Finite-rank conformal quantum mechanics

TL;DR

The paper analyzes the simplest conformal quantum theories in one dimension with finite-dimensional state spaces using Segal's functorial QFT framework. It shows that conformal invariance forces the Hamiltonian to have zero spectrum, yielding a finite, discrete moduli space and correlation functions that are homogeneous polynomials constrained by one-dimensional Ward identities. The authors classify conformal Hamiltonians via their Jordan/Young diagram structure and examine the resulting observable algebra, highlighting a nontrivial Δ=0 topological sector. They propose extending the analysis to non-diagonalizable dilation generators (analogous to logarithmic CFT) and outline future work on Ward identities in that setting.

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.

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Theorems & Definitions (38)

• Definition 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Example 2.5
• Proposition 2.6
• Definition 2.7
• Remark 2.8
• Proposition 2.9
• Example 2.10