Quantum dilogarithm

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

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

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

$\varphi (u)\varphi (v)=\varphi (u+v),$

Faddeev-Volkov's identity

$\varphi (v)\varphi (u)=\varphi (u+v-vu),$

and Faddeev-Kashaev's identity

$\varphi (v)\varphi (u)=\varphi (u)\varphi (-vu)\varphi (v).$

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

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

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

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:

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

Ludvig Faddeev discovered the quantum pentagon identity:

${\mathrm{\Phi }}_b(\hat{p}){\mathrm{\Phi }}_b(\hat{q})={\mathrm{\Phi }}_b(\hat{q}){\mathrm{\Phi }}_b(\hat{p}+\hat{q}){\mathrm{\Phi }}_b(\hat{p}),$

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

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

and the inversion relation

${\mathrm{\Phi }}_b(x){\mathrm{\Phi }}_b(-x)={\mathrm{\Phi }}_b(0{)}^2{e}^{\pi i{x}^2},{\mathrm{\Phi }}_b(0)=e^{\frac{\pi i}{24}(b^2+b^{-2})}.$

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

The precise relationship between the q-exponential and ${\mathrm{\Phi }}_b$ is expressed by the equality

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

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

References

• 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. doi:10.1007/BF01872779. Bibcode: 1995LMaPh..34..249F. 
• Faddeev, L. D.; Kashaev, R. M. (1994). "Quantum dilogarithm". Modern Physics Letters A 9 (5): 427–434. doi:10.1142/S0217732394000447. Bibcode: 1994MPLA....9..427F. 

• 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. doi:10.1016/0370-2693(93)91618-W. Bibcode: 1993PhLB..315..311F. 

• Kirillov, A. N. (1995). "Dilogarithm identities". Progress of Theoretical Physics Supplement 118: 61–142. doi:10.1143/PTPS.118.61. Bibcode: 1995PThPS.118...61K. 

• 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. Bibcode: 2000RvMaP..12..873W. 

External links

• quantum dilogarithm in nLab