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In mathematics, the ''' quantum dilogarithm''' also known as '''[[Q-exponential|q-exponential]]''' is a [[special function]] defined by the formula

:<math>
\phi(x)\equiv(x;q)_\infty=\prod_{n=0}^\infty (1-xq^n),\quad |q|<1
</math>

Thus in the notation of the page on [[Q-exponential|q-exponential]] mentioned above, <math>\phi(x)=E_q(x)^{-1}</math> .

Let <math>u,v</math> be “q-commuting variables”, that is elements of a suitable
noncommutative algebra satisfying Weyl’s relation <math>uv=qvu</math>. Then, the quantum dilogarithm
satisfies Schützenberger’s identity

:<math>
\phi(u) \phi(v)=\phi(u + v)
</math>

Faddeev-Volkov's identity

:<math>
\phi(v) \phi(u)=\phi(u +v -vu)
</math>

and Faddeev-Kashaev's identity

:<math>
\phi(v) \phi(u)=\phi(u)\phi(-vu)\phi(v)
</math>

The latter is known to be a quantum generalization of Roger's five term dilogarithm identity.

'''Faddeev's quantum dilogarithm''' <math>\Phi_b(w)</math> is defined by the following formula:

: <math>\Phi_b(z)=\exp
\left(
\frac{1}{4}\int_C
\frac{e^{-2\sqrt{-1} zw }}
{\sinh (wb) \sinh (w/b) }
\frac{dw}{w}
\right)</math>

where the contour of integration <math>C </math> goes along the real axis outside a small neighborhood of the origin and deviates into the [[upper half-plane]] near the origin. [[Ludvig Faddeev]] discovered the quantum pentagon identity:

: <math>\Phi_b(\hat p)\Phi_b(\hat q)
=
\Phi_b(\hat q)
\Phi_b(\hat p+ \hat q)
\Phi_b(\hat p)
</math>
where <math>\hat p</math> and <math>\hat q</math> are (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation

:<math>[\hat p,\hat q]=\frac1{2\pi\sqrt{-1}}.</math>

The quantum dilogarithm finds applications in [[mathematical physics]], [[quantum topology]], [[cluster algebra]] theory.

The precise relationship between the [[Q-exponential|q-exponential]] and <math>\Phi_b</math> is expressed by the equality

:<math>\Phi_b(z)=\frac{E_{e^{2\pi ib^2}}(-e^{\pi ib^2+2\pi zb})}{E_{e^{-2\pi i/b^2}}(-e^{-\pi i/b^2+2\pi z/b})}</math>

valid for Im <math>b^2>0</math>.

== References ==

*{{cite arxiv
| last = Faddeev | first = L. D.
| year = 1994
| title = Current-Like Variables in Massive and Massless Integrable Models
| class =hep-th
| eprint = hep-th/9408041
}}
*{{cite journal
| last = Faddeev | first = L. D.
| year = 1995
| journal = [[Letters in Mathematical Physics]]
| title = Discrete Heisenberg-Weyl group and modular group
| volume = 34 | issue = 3 | pages = 249–254
| arxiv = hep-th/9504111
| bibcode = 1995LMaPh..34..249F
| doi = 10.1007/BF01872779
| mr = 1345554
}}
*{{cite journal
| last1=Faddeev | first1=L. D.
| last2=Kashaev | first2=R. M.
| year=1994
| title=Quantum dilogarithm
| journal=[[Modern Physics Letters A]]
| volume=9 | issue=5 | pages=427–434
| arxiv= hep-th/9310070
| bibcode= 1994MPLA....9..427F
| doi=10.1142/S0217732394000447
| mr=1264393
}}

*{{cite journal
| last1=Faddeev | first1=L. D.
| last2=Volkov| first2= A. Yu.
| year= 1993
| title= Abelian current algebra and the Virasoro algebra on the lattice
| journal= [[Physics Letters B]]
| volume= 315 | issue=3–4 | pages=311–318
| arxiv= hep-th/9307048
| bibcode= 1993PhLB..315..311F
| doi= 10.1016/0370-2693(93)91618-W
}}

*{{cite journal
| last1=Kirillov | first1=A. N.
| year=1995
| title=Dilogarithm identities
| journal=[[Progress of Theoretical Physics Supplement]]
| volume=118 |issue= | pages=61–142
| arxiv=hep-th/9408113
| bibcode= 1995PThPS.118...61K
| doi=10.1143/PTPS.118.61
| mr=1356515
}}

*{{cite journal
| last = Schützenberger | first = M. P.
| year = 1953
| title = Une interprétation de certaines solutions de l'équation fonctionnelle: F (x + y) = F (x)F (y)
| journal = [[Comptes rendus de l'Académie des Sciences de Paris]]
| volume = 236 |issue= | pages = 352–353
| arxiv=
| bibcode=
| doi=
}}

== External links ==
* {{nlab|id=quantum+dilogarithm|title=quantum dilogarithm}}



{{DEFAULTSORT:Faddeev's Quantum Dilogarithm}}
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[[Category:Special functions]]

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