[Paper Review] Quantum Mechanics in the Infrared

Djordje Radicevic|arXiv (Cornell University)|Aug 25, 2016

Quantum chaos and dynamical systems21 references5 citations

TL;DR

This paper introduces an algebraic renormalization group (RG) framework for quantum mechanics on flat target spaces, using Hamiltonian-based coarse-graining via group structure to flow to long times. It shows that the infrared fixed point is a classical probability theory, with entropy of the resulting distribution serving as a diagnostic for quantum ergodicity—offering a new, non-perturbative tool to study long-time quantum dynamics and chaos.

ABSTRACT

This paper presents an algebraic formulation of the renormalization group flow in quantum mechanics on flat target spaces. For any interacting quantum mechanical theory, the fixed point of this flow is a theory of classical probability, not a different effective quantum mechanics. Each energy eigenstate of the UV Hamiltonian flows to a probability distribution whose entropy is a natural diagnostic of quantum ergodicity of the original state. These conclusions are supported by various examples worked out in some detail.

Motivation & Objective

• To develop a systematic renormalization group procedure for quantum mechanics that captures long-time dynamics, distinct from standard Wilsonian RG.
• To understand the infrared (IR) limit of quantum systems, particularly how quantum states evolve under long-time coarse-graining.
• To identify whether the IR limit is an effective quantum theory or a classical statistical theory, especially in chaotic systems.
• To introduce a new diagnostic—entropy of the IR probability distribution—for quantum ergodicity, grounded in algebraic decimation.
• To explore the role of group structure in defining observables and coarse-graining, especially in systems with finite Hilbert space dimensions.

Proposed method

• Formulates a Hamiltonian-based RG flow using algebraic decimation, where observables are coarse-grained by restricting to subalgebras of the original algebra.
• Applies the method to quantum mechanics on flat group manifolds (e.g., tori), leveraging group structure to define consistent time-discretized coarse-graining.
• Uses the path integral formulation as motivation but avoids direct path integral methods, instead focusing on operator algebra and spectral decomposition.
• Defines the IR limit as the result of repeated decimation, leading to a classical probability distribution over energy eigenstates.
• Computes the entropy of the resulting probability distribution as a measure of quantum ergodicity, using the overlap function between initial states and energy eigenstates.
• Extends the framework to finite-dimensional Hilbert spaces, particularly those of dimension $N = 2^n$, and generalizes to prime $N$ via single-step decimation to classical regime.

Experimental results

Research questions

• RQ1 What is the infrared fixed point of a generic interacting quantum mechanical system under long-time coarse-graining?
• RQ2 Can a systematic renormalization group flow be defined in quantum mechanics that avoids the limitations of standard Wilsonian RG in 0+1 dimensions?
• RQ3 Does the long-time limit of quantum dynamics lead to an effective quantum theory or a classical statistical theory?
• RQ4 How can quantum ergodicity be diagnosed in a way that is intrinsic to the system's algebraic structure and not dependent on semiclassical approximations?
• RQ5 What is the role of group structure in enabling consistent coarse-graining and defining the IR limit in quantum mechanics?

Key findings

• The infrared fixed point of any interacting quantum mechanical system on a flat target space is not another quantum theory, but a classical theory of probability.
• The entropy of the resulting probability distribution in the IR is a natural and intrinsic diagnostic for quantum ergodicity of the original state.
• For Hilbert spaces of dimension $N = 2^n$, repeated algebraic decimation leads to the classical probability regime, with entropy computable via equation (7).
• When $N$ is prime, only a single decimation step is needed to reach the classical probability regime, independent of $N$.
• The framework naturally identifies Majorana fermions as the algebraic remnant of $bZ_2$ quantum mechanics after one decimation, corresponding to classical bit-like degrees of freedom.
• The method avoids path integral singularities and provides a non-perturbative, algebraic alternative to conventional RG, particularly effective in systems with compact, finite-dimensional Hilbert spaces.