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

2011년 1월 26일 (수) 12:46 판

introduction

scaling and power law
scale invariance and conformal invariance

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

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 (x^p-1)/p = log x
which shows that logs are a special case of power law functions with power 0.

correlation at large distance

appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature.

basics of magnetism

the critical exponent describes the behavior of physical quantities around the critical temperature
e.g. magnetization \(M\sim (T-T_C)^{1/8}\)
magnetization and susceptibility can be written as correlation functions
there are six critical exponent for Ising model
2 microscopit critical exponents
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

books

expositions

Scale Invariance of power law functions

articles

question and answers(Math Overflow)

blogs

구글 블로그 검색
- http://blogsearch.google.com/blogsearch?q=
- http://blogsearch.google.com/blogsearch?q=
http://ncatlab.org/nlab/show/HomePage

experts on the field

http://arxiv.org/

@@ 19번째 줄: / 19번째 줄: @@
   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 (x^p-1)/p = log x<br> which shows that logs are a special case of power law functions with power 0.
+<h5>correlation at large distance</h5>
 appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature.
@@ 25번째 줄: / 31번째 줄: @@
+*  the critical exponent describes the behavior of physical quantities around the critical temperature<br> e.g. magnetization <math>M\sim (T-T_C)^{1/8}</math><br>
+* magnetization and susceptibility can be written as '''correlation functions'''
+* there are six critical exponent for Ising model
+* 2 microscopit critical exponents
+*  large distance behavior of spin at criticality <math>\eta=1/4</math><br><math><\sigma_{i}\sigma_{i+n}>=\frac{1}{|n|^{\eta}}</math><br>
+*  correlation length critivel exponent <math>\nu=1</math><br><math><\epsilon_{i}\epsilon_{i+n}>=\frac{1}{|n|^{4-\frac{2}{\nu}}}</math><br>