# Conformal quantum mechanics and holography in noncommutative space-time

## TL;DR

This paper investigates how noncommutative space-time affects conformal quantum mechanics, revealing deformations in symmetry, correlation functions, and holographic duals, and showing noncommutativity can mitigate quantum instabilities.

## Contribution

It demonstrates the first-order effects of noncommutativity on CQM symmetry, correlation functions, and holography, including scale invariance breaking and stability improvements.

## Key findings

- Deformation of coupling in CQM due to noncommutativity
- Deformation of two and three point correlation functions
- Noncommutative corrections to the Breitenlohner-Freedman bound

## Abstract

We analyze the effects of noncommutativity in conformal quantum mechanics (CQM) using the $\kappa$-deformed space-time as a prototype. Upto the first order in the deformation parameter, the symmetry structure of the CQM algebra is preserved but the coupling in a canonical model of the CQM gets deformed. We show that the boundary conditions that ensure a unitary time evolution in the noncommutative CQM can break the scale invariance, leading to a quantum mechanical scaling anomaly. We calculate the scaling dimensions of the two and three point functions in the noncommutative CQM which are shown to be deformed. The $AdS_2/CFT_1$ duality for the CQM suggests that the corresponding correlation functions in the holographic duals are modified. In addition, the Breitenlohner-Freedman bound also picks up a noncommutative correction. The strongly attractive regime of a canonical model of the CQM exhibit quantum instability. We show that the noncommutativity softens this singular behaviour and its implications for the corresponding holographic duals are discussed.