Contents

# Contents

## 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

$d$partition function in $d$-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin \to KO$
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory

## History

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

## References

### General

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

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)

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

### 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

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