nLab fermion

Idea

A fermion is a particle/quantum field that obeys Fermi-Dirac statistics? (Pauli exclusion principle). By the spin-statistics theorem, this is the same thing as a particle whose spin is half an integer but not itself an integer.

In the standard model of particle physics and the standard model of cosmology fermions are the matter constituents of the observable universe, as opposed to bosons, which are the quanta of force gauge fields.

The formalization of fermions is in supergeometry: fermion fields are sections of spinor bundles with odd-degree fibers in supergeometry. This is exhibited by what is called Fermi-Dirac statistics? and the Pauli exclusion principle.

Examples

flavors of fundamental fermions in the standard model of particle physics:
generation of fermions	1st generation	2nd generation	3d generation
quarks ($q$)
up-type	up quark ($u$)	charm quark ($c$)	top quark ($t$)
down-type	down quark ($d$)	strange quark ($s$)	bottom quark ($b$)
leptons
charged	electron	muon	tauon
neutral	electron neutrino	muon neutrino	tau neutrino
bound states:
mesons	light mesons: pion ($u d$) ρ-meson ($u d$) ω-meson ($u d$) f1-meson a1-meson	strange-mesons: ϕ-meson ($s \bar s$), kaon, K*-meson ($u s$, $d s$) eta-meson ($u u + d d + s s$) charmed heavy mesons: D-meson ($u c$, $d c$, $s c$) J/ψ-meson ($c \bar c$)	bottom heavy mesons: B-meson ($q b$) ϒ-meson ($b \bar b$)
baryons	nucleons: proton $(u u d)$ neutron $(u d d)$

Related entries

• Pauli exclusion principle, degeneracy pressure

• chiral fermion

• particle, matter

• boson, fermion, anyon

• parafermion

• fermionic modality

• Pauli exclusion principle

• fermion sea

• Landau's Fermi liquid model

• spin-statistics theorem

• spinors in Yang-Mills theory

• generation of fermions

• fermionic path integral, Pfaffian line bundle

• fermion anomaly

References

General

Fermions are named after Enrico Fermi.

Supergeometry of fermion fields

Discussion of the classical mechanics of the spinning particle or of classical field theory with fermion fields (possibly but not necessarily super-symmetric) as taking place in supergeometry:

via (possibly infinite-dimensional) supermanifolds:

• Felix A. Berezin, M. S. Marinov: Particle Spin Dynamics as the Grassmann Variant of Classical Mechanics, Annals of Physics 104 2 (1977) 336-362 [doi:10.1016/0003-4916(77)90335-9, pdf, pdf]

  reprinted in Appendix I of: Alexandre A. Kirillov (ed.): Introduction to Superanalysis, Mathematical Physics and Applied Mathematics 9, Springer (1987) [doi:10.1007/978-94-017-1963-6]

• Thomas Schmitt: The Cauchy Problem for Classical Field Equations with Ghost and Fermion Fields [arXiv:hep-th/9607133]

• Thomas Schmitt: Supergeometry and Quantum Field Theory, or: What is a Classical Configuration?, Rev. Math. Phys. 9 (1997) 993-1052 [doi:10.1142/S0129055X97000348, arXiv:hep-th/9607132].

• Thomas Schmitt: Supermanifolds of classical solutions for Lagrangian field models with ghost and fermion fields, Sfb 288 Preprint No. 270 [hep-th/9707104, inspire:445574]

• Daniel Freed, What are fermions?, Lecture 1 in: Five lectures on supersymmetry, AMS (1999) [ISBN:978-0-8218-1953-1, spire:517862]

• Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily, chapter 3 of: Advanced classical field theory, World Scientific (2009) [doi:10.1142/7189]

• Gennadi Sardanashvily, Grassmann-graded Lagrangian theory of even and odd variables, [arXiv:1206.2508]

• Gennadi Sardanashvily W. Wachowski: SUSY gauge theory on graded manifolds [arXiv:1406.6318, spire:1302860]

• Viola Gattus, Apostolos Pilaftsis, Supergeometric Approach to Quantum Field Theory, CORFU2023, PoS 463 (2024) 156 [doi:10.22323/1.463.0156, arXiv:2404.13107]

• Viola Gattus, Apostolos Pilaftsis: Supergeometric Quantum Effective Action [arXiv:2406.13594]

and more generally via smooth super sets:

• Urs Schreiber: §3.1 in: Higher Prequantum Geometry, chapter in: New Spaces for Mathematics and Physics, Cambridge University Press (2021) [doi:10.1017/9781108854399.008, arXiv:1601.05956]

Discussion with focus on supersymmetry:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section II.2.4 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, chII.2: pdf]

• Pierre Deligne, Daniel Freed: Supersolutions, in: Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence (1999) 357-366 [arXiv:hep-th/9901094, ISBN:978-0-8218-2014-8, web version]

• Daniel Freed, Classical field theory and Supersymmetry, IAS/Park City Mathematics Series 11 (2001) [pdf, pdf]

and specifically in the context of super- string theory (regarding worldsheets as super Riemann surfaces):

• Eric D'Hoker, Duong Phong, Complex geometry and supergeometry, Current Developments in Mathematics 2005 (2007) 1-40 [arXiv:hep-th/0512197, euclid.cdm/1223654523]