In mathematics, the quantum dilogarithm also known as q-exponential is a special function defined by the formula
ϕ ( x ) ≡ ( x ; q ) ∞ = ∏ n = 0 ∞ ( 1 − x q n ) , | q | < 1 Thus in the notation of the page on q-exponential mentioned above, ϕ ( x ) = E q ( x ) .
Let u , v be “q-commuting variables”, that is elements of a suitable noncommutative algebra satisfying Weyl’s relation u v = q v u . Then, the quantum dilogarithm satisfies Schützenberger’s identity
ϕ ( u ) ϕ ( v ) = ϕ ( u + v ) Faddeev-Volkov's identity
ϕ ( v ) ϕ ( u ) = ϕ ( u + v − v u ) and Faddeev-Kashaev's identity
ϕ ( v ) ϕ ( u ) = ϕ ( u ) ϕ ( − v u ) ϕ ( v ) The latter is known to be a quantum generalization of Roger's five term dilogarithm identity.
Faddeev's quantum dilogarithm Φ b ( w ) is defined by the following formula:
Φ b ( z ) = exp ( 1 4 ∫ C e − 2 i z w sinh ( w b ) sinh ( w / b ) d w w ) 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:
Φ b ( x ) = exp ( i 2 π ∫ R log ( 1 + e t b 2 + 2 π b x ) 1 + e t d t ) . Ludvig Faddeev discovered the quantum pentagon identity:
Φ b ( p ^ ) Φ b ( q ^ ) = Φ b ( q ^ ) Φ b ( p ^ + q ^ ) Φ b ( p ^ ) where p ^ and q ^ are self-adjoint (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation
[ p ^ , q ^ ] = 1 2 π i , and the inversion relation
Φ b ( x ) Φ b ( − x ) = Φ b ( 0 ) 2 e π i x 2 , Φ b ( 0 ) = e π 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 Φ b is expressed by the equality
Φ b ( z ) = E e 2 π i b 2 ( − e π i b 2 + 2 π z b ) E e − 2 π i / b 2 ( − e − π i / b 2 + 2 π z / b ) valid for Im b 2 > 0 .
