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 (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]

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):

• Michael G. G. Laidlaw, Cécile Morette DeWitt, Feynman Functional Integrals for Systems of Indistinguishable Particles, Phys. Rev. D 3 6 (1971) 1375-1378 [doi:10.1103/PhysRevD.3.1375]

• A. P. Balachandran, A. Daughton, Z. C. Gu, Giuseppe Marmo, Rafael D. Sorkin Spin statistics theorems without relativity or field theory, Int. J. Mod. Phys. A 8 (1993) 2993-3044 [doi:10.1142/S0217751X93001223]

• 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]

  (cf. also the Atiyah-Sutcliffe conjecture)

• Michael V. Berry, Jonathan M. Robbins: Quantum indistinguishability: alternative constructions of the transported basis, J. Phys. A: Math. Gen. 33 (2000) L207 [doi:10.1088/0305-4470/33/24/101, pdf]

• Murray Peshkin: Spin and Statistics in Nonrelativistic Quantum Mechanics: The Zero Spin Case, Phys. Rev.A 67 (2003) 042102 [doi:10.1103/PhysRevA.67.042102, arXiv:quant-ph/0207017]

• Charis Anastopoulos, International Journal of Modern Physics A 19 05 (2004) 655-676 [doi:10.1142/S0217751X04017860, arXiv:quant-ph/0110169]

  (via geometric quantization)

• Nikolaos A. Papadopoulos, Mario Paschke, Andrés F. Reyes-Lega, F. Florian Scheck, The spin-statistics relation in nonrelativistic quantum mechanics and projective modules, Annales Mathematiques Blaise Pascal 11 (2004) 205-220 [doi:10.5802/ambp.193, arXiv:quant-ph/0608125, numdam:AMBP_2004__11_2_205_0/]

• J. M. Harrison, Jonathan M. Robbins, Quantum indistinguishability from general representations of $SU(2n)$, J. Math. Phys. 45 (2004) 1332–1358 [doi:10.1063/1.1666979, arXiv:math-ph/0302037]

• Bernd Kuckert, Spin and statistics in nonrelativistic quantum mechanics, I, Physics Letters A 322 1–2 (2004) 1–2 47-53 [doi:10.1016/j.physleta.2003.12.051, arXiv:quant-ph/0208151]

• Bernd Kuckert, Jens Mund, Spin & Statistics in Nonrelativistic Quantum Mechanics, II, Ann. Phys. 517 5 (2005) 309-311 [doi:10.1002/andp.200410129, arXiv:quant-ph/0411197]

• Murray Peshkin: Spin-Zero Particles must be Bosons: A New Proof within Nonrelativistic Quantum Mechanics, Found Phys 36 (2006) 19–29 [doi:10.1007/s10701-005-9011-2]

• Nikolaos A. Papadopoulos Andrés F. Reyes-Lega: On the Geometry of the Berry-Robbins Approach to Spin-Statistics, Found Phys 40 (2010) 829-851 [doi:10.1007/s10701-009-9365-y, arXiv:0910.1659]

• Andrés F. Reyes-Lega, Carlos Benavides, Remarks on the Configuration Space Approach to Spin-Statistics, Found Phys 40 (2010) 1004-1029 [doi:10.1007/s10701-009-9397-3, arXiv:0911.0579]

• Andrés F. Reyes-Lega: On the geometry of quantum indistinguishability, J. Phys. A: Math. Theor. 44 33 (2011) 325308 [doi:10.1088/1751-8113/44/32/325308, arXiv:1112.6300]

• Jonathan Bain: Non-RQFT Derivations of CPT Invariance and the Spin–Statistics Connection, Chapter 4 in: CPT Invariance and the Spin-Statistics Connection, Oxford University Press (2016) [doi:10.1093/acprof:oso/9780198728801.003.0005, ISBN:9780198728801]

• Michael Berry, Jonathan Robbins: Quantum Indistinguishability: Spin-statistics without Relativity or Field Theory?, in: A Half-Century of Physical Asymptotics and Other Diversions – Selected Works by Michael Berry, World Scientific (2017) 108-120 [doi:10.1142/9789813221215_0009, pdf]

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]