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Exchanging two identical particles

S. J. van Enk

Published Oct 11, 2018• 0 citations
OpenAlex

Abstract

The phase factor $(-1)^{2s}$ that features in the exchange symmetry for identical spin-$s$ fermions or bosons contributes to but is not identical to the phase factor one may observe when one actually exchanges two such particles. The observable phase contains {\em single-particle} geometric and dynamical phases as well, induced by both spin and spatial exchange transformations. Extending the analysis to (abelian) anyons by incorporating the Aharonov-Bohm effect shows that the topological phase two anyons pick up under an interchange is at bottom a single-particle effect, in contrast to the fermion/boson exchange phase factor $(-1)^{2s}$, which is a two-particle effect.

Related papers

Exchanging identical particles and topological quantum computing

S. J. van Enk

The phase factor $(-1)^{2s}$ that features in the exchange symmetry for identical spin-$s$ fermions or bosons is not simply and automatically equal to the phase factor one can observe in an interference experiment that involves physically exchanging two such particles. The observable phase contains, in general, single-particle geometric and dynamical phases as well, induced by both spin and spatial exchange transformations. By extending the analysis to (non-abelian) anyons it is argued that, similarly, there are single-anyon geometric and dynamical contributions in addition to purely topological unitary transformations that accompany physical exchanges of anyons. Work remains to be done in order to demonstrate---if it is still true---that those additional contributions to the gates in anyonic topological quantum computers do not destroy the inherent robustness of the ideal gates. This negative result is described most clearly in terms of the Berry matrix.

arXiv:1810.05208•Oct 11, 2018•2 citations

Fundamental Dual-Phase Theory of Identical Particles: The Existence Conditions for Non-Abelian Anyons in 2D Topology (Series II)

Jian-Hua Wang

This work uncovers an essential dual-phase structure of identical particles, revealing a fundamental limitation in existing quantum statistical theories. For a considerable period, it was commonly assumed that an interchange of two particles would be accompanied by a fixed phase factor. Here, the author discovers that the exchange process inherently involves two distinct phases: a relative phase α and an exchange phase β, rigorously related through α = β-1 = e±iθ. Specifically, α = β = 1 characterizes bosons, while α = β = -1 describes fermions. This finding necessitates a radical change in handling identical particles. Within this new framework, clockwise exchange yields α = eiθ and β = e-iθ, while counter-clockwise exchange gives α = e-iθ and β = eiθ. Crucially, both scenarios, despite their topological differences, are physically equivalent, governed by the universal constraint αβ = 1 for all identical particles. This framework not only fundamentally changes the conceptual basis of quantum statistics but also provides critical theoretical constraints on the existence conditions of non-Abelian anyons in two-dimensional topological systems.

•Oct 3, 2025•0 citations

Introduction

Jiannis K. Pachos

Symmetries play a central role in physics. They dictate what one can change in a physical system without affecting any of its properties. You might have encountered symmetries like translational symmetry, where a system remains unchanged if it is spatially translated by an arbitrary distance. A system with rotational symmetry, however, is invariant under rotations. Some symmetries, like the ones mentioned above, give information about the structure of the system. Others have to do with the more fundamental physical framework that we adopt. An example for this is the invariance under Lorentz transformations in relativistic physics. Other types of symmetries can be even more subtle. For example, it is rather self-evident that physics should remain unchanged if we exchange two identical point-like particles. Nevertheless, this fundamental property that we call statistical symmetry gives rise to rich and beautiful physics. In three spatial dimensions it dictates the existence of bosons and fermions. These are particles with very different quantum mechanical properties. Their wave function acquires a +1 or a -1 phase, respectively, whenever two particles are exchanged. A direct consequence of this is that bosons can actually occupy the same state. In contrast, fermions can only be stacked together with each particle occupying a different state. When one considers two spatial dimensions, a wide variety of statistical behaviours is possible. Apart from bosonic and fermionic behaviours, arbitrary phase factors, or even non-trivial unitary evolutions, can be obtained when two particles are exchanged (Leinaas and Myrheim, 1977).

Cambridge University Press eBooks•Apr 12, 2012•0 citations

Non-Abelian Anyons: Statistical Repulsion and Topological Quantum Computation

Viktor Qvarfordt

As opposed to classical mechanics, quantum mechanical particles can be truly identical and lead to new and interesting phenomena. Identical particles can be of different types, determined by their exchange symmetry, which in turn gives rise to statistical repulsion. The exchange symmetry is given by a representation of the exchange group; the fundamental group of the configuration space of identical particles. In three dimensions the exchange group is the permutation group and there are only two types of identical particles; bosons and fermions. While any number of bosons can be at the same place or in the same state, fermions repel each other. In two dimensions the exchange group is the braid group and essentially any exchange symmetry is allowed, such particles are called anyons. Abelian anyons are described by abelian representations of the exchange group and can be seen as giving a continuous interpolation between bosons and fermions. Non-abelian anyons are much more complex and their statistical repulsion is yet largely unexplored. We use the framework of modular tensor categories to show how the statistical repulsion of non-abelian anyons depends on the exchange symmetry. The Fibonacci anyon model is studied, for which explicit results are obtained. We also show how Fibonacci anyons can be used to implement topological quantum computation, providing topologically stable quantum information encoded in the state of non-abelian anyons that can be manipulated via the non-abelian exchange symmetry.

•Jan 1, 2017•3 citations

Aharonov-Bohm Effect and the Supersymmetry of Identical Anyons

Vít Jakubský

We briefly review the relation between the Aharonov-Bohm effect and the dynamical realization of anyons. We show how the particular symmetries of the Aharonov-Bohm model give rise to the (nonlinear) supersymmetry of the two-body system of identical anyons.

Acta Polytechnica•Jan 5, 2010•0 citations

Statistics of Indistinguishable Particles

C. Wittig

The wave function of a system containing identical particles takes into account the relationship between a particle's intrinsic spin and its statistical property. Specifically, the exchange of two identical particles having odd-half-integer spin results in the wave function changing sign, whereas the exchange of two identical particles having integer spin is accompanied by no such sign change. This is embodied in a term (-1)(2s), which has the value +1 for integer s (bosons), and -1 for odd-half-integer s (fermions), where s is the particle spin. All of this is well-known. In the nonrelativistic limit, a detailed consideration of the exchange of two identical particles shows that exchange is accompanied by a 2pi reorientation that yields the (-1)(2s) term. The same bookkeeping is applicable to the relativistic case described by the proper orthochronous Lorentz group, because any proper orthochronous Lorentz transformation can be expressed as the product of spatial rotations and a boost along the direction of motion.

The Journal of Physical Chemistry A•Mar 19, 2009•2 citations
The phase factor (−1)2s(-1)^{2s}(−1)2s that features in the exchange symmetry for identical spin-sss fermions or bosons contributes to but is not identical to the phase factor one may observe when one actually exchanges two such particles. The observable phase contains {\em single-particle} geometric and dynamical phases as well, induced by both spin and spatial exchange transformations. Extending the analysis to (abelian) anyons by incorporating the Aharonov-Bohm effect shows that the topological phase two anyons pick up under an interchange is at bottom a single-particle effect, in contrast to the fermion/boson exchange phase factor (−1)2s(-1)^{2s}(−1)2s, which is a two-particle effect.
The phase factor (−1)2s(-1)^{2s}(−1)2s that features in the exchange symmetry for identical spin-sss fermions or bosons is not simply and automatically equal to the phase factor one can observe in an interference experiment that involves physically exchanging two such particles. The observable phase contains, in general, single-particle geometric and dynamical phases as well, induced by both spin and spatial exchange transformations. By extending the analysis to (non-abelian) anyons it is argued that, similarly, there are single-anyon geometric and dynamical contributions in addition to purely topological unitary transformations that accompany physical exchanges of anyons. Work remains to be done in order to demonstrate---if it is still true---that those additional contributions to the gates in anyonic topological quantum computers do not destroy the inherent robustness of the ideal gates. This negative result is described most clearly in terms of the Berry matrix.
This work uncovers an essential dual-phase structure of identical particles, revealing a fundamental limitation in existing quantum statistical theories. For a considerable period, it was commonly assumed that an interchange of two particles would be accompanied by a fixed phase factor. Here, the author discovers that the exchange process inherently involves two distinct phases: a relative phase α and an exchange phase β, rigorously related through α = β-1 = e±iθ. Specifically, α = β = 1 characterizes bosons, while α = β = -1 describes fermions. This finding necessitates a radical change in handling identical particles. Within this new framework, clockwise exchange yields α = eiθ and β = e-iθ, while counter-clockwise exchange gives α = e-iθ and β = eiθ. Crucially, both scenarios, despite their topological differences, are physically equivalent, governed by the universal constraint αβ = 1 for all identical particles. This framework not only fundamentally changes the conceptual basis of quantum statistics but also provides critical theoretical constraints on the existence conditions of non-Abelian anyons in two-dimensional topological systems.
Symmetries play a central role in physics. They dictate what one can change in a physical system without affecting any of its properties. You might have encountered symmetries like translational symmetry, where a system remains unchanged if it is spatially translated by an arbitrary distance. A system with rotational symmetry, however, is invariant under rotations. Some symmetries, like the ones mentioned above, give information about the structure of the system. Others have to do with the more fundamental physical framework that we adopt. An example for this is the invariance under Lorentz transformations in relativistic physics. Other types of symmetries can be even more subtle. For example, it is rather self-evident that physics should remain unchanged if we exchange two identical point-like particles. Nevertheless, this fundamental property that we call statistical symmetry gives rise to rich and beautiful physics. In three spatial dimensions it dictates the existence of bosons and fermions. These are particles with very different quantum mechanical properties. Their wave function acquires a +1 or a -1 phase, respectively, whenever two particles are exchanged. A direct consequence of this is that bosons can actually occupy the same state. In contrast, fermions can only be stacked together with each particle occupying a different state. When one considers two spatial dimensions, a wide variety of statistical behaviours is possible. Apart from bosonic and fermionic behaviours, arbitrary phase factors, or even non-trivial unitary evolutions, can be obtained when two particles are exchanged (Leinaas and Myrheim, 1977).
As opposed to classical mechanics, quantum mechanical particles can be truly identical and lead to new and interesting phenomena. Identical particles can be of different types, determined by their exchange symmetry, which in turn gives rise to statistical repulsion. The exchange symmetry is given by a representation of the exchange group; the fundamental group of the configuration space of identical particles. In three dimensions the exchange group is the permutation group and there are only two types of identical particles; bosons and fermions. While any number of bosons can be at the same place or in the same state, fermions repel each other. In two dimensions the exchange group is the braid group and essentially any exchange symmetry is allowed, such particles are called anyons. Abelian anyons are described by abelian representations of the exchange group and can be seen as giving a continuous interpolation between bosons and fermions. Non-abelian anyons are much more complex and their statistical repulsion is yet largely unexplored. We use the framework of modular tensor categories to show how the statistical repulsion of non-abelian anyons depends on the exchange symmetry. The Fibonacci anyon model is studied, for which explicit results are obtained. We also show how Fibonacci anyons can be used to implement topological quantum computation, providing topologically stable quantum information encoded in the state of non-abelian anyons that can be manipulated via the non-abelian exchange symmetry.
The wave function of a system containing identical particles takes into account the relationship between a particle's intrinsic spin and its statistical property. Specifically, the exchange of two identical particles having odd-half-integer spin results in the wave function changing sign, whereas the exchange of two identical particles having integer spin is accompanied by no such sign change. This is embodied in a term (-1)(2s), which has the value +1 for integer s (bosons), and -1 for odd-half-integer s (fermions), where s is the particle spin. All of this is well-known. In the nonrelativistic limit, a detailed consideration of the exchange of two identical particles shows that exchange is accompanied by a 2pi reorientation that yields the (-1)(2s) term. The same bookkeeping is applicable to the relativistic case described by the proper orthochronous Lorentz group, because any proper orthochronous Lorentz transformation can be expressed as the product of spatial rotations and a boost along the direction of motion.