Finite-rank conformal quantum mechanics

TL;DR

This paper classifies finite-rank conformal Hamiltonians in one-dimensional conformal field theories, showing their correlation functions are polynomial functions determined by geometric data and conformal Ward identities.

Contribution

It provides a complete classification of finite-rank conformal Hamiltonians in 1D CFTs and characterizes their correlation functions as homogeneous polynomials.

Findings

Correlation functions are polynomial functions of geometric data.

Conformal Ward identities determine the scaling behavior of correlators.

Complete classification of finite-rank conformal Hamiltonians in 1D CFTs.

Abstract

In this work, we study the simplest example of the landscape of conformal field theories: one-dimensional CFTs with finite-dimensional state space. Following the definition of quantum field theory given by G. Segal, we formulate the condition under which a one-dimensional QFT (quantum mechanics) possesses conformal symmetry, and we give a complete classification of conformal Hamiltonians with finite rank. It turns out that correlation functions in such theories are polynomial functions of the underlying geometric data. Moreover, the one-dimensional conformal Ward identities determine their scaling behavior, so that the correlators of the conformal observables are, in fact, homogeneous polynomials.