nLab spin-statistics theorem

Contents

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

In quantum field theory on Minkowski space all fields must transform according to a definite (finite dimensional) representation of the universal cover of the Poincare group, which determines the spin of each field. From the representation theory of the Poincare group it is known that the spin ss is a number s=n2s = \frac{n}{2} with nn \in \mathbb{N}.

On the other hand, if we take fields to be pointwise localized in the sense of the Wightman axioms, then the locality axiom (also known as Einstein microcausality ) says that spacelike separated field operators either commute or anticommute: Two Fermionic fields anticommute, two Bosonic fields commute, a Fermionic and a Bosonic field commute.

The spin-statistics theorem states that fields with integer spin ss (n is even) are Bosonic fields, fields with half-integer spin (n is uneven) are Fermionic fields. A better name for the theorem would therefore be spin- commutation theorem, the name spin- statistics theorem stems from the fact that Bosons (the particles associated to Bosonic fields) are social, multiple particles can exist in the same quantum state, while Fermions are not social: The Pauli exclusion principle says maximally one Fermion can exist in a given quantum state. This leads to different partition functions in statistical mechanics of systems consisting of Bosons only and of Fermions only, hence the name.

The statement and proof of the theorem depend on the framework for quantum field theory that is used, therefore there are, strictly speaking, several versions of the spin-statistics theorem, but the physical interpretation is always the same.

Statement

In Algebraic quantum field theory

In the Haag-Kastler approach (“algebraic quantum field theory”) the Bisognano-Wichmann theorem states a relation of the representation of the Poincare group on certain local algebras of local nets of algebras of observables and their modular groups. If this relation holds for a given net, then this net is said to fulfill the Bisognano-Wichmann property.

(GuidoLongo 94, Verch 01)

References

The original formulation of the spin-statistics theorem:

An argument via Dirac charge quantization is offered in:

Monographs:

A statement and proof of both a spin-statistics and a PCT theorem in the axiomatics of algebraic quantum field theory:

Discussion/proof of the spin-statistics theorem for non-relativistic particles via the topology/homotopy theory of their configuration spaces of points (cf. also braid group statistics):

Generalization for AQFT on curved spacetime is in

General review:

Brief exposition:

  • John Baez, Spin, Statistics, CPT and All That Jazz (2012) [web]

Discussion relating the spin-statistics theorem to extended topological field theory, categorification (via 2-rings) and Deligne's theorem on tensor categories:

and via hermitian functorial field theory:

Last revised on August 9, 2024 at 10:09:16. See the history of this page for a list of all contributions to it.