Quantum dilogarithm

In mathematics, the quantum dilogarithm also known as q-exponential is a special function defined by the formula

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

Thus in the notation of the page on q-exponential mentioned above, ${\displaystyle \phi (x)=E_{q}(x)}$ .

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 Roger's five term dilogarithm identity.

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

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

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:

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

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

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

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 $\Phi _{b}$ 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 Im $b^{2}>0$.

References

• Faddeev, L. D. (1994). "Current-Like Variables in Massive and Massless Integrable Models". arXiv:hep-th/9408041 [hep-th].
• 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.
• 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.

• 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.

• 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.

• 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.

External links

• quantum dilogarithm in nLab