Quantum dilogarithm

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

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

It is the same as the q-exponential function ${\displaystyle e_{q}(x)}$.

Let ${\displaystyle u,v}$ be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl's relation ${\displaystyle uv=qvu}$. Then, the quantum dilogarithm satisfies Schützenberger's identity

${\displaystyle \phi (u)\phi (v)=\phi (u+v),}$

Faddeev-Volkov's identity

${\displaystyle \phi (v)\phi (u)=\phi (u+v-vu),}$

and Faddeev-Kashaev's identity

${\displaystyle \phi (v)\phi (u)=\phi (u)\phi (-vu)\phi (v).}$

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

Faddeev's quantum dilogarithm ${\displaystyle \Phi _{b}(w)}$ is defined by the following formula:

${\displaystyle \Phi _{b}(z)=\exp \left({\frac {1}{4}}\int _{C}{\frac {e^{-2izw}}{\sinh(wb)\sinh(w/b)}}{\frac {dw}{w}}\right),}$

where the contour of integration ${\displaystyle C}$ 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:

${\displaystyle \Phi _{b}(x)=\exp \left({\frac {i}{2\pi }}\int _{\mathbb {R} }{\frac {\log(1+e^{tb^{2}+2\pi bx})}{1+e^{t}}}\,dt\right).}$

Ludvig Faddeev discovered the quantum pentagon identity:

${\displaystyle \Phi _{b}({\hat {p}})\Phi _{b}({\hat {q}})=\Phi _{b}({\hat {q}})\Phi _{b}({\hat {p}}+{\hat {q}})\Phi _{b}({\hat {p}}),}$

where ${\displaystyle {\hat {p}}}$ and ${\displaystyle {\hat {q}}}$ are self-adjoint (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation

${\displaystyle [{\hat {p}},{\hat {q}}]={\frac {1}{2\pi i}}}$

and the inversion relation

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

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

The precise relationship between the q-exponential and ${\displaystyle \Phi _{b}}$ is expressed by the equality

${\displaystyle \Phi _{b}(z)={\frac {E_{e^{2\pi ib^{2}}}(-e^{\pi ib^{2}+2\pi zb})}{E_{e^{-2\pi i/b^{2}}}(-e^{-\pi i/b^{2}+2\pi z/b})}},}$

valid for ${\displaystyle \operatorname {Im} b^{2}>0}$.

• Faddeev, L. D. (1994). "Current-Like Variables in Massive and Massless Integrable Models". arXiv:hep-th/9408041.
• Faddeev, L. D. (1995). "Discrete Heisenberg-Weyl group and modular group". Letters in Mathematical Physics. 34 (3): 249–254. arXiv:hep-th/9504111. Bibcode:1995LMaPh..34..249F. doi:10.1007/BF01872779. MR 1345554. S2CID 119435070.
• Faddeev, L. D.; Kashaev, R. M. (1994). "Quantum dilogarithm". Modern Physics Letters A. 9 (5): 427–434. arXiv:hep-th/9310070. Bibcode:1994MPLA....9..427F. doi:10.1142/S0217732394000447. MR 1264393. S2CID 6172445.

• Faddeev, L. D.; Volkov, A. Yu. (1993). "Abelian current algebra and the Virasoro algebra on the lattice". Physics Letters B. 315 (3–4): 311–318. arXiv:hep-th/9307048. Bibcode:1993PhLB..315..311F. doi:10.1016/0370-2693(93)91618-W. S2CID 10294434.

• Kirillov, A. N. (1995). "Dilogarithm identities". Progress of Theoretical Physics Supplement. 118: 61–142. arXiv:hep-th/9408113. Bibcode:1995PThPS.118...61K. doi:10.1143/PTPS.118.61. MR 1356515. S2CID 119177149.

• Schützenberger, M. P. (1953). "Une interprétation de certaines solutions de l'équation fonctionnelle: F (x + y) = F (x)F (y)". Comptes Rendus de l'Académie des Sciences de Paris. 236: 352–353.

• Woronowicz, S. L. (2000). "Quantum exponential function". Reviews in Mathematical Physics. 12 (6): 873–920. Bibcode:2000RvMaP..12..873W. doi:10.1142/S0129055X00000344. MR 1770545.

• quantum dilogarithm at the nLab