Cyclic Quantum Dilogarithm and Shift Operator

ar X iv : h ep - t h / 94 10 04 5 v 1 8 O ct 1 99 4 Weyl Pair , Current Algebra and Shift

The Abelian current algebra on the lattice is given from a series of the independent Weyl pairs and the shift operator is constructed by this algebra. So the realization of the operators of the braid group is obtained. For |q| = 1 the shift operator is the product of the theta functions of the generators w n of the current algebra. For |q| = 1 it can be expressed by the quantum dilogarithm of w...

A proof of the pentagon relation for the quantum dilogarithm

1 Introduction The quantum dilogarithm function is given by the following integral: Φ h (z) := exp − 1 4 Ω e −ipz sh(πp)sh(πhp) dp p , sh(p) = e p − e −p 2. Here Ω is a path from −∞ to +∞ making a little half circle going over the zero. So the integral is convergent. It goes back to Barnes [Ba], and appeared in many papers during the last 30 years: • The function Φ h (z) is meromorphic with pol...

Pentagon Relation for the Quantum Dilogarithm and Quantized M

SHIFT OPERATOR FOR PERIODICALLY CORRELATED PROCESSES

The existence of shift for periodically correlated processes and its boundedness are investigated. Spectral criteria for these non-stationary processes to have such shifts are obtained.

Hyperbolic Structure Arising from a Knot Invariant

We study the knot invariant based on the quantum dilogarithm function. This invariant can be regarded as a non-compact analogue of Kashaev’s invariant, or the colored Jones invariant, and is defined by an integral form. The 3-dimensional picture of our invariant originates from the pentagon identity of the quantum dilogarithm function, and we show that the hyperbolicity consistency conditions i...