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arXiv:2111.03648

Identity and difference: how topology helps to understand quantum indiscernability

Amaury Mouchet

Published Nov 4, 2021• 0 citations
arXivHTMLPDF

Abstract

This contribution, to be published in Imagine Math 8 to celebrate Michele Emmer's 75th birthday, can be seen as the second part of my previous considerations on the relationships between topology and physics (Mouchet, 2018). Nevertheless, the present work can be read independently. The following mainly focusses on the connection between topology and quantum statistics. I will try to explain to the non specialist how Feynman's interpretation of quantum processes through interference of classical paths (path integrals formulation), makes the dichotomy between bosons and fermions quite natural in three spatial dimensions. In (effective) two dimensions, the recent experimental evidence of intermediate statistics (anyons) (Bartolomei et al. 2020) comfort that topology (of the braids) provides a fertile soil for our understanding of quantum particles.

DOI
10.48550/arxiv.2111.03648

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This contribution, to be published in Imagine Math 8 to celebrate Michele Emmer's 75th birthday, can be seen as the second part of my previous considerations on the relationships between topology and physics (Mouchet, 2018). Nevertheless, the present work can be read independently. The following mainly focusses on the connection between topology and quantum statistics. I will try to explain to the non specialist how Feynman's interpretation of quantum processes through interference of classical paths (path integrals formulation), makes the dichotomy between bosons and fermions quite natural in three spatial dimensions. In (effective) two dimensions, the recent experimental evidence of intermediate statistics (anyons) (Bartolomei et al. 2020) comfort that topology (of the braids) provides a fertile soil for our understanding of quantum particles.
Abstract We introduce the idea of the path integral and use it to study the properties of identical particles under exhange. We explain why the topology of paths through space-time, the fundamental group of the configuration space (the braid group in 2+1 dimensions and the permutation group in 3+1 dimensions) determines the possible particle types. We argue that in 3+1 dimensions one only has bosons and fermions but in 2+1 dimensions other particles, known as anyons, can exist. We describe both the abelian and nonabelian type of anyon and argue why nonabelian anyons could be useful for quantum computation. We introduce the notion of the “quantum dimension” of an anyon which describes the size of the many anyon Hilbert space.
These notes review a description of quantum mechanics in terms of the topology of spaces, basing on the axioms of Topological Quantum Field Theory and path integral formalism. In this description quantum states and operators are encoded by the topology of spaces that are used as modules to build the quantum mechanical model, while expectation values and probabilities are given by topological invariants of spaces, knots and links. The notes focus on the specific way the topology encodes quantum mechanical features, or, equivalently, on how these features can be controlled through the topology. A topological classification of entanglement is discussed, as well as properties of entanglement entropy and basic quantum protocols. The primary aim is to build a less conventional diagrammatic intuition about quantum mechanics, expanding the paradigm of ``Quantum Picturalism".
Background This essay argues that the boson/fermion dichotomy of quantum statistics is not a law of thought but a topological constraint of living in at least three macroscopic spatial dimensions. The exchange phase of two identical particles is ±1 in any space of dimension three or more; only in two dimensions does the braid group allow the continuous anyonic phases of the fractional quantum Hall effect. Content The essay is the popular companion to the author's research paper, The Boson/Fermion Distinction: Spin-Statistics as Structural Invariant (doi:10.5281/zenodo.21938971). It walks the non-specialist through the configuration-space argument (Leinaas-Myrheim), the existence of anyons (Wilczek; Nakamura et al. 2020; Kundu et al. 2023), the Z2 structure of rotation groups in all dimensions d>=3, and the anthropic account of why we observe three large spatial dimensions. Usage Notes The full text is provided as a single Markdown file. Inline citations and a reference list are included; all references were verified against Crossref on 2026-08-14.
Quantum entanglement, a cornerstone of quantum mechanics, presents a unique opportunity to explore the fundamental principles governing topology. This paper investigates the potential of utilizing entanglement to construct and analyze topological spaces, focusing on the influence of entanglement on topological structure and probability distributions. We propose a novel approach that leverages the probabilistic nature of entanglement to generate novel topological properties, ultimately aiming to provide insights into the evolution of complex systems. The core claim is that quantum entanglement, when incorporated into probability geometry, can lead to the discovery of new topological characteristics and a deeper understanding of the interplay between quantum mechanics and topology.
Anyons obeying fractional exchange statistics arise naturally in two dimensions: Hard-core two-body constraints make the configuration space of particles not simply-connected. The braid group describes how topologically-inequivalent exchange paths can be associated to non-trivial geometric phases for abelian anyons. Braid-anyon exchange statistics can also be found in one dimension (1D), but this requires broken Galilean invariance to distinguish different ways for two anyons to exchange. However, recently it was shown that an alternative form of exchange statistics can occur in 1D because hard-core three-body constraints also make the configuration space not simply-connected. Instead of the braid group, the topology of exchange paths and their associated non-trivial geometric phases are described by the traid group. In this article we propose a first concrete model realizing this alternative form of anyonic exchange statistics. Starting from a bosonic lattice model that implements the desired geometric phases with number-dependent Peierls phases, we then define anyonic operators so that the kinetic energy term in the Hamiltonian becomes local and quadratic with respect to them. The ground-state of this traid-anyon-Hubbard model exhibits several indications of exchange statistics intermediate between bosons and fermions, as well as signs of emergent approximate Haldane exclusion statistics. The continuum limit results in a Galilean invariant Hamiltonian with eigenstates that correspond to previously constructed continuum wave functions for traid anyons. This provides not only an a-posteriori justification of our lattice model, but also shows that our construction serves as an intuitive approach to traid anyons, i.e. anyons intrinsic to 1D.