Quantum dilogarithm explained

In mathematics, the quantum dilogarithm is a special function defined by the formula

\phi(x)\equiv(x;q) _{infty=\prod} ^{inf ty} ^{n =0} ( 1- xq ^{n),} |q|< 1

It is the same as the q-exponential function

e _{q(x)} .

Let

u,v be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl's relation uv = qvu . Then, the quantum dilogarithm satisfies Schützenberger's identity\phi(u) \phi(v) = \phi(u + v), Faddeev-Volkov's identity\phi(v) \phi(u) = \phi(u + v - vu), and Faddeev-Kashaev's identity\phi(v)\phi(u) = \phi(u)\phi( - vu)\phi(v).

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

Faddeev's quantum dilogarithm

\Phi _{b(w)} is defined by the following formula:\Phi _{b(z) = \exp \left( (1)/(4)} \int _{C (e ^{-2i})/(\sinh (wb) \sinh (w/b))} (dw)/(w) \right),

where the contour of integration

C goes along the real axis outside a small neighborhood of the origin and deviates into the upper half-plane near the origin. The same function can be described by the integral formula of Woronowicz:\Phi _{b(x) = \exp\left( (i)/(2 \pi)} \int _{R} (log ( 1+ e ^{tb ^{2+2\pi} b x} ))/(1+ e ^{t}) dt\right).

Ludvig Faddeev discovered the quantum pentagon identity:

\Phi _{b(\hat} p)\Phi _{b(\hat} q) = \Phi _{b(\hat} q) \Phi _{b(\hat} p + \hat q) \Phi _{b(\hat} p), where \hat p and \hat q are self-adjoint (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation[\hat p,\hat q] = (1 { 2 \pi)/(i})

and the inversion relation

\Phi _{b(x)\Phi} _{b(-x)=\Phi} ^{2} ^{b( 0 )} e ^{\pi ix ^{2}} , \Phi ^{(\pi i)/(\left(b ^{2+b} ^{-2} \right) 24)} _{b( 0 ) = e} .

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

The precise relationship between the q-exponential and

\Phi _{b} is expressed by the equality\Phi _{b(z) = (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} ))} ,

valid for

\operatorname{Im} b ^{2>0} .

References

• Faddeev . L. D. . 1994 . Current-Like Variables in Massive and Massless Integrable Models . hep-th/9408041.
• Faddeev . L. D. . 1995 . . Discrete Heisenberg-Weyl group and modular group . 34 . 3 . 249–254 . hep-th/9504111 . 1995LMaPh..34..249F . 10.1007/BF01872779 . 1345554. 119435070 .
• Faddeev . L. D. . Kashaev . R. M. . 1994 . Quantum dilogarithm . . 9 . 5 . 427–434 . hep-th/9310070 . 1994MPLA....9..427F . 10.1142/S0217732394000447 . 1264393. 6172445 .
• Faddeev . L. D. . Volkov. A. Yu. . 1993 . Abelian current algebra and the Virasoro algebra on the lattice . . 315 . 3–4 . 311–318 . hep-th/9307048 . 1993PhLB..315..311F . 10.1016/0370-2693(93)91618-W. 10294434 .
• Kirillov . A. N. . 1995 . Dilogarithm identities . . 118 . 61–142 . hep-th/9408113 . 1995PThPS.118...61K . 10.1143/PTPS.118.61 . 1356515. 119177149 .
• Schützenberger . M. P. . 1953 . Une interprétation de certaines solutions de l'équation fonctionnelle: F (x + y) = F (x)F (y) . . 236 . 352–353.
• Woronowicz . S. L.. 2000. Quantum exponential function. Reviews in Mathematical Physics. 12 . 6 . 873–920. 10.1142/S0129055X00000344. 1770545. 2000RvMaP..12..873W .