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Quantum dilogarithm
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<p>In mathematics, the quantum dilogarithm is a <a href="/facts/Special_function/JETSeucA">special function</a> defined by the formula </p> 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} <p>It is the same as the <a href="/facts/Q-exponential/t9gJxytF"><i>q</i>-exponential</a> function e q ( x ) {\displaystyle e_{q}(x)} . </p><p>Let u , v {\displaystyle u,v} be "<i>q</i>-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 </p> ϕ ( u ) ϕ ( v ) = ϕ ( u + v ) , {\displaystyle \phi (u)\phi (v)=\phi (u+v),} <p>Faddeev-Volkov's identity </p> ϕ ( v ) ϕ ( u ) = ϕ ( u + v − v u ) , {\displaystyle \phi (v)\phi (u)=\phi (u+v-vu),} <p>and Faddeev-Kashaev's identity </p> ϕ ( v ) ϕ ( u ) = ϕ ( u ) ϕ ( − v u ) ϕ ( v ) . {\displaystyle \phi (v)\phi (u)=\phi (u)\phi (-vu)\phi (v).} <p>The latter is known to be a quantum generalization of <a href="/facts/Dilogarithm/lsnCj3Gb">Rogers' five term dilogarithm identity</a>. </p><p>Faddeev's quantum dilogarithm Φ b ( w ) {\displaystyle \Phi _{b}(w)} is defined by the following formula: </p> 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),} <p>where the contour of integration C {\displaystyle C} goes along the real axis outside a small neighborhood of the origin and deviates into the <a href="/facts/Upper_half-plane/7fPqrowr">upper half-plane</a> near the origin. The same function can be described by the integral formula of Woronowicz: </p> 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).} <p><a href="/facts/Ludvig_Faddeev/gR8t21SP">Ludvig Faddeev</a> discovered the quantum pentagon identity: </p> 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}}),} <p>where p ^ {\displaystyle {\hat {p}}} and q ^ {\displaystyle {\hat {q}}} are <a href="/facts/Self-adjoint/rKqGtlpy">self-adjoint</a> (normalized) quantum mechanical momentum and position operators satisfying Heisenberg's commutation relation </p> [ p ^ , q ^ ] = 1 2 π i {\displaystyle [{\hat {p}},{\hat {q}}]={\frac {1}{2\pi i}}} <p>and the inversion relation </p> 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)}.} <p>The quantum dilogarithm finds applications in <a href="/facts/Mathematical_physics/sLKg8T1G">mathematical physics</a>, <a href="/facts/Quantum_topology/Bw33oFLk">quantum topology</a>, <a href="/facts/Cluster_algebra/FQVr3Wcp">cluster algebra</a> theory. </p><p>The precise relationship between the <i>q</i>-exponential and Φ b {\displaystyle \Phi _{b}} is expressed by the equality </p> 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})}},} <p>valid for Im b 2 > 0 {\displaystyle \operatorname {Im} b^{2}>0} . </p> <ul><li>Faddeev, L. D. (1994). "Current-Like Variables in Massive and Massless Integrable Models". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/hep-th/9408041">hep-th/9408041</a>.</li> <li>Faddeev, L. D. (1995). "Discrete Heisenberg-Weyl group and modular group". <i><a href="/facts/Letters_in_Mathematical_Physics/XA9EgydV">Letters in Mathematical Physics</a></i>. 34 (3): 249–254. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/hep-th/9504111">hep-th/9504111</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1995LMaPh..34..249F">1995LMaPh..34..249F</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01872779">10.1007/BF01872779</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1345554">1345554</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119435070">119435070</a>.</li> <li>Faddeev, L. D.; Kashaev, R. M. (1994). "Quantum dilogarithm". <i><a href="/facts/Modern_Physics_Letters_A/X1RBtMBg">Modern Physics Letters A</a></i>. 9 (5): 427–434. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/hep-th/9310070">hep-th/9310070</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1994MPLA....9..427F">1994MPLA....9..427F</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2FS0217732394000447">10.1142/S0217732394000447</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1264393">1264393</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:6172445">6172445</a>.</li></ul> <ul><li>Faddeev, L. D.; Volkov, A. Yu. (1993). "Abelian current algebra and the Virasoro algebra on the lattice". <i><a href="/facts/Physics_Letters_B/6EkxhxM5">Physics Letters B</a></i>. 315 (3–4): 311–318. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/hep-th/9307048">hep-th/9307048</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1993PhLB..315..311F">1993PhLB..315..311F</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0370-2693%2893%2991618-W">10.1016/0370-2693(93)91618-W</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:10294434">10294434</a>.</li></ul> <ul><li>Kirillov, A. N. (1995). "Dilogarithm identities". <i><a href="/facts/Progress_of_Theoretical_Physics_Supplement/IU51VW29">Progress of Theoretical Physics Supplement</a></i>. 118: 61–142. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/hep-th/9408113">hep-th/9408113</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1995PThPS.118...61K">1995PThPS.118...61K</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1143%2FPTPS.118.61">10.1143/PTPS.118.61</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1356515">1356515</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119177149">119177149</a>.</li></ul> <ul><li>Schützenberger, M. P. (1953). "Une interprétation de certaines solutions de l'équation fonctionnelle: F (x + y) = F (x)F (y)". <i><a href="/facts/Comptes_Rendus_de_l%27Acad%C3%A9mie_des_Sciences_de_Paris/9yR9XTN5">Comptes Rendus de l'Académie des Sciences de Paris</a></i>. 236: 352–353.</li></ul> <ul><li>Woronowicz, S. L. (2000). "Quantum exponential function". <i><a href="/facts/Reviews_in_Mathematical_Physics/LrNvFXoF">Reviews in Mathematical Physics</a></i>. 12 (6): 873–920. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2000RvMaP..12..873W">2000RvMaP..12..873W</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2FS0129055X00000344">10.1142/S0129055X00000344</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1770545">1770545</a>.</li></ul>