spin-statistics theorem in nLab

Contents

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 n∈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)

…

• Atiyah-Sutcliffe conjecture

• PCT theorem

References

The original formulation of the spin-statistics theorem:

• Markus Fierz, Über die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin, Helvetica Physica Acta. 12 1 (1939) 3-37 [doi:10.5169/seals-110930, pdf]

• Wolfgang Pauli, The connection between spin and statistics, Phys. Rev. 58 (1940) 716-722 [doi:10.1103/PhysRev.58.716]

An argument via Dirac charge quantization is offered in:

• Richard Feynman, pp. 48 in: The reason for antiparticles, in: Richard Feynman and Steven Weinberg: Elementary Particles and the Laws of Physics: The 1986 Dirac Memorial Lectures, Cambridge U. Press (1987) [doi:10.1017/CBO9781107590076]

Monographs:

• Raymond F. Streater, Arthur S. Wightman, PCT, Spin and Statistics, and All That, Princeton University Press (1989, 2000) [ISBN:9780691070629, jstor:j.ctt1cx3vcq]

• Franco Strocchi, §4.2 in: An Introduction to Non-Perturbative Foundations of Quantum Field Theory, Oxford University Press (2013) [doi:10.1093/acprof:oso/9780199671571.001.0001]

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

• Daniele Guido, Roberto Longo, An Algebraic Spin and Statistics Theorem, Commun. Math.Phys. 172 (1995) 517-534 [arXiv:funct-an/9406005, doi:10.1007/BF02101806]

Generalization for AQFT on curved spacetime is in

• Rainer Verch, A spin-statistics theorem for quantum fields on curved spacetime manifolds in a generally covariant framework, Communications in Mathematical Physics, 223 2 (2001) 261-288 [arXiv:math-ph/0102035, doi:10.1007/s002200100526]

General review:

• Bernd Kuckert, The Classical and Quantum Roots of Pauli’s Spin-statistics Relation, in: Quantum Analogues: From Phase Transitions to Black Holes and Cosmology, Lecture Notes in Physics 718, Springer (2007) 207-228 [doi:10.1007/3-540-70859-6_8]

• Jonathan Bain: CPT Invariance and the Spin-Statistics Connection, Oxford University Press (2016) [ISBN:9780198728801, doi:10.1093/acprof:oso/9780198728801.001.0001]

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:

• Theo Johnson-Freyd, Spin, statistics, orientations, unitarity, Algebraic and Geometric Topology [arXiv:1507.06297]

and via hermitian functorial field theory:

• Lukas Müller, Luuk Stehouwer, Reflection Structures and Spin Statistics in Low Dimensions [arXiv:2301.06664]

• Luuk Stehouwer, The categorical spin-statistics theorem [arXiv:2403.02282]

• Cameron Krulewski, Lukas Müller, Luuk Stehouwer: A Higher Spin Statistics Theorem for Invertible Quantum Field Theories [arXiv:2408.03981]