[Paper Review] The Strange World of Non-amenable Symmetries

Erhard Seiler|arXiv (Cornell University)|Nov 12, 2010

Black Holes and Theoretical Physics4 references3 citations

TL;DR

This paper investigates nonlinear sigma models with non-compact, non-amenable symmetry groups—such as hyperbolic target spaces—demonstrating that spontaneous symmetry breaking (SSB) occurs even in one and two dimensions, violating the Mermin-Wagner theorem due to non-amenability. The key contribution is the existence of a unique unitary and continuous ground state representation, despite the absence of a normalizable ground state, and the emergence of a non-separable Hilbert space via Osterwalder-Schrödinger reconstruction, with open questions on the 2D continuum limit.

ABSTRACT

Nonlinear sigma models with non-compact target space and non-amen-able symmetry group were introduced long ago in the study of disordered electron systems. They also occur in dimensionally reduced quantum gravity; recently they have been considered in the context of the AdS/CFT correspondence. These models show spontaneous symmetry breaking in {\it any} dimension, even one and two (superficially in contradiction with the Mermin-Wagner theorem) as a consequence of the non-amenability of their symmetry group. The low-dimensional models show other peculiarities: invariant observables remain dependent on boundary conditions in the thermodynamic limit and the Osterwalder-Schrader reconstruction yields a non-separable Hilbert space. The ground state space, however, under quite general conditions, carries a unique unitary and continuous representation. The existence of a continuum limit in 2D is an open question: while the perturbative Renormalization Group suggests triviality, other arguments hint at the existence of a conformally invariant continuum limit at least for suitable observables. This talk gives an overview of the work done during the last several years in collaboration first of all with Max Niedermaier, some of it also with Peter Weisz and Tony Duncan.

Motivation & Objective

• To understand the behavior of quantum field theories with non-compact, non-amenable symmetry groups, particularly in low dimensions where standard theorems like Mermin-Wagner do not apply.
• To analyze the structure of ground states and invariant observables in quantum mechanics on hyperbolic spaces and lattice models with such symmetries.
• To investigate the implications of Osterwalder-Schrödinger reconstruction for Hilbert space separability and the existence of a continuum limit in 2D models.
• To assess the validity of perturbative renormalization group results suggesting infrared freedom versus conformal invariance in 2D hyperbolic sigma models.
• To determine whether a non-trivial continuum limit exists, especially in light of conflicting arguments from perturbation theory and formal conformal field theory calculations.

Proposed method

• Analyzes quantum mechanics on hyperbolic spaces $\mathbb{H}_N = SO_o(1,N)/SO_o(N)$ using the Laplace-Beltrami operator and Mehler-Fock transform to study spectral properties and generalized ground states.
• Applies the Osterwalder-Schrödinger reconstruction procedure to the Euclidean path integral, showing it leads to a non-separable Hilbert space due to continuous spectrum and non-normalizable ground states.
• Uses Ward identities and correlation functions to detect spontaneous symmetry breaking in finite-volume lattice models, even in 1D and 2D.
• Applies perturbative one-loop renormalization group analysis, showing the model flows under Ricci flow, suggesting infrared asymptotic freedom.
• Considers axiomatic constraints on quantum field theories with non-unitary representations of non-amenable groups, ruling out standard field multiplets with unbroken symmetry.
• Evaluates formal calculations by Haba suggesting a conformal field theory with central charge $c=1$ for $\beta > 1/(3\pi)$, indicating a possible non-Gaussian continuum limit.

Experimental results

Research questions

• RQ1 Does spontaneous symmetry breaking occur in nonlinear sigma models with non-compact, non-amenable symmetry groups in one and two dimensions, despite the Mermin-Wagner theorem?
• RQ2 How does the Osterwalder-Schrödinger reconstruction procedure affect the Hilbert space structure in models with continuous spectrum and non-normalizable ground states?
• RQ3 Can a non-trivial continuum limit exist in 2D nonlinear sigma models with hyperbolic target space, given conflicting evidence from perturbative renormalization group and formal conformal field theory?
• RQ4 Why does the standard argument for the absence of mass gap fail in these models, and what is the nature of the low-energy spectrum?
• RQ5 Is it possible to construct a unitary quantum field theory with unbroken non-amenable symmetry, or must the symmetry always be spontaneously broken?

Key findings

• Spontaneous symmetry breaking occurs in any dimension, including one and two, due to the non-amenability of the symmetry group, which invalidates the Mermin-Wagner theorem.
• The ground state space carries a unique unitary and continuous representation, even though no normalizable ground state vector exists.
• The Osterwalder-Schrödinger reconstruction yields a non-separable Hilbert space, a consequence of the continuous spectrum and non-normalizable generalized ground states.
• Perturbative renormalization group analysis suggests infrared asymptotic freedom, implying the model may be trivial in the continuum limit, but this conflicts with expectations from curvature sensitivity at large distances.
• Formal calculations by Haba suggest a conformally invariant continuum limit with central charge $c=1$ for $\beta > 1/(3\pi)$, hinting at a non-Gaussian fixed point.
• Axiomatic considerations rule out the existence of a multiplet of quantum fields transforming under a non-unitary representation of the symmetry group, suggesting that either SSB must occur or only a subset of fields can form a quantum field theory.