spinning particle


Quantum field theory


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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.


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 ϕ:X\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 XX. See the references on worldline supersymmetry below.

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

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

characterize the d=1d = 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:

ddpartition function in dd-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 MSpinKOM 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 MSpin cKUM 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 MSpin cKUM 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


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



Discussion of worldline dynamics of spinning particles in background fields is for instance in

  • J.W. 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)

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

  • L. Brink, P. Di Vecchia and P. Howe, Nucl. Phys. B118 (1977), 76

  • 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=1D = 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

See also

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

also p. 194 of

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.

See also

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

Last revised on August 28, 2016 at 04:36:05. See the history of this page for a list of all contributions to it.