Quantum Dilogarithm

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

:$\phi(x)\equiv(x;q)_\infty=\prod_^\infty (1-xq^n),\quad , q, <1$

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
:$\phi(u) \phi(v)=\phi(u + v),$
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 Rogers' five term dilogarithm identity.

Faddeev's quantum dilogarithm $\Phi_b(w)$ is defined by the following formula:

: $\Phi_b(z)=\exp \left( \frac\int_C \frac \frac \right),$

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:

: $\Phi_b(x)=\exp\left(\frac\int_\frac\,dt\right).$

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

and the inversion relation

: $\Phi_b(x)\Phi_b(-x)=\Phi_b(0)^2 e^,\quad \Phi_b(0)=e^.$

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

:$\Phi_b(z)=\frac,$

valid for $\operatorname b^2>0$.

References

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External links

* {{nlab, id=quantum+dilogarithm, title=quantum dilogarithm

Special functions
Q-analogs