Quantum dilogarithm
In mathematics, the quantum dilogarithm is a special function defined by the formula
It is the same as the q-exponential function.
Let be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl’s relation. Then, the quantum dilogarithm satisfies Schützenberger’s identity
Faddeev-Volkov's identity
and Faddeev-Kashaev's identity
The latter is known to be a quantum generalization of Rogers' five term dilogarithm identity.
Faddeev's quantum dilogarithm is defined by the following formula:
where the contour of integration 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:
Ludvig Faddeev discovered the quantum pentagon identity:
where and are self-adjoint quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation
and the inversion relation
The quantum dilogarithm finds applications in mathematical physics, quantum topology, cluster algebra theory.
The precise relationship between the q-exponential and is expressed by the equality
valid for.
