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:

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

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

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External links

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