A relativistic wave equation for the Skyrmion
S.G. Rajeev
Department of Physics and Astronomy, Department of Mathematics, University of Rochester, Rochester, New York 14627, United States
Received 5 February 2008; accepted 27 February 2008 Available online 6 March 2008
Abstract
We propose a relativistically invariant wave equation for the Skyrme soliton. It is a differential equation on the spaceR1;3
S3which is invariant under the Lorentz group and isospin. The internal variable valued inSUð2Þ S3describes the orientation of the soliton. The mass of a particle of spin
and isospin both equal to j¼1 2;
3
2. . . is predicted to be M¼m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þK2jðjþ1Þ
1þK1jðjþ1Þ
q
which agrees with the
known spectrum for low angular momentum. The iso-scalar magnetic moment is predicted to be
K1
4mR, whereRis the spin. Ó2008 Published by Elsevier Inc.
PACS: 12.39.Dc; 03.65.Pm; 13.40.Em
Keywords: Skyrmion; Wave equation; Relativistic; Magnetic moment
1. Introduction
Relativistic wave equations for extended objects, as opposed to point-like particles, are hard to find. Perhaps the most well-known such theories of this type are string and mem-brane theories. Yet we know that many relativistic field theories have soliton solutions. There ought to be relativistic wave equations that describe their collective quantum
motion. The most physically interesting of these is the Skyrme soliton (Skyrmion)[1–4]
0003-4916/$ - see front matterÓ2008 Published by Elsevier Inc. doi:10.1016/j.aop.2008.02.005
E-mail address:rajeev@pas.rochester.edu
Annals of Physics 323 (2008) 2873–2880
which describes the baryon (such as a proton) as a topologically non-trivial configuration of pi mesons.
There is a quantum theory[5,6]of the collective excitations of this solution, which
pre-dicts a mass spectrum
M ¼m 1þ1
2Kjðjþ1Þ
ð1Þ
whereK is inversely proportional to the moment of inertia. If the number of colorsNcof
the underlying fundamental theory of strong interactions (QCD) is odd, the angular momentum must take on half-odd-integer values
j¼1 2;
3
2;. . . ð2Þ
IfNcis even, the values ofjare integers 1;2;. . .. In either case, each such state also carries
an isospinI¼j. Of course, once interactions with the pion waves are allowed, the higher
energy states ought to be unstable against decay to the ground state, the nucleon. Since the soliton arises in a relativistic theory, we should expect that there is some rel-ativistically invariant wave equation that describes all of these excitations. In the usual the-ory of collective excitations, the soliton is thought of as a heavy particle, with only small changes in energy allowed from its ground states. Thus the soliton must have small kinetic energy (compared to rest-mass). Also, it must have small angular momentum, so that the velocity of rotation at any point in its interior is small compared to the velocity of light. For large angular momentum, the soliton bulges out in the middle due to the centrifugal force; its moment of inertia increases, so that the energy rises slower than quadratically for
largej. It is interesting to ask if there is a relativistic wave equation that predicts the above
spectrum for small angular momentum, and incorporates this effect for largej.
There were many attempts to find physically interesting relativistic wave equations for
particles with internal degrees of freedom[7]. They were motivated largely by the desire to
find a relativistic wave equation with positive energy spectrum. Although none of them turned to be of use in physics, these investigations led to profound developments by
Har-ish-Chandra in the representation theory of semi-simple groups[8]. Our investigation can
be considered a continuation of the original physical theme. We will not insist on having positive energy though, as it has turned out that this is not a physically realistic restriction. Although we do not address this issue here, we expect that the problem of negative ener-gies is solved by second quantization, as with all the other successful wave equations.
In 2+1 dimensions, Jackiw and Nair[9]have found relativistic wave equations for
par-ticles with fractional spin.
The wave equation we propose is a differential equation for a complex-valued function onR1;3S3
. The internal degree of freedomS3leads to spin as well as isospin. There could
be an entirely different application of this idea to Kaluza–Klein theories, in whichS3
rep-resents compactified dimensions of space-time. The spectrum it predicts does not, how-ever, look like that of elementary particles we know currently: quarks and leptons.
2. The non-relativistic wave equation
In the Skyrme model the instantaneous wave function of the baryon is a function of the
space, time, andg2SUð2Þwhich describes its orientation in internal (isospin) space. If the
baryon is a boson (resp. fermion), the wave function is invariant (resp. changes sign) under
reflections in SU(2). Define the space of spinors1
VL2ðSUð2ÞÞ ¼ fv:SUð2Þ !CjvðgÞ ¼ vðgÞg ð3Þ
Vcarries a representation of SU(2)SU(2), the left and right actions of SU(2) on itself.
The wave function depends on space-time and takes values inV
w:R1;3!V: ð4Þ
ThusVplays a role analogous to the space of spinors in Dirac’s theory. The rotation
group of space (generating angular momentum) acts on the left:
ðx;gÞ ! ðRðhÞx;hgÞ;h2SUð2Þ: ð5Þ
where R:SU(2)?SO(3) is the usual homomorphism into the rotation group. Isospin acts
on the other side ofg and leaves the positionxinvariant:
ðx;gÞ ! ðx;gh1Þ: ð6Þ
Using the Peter–Weyl theorem of harmonic analysis, we can expand this wave function
in terms of irreducible representations of SU(2)SU(2):
wðx;gÞ ¼ X
j2N
wjmm0djmm0ðgÞ: ð7Þ
We denote byNthe setsf1;2;. . .gandf12;
3
2;. . .g, respectively: half-integer values forV
and integer values forVþ. Also,m;m0¼ j;jþ1;. . .janddjmm0ðgÞare the representation
matrices for the spinjrepresentation of SU(2). Thus we see that the squares of isospin and
spin have equal magnitudesjðjþ1Þin each irreducible component.
The rotational energy of a baryon at rest is given in terms of the Laplace operator on
SU(2) asm
2KDg, wheremis the mass of the lightest baryon andKis inversely proportional
to the moment of inertia of the soliton. In the units we use,h¼c¼1 andK is a
dimen-sionless quantity. Recall that in terms of the generators of the left and right action of SU(2) on itself,
Dg¼R2L ¼R
2
R: ð8Þ
Thus, in each irreducible component the rotational energy has eigenvalues m
2Kjðjþ1Þ: ð9Þ
The Schrodinger equation for the free Skyrmion is then
iow ot ¼
1
2mDþ
m
2KDg
w ð10Þ
1
A more intrinsic point of view is thatVis the space of sections of a complex line bundle on SO(3)=SU(2)/Z2. There are exactly two such line bundles: the trivial one who sections areVþdefined above and the non-trivial one
whereD¼ r2is the Laplace operator on spaceR3. This equation has too much
symme-try: it is invariant under separate left and right actions of SU(2) ongas well as under the
SO(3) action on x. In other words it conserves the orbital angular momentum and spin
separately. This is also true of the non-relativistic free (Pauli) wave equation for the elec-tron. The correct relativistic wave equation should break this unwanted symmetry so that only the sum of orbital and spin angular momentum is conserved, as for the Dirac equa-tion.
3. The SL(2,C) representation onV%
The universal cover of the Lorentz group, SL(2,C), is the complexification of SU(2). Hence, a representation of the Lie algebra su(2) on any complex vector space can be extended to a representation of sl(2,C).To illustrate this idea, if we were to do this exten-sion to the defining representation of SU(2), we will get the two dimenexten-sional (Pauli) spinor representation of SL(2,C).
Explicitly, if the su(2) representation is given by Hermitian operatorsR, we will have
½R1;R2 ¼
ffiffiffiffiffiffiffi
1 p
R3; ½R2;R3 ¼
ffiffiffiffiffiffiffi
1 p
R1; ½R3;R1 ¼
ffiffiffiffiffiffiffi
1 p
R2 ð11Þ
or more succinctly
½Ri;Rj ¼ ffiffiffiffiffiffiffi
1 p
ijkRk: ð12Þ
The six generators of the sl(2,C) action are then given by
R01¼ iR1; R02¼ iR2; R03¼ iR3 ð13Þ
for the boost and
R23¼R1; R31¼R2; R12¼R3 ð14Þ
for the rotation generators. They satisfy the sl(2,C) relations, as can be directly verified:
½Rlm;Rqr ¼ ffiffiffiffiffiffiffi
1 p
½glqRmrgmqRlrglrRmqþgmrRlq: ð15Þ
In our case this is an infinite dimensional non-unitary representation of the Lorentz group which commutes with isospin; i.e., the right action of SU(2).
4. Relativistic invariance
Let us see how the above wave Eq.(10)can be turned into a relativistically invariant
one. The obvious guess is the Klein–Gordon type equation on functionsw:R1;3!V
,
½ololþm2KDgþm2w¼0: ð16Þ
This gives the mass shell condition:
plpl¼m2þm2Kjðjþ1Þ; j2N
: ð17Þ
As is common in relativistic wave equations, the energy can be positive or negative:
E¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2þm2þm2Kjðjþ1Þ: ð18Þ
In the non-relativistic limit, kinetic energies due to translation and rotation are both small compared to the rest-mass:
E¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2þm2þm2Kjðjþ1Þ’mþm
2Kjðjþ1Þ þ
p2
2m ð19Þ
thus recovering the earlier formula, except for the addition of the rest mass to the energy
But again this equation has too much invariance. It is invariant under
SOð1;3Þ SULð2Þ SURð2Þ. The invariance should under SOð1;3Þ SURð2Þ. Unlike in
the non-relativistic case, this error is not forgivable: only orbital plus spin angular momen-tum can be conserved in a relativistic theory, not each separately. There must be a
simul-taneous action of SO(1,3) on space-time and SL(2,C) on S3SUð2Þwhich ties Lorentz
transformations to the left action on SU(2).
To break the symmetry, recall the Pauli–Lubansky vector
Wl¼
1 2lmqrp
m
Rqr: ð20Þ
This is a Lorentz-vector only under thesimultaneousaction on space-time andV. So we
can add this to the wave equation to get an equation with the right symmetry:
½plplþK
1WlWlþm2K2Dgþm2w¼0: ð21Þ
The dimensionless constantsK1;2 will be determined by physical properties of the soliton.
In the rest-framep’ ðM;0;0;0ÞandW ’ ð0;MRÞand
WlWl ’ M2jðjþ1Þ; ð22Þ
since R generates the left action of the SU(2) Lie algebra on SU(2). So, the rest energy
spectrum is now given by
M2M2K
1jðjþ1Þ þm2K2jðjþ1Þ þm2¼0 ð23Þ
or
M ¼ m
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þK2jðjþ1Þ
1þK1jðjþ1Þ
s
: ð24Þ
Of course, each eigen-space with fixed jhas degeneracy 2jþ1 as it carries also a
rep-resentation ofSUð2ÞR with isospinj. For small isospin and spin,
M ’mþm
2Kjðjþ1Þ; K¼K2K1: ð25Þ
in agreement with the non-relativistic spectrum.
If we want the mass to remain finite for largej, we must impose
K2>K1>0: ð26Þ
In this case, as angular momentum grows, the energy tends to a constant:
E!m
ffiffiffiffiffiffi
K2 K1
r
This can be interpreted as due to a deformation of the rotating soliton that increases its
moment of inertia: it bulges out around its middle (becomes an oblate spheroid)[10]. Our
spectrum says essentially that for large angular momentum the moment of inertia becomes very large.
More realistically, we can add higher derivative terms proportional toðDgÞ
2
to make the
energy go to infinity asj! 1: such ambiguities always exist in an effective theory. After
coupling to pions, it is quite possible that for large enough angular momentum the energy
of the Skyrmion is complex: it decays by emitting pions[10].
We need to know one extra piece of information about the Skyrmion, in addition to the
low energy spectrum, to determine the two parametersK1;2. The magnetic moment gives
this piece.
5. Electromagnetic coupling
It is an interesting experimental fact that the proton and the neutron do not have the value of magnetic moment predicted by the Dirac equation through minimal coupling. The gyromagnetic ratio of the proton is not two. And the neutron, as a charge zero par-ticle should not have a magnetic moment at all, according to the minimally coupled Dirac equation. This can be understood in the quark model. Can our relativistic wave equation, with minimal coupling, predict the magnetic moment in the soliton picture?
Recall that in the standard model of elementary particles, the electric charge of a had-ron is isospin plus half the baryon number:
Q¼I3þ B
2: ð28Þ
The isovector part seems to be more subtle to understand in the Skyrme model[11]. We
will restrict ourselves to the iso-scalar component, given by the minimal coupling rule
pl!pl¼pleAl: ð29Þ
Applied to the wave equation, this ought to tell us about the iso-scalar magnetic moment
of the Skyrmion. We will see that the answer is proportional to the constantK1.
We get
½plplþK1xlxlþm2K2Dgþm2w¼0; xl¼
1
2lmqrp
mRqr: ð30Þ
If we take the non-relativistic limit in the presence of a weak constant magnetic field,
x0¼pR; x p0R ð31Þ
xlxl ðp:RÞ2p20R 2
¼ p2
0R
2þ1
4½pi;pj½Ri;Rj þ 1
4½pi;pjþ½Ri;Rjþ
¼ 1
2RBþ
1
4½pi;pjþ½Ri;Rjþp 2
0R
2: ð32Þ
So that the magnetic energy in the NR approximation is K1
4mRB. This gives the
c¼ K1
4m: ð33Þ
Thus the low energy spectrum plus the magnetic moment determine the masses of all the Skyrmion states.
6. The complex analytic point of view
We now describe another way to think of the representation of SL(2,C) onV, which
does not make much difference to the physics, but which is interesting mathematically.
There is a representation of SL(2,C)SU(2) on complex analytic functions on SL(2,C):
/ðcÞ7!/ðkch1Þ; k2SLð2;CÞ;h2SUð2Þ: ð34Þ
Now, every analytic function/:SLð2;CÞ !C can be obtained uniquely[12,13]as the
continuation of a function won SU(2). It is clear that even (resp. odd) go over to even
(odd functions) functions:
wðgÞ ¼ wðgÞ ()/ðcÞ ¼ /ðcÞ ð35Þ
and that this condition is preserved by the action of SL(2,C)SU(2). Thus we can
iden-tifyV with the space of analytic functions satisfying the above condition. Thus we get a
representation of SL(2,C) onV.
This is analogous to the way that the space of functions on the real line can be identified with holomorphic functions on the complex plane: the well-known Bargmann
correspon-dence useful in studying coherent states [14]. Indeed SL(2,C) is diffeomorphic to the
co-tangent space of S3. Thus we are thinking of wave functions as analytic functions on
the phase space instead of as complex-valued functions on the configuration space, just as with coherent states.
There is a unique (up to multiplication) Riemannian metric that is invariant under the
SL(2,C)SU(2) action. This is not the familiar metric on a simple Lie group that is invariant
under the left and right actions: on a non-compact group like SL(2,C) such a metric would not be positive. If we take the positive quadratic form in left-invariant vector fields, we will get
instead a Riemann metric invariant under SL(2,C)LSU(2)R. The Laplacian of this metric
is, up to a constant multiple, the operatorDgonVunder the above identification.
7. Some red herrings and future directions
Is it possible to have an action of SL(2,C)SU(2) onS3without the above analytic
con-tinuation to SL(2,C)? There are indeed ways that SL(2,C) can act onS3itself. They have
all been classified[15], but none commute with the right action of SU(2). Being not
invari-ant under isospin, they are not useful to us.
In the absence of the term withK2, the mass of particles would have decreased with
angular momentum, a physically unacceptable answer. This is typical of the wave equa-tions studied by Bhabha and Harish-Chandra.
The argument for adding the Pauli–Lubansky term to break an unwanted symmetry is
reminiscent of Witten’s argument[3]for the WZW term in the QCD effective action. But,
vanishes identically for the two-flavor case we are considering here. We hope to return to the question of a relativistic wave equation for the SU(3)-symmetric Skyrme model later. We were unable to find a first order wave equation analogous to the Dirac equation for
our case. However, Feynman and Gell-Mann pointed out[16] years ago that the Dirac
equation can be recast as a second order equation for a Pauli spinor (which has the half the components of a Dirac spinor, thereby compensating for the increase in order of the equation). Our equation could be viewed as a generalization to higher spins of this form of Dirac’s equation.
There is a simple Lagrangian that leads to the above wave equation. This could be use-ful in pursuing second quantization, which is needed to remove negative energy states.
We would like to understand the isovector part of magnetic moments as well. Also, cou-pling to pions and the nature of chiral symmetry need to be understood. Although our wave equation has a term which is a fourth order differential operator, in the space-time co-ordinates it is still second order. It would be interesting to determine whether the Cau-chy problem is well-posed for this equation.
Acknowledgments
I thank L. Gross, B. Hall, A. Jordan and M. Paranjape for discussions. This work was supported in part by the Department of Energy under the contract number DE-FG02-91ER40685.
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