Quantum dilogarithm

Quantum dilogarithm is a special function defined by the formula

\phi (x)\equiv (x;q)_{\infty }=\prod _{n=0}^{\infty }(1-xq^{n}),\quad |q|<1

{\ displaystyle \ phi (x) \ equiv (x; q) _ {\ infty} = \ prod _ {n = 0} ^ {\ infty} (1-xq ^ {n}), \ quad | q | < one}

{\ displaystyle \ phi (x) \ equiv (x; q) _ {\ infty} = \ prod _ {n = 0} ^ {\ infty} (1-xq ^ {n}), \ quad | q | < one}

In terms of the we have $\phi (x)=E_{q}(x)$ ${\ displaystyle \ phi (x) = E_ {q} (x)}$ ${\ displaystyle \ phi (x) = E_ {q} (x)}$ .

Let be $u, v$ ${\ displaystyle u, v}$ $u, v$ - “q-commuting variables” that are elements of some non-commutative algebra and satisfy the Weyl relation $uv=qvu$ ${\ displaystyle uv = qvu}$ ${\ displaystyle uv = qvu}$ ^[1] . Then the quantum dilogarithm satisfies the Schutzenberger identity ^[2]

\phi (u)\phi (v)=\phi (u+v)

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

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

Faddeev – Volkov identity ^[3]

\phi (v)\phi (u)=\phi (u+v-vu)

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

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

and the identity of Faddeev - Kashaev ^[4]

\phi (v)\phi (u)=\phi (u)\phi (-vu)\phi (v)

{\ displaystyle \ phi (v) \ phi (u) = \ phi (u) \ phi (-vu) \ phi (v)}

{\ displaystyle \ phi (v) \ phi (u) = \ phi (u) \ phi (-vu) \ phi (v)}

The last identity is a quantum generalization of the five-member Rogers identity.

Faddeev's quantum dilogarithm $\Phi _{b}(w)$ ${\ displaystyle \ Phi _ {b} (w)}$ ${\ displaystyle \ Phi _ {b} (w)}$ defined by the following formula:

\Phi _{b}(z)=\exp \left({\frac {1}{4}}\int _{C}{\frac {e^{-2izw}}{\sinh(wb)\sinh(w/b)}}{\frac {\operatorname {d} \!w}{w}}\right)

{\ displaystyle \ Phi _ {b} (z) = \ exp \ left ({\ frac {1} {4}} \ int _ {C} {\ frac {e ^ {- 2izw}} {\ sinh (wb ) \ sinh (w / b)}} {\ frac {\ operatorname {d} \! w} {w}} \ right)}

{\ displaystyle \ Phi _ {b} (z) = \ exp \ left ({\ frac {1} {4}} \ int _ {C} {\ frac {e ^ {- 2izw}} {\ sinh (wb ) \ sinh (w / b)}} {\ frac {\ operatorname {d} \! w} {w}} \ right)}

,

where is the integration loop $C$ ${\ displaystyle C}$ $C$ bypasses the singularity at t = 0 from above ^[5] . The same function can be described using the Voronovich integral formula

\Phi _{b}(x)=\exp \left({\frac {i}{2\pi }}\int _{\mathbb {R} }{\frac {\log(1+e^{tb^{2}+2\pi bx})}{1+e^{t}}}\operatorname {d} \!t\right).

{\ 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}}} \ operatorname {d} \! t \ right).}

{\ 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}}} \ operatorname {d} \! t \ right).}

Ludwig Dmitrievich Faddeev discovered a five-term quantum identity

\Phi _{b}({\hat {p}})\Phi _{b}({\hat {q}})=\Phi _{b}({\hat {q}})\Phi _{b}({\hat {p}}+{\hat {q}})\Phi _{b}({\hat {p}})

{\ displaystyle \ Phi _ {b} ({\ hat {p}}) \ Phi _ {b} ({\ hat {q}}) = \ Phi _ {b} ({\ hat {q}}) \ Phi _ {b} ({\ hat {p}} + {\ hat {q}}) \ Phi _ {b} ({\ hat {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 ${\hat {p}}$ ${\ displaystyle {\ hat {p}}}$ ${\ displaystyle {\ hat {p}}}$ and ${\hat {q}}$ ${\ displaystyle {\ hat {q}}}$ ${\ displaystyle {\ hat {q}}}$ - - normalized quantum mechanical momentum operators and positions satisfying the Heisenberg uncertainty relation

[{\hat {p}},{\hat {q}}]={\frac {1}{2\pi i}},

{\ displaystyle [{\ hat {p}}, {\ hat {q}}] = {\ frac {1} {2 \ pi i}},}

{\ displaystyle [{\ hat {p}}, {\ hat {q}}] = {\ frac {1} {2 \ pi i}},}

and inverse relation

\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)}.

{\ 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)}.}

{\ 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 application in mathematical physics , and the theory of .

The exact relationship between the and $\Phi _{b}$ ${\ displaystyle \ Phi _ {b}}$ ${\ displaystyle \ Phi _ {b}}$ expressed by identity

\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})}}

{\ 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})}}}

{\ 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})}}}

,

which is executed at Im $b^{2}>0$ ${\ displaystyle b ^ {2}> 0}$ ${\ displaystyle b ^ {2}> 0}$ .

Notes

↑ Faddeev, 2011 , p. 65.
↑ The writing of Schützenberger is taken from Faddeev’s article.
↑ Faddeev, 2011 , p. 65, formula (4).
↑ Faddeev, 2011 , p. 65-66, formula (5).
↑ Faddeev, 2011 , p. 67, formula (13).

Literature

Faddeev L.D. Volkov Pentagon for a modular quantum dilogarithm // Funkts. analysis and its application .. - 2011. - T. 45 , no. 4 . - S. 65–71 .
Faddeev LD Currentlike variables in massive and massless integrable models // Quantum groups and their applications in physics (Varenna, 1994), Proc. Internat. School Phys. Enrico Fermi, 127, IOS. - Amsterdam, 1996 .-- pp. 117–135.

Faddeev LD Discrete Heisenberg-Weyl group and modular group // Letters in Mathematical Physics . - 1995. - T. 34 , no. 3 . - S. 249–254 . - DOI : 10.1007 / BF01872779 . - . - arXiv : hep-th / 9504111 .
Faddeev LD, Kashaev RM Quantum dilogarithm // Modern Physics Letters A. - 1994. - T. 9 , no. 5 . - S. 427-434 . - DOI : 10.1142 / S0217732394000447 . - . - arXiv : hep-th / 9310070 .

Faddeev LD, Volkov A. Yu. Abelian current algebra and the Virasoro algebra on the lattice // Physics Letters B. - 1993.- T. 315 , no. 3-4 . - S. 311-318 . - DOI : 10.1016 / 0370-2693 (93) 91618-W . - . - arXiv : hep-th / 9307048 .

Kirillov AN Dilogarithm identities // Progress of Theoretical Physics Supplement . - 1995 .-- T. 118 . - S. 61–142 . - DOI : 10.1143 / PTPS.118.61 . - . - arXiv : hep-th / 9408113 .

Schützenberger MP 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 . - 1953. - T. 236 . - S. 352–353 .

Woronowicz SL Quantum exponential function // Reviews in Mathematical Physics . - 2000. - T. 12 , no. 6 . - S. 873–920 . - DOI : 10.1142 / S0129055X00000344 . - .

Links

quantum dilogarithm on nLab

[_9269f8b801a51b68-1] Faddeev, 2011 , p. 65.

[2] The writing of Schützenberger is taken from Faddeev’s article.

[_81d6acfb5ce7135c-3] Faddeev, 2011 , p. 65, formula (4).

[_caa453e68c1787fe-4] Faddeev, 2011 , p. 65-66, formula (5).

[_33ae3762338e5c3c-5] Faddeev, 2011 , p. 67, formula (13).

Quantum dilogarithm

Notes

Literature

Links

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