Quantum dilogarithm
In mathematics, the quantum dilogarithm is a special function defined by the formula
- [math]\displaystyle{ \phi(x)\equiv(x;q)_\infty=\prod_{n=0}^\infty (1-xq^n),\quad |q|\lt 1 }[/math]
It is the same as the q-exponential function [math]\displaystyle{ e_q(x) }[/math].
Let [math]\displaystyle{ u,v }[/math] be "q-commuting variables", that is elements of a suitable noncommutative algebra satisfying Weyl's relation [math]\displaystyle{ uv=qvu }[/math]. Then, the quantum dilogarithm satisfies Schützenberger's identity
- [math]\displaystyle{ \phi(u) \phi(v)=\phi(u + v), }[/math]
Faddeev-Volkov's identity
- [math]\displaystyle{ \phi(v) \phi(u)=\phi(u +v -vu), }[/math]
and Faddeev-Kashaev's identity
- [math]\displaystyle{ \phi(v)\phi(u)=\phi(u)\phi(-vu)\phi(v). }[/math]
The latter is known to be a quantum generalization of Rogers' five term dilogarithm identity.
Faddeev's quantum dilogarithm [math]\displaystyle{ \Phi_b(w) }[/math] is defined by the following formula:
- [math]\displaystyle{ \Phi_b(z)=\exp \left( \frac{1}{4}\int_C \frac{e^{-2i zw }} {\sinh (wb) \sinh (w/b) } \frac{dw}{w} \right), }[/math]
where the contour of integration [math]\displaystyle{ C }[/math] 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:
- [math]\displaystyle{ \Phi_b(x)=\exp\left(\frac{i}{2\pi}\int_{\mathbb R}\frac{\log(1+e^{tb^2+2\pi b x})}{1+e^{t}}\,dt\right). }[/math]
Ludvig Faddeev discovered the quantum pentagon identity:
- [math]\displaystyle{ \Phi_b(\hat p)\Phi_b(\hat q) = \Phi_b(\hat q) \Phi_b(\hat p+ \hat q) \Phi_b(\hat p), }[/math]
where [math]\displaystyle{ \hat p }[/math] and [math]\displaystyle{ \hat q }[/math] are self-adjoint (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation
- [math]\displaystyle{ [\hat p,\hat q]=\frac1{2\pi i} }[/math]
and the inversion relation
- [math]\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)}. }[/math]
The quantum dilogarithm finds applications in mathematical physics, quantum topology, cluster algebra theory.
The precise relationship between the q-exponential and [math]\displaystyle{ \Phi_b }[/math] is expressed by the equality
- [math]\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})}, }[/math]
valid for [math]\displaystyle{ \operatorname{Im} b^2\gt 0 }[/math].
References
- Faddeev, L. D. (1994). "Current-Like Variables in Massive and Massless Integrable Models". arXiv:hep-th/9408041.
- Faddeev, L. D. (1995). "Discrete Heisenberg-Weyl group and modular group". Letters in Mathematical Physics 34 (3): 249–254. doi:10.1007/BF01872779. Bibcode: 1995LMaPh..34..249F.
- Faddeev, L. D.; Kashaev, R. M. (1994). "Quantum dilogarithm". Modern Physics Letters A 9 (5): 427–434. doi:10.1142/S0217732394000447. Bibcode: 1994MPLA....9..427F.
- Faddeev, L. D.; Volkov, A. Yu. (1993). "Abelian current algebra and the Virasoro algebra on the lattice". Physics Letters B 315 (3–4): 311–318. doi:10.1016/0370-2693(93)91618-W. Bibcode: 1993PhLB..315..311F.
- Kirillov, A. N. (1995). "Dilogarithm identities". Progress of Theoretical Physics Supplement 118: 61–142. doi:10.1143/PTPS.118.61. Bibcode: 1995PThPS.118...61K.
- Schützenberger, M. P. (1953). "Une interprétation de certaines solutions de l'équation fonctionnelle: F (x + y) = F (x)F (y)". Comptes Rendus de l'Académie des Sciences de Paris 236: 352–353.
- Woronowicz, S. L. (2000). "Quantum exponential function". Reviews in Mathematical Physics 12 (6): 873–920. doi:10.1142/S0129055X00000344. Bibcode: 2000RvMaP..12..873W.
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