Quantum dilogarithm

introduction

• Notes on Construction of the Knot Invariant from Quantum Dilogarithm Function
  

 

 

quantum plane

• also called the Weyl algebra
  

• noncommutative geometry
  
• \(uv=qvu\)
  

 

 

q-integral (Jackson integral)

• \(0<q<1\)에 대하여 다음과 같이 정의
  \(\int_0^a f(x) d_q x = a(1-q)\sum_{k=0}^{\infty}q^k f(aq^k )\)
  \(\int_0^{\infty} f(x) d_q x =(1-q)\sum_{k=-\infty}^{\infty}q^k f(aq^k )\)
  
• \(q\to 1\) 이면, \(\int_0^a f(x) d_q x \to \int_0^a f(x) dx \)
  

 

 

quantum dilogarithm

\(\Psi(z)=\prod_{n=0}^{\infty}(1-zq^n)=\sum_{n\geq 0}\frac{(-1)^nq^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\)

\(\Psi(z)=\exp(\frac{\operatorname{Li}_{2,q}(z)}{q-1})\)

\(\operatorname{Li}_{2,q}(z) = -\int_0^z{{\ln (1-t)}\over t} d_{q}t \)

\(\operatorname{Li}_2(z) = -\int_0^z{{\ln (1-t)}\over t} dt \)

 

 

asymptotics 

• \(q=e^{-t}\) and as the t goes 0 (i.e. as q goes to 1)
  

\(\sum_{n=0}^{\infty}\frac{q^{\frac{A}{2}n^2+cn}}{(q)_n}\sim\exp(\frac{C}{t})\)

where C= sum of Rogers dilogarithms

 

 

quantum 5-term relation

• In Weyl algebra, the following identity holds
  \((v)_{\infty}(u)_{\infty}=(u)_{\infty}(-vu)_{\infty}(v)_{\infty}\)
  
• manufacturing matrices from lower ranks
  

 

 

 

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 ofthe simplicial (combinatorial) version of the three-dimensional TQFT
  
• this invariant is in fact a quantum generalization of thehyperbolic 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.
  

 

 

history

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

 

 

related items

• Boson and Fermion summation form
  
• asymptotic analysis of basic hypergeometric series
  
• Quantum groups
  
• Kashaev's volume Conjecture
  

 

 

encyclopedia

• q-적분
• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.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=

[[4909919|]]

 

 

articles

• Quantum dilogarithm.
  
  • Wadim Zudilin, Preprint, Bonn and Moscow (2006)
• Notes on Construction of the Knot Invariant from Quantum Dilogarithm Function
• The hyperbolic volume of knots from quantum dilogarithm
  
  • R. M. Kashaev, 1996
• Remarks on the quantum dilogarithm
  
  • V V Bazhanov and N Yu Reshetikhin, 1995 J. Phys. A: Math. Gen. 28 2217
• [Kashaev1995]A link invariant from quantum dilogarithm
  
  • Kashaev, R. M., Modern Phys. Lett. A 10 (1995), 1409–1418

• Quantum Dilogarithm as a 6j-Symbol
  
  • R. M. Kashaev, MPLA Volume: 9, Issue: 40(1994) pp. 3757-3768
    
• Quantum Dilogarithm
  
  • L.D.Fadeev and R.M.Kashaev, Mod. Phys. Lett. A. 9 (1994) p.427–434
• http://ncatlab.org/nlab/show/quantum+dilogarithm

• 논문정리
• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html[1]
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• http://dx.doi.org/10.1023/A:1007364912784

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• The On-Line Encyclopedia of Integer Sequences