Generalized Particle Statistics  

In General > s.a. cosmological constant; fock space; information; particle statistics.
* Idea: Statistics is usually dictated by representations of the permutation group; However, examples of non-permutation group statistics are known from anyons in 2D and from \(\mathbb Z\)n, cyclic statistics for a certain non-gravitational system.
@ General references: Fivel PRL(90); Chen et al MPLA(96); Medvedev PRL(97) [ambiguous statistics]; Greenberg qp/99; Polychronakos ht/99-ln [1D]; Greenberg in(00)ht [rev]; Marcinek m.QA/01 [Fock space]; Marcinek in(03)m.QA/04 [categorical approach]; Greenberg in(09)-a0804 [rev]; Swain IJMPD(08)-a0805 [quantum-gravity effects]; Arzano & Benedetti IJMPA(09)-a0809 [momentum-dependent "rainbow statistics" in non-commutative field theory]; Cattani & Bassalo a0903; Lavagno & Narayana Swami PhyA(10) [and deformed algebras]; Dahlsten et al a1307 [in generalized quantum theory]; Goyal a1309 [no generalized statistics]; Neori a1603-PhD [anyons and the symmetrization postulate]; Wang et al AP(19)-a1901 [and spacetime topology].
@ Examples: Greenberg PRL(90) [infinite statistics]; Balachandran et al MPLA(01)ht/00 [geons in 2+1 Chern-Simons theory]; Surya JMP(04)ht/03 [cyclic statistics]; Baez et al ATMP(07)gq/06 [loop defects in BF theory]; Salvitti CMP(07) [2D massive Dirac fields]; Niven & Grendar PLA(09); Maslov TMP(09) [generalied Bose-Einstein distribution]; Bagarello RPMP(11)-a1106, JMP(13)-a1309 [pseudo-bosons]; Matthews et al SRep(13)-a1106 [simulations with entangled photons]; Lundholm & Solovej AHP(14)-a1301 [intermediate and fractional statistics, Lieb-Thirring inequalities]; Palev a1412-proc [A-, B-, C- and D- (super)statistics]; Hoyuelos JSM(18)-a1802 [general statistics, ewkons]; Ramakrishna a2005 [interpolating algebra]; > s.a. non-commutative geometry.
@ Fermions: Niemi & Semenoff PLB(84), PRP(86) [fractional fermion number]; Arik & Tekin JPA(02); Narayana Swami qp/05, Conroy et al PLA(10) [q-deformed]; Treumann a1305 [fermionic fractional statistics].
@ Phenomenology: Marinho & Brito a1907 [q-deformed statistics and thermoelectric properties of solids].

Fractional Statistics in 2+1 Dimensions > s.a. Anyons [including 3D]; chern-simons field theories; photons; supersymmetric theories.
* Idea: Objects with intermediate statistics, arising in some 2D systems, because particle world-lines may braid; Wave functions may change by any real phase under particle exchange; They belong to a 1D representation of the braid group.
* Features: Fractional statistics can be exchanged for extra charges/fluxes in 2D; They imply P and T violation; They do not violate the spin-statistics theorem, because in 2D spin is not quantized.
* Quons: Elementary excitations of fields with intermediate statistics, particles characterized by a parameter q which permits smooth interpolation between Bose and Fermi statistics; q = 1 gives bosons, q = −1 gives fermions.
* Simplest type: Semion (the phase changes by π/2; ground state probably superfluid – superconducting if charged).
* History: Proposed by F Wilczek in 1982; Applications in the fractal quantum Hall effect, high-\(T_{\rm c}\) superconductivity, and edge conduction modes of 2D insulators.
@ I: Khurana PT(89)nov; Canright & Girvin Sci(90)mar; Wilczek PW(91)jan, SA(91)may.
@ General references: Leinaas & Myrheim NCB(77); Sorkin PRD(83); Wu PRL(84); Wu PRL(84) [many-body wave functions]; Haldane & Wu PRL(85) [for vortices in 2D superfluids]; Goldin in(87); Mackenzie & Wilczek IJMPA(88); Semenoff PRL(88); Lavenda & Dunning-Davies JMP(89); Wetterich NPB(89); Imbo et al PLB(90); Aneziris et al IJMPA(91) [1D]; Haldane PRL(91); Hessling & Tscheuschner IJTP(91); Forte RMP(92); Gamboa IJMPA(92); Canright & Johnson JPA(94); Goldin & Sharp PRL(96); Tang & Finkelstein ht/96/PRD; Delves et al PRS(97); Hagen PLB(99)ht [Pauli term]; Khare 05 [text]; Negro et al JMP(06)mp/05 [formalism]; Lima & Landim EPL(06)ht [fractional spin]; Wilczek in(09)-a0812 [rev]; Fitzpatrick et al a1205; Vleeshouwers & Gritsev a2012 [topological field theory approach].
@ Quons: Goodison & Toms PLA(94) [canonical partition function]; Greenberg & Hilborn FP(99)ht/98; Chow & Greenberg PLA(01)ht/00 [in relativistic quantum theory]; Jackson & Hogan IJMPD(08)-ht/07 [and the cosmological constant].
@ Models, phenomenology: Fendley & Fradkin PRB(05)cm [non-Abelian statistics]; Bishara et al PRB(09) + Moore Phy(09); Shtengel Phy(10); Bonderson et al PRB(11) + Wilczek Phy(11) [Hall effect]; Klinovaja & Loss PRL(13)-a1301; Levin PRX(13) [edge conduction modes in 2D insulators].
@ Related topics: Müller ZPC(90) [2D, lattice]; Acharya & Narayana Swami JPA(94) [statistical mechanics], JPA(04) [and detailed balance]; Isakov et al PLA(96) [thermodynamics]; Ramanathan PS(99) [Laughlin liquids]; Pachos AP(07) [lattice]; Sree Ranjani et al AP(09)-a0812 [in 1D three-particle Calogero model]; Freedman & Levaillant a1501 [measuring topological charge]; > s.a. carbon [graphene]; quantum computation [topological]; quantum oscillators.

Parastatistics, Paraparticles > s.a. Bosons [bosonization]; path integrals.
* Idea: They can arise only if 3 or more particles are present (but in generally covariant theories, new possibilities arise even with only two particles); They correspond to higher than 1D representations of the permutation group.
* Para-Fermi: At most p particles (p ∈ \(\mathbb N\)) may occupy a quantum state, antisymmetric; The ordinary case is p = 1.
* Para-Bose: Similar to para-Fermi, but different symmetry under interchange.
@ General references: Green PR(53) [proposal]; Ohnuki & Kamefuchi 82 [and quantum field theory]; Meljanac et al MPLA(98) [as triple operator algebras]; Stoilova & Van der Jeugt JMP(05), JMP(05)mp [and Lie (super)algebras]; Maslov TMP(07); Nelson a1912; Stoilova & Van der Jeugt PLA(20)-a2005 [statistical mechanics, thermodynamics].
@ Examples: Greenberg PRL(64) [quarks]; Ringwood & Woodward PRL(84) [monopoles]; Cobanera & Ortíz PRA(14)-a1307 [Fock parafermions].
@ Parafermions: Campoamor-Stursberg & Rausch de Traubenberg AIP(10)-a0910 [and ternary superspaces]; Dovgard & Gepner PLB(10) [non-abelian].
@ Related topics: Aneziris et al IJMPA(89), MPLA(89) [and general covariance]; Govorkov TMP(94) [non-existence]; Tamura & Ito JMP(07) [and random point fields]; Tichy & Mølmer PRA(17)-a1702 [immanons].

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