If you haven't found something strange: Quantum Dilogarithm

http://iopscience.iop.org/0305-4470/28/8/014/pdf/0305-4470_28_8_014.pdf

The boring dilogarithm is
$Li_s(x)=\sum_{k=1}^\infty \frac{z^k}{k^s}$
with s replaced by 2. Fine, I'll admit its not really boring. Polylogs have a lot going on. They come up in the integrals for Fermi-Dirac and Bose-Einstein distributions. In addition the monodromy group for the dilog is the Heisenberg group.

But we can deform it while still retaining nice identities like the pentagon identity:

$Li_2(x)+Li_2(y)-Li_2(xy)=Li_2(\frac{x-xy}{1-xy})+Li_2(\frac{y-xy}{1-xy})+log(\frac{1-x}{1-xy})log(\frac{1-y}{1-xy})$

We need to use

$\psi(x)=\prod_{n=1}^\infty (1-x \theta^n)$

This function is related to the dilogarithm as

$\frac{1}{\sqrt{1-x}} exp(Li_2(x)/\epsilon)$

where $\theta=e^\epsilon$

Actually this is true to order epsilon.

Let's let U and V be translation operators in position and momentum space.

$U=xe^{\alpha q} \; V=ye^{\beta p}$

They will $\theta = e^{i \alpha \beta \bar{h}}$ commute.

We end up getting
$\psi(U)\psi(V)=\psi(U+V) \;\;\; \psi(V)\psi(U)=\psi(U-UV+V)=\psi(U)\psi(-UV)\psi(V)$

The second identity is a deformed version of the pentagon identity. We can see that if we try taking the classical $\bar{h}\to0$ limit in order to do stationary phase approximation.

These kinds of equations were found by Baxter in some 3D integrable systems.

If you haven't found something strange

Thursday, December 8, 2011

Quantum Dilogarithm

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