Table Of ContentIL NUOVOCIMENTO Vol. ?, N. ? ?
Spinning particles in General Relativity
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F. Cianfrani( ) and G. Montani( )( )
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( ) ICRA-International Center for Relativistic Astrophysics
n Dipartimento di Fisica (G9), Universit`a di Roma, “La Sapienza”,
a Piazzale Aldo Moro 5, 00185 Rome, Italy.
J 2
( ) ENEA C.R. Frascati (Dipartimento F.T.P.),
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via Enrico Fermi 45, 00044 Frascati, Rome, Italy.
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Summary.—Weanalyzethebehaviorofaspinningparticleingravity,bothfroma
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quantumandaclassical pointof view. Weinferthat,sincetheinteraction between
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the space-time curvature and a spinning test particle is expected, then the main
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0 featuresofsuchaninteractioncangetlightonwhichdegreesoffreedom havephys-
/ ical meaning in a quantum gravity theory with fermions. Finally, the dimensional
c reduction of Papapetrou equations is performed in a 5-dimensional Kaluza-Klein
q
background and Dixon-Souriau results for the motion of a charged spinning body
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r are obtained.
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1. – Quantum features of the interaction between spin and gravity
The interaction between gravity and fundamental particles is still an open issue of
our knowledge. Even if the final answer to this problem must be given by a quantum
formulation for the gravitational field, nevertheless we expect that a good effective de-
scription(forenergyscalesmuchlowerthatPlanck’sone)couldcomeoutfromthestudy
of particles dynamics living on a curved space-time. But also this is a highly non-trivial
task, because, in a classical picture a free moving particle follows geodesics lines, but
there are no unambiguous indications for the motion of spinning particles; in fact, in
this last case, the classical dynamics must be inferred as classical limit of a relativistic
quantum mechanical equation.
In this respect, let us consider the Dirac equation on a curved space-time
i
(1) γµD ψ =0 D ψ = ∂ ωabΣ ψ
µ µ (cid:18) µ− 2 µ ab(cid:19)
(cid:13)c Societa`ItalianadiFisica 1
2 F.CIANFRANIand G.MONTANI
being ωab spin connections and γµ Dirac matrices on the given space-time. If we square
µ
the Dirac operator, the following second order equation arises
1
(2) gµνD D ψ Rψ=0
µ ν
− 4
being R the scalar curvature associated to the manifold. Therefore, we expect a non-
trivialspin-curvaturecoupling,sincethelastequationdifferssignificantlyfromtheKlein-
Gordonone(withanon-minimalcurvaturecoupling),describingthedynamicsofaspin-
less particle
1
(3) gµν∂ ∂ φ ξRφ=0
µ ν
− 4
because of the presence of spinor connections into covariant derivatives D .
µ
Furthermore, the back-reaction of spinors on the space-time outlines that they have a
direct effect on the geometry, in terms of the appearance of a non-vanishing torsion: by
varyingthe Einstein-Diracaction(written inaPalatini-likeformulation)withrespectto
ωab, the following second Cartan structure equation is obtained
µ
1
(4) ∂ ea +ωa eb = ǫa ebecJd Ja =ψ¯γaγ ψ,
[µ ν] [µ|b| ν] 4 bcd ν µ A A 5
admitting the following solution ωab =0ωab+ 1ǫab ecJd, with 0ωab =eb eaν.
µ µ 4 cd µ A µ ν∇µ
Therefore, we can rewrite the Einstein-Dirac action as follows
1 3
(5) S = R+iψ¯γµ(0)D ψ+c.c. η JaJb ed4x;
−2Z (cid:18) µ − 8 ab A A(cid:19)
we emphasize the appearance in this reduced action of an interaction term containing
four fermions, which, in general, provides non renormalizable contributions to quantum
amplitudes.
Thus, the analysis abovesinglets out a peculiar property of spinorsback-reactiononthe
space-time; in fact, differently from ordinary matter, spinors induce not only a small
change to the metric field, but also a non-vanishing torsion field; therefore, the classical
limit of spinors dynamics is expected to be no longer a geodesics line, as far as a spin
notion is recovered. Of course, the spin-curvature coupling can survive in a classical
limit only if the metric field changes significantly on the Compton scale of fermions; in
this case, an effective classical theory describing the non geodesics motion of a spinning
particle can have a predictive character. However, we emphasize that, taking such kind
of classical limit contains a good degree of ambiguity and a unique classical effective
theory does not arise.
2. – Classical formulation for spinning particles
The result above presented, about the relation between spin and gravity, has its
classical counterpart in the effect which the presence of an angular momentum has on
the dynamics of a macroscopicbody. In particular, the well-knownPapapetrouwork [1]
was devotedto study the motion ofa body in GeneralRelativity ona fixedbackground.
In particular, in his analysis Papapetrou performed a multipoles expansion such that, if
SPINNINGPARTICLESINGENERALRELATIVITY 3
the first two moments are the only non-vanishing ones, then a characterization of the
internal structure of the body takes place in terms of what he called the spin tensor
(indeed this notion is recovered as a point-like limit of a classical angular momentum),
i.e.
(6) Sµν = δxµTν0 δxνTµ0.
Z −Z
τ τ
withτ denotingthe hypersurfaceatequaltime, Tµν theenergymomentumtensorofthe
body and δxµ the displacement with respect to the center of mass.
So Papapetrou recognized the pole-dipole approximation as able to describe a spinning
body coupled to gravity and he found the following system of dynamical equations
D Pµ = 1R µSνρuσ
Ds 2 νρσ
(7) DDsSµν =Pµuν −Pνuµ
Pµ =muµ DSµνu
− Ds ν
being uµ the 4-velocity, m the rest mass of the body and R µ the Riemann tensor.
νρσ
Thus, a spinning body has a recognizable interaction with the gravitational curvature,
through its spin tensor, and it does not follow a geodesics motion; in this scheme, the
particle retains a test character, but it acquires an internal structure, due to its spin,
which prevents it a free motion even in absence of “external” fields.
To closed the system (7), we must add one of the following additional conditions
(8) Sµ0 =0 Sµνu =0 SµνP =0
ν ν
known in literature as the Papapetrou, Pirani and Tulczyjew conditions, respectively.
Intheweakfieldapproximation,theseconditionsprovideequivalentresultsandtherefore
current experiments cannot select among them.
Inthisscenario,theintroductionofanexternalelectro-magneticfieldF isduetoDixon
µν
and Souriau [2, 3], through the following extension of the Papapetrou paradigm
D Pµ = 1R µSνρuσ+qFµuρ+ 1Mσν µF
Ds 2 νρσ ρ 2 ∇ σν
(9) DDsSµν =Pµuν −Pνuµ−MµρFρν −MνρFρµ
ddqs =0
q and Mµν being the electric charge and the electro-magnetic moment, respectively.
Our aim is to show that, for a particle of charge q and mass m, the Dixon-Souriau
equations canbe obtainedas dimensionalreductionofthe system(7)in a 5-dimensional
Kaluza-Kleinbackground,soemphasizingthatthegeometrizationoftheelectro-magnetic
interaction preserves particles dynamics, up to the dipole order.
Thespace-timeoftheKaluza-Klein[4,5]theoryisthe directsumofa4-dimensionalone
V4 and of an extra-dimension compactified to a circle of Planck length scale, such that
4 F.CIANFRANIand G.MONTANI
it is un-observable. The corresponding metric field takes the form
g e2k2A A ekA
(10) jAB =(cid:18) µν − ekAν µ ν − 1 µ (cid:19)
− −
g = g (xρ) being the 4-dimensional metric, while A = A (xρ) can be interpret as
µν µν µ µ
the electro-magnetic vector potential. In fact, if we split the Einstein-Hilbert action in
such a space-time
c4
(11) S = j(5)Rd4xdx5
−16π(5)GZV4⊗S1p−
the Maxwell Lagrangiandensity for A comes out
µ
c3 2πR e2k2
(12) S = √ g R+ F F gµρgνσ d4x.
µν ρσ
−16π (5)G ZV4 − (cid:20) 4 (cid:21)
In this sense, we recognize electro-magnetic degrees of freedom as components of the
5-dimensional metric.
Withinsuchaspace-timepicture,onecanshow[6,7]thatthedimensionalreductionofa
geodesics motion gives the right trajectory followed by charge test-particles. Let us now
consider Papapetrouequations in a KK space-time, i.e. in (7) we replace indices µ,ν by
Ω,Π=0,1,2,3,5and the spin tensor Sµν by the corresponding 5-dimensionalone ΣΩΠ.
Once the following identifications stand
(13) Σµν =Sµν S =Σ q =2m√Gu
µ 5µ 5
1
(14) Mµν = ek(Sµνu +uµSν uνSµ)
5
2 −
we recoverDixon-Souriauequations [6, 7], with the extra-components of the spin tensor
(Σ ) describing a non-vanishing electric dipole moment.
5µ
3. – Ashtekar formalism and chirality eigenstates
The relation between spinning particles and gravity can have very interesting conse-
quences in our understanding of the physical degrees of freedom concerning a quantum
gravity theory in presence of fermions.
LetusconsidertheframeworkofAshtekarformulationforthegravitationalfielddynam-
ics [8] [9], which is the most promising tool for a consistent Quantum Gravity theory;
it is based on rewriting the Einstein-Hilbert action in terms of Ashtekar connections Ak
µ
and A∗k, i.e.
µ
i i
(15) Ak =ω0k ǫkωij A∗k =ω0k+ ǫkωij
µ µ − 2 ij µ µ µ 2 ij µ
which are the self-dual and antiself-dual parts of the Lorentz connection, respectively.
Hence, the gravitationalactioncanbe splitintwoindependent terms,onewrittenin Ak
µ
SPINNINGPARTICLESINGENERALRELATIVITY 5
and one in A∗k, as follows
µ
1
(16) S = eµeνRabed4x= eµeν(Fk +F∗k)ed4x
2Z a b µν Z a b µν µν
being Fk and F∗k the curvature of Ashtekar connections
µν µν
i i
(17) Fk =∂ Ak ∂ Ak ǫkAiAj F∗k =∂ A∗k ∂ A∗k+ ǫkA∗iA∗j.
µν µ ν − ν µ− 2 ij µ ν µν µ ν − ν µ 2 ij µ ν
The coupling with spinors suggests a physical interpretation of such connections; by
well-known properties of Dirac matricies, we have
i i
(ψ¯γµΣ ψ+ψ¯Σ γµψ)ωab = (ψ¯γµσ ψ +ψ¯ σ γµψ)Ak (ψ¯γµσ ψ +ψ¯ σ γµψ)A∗k
2(cid:20) ab ab µ (cid:21) 2(cid:20) k R L k µ− k L R k µ (cid:21)
therefore Ak turns out to be the connection associated to right-handed particles and
µ
antiparticles, and A∗k has analogous role for left-handed ones. Hence, until P and
µ
CP invariance holds, we have a full correspondence,through these symmetries, between
currentsassociatedtoAk andA∗k;but,assoonasP andCP aresimultaneouslyviolated,
µ µ
thesecurrentswouldacquireanindependentphysicalnature. Asarelevantconsequence,
inthis contextbasicconnectionvariablesmustbe identifiedinAk andA∗k,respectively.
µ µ
From a physical point of view, this would imply that the SU(2) group becomes a more
fundamental symmetry with respect to the Lorentz one. But here the question about
why experiments provide evidence for the Lorentz symmetry, instead of a fundamental
SU(2) one, would naturally arise. In order to precise the physical characterization of
such fundamental role played by the SU(2) group, we propose the following symmetry
breaking scenario. Before particles acquire masses, ψ and ψ are independent and
L R
there is no objection in taking Ak, A∗k as different physical fields, if CP is violated (P
µ µ
violationsbeingprovidedbythe standardElectro-Weakmodelchirality). But,whenthe
energydropsdowntoscalesatwhichthespontaneoussymmetrybreakingprocessoccurs,
thenwegetmassiveparticles,whichmeansthatthedynamicsofψ andψ iscorrelated,
L R
soanewLorentzconnectionωab arises. Ofcourse,thepresentobservationofthe leptons
µ
asindividualparticlesrequires,inthisscheme,afundamentalinteraction,whichconfines
their left and right components. Therefore,just at sufficiently high energies we can deal
with independent Ashtekar connections and independent left and right fermions.
REFERENCES
[1] Papapetrou A.,Proc. Roy. Soc. London, A209 (1951) 248
[2] Dixon W. G.,Il Nuovo Cimento, A XXXIV, no 2 (1964) 317
[3] Souriau J. M.,Ann. Inst. H. Poincar, A XX, no 4 (1974) 22
[4] Kaluza T.,Sitzungseber. Press. Akad. Wiss. Phys. Math. Klasse, K1 (1921) 966
[5] Klein O.,Nature, 118 (1926) 516
[6] Cianfrani F. and Marrocco A. and Montani G., Int. J. Mod. Phys, D14, 7 (2005)
1195
[7] Cianfrani F. and Milillo I. and Montani G., Dixon-Souriau equations from a 5-
dimensional spinning particle in a Kaluza-Klein framework, submitted to Phys. Lett. A
6 F.CIANFRANIand G.MONTANI
[8] Ashtekar A., Phys. Rev. D, 36 (1987) 1587
[9] Ashtekar A., Phys. Rev. Lett., 57 (1986) 2244
