Canonical quantization of a massive particle on $AdS_3$

TL;DR

This paper performs a detailed canonical quantization of a massive particle on $AdS_3$, revealing different quantum theories depending on the order of constraints, and connects these results with quantum field theory and stability bounds.

Contribution

It explicitly constructs the symplectic structure and canonical coordinates for a particle on $AdS_3$, and compares two quantization approaches, highlighting their differences and physical implications.

Findings

Different quantum theories arise depending on the order of imposing constraints.

The Hilbert space decomposes into irreducible $sl(2,R)$ representations.

One quantum theory aligns with the Breitenlohner-Freedman bound.

Abstract

The classical theory for a massive free particle moving on the group manifold $AdS_3 \cong SL(2, \mathbb{R})$ is analysed in detail. In particular a symplectic structure and two different sets of canonical coordinates are explicitly found, corresponding to the Cartan and Iwasawa decomposition of the group. Canonical quantization is performed in two different ways; by imposing the future-directed constraint before and after quantization. It is found that this leads to different quantum theories. The Hilbert space of either theory decomposes into the sum of certain irreducible representations of $sl(2,\mathbb{R}) \oplus sl(2,\mathbb{R})$; however, depending on how the constraint is imposed we get different representations. Quantization of the mass occurs, although a continuum exists in the unconstrained theory corresponding to particles that can reverse their direction in time. A quantization in terms of the ``chiral'' variables of the theory is also carried out giving the same results. Comparisons are made between QFT in $AdS_3$ and the quantum mechanics derived, and it is found that one of the quantum theories is consistent with the Breitenlohner-Freedman bound.