 

 

books

 

 

 

expositions

 

 

 

articles

 

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

experts on the field

 

 

links
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