nLab spinning particle

Idea

The spinning relativistic particle is a variant of the plain relativistic particle which has an “internal degree of freedom” called spin: it is a spinor , a fermion. Examples that appear in the standard model of particle physics are electrons, and quarks.

As a 1-dimensional sigma-model, the spinning relativistic particle is like the relativistic particle but with fermion fields on the worldline. This worldline action always happens to have worldline supersymmetry, entirely independent of whether there is any supersymmetry on target spacetime.

Properties

Worldline supersymmetry

We discuss how spinning particles automatically have supersymmetry in their worldline formalism. For more see the references below for more and see also at string theory FAQ – Does string theory predict supersymmetry?.

In the Polyakov action-formulation the action functional of the relativistic particle sigma model on the 1-dimensional worldline is actually 1-dimensional gravity coupled to “worldline matter fields”, where the latter are the embedding fields $\phi : \mathbb{R} \to X$ into the target space.

It turns out that the generalization of this 1-dimensional gravity action to supergravity yields the action functional that describes ordinary Dirac spinors – spinning particles like electrons – propagating on target space $X$. See the references on worldline supersymmetry below.

The worldline supersymmetry of fermions comes down to the fact that their Hamiltonian(-constraint) operator $H$ has a square root: the Dirac operator $D$. Its defining equations

$$[D,D] = 2 D^2 = H\,,\;\;\; [H,H] = 0\;,\;\; [D,H] = 0$$

characterize the $d = 1$ translation super-Lie algebra.

For appreciating this fact it is important to keep the ingredients of sigma-model theory sorted out correctly: a supersymmetric theory on the worldline describes a spinning particle on some spacetime coupled to some background gauge fields. That background geometry need not have a “global supersymmetry” (a covariant constant spinor), hence under second quantization the perturbation theory on target space induced by the worldline theory need not have any global supersymmetries (in particular no superpartners to the effective particle excitations). What will happen, though, is that the full target space theory induced under second quantization will be a supergravity theory on target space. Some of its solutions may have covariantly constant spinors (and hence global supersymmetry), but generically they will not, just like the generic solution to ordinary Einstein equations does not have a Killing vector.

Paritition function

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	endpoint of 2d Poisson-Chern-Simons theory string	Spin^c Dirac operator twisted by prequantum line bundle	space of quantum states of boundary phase space/Poisson manifold	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-theory

History

That some particles have a property called spin was found in 1922 in the Stern-Gerlach experiment.

Related concepts

• fermion, spinor

• sigma model, brane

  • relativistic particle, spinning particle, superparticle

    • A-hat genus, (1,1)-dimensional Euclidean field theories and K-theory

  • string, spinning string, superstring

    string theory FAQ – Does string theory predict supersymmetry?

  • membrane

References

General

Lecture notes in view of application to perturbative quantum field theory via worldline formalism:

• Olindo Corradini, Christian Schubert, James P. Edwards, Naser Ahmadiniaz, Spinning Particles in Quantum Mechanics and Quantum Field Theory (arXiv:1512.08694)

Discussion that cancellation of the quantum anomaly of the spinning particle precisely requires Spin-structure on its target spacetime:

• Edward Witten, Global anomalies in string theory, in: W. Bardeen and A. White (eds.). Symposium on Anomalies, Geometry, Topology, pp. 61–99. World Scientific, 1985 (pdf, spire:214913)

Discussion of worldline dynamics of spinning particles in background fields:

• Jan-Willem van Holten, Relativistic Dynamics of Spin in Strong External Fields [arXiv:hep-th/9303124]

• A. Pomeranskii, R A Sen’kov, I.B. Khriplovich, Spinning relativistic particles in external fields Acta Physica Polonica B Proceedings Supplement Vol. 1 (2008) (pdf)

• Krzysztof Andrzejewski, Cezary Gonera, Joanna Goner, Piotr Kosinski, Pawel Maslanka, Spinning particles, coadjoint orbits and Hamiltonian formalism (arXiv:2008.09478)

Discussion via coadjoint orbits:

• Thomas Basile, Euihun Joung, TaeHwan Oh, Manifestly Covariant Worldline Actions from Coadjoint Orbits. Part I: Generalities and Vectorial Descriptions [arXiv:2307.13644]

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]

Worldline supersymmetry

Among the early references that describe the observation that the supersymmetric extension of the worldline theory of the relativistic particle describes ordinary Dirac fermions are

• F.A. Berezin and M.S. Marinov, Ann. Phys. (NY) 104 (1977) 336

• Lars Brink, Paolo Di Vecchia, Paul Howe, A Lagrangian formulation of the classical and quantum dynamics of spinning particles, Nucl. Phys. B 118 (1977) 76 [doi:10.1016/0550-3213(77)90364-9]

• R. Casalbuoni, Phys. Lett. B62 (1976), 49

• A. Barducci, R. Casalbuoni and L. Lusanna, Nuov. Cim. 35A (1976), 377 Nucl. Phys. B124 (1977), 93; id. 521

A survey of constructions of worldline supersymmetric action functional for spinning particles in various background fields is given in

• J. van Holten, $D = 1$ Supergravity and Spinning Particles (pdf)

An argument that for arbitrary backgrounds the spinning particle’s worldline action is supersymmetric is given in

• Darius Gagné, Worldline Supersymmetry and Dimensional Reduction (arXiv:hep-th/9604149)

Textbook surveys of worline supersymmetry include

the beginning of section 14.1.1 in

• Tomás Ortín, Gravity and strings Cambridge University Press (2004)

Derivation of the supersymmetric worldline action of the spinning particle in the worldline formalism of QFT scattering amplitudes is around (3.6) of

• Matthew Strassler, Field Theory Without Feynman Diagrams: One-Loop Effective Actions, Nucl. Phys. B385:145-184,1992 (arXiv:hep-ph/9205205)

See also:

• Dmitri P. Sorokin, Vladimir I. Tkach, Dmitrij V. Volkov, Aleksandr A. Zheltukhin, From the Superparticle Siegel Symmetry to the Spinning Particle Proper Time Supersymmetry, Phys. Lett. B 216 (1989) 302-306 [doi:10.1016/0370-2693(89)91119-2]

• Rachel H. Rietdijk, Spinning particles in general relativity Theoretical and Mathematical Physics, Vol. 98, No. 3 (1994)

• Warren Siegel, p. 194 of: Fields (arXiv:hep-th/9912205)

  There, first exercise IIIB1.3 gives the action expanded out in all components, and then the following exercise IIIB1.4 gives the reformulation in superfield formalism which makes manifest that the action is invariant under worldline supersymmetry.

• Roberto Casalbuoni, Joaquim Gomis, Kiyoshi Kamimura, Giorgio Longhi, Space-time Vector Supersymmetry and Massive Spinning Particle (web)

• E. Boffo, Particles in the superworldline and BRST [arXiv:2207.02041]