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