Menu
Home Explore People Places Arts History Plants & Animals Science Life & Culture Technology
On this page
Quantum dilogarithm

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

ϕ ( x ) ≡ ( x ; q ) ∞ = ∏ n = 0 ∞ ( 1 − x q n ) , | q | < 1 {\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 e q ( x ) {\displaystyle e_{q}(x)} .

Let u , v {\displaystyle u,v} be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl's relation u v = q v u {\displaystyle uv=qvu} . Then, the quantum dilogarithm satisfies Schützenberger's identity

ϕ ( u ) ϕ ( v ) = ϕ ( u + v ) , {\displaystyle \phi (u)\phi (v)=\phi (u+v),}

Faddeev-Volkov's identity

ϕ ( v ) ϕ ( u ) = ϕ ( u + v − v u ) , {\displaystyle \phi (v)\phi (u)=\phi (u+v-vu),}

and Faddeev-Kashaev's identity

ϕ ( v ) ϕ ( u ) = ϕ ( u ) ϕ ( − v u ) ϕ ( v ) . {\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 Φ b ( w ) {\displaystyle \Phi _{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 ) , {\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 C {\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:

Φ b ( x ) = exp ⁡ ( i 2 π ∫ R log ⁡ ( 1 + e t b 2 + 2 π b x ) 1 + e t d t ) . {\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:

Φ b ( p ^ ) Φ b ( q ^ ) = Φ b ( q ^ ) Φ b ( p ^ + q ^ ) Φ b ( p ^ ) , {\displaystyle \Phi _{b}({\hat {p}})\Phi _{b}({\hat {q}})=\Phi _{b}({\hat {q}})\Phi _{b}({\hat {p}}+{\hat {q}})\Phi _{b}({\hat {p}}),}

where p ^ {\displaystyle {\hat {p}}} and q ^ {\displaystyle {\hat {q}}} are self-adjoint (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation

[ p ^ , q ^ ] = 1 2 π i {\displaystyle [{\hat {p}},{\hat {q}}]={\frac {1}{2\pi 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 ) . {\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 Φ b {\displaystyle \Phi _{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 ) , {\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 Im ⁡ b 2 > 0 {\displaystyle \operatorname {Im} b^{2}>0} .

We don't have any images related to Quantum dilogarithm yet.
We don't have any YouTube videos related to Quantum dilogarithm yet.
We don't have any PDF documents related to Quantum dilogarithm yet.
We don't have any Books related to Quantum dilogarithm yet.
We don't have any archived web articles related to Quantum dilogarithm yet.