nLab parastatistics

Idea

Parastatistics is an exotic kind of particle statistics where permutations of “indistinguishable particles” transform their space of quantum states by higher dimensional irreducible linear representations of the symmetric group (Messiah & Greenberg 1964, cf. Halvorson 2007 §14.4.2-4, Wang & Hazzard 2025, below (4)).

In generality of its transformation properties, parastatistics is in-between ordinary boson/fermion-statistics and the yet more general braid group statistics. But while braid group statistics is usually associated only with particles (“anyons”) that are restricted to an effectively 2-dimensional surface, parastatistics – to the extent that it applies at all – is thought to apply in principle in any dimension. (Concretely for $d = 3$ see Freedman, Hastings, Nayak, Qi, Walker & Wang 2011)

Related concepts

• bosons, fermions

• braid group statistics, anyons

References

General

The notion originates with “triple commutator”-relations motivated in a field theoretic/second quantized context:

• H. S. Green: A generalized method of field quantization, Phys. Rev. 90 (1953) 270-273 [doi:10.1103/PhysRev.90.270]

• O. W. Greenberg, Albert M. L. Messiah: Selection Rules for Parafields and the Absence of Para Particles in Nature, Phys. Rev. 138 (1965) B1155 [doi:10.1103/PhysRev.138.B1155]

Proof that the triple-bracket algebra of Green 1953 — for $m$ parafermionic and $n$ parabosonic degrees of freedom — is isomorphic to the orthosymplectic super Lie algebra $\mathfrak{osp}_{1 + 2m\vert 2n}$:

• Tchavdar D. Palev: Para‐Bose and para‐Fermi operators as generators of orthosymplectic Lie superalgebras, J. Math. Phys. 23 (1982) 1100–1102 [doi:10.1063/1.525474]

The alternative “first quantized” perspective (at fixed particle number) and with it the relation to the representation theory of the symmetric group originates with:

• Albert M. L. Messiah, O. W. Greenberg: Symmetrization Postulate and Its Experimental Foundation, Phys. Rev. 136 (1964) B248 [doi:10.1103/PhysRev.136.B248]

  (where irreps of the symmetric group are called “generalized rays”)

• James B. Hartle, John R. Taylor: Quantum Mechanics of Paraparticles, Phys. Rev. 178 (1969) 2043 [doi:10.1103/PhysRev.178.2043]

• Robert H. Stolt, John R. Taylor: Correspondence between the first- and second-quantized theories of paraparticles, Nuclear Physics B 19 1 (1970) 1-19 [doi:10.1016/0550-3213(70)90024-6]

• James B. Hartle, Robert H. Stolt, John R. Taylor: Paraparticles of Infinite Order, Phys. Rev. D 2 (1970) 1759 [doi:10.1016/0550-3213(70)90024-6]

On a relation between the triple-bracket algebra of Green 1953 to the representation theory of the symmetric group:

• Jean-Louis Loday, Todor Popov: Parastatistics algebra and super semistandard Young tableaux, in: V.K. Dobrev et al. (eds.), Lie Theory and Its Applications in Physics VII, Heron Press, Sofia (2008) [arXiv:0711.3648, pdf]

Review:

• Vo-Dai Thien: First and second quantization theories of parastatistics, PhD thesis, McMaster (1972) [pdf]

• Y. Ohnuki, S. Kamefuchi: Quantum field theory and parastatistics, Springer (1982) [ISBN:9783642686245]

• Wikipedia: Parastatistics,

• Wikipedia: Klein transformation

Discussion via algebraic quantum field theory and DHR superselection theory:

• Sergio Doplicher, Rudolf Haag, John E. Roberts: Local observables and particle statistics I, Commun.Math. Phys. 23 (1971) 199–230 [doi:10.1007/BF01877742, euclid:cmp/1103857630]

• Sergio Doplicher, Rudolf Haag, John E. Roberts: Local observables and particle statistics I, Commun.Math. Phys. 35 (1974) 49–85 [doi:10.1007/BF01646454]

  (cf. DHR superselection theory)

• Hans Halvorson: Statistics, permutation symmetry, and identical particles, Section 11.4 in: Algebraic Quantum Field Theory, in Philosophy of Physics, Handbook of the Philosophy of Science (2007) 731-864 [doi:10.1016/B978-044451560-5/50011-7, arXiv:math-ph/0602036]

Further discussion:

• F. Duncan M. Haldane: “Fractional statistics” in arbitrary dimensions: A generalization of the Pauli principle, Phys. Rev. Lett. 67 (1991) 937-940 [doi:10.1103/PhysRevLett.67.937, inSpire:321137]

• Fred Cooper, Avinash Khare, Uday Sukhatme: Parasupersymmetry in quantum mechanics, J. Math. Phys. 34 (1993) 1277–1294 [doi:10.1063/1.530209]

  (para-supersymmetric quantum mechanics)

• Alexios P. Polychronakos: Path Integrals and Parastatistics, Nucl. Phys. B 474 (1996) 529-539 [doi:10.1016/0550-3213(96)00277-5,arXiv:hep-th/9603179]

• Lee Brekke, Sterrett J. Collins, Tom D. Imbo, Nonabelian vortices on surfaces and their statistics [hep-th/9701056]

  If we could “turn off” all interactions in the universe which couple to spin, then different spin states would be in principle indistinguishable (although still not identical), and the particles would obey parastatistics at the fundamental level. We can play the same game with color, isospin and numerous other internal degrees freedom. (Indeed, the parastatistical treatment of quarks [3] predates the introduction of color [4].)

• Weimin Yang, Sicong Jing, Graded Lie algebra generating of parastatistical algebraic structure [math-ph/0212009]

• Boyka Aneva, Todor Popov: Hopf structure and Green ansatz of deformed parastatistics algebras, J. Phys. A: Math. Gen. 38 (2005) 6473 [doi:10.1088/0305-4470/38/29/004]

• K. Kanakoglou, C. Daskaloyannis: Paraboson quotients. A braided look at Green ansatz and a generalization, J. Math. Phys. 48 (2007) 113516 [arXiv:0901.4320, doi:10.1063/1.2816258]

• N.I. Stoilova, J. Van der Jeugt: Partition functions and thermodynamic properties of paraboson and parafermion systems, Physics Letters A 384 21 (2020) 126421 [doi:10.1016/j.physleta.2020.126421]

• Zhiyuan Wang: Parastatistics and a secret communication challenge [arXiv:2412.13360]

In quantum information theory

On permutation representations, such as arising in parastatistics, as quantum gates for quantum computing akin to anyon-based topological quantum computing:

• Stephen P. Jordan, pp 108 in: Quantum Computation Beyond the Circuit Model, PhD thesis, MIT (2010) [arXiv:0809.2307, pdf]

• Stephen P. Jordan: Permutational Quantum Computing, Quantum Information and Computation 10 (2010) 470 [arXiv:0906.2508, doi:10.26421/QIC10.5-6-7]

• Stephen P. Jordan: Permutational Quantum Computation, talk at Complexity Resources in Physical Computation, Oxford (2009) [pdf]

• Michel Planat, Rukhsan Ul Haq: The Magic of Universal Quantum Computing with Permutations, Advances in Mathematical Physics 2017 287862 (2017) [doi:10.1155/2017/5287862]

In the context of quantum complexity theory:

• Scott Aaronson, Adam Bouland, Greg Kuperberg, Saeed Mehraban: The computational complexity of ball permutations, STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (2017) 317-327 [doi:10.1145/3055399.3055453]

Application to quantum machine learning:

• Han Zheng, Zimu Li, Junyu Liu, Sergii Strelchuk, Risi Kondor: Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ansätze, PRX Quantum 4 020327 (2023) [arXiv:2112.07611, doi:10.1103/PRXQuantum.4.020327]

• Han Zheng, Zimu Li, Junyu Liu, Sergii Strelchuk, Risi Kondor: On the Super-exponential Quantum Speedup of Equivariant Quantum Machine Learning Algorithms with $SU(d)$ Symmetry, presentation at TQC 2022 [arXiv:2207.07250]

Concrete lattice models for a kind of parastatistics in 3d:

• Michael Freedman, Matthew B. Hastings, Chetan Nayak, Xiao-Liang Qi, Kevin Walker, Zhenghan Wang: Projective Ribbon Permutation Statistics: a Remnant of non-Abelian Braiding in Higher Dimensions, Phys. Rev. B 83 115132 (2011) [doi:10.1103/PhysRevB.83.115132, arXiv:1005.0583]

Claim of parastatistical quasiparticles emerging in low dimensional quantum spin systems:

• Zhiyuan Wang, Kaden R. A. Hazzard: Particle exchange statistics beyond fermions and bosons, Nature 637 (2025) 314-318 [arXiv:2308.05203, doi:10.1038/s41586-024-08262-7]

category: physics