"Quantum dilogarithm"의 두 판 사이의 차이

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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">introduction[http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/64430/1/1172-4.pdf ]</h5>
 
<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 3.42em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">introduction[http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/64430/1/1172-4.pdf ]</h5>
  
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* [http://pythagoras0.springnote.com/pages/7978406 양자 다이로그 함수(quantum dilogarithm)]
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<h5 style="margin: 0px; line-height: 2em;">quantum plane</h5>
 
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*  also called the Weyl algebra<br>
 
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*  noncommutative geometry<br>
 
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* <math>uv=qvu</math><br>
 
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<h5 style="background-position: 0px 100%; font-size: 1.16em; margin: 0px; color: rgb(34, 61, 103); line-height: 2em; font-family: 'malgun gothic',dotum,gulim,sans-serif;">q-integral (Jackson integral)</h5>
 
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* <math>0<q<1</math>에 대하여 다음과 같이 정의<br><math>\int_0^a f(x) d_q x = a(1-q)\sum_{k=0}^{\infty}q^k f(aq^k )</math><br><math>\int_0^{\infty} f(x) d_q x =(1-q)\sum_{k=-\infty}^{\infty}q^k f(aq^k )</math><br>
 
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* <math>q\to 1</math> 이면, <math>\int_0^a f(x) d_q x \to  \int_0^a f(x) dx </math><br>
 
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<h5 style="margin: 0px; line-height: 2em;">quantum dilogarithm</h5>
 
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<math>\Psi(z)=\prod_{n=0}^{\infty}(1-zq^n)=\sum_{n\geq 0}\frac{(-1)^nq^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n</math>
 
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<math>\Psi(z)=\exp(\frac{\operatorname{Li}_{2,q}(z)}{q-1})</math>
 
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<math>\operatorname{Li}_{2,q}(z) = -\int_0^z{{\ln (1-t)}\over t} d_{q}t </math>
 
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<math>\operatorname{Li}_2(z) = -\int_0^z{{\ln (1-t)}\over t} dt </math>
 
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<h5 style="margin: 0px; line-height: 2em;">asymptotics </h5>
 
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* <math>q=e^{-t}</math> and as the t goes 0 (i.e. as q goes to 1)<br>
 
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<math>\sum_{n=0}^{\infty}\frac{q^{\frac{A}{2}n^2+cn}}{(q)_n}\sim\exp(\frac{C}{t})</math>
 
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where C= sum of Rogers dilogarithms
 
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<h5 style="margin: 0px; line-height: 2em;">quantum 5-term relation</h5>
 
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*  In Weyl algebra, the following identity holds<br><math>(v)_{\infty}(u)_{\infty}=(u)_{\infty}(-vu)_{\infty}(v)_{\infty}</math><br>
 
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* [[1 manufacturing matrices from lower ranks|manufacturing matrices from lower ranks]]<br>
 
  
 
 
 
 

2011년 6월 30일 (목) 05:46 판

introduction[1]

 

 

 

Knot and invariants from quantum dilogarithm
  • [Kashaev1995] 
  • a link invariant, depending on a positive integer parameter N, has been defined via three-dimensional interpretation of the cyclic quantum dilogarithm
  • The construction can be considered as an example of the simplicial (combinatorial) version of the three-dimensional TQFT
  • this invariant is in fact a quantum generalization of the hyperbolic volume invariant.
  • It is possible that the simplicialTQFT, defined in terms of the cyclic quantum dilogarithm, can be associated with quantum 2 + 1-dimensional gravity.

 

 

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