nLab spin-statistics theorem

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 $s$ is a number $s = \frac{n}{2}$ with $n \in \mathbb{N}$.

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 $s$ (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)

…

Related concepts

• 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): 3–37. (1939) doi:10.5169/seals-110930

• Wolfgang Pauli, The connection between spin and statistics, Phys. Rev. 58, 716–722 (1940)

Textbook accounts:

• 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]

The proof, which goes back originally to Fermi. There is also a more intuitive approach based on topology. One can see hints of it in Feynman’s lecture here:

• Richard Feynman and Steven Weinberg, Elementary Particles and the Laws of Physics , the 1986 Dirac Memorial Lectures, Cambridge U. Press, Cambridge, 1987.

Exposition:

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

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

• Daniele Guido, Roberto Longo, An Algebraic Spin and Statistics Theorem (arXiv:funct-an/9406005)

Geometric proofs of spin-statistics via configuration spaces of points (related to the Atiyah-Sutcliffe conjecture):

• Michael V. Berry, Jonathan M. Robbins, Indistinguishability for quantum particles: spin, statistics and the geometric phase, Proceedings of the Royal Society A 453 1963 (1997) 1771-1790 (doi:10.1098/rspa.1997.0096)

• A.F. Reyes-Lega, C. Benavides, Remarks on the Configuration Space Approach to Spin-Statistics, Found Phys (2010) 40: 1004-1029 (arXiv:0911.0579)

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 October 2001, Volume 223, Issue 2, pp 261–288 (arXiv:math-ph/0102035, doi:10.1007/s002200100526)

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

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