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, $\phi(x)=E_q(x)$ .

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:

$\Phi_b(z)=\exp \left( \frac{1}{4}\int_C \frac{e^{-2i zw }} {\sinh (wb) \sinh (w/b) } \frac{\operatorname{d}\! w}{w} \right)$

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:

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

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

$[\hat p,\hat q]=\frac1{2\pi i},$

and the inversion relation

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

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

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

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. doi:10.1142/S0129055X00000344. MR 1770545.

External links

• quantum dilogarithm in nLab