Quantum dilogarithm
In mathematics, the quantum dilogarithm also known as q-exponential is a special function defined by the formula
Thus in the notation of the page on q-exponential mentioned above, .
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 Roger's 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 (normalized) 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 Im .
References
- Faddeev, L. D. (1994). "Current-Like Variables in Massive and Massless Integrable Models". arXiv:hep-th/9408041.
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ignored (help) - Faddeev, L. D. (1995). "Discrete Heisenberg-Weyl group and modular group". Letters in Mathematical Physics. 34 (3): 249–254. arXiv:hep-th/9504111. Bibcode:1995LMaPh..34..249F. doi:10.1007/BF01872779. MR 1345554.
- Faddeev, L. D.; Kashaev, R. M. (1994). "Quantum dilogarithm". Modern Physics Letters A. 9 (5): 427–434. arXiv:hep-th/9310070. Bibcode:1994MPLA....9..427F. doi:10.1142/S0217732394000447. MR 1264393.
- 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. arXiv:hep-th/9307048. Bibcode:1993PhLB..315..311F. doi:10.1016/0370-2693(93)91618-W.
- Kirillov, A. N. (1995). "Dilogarithm identities". Progress of Theoretical Physics Supplement. 118: 61–142. arXiv:hep-th/9408113. Bibcode:1995PThPS.118...61K. doi:10.1143/PTPS.118.61. MR 1356515.
- 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. Bibcode:2000RvMaP..12..873W. doi:10.1142/S0129055X00000344. MR 1770545.
External links
- quantum dilogarithm at the nLab