AdS2/CFT1, Whittaker vector and Wheeler-De Witt equation

TL;DR

This paper explores the connection between conformal quantum mechanics, the Wheeler-DeWitt equation, and the AdS2/CFT1 correspondence, revealing links to Liouville theory and the Riemann zeros.

Contribution

It demonstrates that expectation values in conformal quantum mechanics form solutions to the Wheeler-DeWitt equation and relate to the AdS2 partition function without relying on a classical Lagrangian.

Findings

The generating function of expectation values solves the Wheeler-DeWitt equation.

It corresponds to the AdS2 partition function as a minisuperspace wave function.

Dilatation expectation values relate to the Riemann zeros' counting function.

Abstract

We study the energy representation of conformal quantum mechanics as the Whittaker vector without specifying classical Lagrangian. We show that a generating function of expectation values among two excited states of the dilatation operator in conformal quantum mechanics is a solution to the Wheeler-DeWitt equation and it corresponds to the AdS2 partition function evaluated as the minisuperspace wave function in Liouville field theory. We also show that the dilatation expectation values in conformal quantum mechanics lead to the asymptotic smoothed counting function of the Riemann zeros.