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Spin and statistics in massive (2+1)-dimensional QED
Volume 214, number 3
PHYSICS LETTERS B
24 November 1988
SPIN AND STATISTICS IN M A S S I V E ( 2 + I ) - D I M E N S I O N A L Q E D T.H. H A N S S O N
l
M. ROt~EK
2 I. Z A H E D
l
Physics Department, State University of New York at Stony Brook, Stony Brook, NY 11794, USA and S.C. Z H A N G 3
Institute for TheoreticalPhysics, Universityof California, Santa Barbara, CA 93106, USA Received 27 June 1988
We reexamine the questions of spin and statistics of nonrelativistic charged particles coupled to a topologicallymassive abelian gauge field. We show that these particles obey fractional statistics and carry an extra (gaugeinvariant) spin so that the generalized spin statistics relation is fulfilled. In particular, if the topologicalmass is obtained from integrating out heavy fermion fields, two flavors of fermions are needed to turn bosons into fermions. We also show, by explicitly considering the Dirac vacuum in the presence of a heavy point particle, that the screeningcharge is exponentiallylocalized and that the fermions do not contribute to the (gauge invariant) spin of the particle.
It was discovered some time ago that in two spatial dimensions, the wave functions of identical particles need not be either totally symmetric or a n t i s y m m e tric. The usual Bose a n d Fermi alternatives are just two special cases of more general fractional statistics where a multiparticle wavefunction changes by a phase e x p ( i 0 ) u n d e r interchange of identical particles [ 1-3]. (We refer to this as 0-statistics.) What makes this possibility so interesting is that there seem to be physical systems where the quasiparticles obey fractional statistics. The prime example is the quasiparticles in Laughlin's wavefunction for the fractional q u a n t u m Hall effect [4,5], but there are also indications that statistics-changing interactions may be i m p o r t a n t for the u n d e r s t a n d i n g of high Tc superconductivity [ 6 ]. In 2 + 1 dimensions, f u n d a m e n t a l bosons or ferm i o n s with certain particular long range interactions ' Supportedin part by the US Department of Energyunder Grant No. DE-FG02-88ER40388. 2 Supported in part by the National Science Foundation under Grant No. PHY85-07627. 3 Supported in part by the National Science Foundation under Grant No. PHY82-17853, supplemented by NASA. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
can alternatively be viewed as free particles obeying 0-statistics. The archetypical nonrelativistic lagrangian which gives rise to 0-statistics
d 0 L = ~ ½mii'i2+ ~ j ~ j - ~ o l j j ,
(1)
where the relative azimuthal angle is given by o~= t a n - l [ ( Y j - Y i ) / x j - x j ) ]. Alternatively this can be cast in the h a m i l t o n i a n form H = ~ ~ m , [ p , - e A ( r , ) ]2,
(2)
where the statistics phase is obtained as a B o h m A h a r o n o v effect originating from the "statistical" gauge field ~l
zne
j#i
(3)
where V(i) is the derivative with respect to ri. Another ~' Note that this hamiltonian does not describe the electromagnetic interaction of a real charge flux bound state [7]. However, with the particular form of the "gauge field" givenby (3) it does describe a system with 0-statistics. 475
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PHYSICS LETTERS B
example of 0-statistics is provided by the solitons of the O (3) nonlinear sigma model with a Hopf term
[8]. It has also been noted that particles with fractional statistics also have fractional spin, so there is a generalized spin statistics connections [2,9]. In this note we consider another class of model hamiltonians which have been claimed to exhibit fractional statistics, namely charged particles coupled to a topologically massive abelian gauge field [10-13]:
5f = ½l ( 0~ + ieqAu)~ol 2 - ½M2~o2 - -~F~,,Fu~ + ½#e~'~l'Al, O,A/,.
(4)
Here q is an integer so the fundamental ~0field has charge qe. The results of our analysis of the spin and statistics of the massive (0 particles in this model are not in agreement with several of the claims made in the literature, and we comment on this at the end. As shown by Redlich [ 14 ], the topological mass term in (4) is generated radiatively at the one loop level if the gauge field is coupled to n species of massive fermions with charge e. It has also been argued that this result is not changed by higher order effects. Thus for # = n e 2 / 4 n a model equivalent to (3) is given by
.Se= ½I (0,, + iqeA,, ) g l 2-
½M2(,o 2 - 1 Fu,, F,,,,
+ L ~,(iO-e4-mi)~ui,
(5)
i=l
where ~'i is the ith flavor fermion field and mi are large masses ~2 In the following we first show that for large M, the charge (qe) excitations of (4) have spin s = q 2/ n and obey 0-statistics with O=2nq2/n which implies the conventional spin statistics connection. We then consider (5) and show that a point charge (qe) will induce an asymmetry in the fermion spectrum corresponding to a localized screening charge - q e . The canonical angular momentum of the fermions is q2/n and originates from zero modes, but there is a compensating piece from the interaction with the ~-~As discussed by Johnson [ 15 ] (4) and ( 5 ) are only equivalent when the fermion masses are large enough to unambiguously define the vacuum states even in the presence of interactions, which requires m~ >> e 2.
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24 November 1988
gauge field giving zero for the fermion part of the gauge invariant angular momentum. We now proceed to demonstrate these assertions. First consider model (4) in the nonrelativistic limit M>> e 2, #. For a single q~quantum we then have to solve for the electromagnetic field in the presence of a point-like source ju=~U°qe6(r). The solution is
[121 #qeK," 2n or# r '),
B=-eiJO'AJ=-
E = - V1 B , #
(6)
where Ko is a modified Bessel function. The angular momentum is carried by the field (ju is a pointlike scalar) and is given by
J= f d2r
8ijriO
(7)
°j ,
where the symmetric and gauge invariant energymomentum tensor is given by
0 u~ = ¼gU~FZ-F~°F" ~.
(8)
Note that the topological mass term does not contribute to the symmetric and gauge invariant stress tensor since it has no dependence on the metric. Substituting (6) and (8) into (7) gives
J= ~ d2r B ( r ' E )
q2
(qe) 2 -
-
-
4n#
-
(9)
n
To analyze the statistics, we must determine the force between two widely separated charges. The hamiltonian corresponding to (4) was constructed in the Coulomb gauge by Deser, Jackiw, and Templeton, who found ~¢t¢=
1(
1
H2+{(
- V2+ #2)~-I-p _V2+#2P
)
1
+ #~iji - V 2 + # 2 H - e °~ijJ{ ...
-
#evj '
_
oj V2 ( _ 72.o1_#2 )/9 ,
(1o)
where ~i= 0i/_x/7~ and (p, f ) is the gauge invariant current of the ~0field..4i=g~a~ and H i = -e°OjHare the transverse vector potential and its conjugate momentum, and 4 ' is related to the original transverse gauge field via a source dependent shift. Eq. (10) clearly shows that the only dynamical degree of freedom related to the gauge field is massive. In addition,
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PHYSICSLETTERSB
however, there is a long range instantaneous interaction coupling the charge directly to the current. For distances much longer than the screening length ( _~ 1 / e2), this is the only remaining interaction, and in the nonrelativistic limit, we get the following hamiltonian in the presence of two ¢0particles:
~(p H=
q2e2* )2r -4- p2 + 4--~ ~5
~--~ +Hr,a,
(11)
where r a n d p are the relative position and its conjugate momentum, and *ri= g;rj. P is the center-of-mass momentum and H~aa is the part of the hamiltonian that depends on the radiation field. Comparison with (2) shows that these particles obey 0-statistics with O= 2nq2/n as claimed above. We should stress that, even though each ~0particle has a charge e and a flux ¢ = e / / x associated with it, it does not act like a charge flux-tube composite in normal QED [2,7,16] (which we refer to as a cyon [ 17 ] ) because the dynamical transverse gauge field in (10) is massive. Consequently, the interchange of two ~0particles gives ½the phase that the interchange of two cyons gives. One can understand this intuitively as follows: Charge can be defined either in terms of the asymptotic electric field through Gauss's law, or in terms of the coupling to a gauge potential. Because the electric field decays exponentially, the ~0 particles have no Gauss law charge; however, the topological mass term is independent of matter fields, and hence does not change the gauge coupling. In contrast, for cyons the two definitions of charge agree. Hence interchanging two ~0 particles gives only the phase from the charges moving around fluxes (since the gauge potential is long range), but no contribution from the fluxes moving around the charges (since the fields are screened), whereas interchanging two cyons gives the sum of these two phases. We now proceed to analyze the problem from the point of view of the heavy fermion lagrangian (5). For sufficiently large m~ the topological mass term is the only contribution to the polarization tensor, and the electromagnetic field in the presence of a point charge is given by (6). We would now like to demonstrate that this solution is self consistent, i.e., that the fermionic vacuum has been polarized so as to give a localized screening charge-qe and zero angular momentum. Since we were not able to solve the Dirac equation analytically in the self-consistent field given
24 November 1988
by (6) we proceed somewhat indirectly. As pointed out above, (6) corresponds to a magnetic flux ~b=qe/ Ix= (2q/n)2n/e, or 2q/n units of flux. There is also a short range electrostatic potential A 0(r) = B (r) //t due to the screened electric charge. (Here we treat the electromagnetic field as a static classical background and impose the Lorentz condition 0uA ~= 0 on the potential, corresponding to Landau gauge for the full quantum theory.) Since the induced charge is expected to depend on the total flux rather than the details of the wavefunctions, we considered the problem of a thin flux tube and a weak box potential:
eel) ¢ eA ( r' a ) = 2--~r~r& - - & eA°( r, a )= - VO(R-r) ,
(12)
where r and a are polar coordinates and R ~ 1/e 2. Even in this simplified potential, the transcendental equations that determine the energy eigenvalues are quite messy. However, for 0 ~<¢ < 1 and for fermion masses going to infinity (i.e. large compared to Vand e 2), the spectrum is given by
T~ ( r, a) =Jt+o( k,pr) e iz" u
l)0,
=J_t_o(k21pr) e " u l < 0 , T~ ( r, o~) =Jt+~( klzpr) e il~ v
(13)
/~l,
=J_/_v ( kztpr ) eim V l< 1,
(14)
for r<~R, and T + ( r , a ) = T T p ( r , a ) = O for r>R. Here + and - stand for positive and negative energy, l is the (integer) Noether orbital angular momentum, and u and v are the upper and lower unit spinors. The eigenvalues are given by E21p.=+_-tx / m 2 + (k~p) 2, where
J/+~( klo, R ) =J_/_o( k2/~,R) = 0 .
(15)
We see that positive and negative energy eigenvalues are matched except for the l= 0 ones, which thus give the whole spectral asymmetry. Using the asymptotic expansion of the Bessel functions we find k,op=(p-t-~q-l¢)Tr/R and kzop=(p+¼-½¢)Tg/R which implies
477
Volume 214, number 3
PHYSICS LETTERS B
q = - ½1im ~ sgn(Etp) exp(--zlElp IR) • ~0
Lp
=-½1im ~ {exp[-T(p+l+½q~)zr ] v~0 p=0
- e x p [ - r ( p + ~ - ½0)n] } = ½0.
(16)
Note that the regularized charge density is explicitly odd under charge conjugation ( q ~ - 0 ) as required. The above calculation is valid for 0~< 0 < 1 but can easily be extended to n ~<0 < n + 1 giving the result ½( 0 - n ). In this case, however, the spectrum even for A o = 0 contains n normalizable b o u n d states with E=m (i.e. zero modes o f liD,rat--m) [18], o f the form ~,~ u exp [if r dr' A , ( r ' ) ], each o f which gives a contribution + ½to the asymmetry, so the final charge is again given by (16). This result agrees with the one obtained for the A o = 0 case by Niemi and Semenoff who used index theorems to show that the induced charge is equal to half the flux [ 19 ]. Our result shows that the A o= 0 result holds even when an electrostatic potential is present and that the spectral asymmetry arises from an infinite n u m b e r o f localized states. To calculate the spin requires more care since it turns out to depend on the wavefunctions. The pure fermionic (Noether) contribution to the angular mom e n t u m (from one flavor) is
jr = f dZr g/t( _ieijriOJ+ ½o.3)~,
(17)
while the interaction term contributes Jim = ~ dZr ~u-i~gj~,m=½~
d2r[~t,~leij r'Aj ,
(18)
since the relevant component of the interaction part of the energy-momentum tensor is O~t = -j°A ( Note that only the sum J=Jf+J~,, is gauge invariant. For the angular m o m e n t u m to be even under charge conjugation ½ [gt*, ~,] must be the induced charge defined by the spectral asymmetry. Since, however, both the induced charge and the vector potential is known for the solution (6) we can evaluate Ji.~ explicitly to get [ 20 ] Ji,t=
f dZreijrB(r)e
jl 01
~-TB(r)=-
q2
n
(19)
This contribution does depend on the wavefunctions as is easily seen by considering the same quantity for the flux-string potential (12), where the result is twice 478
24 November 1988
that of (19). To calculate the remaining piece, Jr, for an arbitrary flux we need the explicit fermion spectrum. We have not been able to do this in general, but we argue that J r = q2/n so the net contribution from the fermions is zero. For integer units o f flux and no electrostatic potential the whole contribution comes from the zero modes. Including the contribution o f the 0th mode (which is not localized), we get J r = (½n)(½~)[½+ 3+...+½(2¢--l)]=~nO2=q2/n. We must include the last zero mode for consistency with our result for the induced charge. Although we have not shown that J~+Ji= 0 for general values of the flux, we conjecture that it should indeed he true, since there is no reason to believe that the spin should be nonanalytic in the flux. Also note that the interesting cases of one and two flavors (corresponding to two and one unit o f flux respectively) is covered by our demonstration. We are also consistent with the result o f Paranjape and Brown who considered the A ° = 0 case [ 2 1 - 2 4 ] ~3 Finally we discuss our results in comparison with earlier work. Model (4) was considered by Pisarski and Rao in ref. [ 13 ]. They derive the interaction ( I 1 ), but conclude that the charge qe excitations obey 0-statistics with 0 = 4rcqZ/n, so we do not agree with their conclusion (their result is correct for cyons, as discussed above). The spin o f the excitations was calculated by Schonfeld in ref. [ 12 ] who obtained zero. He, however, calculated a different quantity, namely the angular m o m e n t u m constructed from the gauge noninvariant and nonsymmetric Noether stress tensor. There is no technical discrepancy between our results, but J as defined in (9) satisfies the generalized spin statistics connection. Hagen considered a model with a topological mass but no F ~ term [20]. This case has no propagating gauge field, and corresponds to the e 2 ~ limit o f model 4. Again there is a long range current-charge interaction which gives rise to a change o f statistics. We can see no reason why our conclusions about spin and statistics should change in the large e 2 limit. We ~3 We stress that all the time we calculate the gauge invariant angular momentum, corresponding to the definition of Wilczek [ 16] and Brown [24] rather than the Noether angular momentum considered by Jackiw and Redlich [25 ], and Paranjape [21 ]. Both definitions are perfectly good, but the gauge invariant notion should be used for discussing the generalized spin statistics connection.
Volume 214, number 3
PHYSICS LETTERS B
t h u s agree w i t h H a g e n ' s e v a l u a t i o n o f the spin o f particles, b u t e x p e c t the statistics to be c h a n g e d so t h a t the g e n e r a l i z e d spin statistics r e l a t i o n is satisfied j u s t as in m o d e l ( 4 ) . In this c o n n e c t i o n we s h o u l d also m e n t i o n a v e r y r e c e n t p a p e r by P o l y a k o v w h o argues t h a t t h e c h a r g e d e x c i t a t i o n s o f a n o n l i n e a r s i g m a m o d e l c o u p l e d to a gauge field w i t h a C h e r n S i m o n t e r m ( b u t again w i t h o u t a n F ~ t e r m ) bec o m e f e r m i o n s for # = 1/87r 2 [26 ]. T o t h e e x t e n t t h a t the e x c i t a t i o n s o f t h e n o n l i n e a r s i g m a m o d e l can be c o n s i d e r e d as n o n r e l a t i v i s t i c , we differ f r o m Polya k o v ' s c o n c l u s i o n in t h a t we n e e d # = 1/27r 2 for the fermion boson transmutation. W h i l e w r i t i n g up this work, we l e a r n e d t h a t A.S. G o l d h a b e r a n d R. M a c k e n z i e w e r e c o m p l e t i n g an article o n c y o n s t h a t agrees w i t h a n d c o m p l e m e n t s o u r conclusions.
Acknowledgement We t h a n k K. J o h n s o n , S. K i v e l s o n , X . G . W e n , A. Zee, a n d A.S. G o l d h a b e r for s t i m u l a t i n g a n d useful discussions.
References
24 November 1988
[2] F. Wilczek, Phys. Rev. Lett. 49 (1982) 957. [3] D.P. Arovas, R. Schrieffer, F. Wilczek and A. Zee, Nucl. Phys. B 251 (1985) 117. [4] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. [5] D.P. Arovas, R. Schrieffer and F. Wilczek, Phys. Rev. Lett. 53 (1984) 722. [6] P.B. Wiegman, Phys. Rev. Len. 60 (1988) 821. [7] Y.S. Wu, Phys. Rev. Lett. 52 (1984) 2103. [8] F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250. [9] D. Thouless and Y.S. Wu, Phys. Rev. B 31 (1985) 1191. [ 10] W. Siegel, Nucl. Phys. B 156 (1979) 135. [ l l ] S . Deser, R. Jackiw and S. Templeton, Ann. Phys. 140 (1982) 372. [ 12 ] J.F. Schonfeld, Nucl. Phys. B 185 ( 1981 ) 157. [ 13] R.D, Pisarski and S. Rao, Phys. Rev. D 32 ( 1985 ) 2081. [ 14] A.N. Redlich, Phys. Rev. D 29 (1984) 2366. [ 15 ] K. Johnson, Proc. 1984 APS meeting (Santa Fe), (World Scientific, Singapore, 1985 ). [ 16 ] F. Wilczek, Phys. Rev. Lett. 48 ( 1982 ) 1144. [ 17] A.S. Goldhaher, Phys. Rev. Lett. 49 (1982) 905. [ 18] R. Jackiw, Phys. Rev. D 29 (1984) 2375. [ 19 ] A.J. Niemi and G.W. Semenoff, Phys. Rev. Lett 51 ( 1983 ) 2077. [20] C.R. Hagen, Phys. Rev. D 31 (1985) 2135. [21 ] M.B. Paranjape, Phys. Rev. Lett. 55 (1985) 2390. [22] M.B. Paranjape, Phys. Rev. D 36 (1987) 3766. [23] M.B. Paranjape, Phys. Rev. Lett. 57 (1986) 500. [24] L.S. Brown, Phys. Rev. Lett. 57 (1986) 499. [25] R. Jackiw and A.N. Redlich, Phys. Rev. Lett. 50 (1983) 555. [26] A.M. Polyakov, Mod. Phys. Lett. 3 (1988) 325.
[ 1 ] J.M. Leinaas and J. Myrheim, Nuovo Cimento B 37 ( 1977 ) 1.
479
PHYSICS LETTERS B
24 November 1988
SPIN AND STATISTICS IN M A S S I V E ( 2 + I ) - D I M E N S I O N A L Q E D T.H. H A N S S O N
l
M. ROt~EK
2 I. Z A H E D
l
Physics Department, State University of New York at Stony Brook, Stony Brook, NY 11794, USA and S.C. Z H A N G 3
Institute for TheoreticalPhysics, Universityof California, Santa Barbara, CA 93106, USA Received 27 June 1988
We reexamine the questions of spin and statistics of nonrelativistic charged particles coupled to a topologicallymassive abelian gauge field. We show that these particles obey fractional statistics and carry an extra (gaugeinvariant) spin so that the generalized spin statistics relation is fulfilled. In particular, if the topologicalmass is obtained from integrating out heavy fermion fields, two flavors of fermions are needed to turn bosons into fermions. We also show, by explicitly considering the Dirac vacuum in the presence of a heavy point particle, that the screeningcharge is exponentiallylocalized and that the fermions do not contribute to the (gauge invariant) spin of the particle.
It was discovered some time ago that in two spatial dimensions, the wave functions of identical particles need not be either totally symmetric or a n t i s y m m e tric. The usual Bose a n d Fermi alternatives are just two special cases of more general fractional statistics where a multiparticle wavefunction changes by a phase e x p ( i 0 ) u n d e r interchange of identical particles [ 1-3]. (We refer to this as 0-statistics.) What makes this possibility so interesting is that there seem to be physical systems where the quasiparticles obey fractional statistics. The prime example is the quasiparticles in Laughlin's wavefunction for the fractional q u a n t u m Hall effect [4,5], but there are also indications that statistics-changing interactions may be i m p o r t a n t for the u n d e r s t a n d i n g of high Tc superconductivity [ 6 ]. In 2 + 1 dimensions, f u n d a m e n t a l bosons or ferm i o n s with certain particular long range interactions ' Supportedin part by the US Department of Energyunder Grant No. DE-FG02-88ER40388. 2 Supported in part by the National Science Foundation under Grant No. PHY85-07627. 3 Supported in part by the National Science Foundation under Grant No. PHY82-17853, supplemented by NASA. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
can alternatively be viewed as free particles obeying 0-statistics. The archetypical nonrelativistic lagrangian which gives rise to 0-statistics
d 0 L = ~ ½mii'i2+ ~ j ~ j - ~ o l j j ,
(1)
where the relative azimuthal angle is given by o~= t a n - l [ ( Y j - Y i ) / x j - x j ) ]. Alternatively this can be cast in the h a m i l t o n i a n form H = ~ ~ m , [ p , - e A ( r , ) ]2,
(2)
where the statistics phase is obtained as a B o h m A h a r o n o v effect originating from the "statistical" gauge field ~l
zne
j#i
(3)
where V(i) is the derivative with respect to ri. Another ~' Note that this hamiltonian does not describe the electromagnetic interaction of a real charge flux bound state [7]. However, with the particular form of the "gauge field" givenby (3) it does describe a system with 0-statistics. 475
Volume 214, number 3
PHYSICS LETTERS B
example of 0-statistics is provided by the solitons of the O (3) nonlinear sigma model with a Hopf term
[8]. It has also been noted that particles with fractional statistics also have fractional spin, so there is a generalized spin statistics connections [2,9]. In this note we consider another class of model hamiltonians which have been claimed to exhibit fractional statistics, namely charged particles coupled to a topologically massive abelian gauge field [10-13]:
5f = ½l ( 0~ + ieqAu)~ol 2 - ½M2~o2 - -~F~,,Fu~ + ½#e~'~l'Al, O,A/,.
(4)
Here q is an integer so the fundamental ~0field has charge qe. The results of our analysis of the spin and statistics of the massive (0 particles in this model are not in agreement with several of the claims made in the literature, and we comment on this at the end. As shown by Redlich [ 14 ], the topological mass term in (4) is generated radiatively at the one loop level if the gauge field is coupled to n species of massive fermions with charge e. It has also been argued that this result is not changed by higher order effects. Thus for # = n e 2 / 4 n a model equivalent to (3) is given by
.Se= ½I (0,, + iqeA,, ) g l 2-
½M2(,o 2 - 1 Fu,, F,,,,
+ L ~,(iO-e4-mi)~ui,
(5)
i=l
where ~'i is the ith flavor fermion field and mi are large masses ~2 In the following we first show that for large M, the charge (qe) excitations of (4) have spin s = q 2/ n and obey 0-statistics with O=2nq2/n which implies the conventional spin statistics connection. We then consider (5) and show that a point charge (qe) will induce an asymmetry in the fermion spectrum corresponding to a localized screening charge - q e . The canonical angular momentum of the fermions is q2/n and originates from zero modes, but there is a compensating piece from the interaction with the ~-~As discussed by Johnson [ 15 ] (4) and ( 5 ) are only equivalent when the fermion masses are large enough to unambiguously define the vacuum states even in the presence of interactions, which requires m~ >> e 2.
476
24 November 1988
gauge field giving zero for the fermion part of the gauge invariant angular momentum. We now proceed to demonstrate these assertions. First consider model (4) in the nonrelativistic limit M>> e 2, #. For a single q~quantum we then have to solve for the electromagnetic field in the presence of a point-like source ju=~U°qe6(r). The solution is
[121 #qeK," 2n or# r '),
B=-eiJO'AJ=-
E = - V1 B , #
(6)
where Ko is a modified Bessel function. The angular momentum is carried by the field (ju is a pointlike scalar) and is given by
J= f d2r
8ijriO
(7)
°j ,
where the symmetric and gauge invariant energymomentum tensor is given by
0 u~ = ¼gU~FZ-F~°F" ~.
(8)
Note that the topological mass term does not contribute to the symmetric and gauge invariant stress tensor since it has no dependence on the metric. Substituting (6) and (8) into (7) gives
J= ~ d2r B ( r ' E )
q2
(qe) 2 -
-
-
4n#
-
(9)
n
To analyze the statistics, we must determine the force between two widely separated charges. The hamiltonian corresponding to (4) was constructed in the Coulomb gauge by Deser, Jackiw, and Templeton, who found ~¢t¢=
1(
1
H2+{(
- V2+ #2)~-I-p _V2+#2P
)
1
+ #~iji - V 2 + # 2 H - e °~ijJ{ ...
-
#evj '
_
oj V2 ( _ 72.o1_#2 )/9 ,
(1o)
where ~i= 0i/_x/7~ and (p, f ) is the gauge invariant current of the ~0field..4i=g~a~ and H i = -e°OjHare the transverse vector potential and its conjugate momentum, and 4 ' is related to the original transverse gauge field via a source dependent shift. Eq. (10) clearly shows that the only dynamical degree of freedom related to the gauge field is massive. In addition,
Volume 214, number 3
PHYSICSLETTERSB
however, there is a long range instantaneous interaction coupling the charge directly to the current. For distances much longer than the screening length ( _~ 1 / e2), this is the only remaining interaction, and in the nonrelativistic limit, we get the following hamiltonian in the presence of two ¢0particles:
~(p H=
q2e2* )2r -4- p2 + 4--~ ~5
~--~ +Hr,a,
(11)
where r a n d p are the relative position and its conjugate momentum, and *ri= g;rj. P is the center-of-mass momentum and H~aa is the part of the hamiltonian that depends on the radiation field. Comparison with (2) shows that these particles obey 0-statistics with O= 2nq2/n as claimed above. We should stress that, even though each ~0particle has a charge e and a flux ¢ = e / / x associated with it, it does not act like a charge flux-tube composite in normal QED [2,7,16] (which we refer to as a cyon [ 17 ] ) because the dynamical transverse gauge field in (10) is massive. Consequently, the interchange of two ~0particles gives ½the phase that the interchange of two cyons gives. One can understand this intuitively as follows: Charge can be defined either in terms of the asymptotic electric field through Gauss's law, or in terms of the coupling to a gauge potential. Because the electric field decays exponentially, the ~0 particles have no Gauss law charge; however, the topological mass term is independent of matter fields, and hence does not change the gauge coupling. In contrast, for cyons the two definitions of charge agree. Hence interchanging two ~0 particles gives only the phase from the charges moving around fluxes (since the gauge potential is long range), but no contribution from the fluxes moving around the charges (since the fields are screened), whereas interchanging two cyons gives the sum of these two phases. We now proceed to analyze the problem from the point of view of the heavy fermion lagrangian (5). For sufficiently large m~ the topological mass term is the only contribution to the polarization tensor, and the electromagnetic field in the presence of a point charge is given by (6). We would now like to demonstrate that this solution is self consistent, i.e., that the fermionic vacuum has been polarized so as to give a localized screening charge-qe and zero angular momentum. Since we were not able to solve the Dirac equation analytically in the self-consistent field given
24 November 1988
by (6) we proceed somewhat indirectly. As pointed out above, (6) corresponds to a magnetic flux ~b=qe/ Ix= (2q/n)2n/e, or 2q/n units of flux. There is also a short range electrostatic potential A 0(r) = B (r) //t due to the screened electric charge. (Here we treat the electromagnetic field as a static classical background and impose the Lorentz condition 0uA ~= 0 on the potential, corresponding to Landau gauge for the full quantum theory.) Since the induced charge is expected to depend on the total flux rather than the details of the wavefunctions, we considered the problem of a thin flux tube and a weak box potential:
eel) ¢ eA ( r' a ) = 2--~r~r& - - & eA°( r, a )= - VO(R-r) ,
(12)
where r and a are polar coordinates and R ~ 1/e 2. Even in this simplified potential, the transcendental equations that determine the energy eigenvalues are quite messy. However, for 0 ~<¢ < 1 and for fermion masses going to infinity (i.e. large compared to Vand e 2), the spectrum is given by
T~ ( r, a) =Jt+o( k,pr) e iz" u
l)0,
=J_t_o(k21pr) e " u l < 0 , T~ ( r, o~) =Jt+~( klzpr) e il~ v
(13)
/~l,
=J_/_v ( kztpr ) eim V l< 1,
(14)
for r<~R, and T + ( r , a ) = T T p ( r , a ) = O for r>R. Here + and - stand for positive and negative energy, l is the (integer) Noether orbital angular momentum, and u and v are the upper and lower unit spinors. The eigenvalues are given by E21p.=+_-tx / m 2 + (k~p) 2, where
J/+~( klo, R ) =J_/_o( k2/~,R) = 0 .
(15)
We see that positive and negative energy eigenvalues are matched except for the l= 0 ones, which thus give the whole spectral asymmetry. Using the asymptotic expansion of the Bessel functions we find k,op=(p-t-~q-l¢)Tr/R and kzop=(p+¼-½¢)Tg/R which implies
477
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PHYSICS LETTERS B
q = - ½1im ~ sgn(Etp) exp(--zlElp IR) • ~0
Lp
=-½1im ~ {exp[-T(p+l+½q~)zr ] v~0 p=0
- e x p [ - r ( p + ~ - ½0)n] } = ½0.
(16)
Note that the regularized charge density is explicitly odd under charge conjugation ( q ~ - 0 ) as required. The above calculation is valid for 0~< 0 < 1 but can easily be extended to n ~<0 < n + 1 giving the result ½( 0 - n ). In this case, however, the spectrum even for A o = 0 contains n normalizable b o u n d states with E=m (i.e. zero modes o f liD,rat--m) [18], o f the form ~,~ u exp [if r dr' A , ( r ' ) ], each o f which gives a contribution + ½to the asymmetry, so the final charge is again given by (16). This result agrees with the one obtained for the A o = 0 case by Niemi and Semenoff who used index theorems to show that the induced charge is equal to half the flux [ 19 ]. Our result shows that the A o= 0 result holds even when an electrostatic potential is present and that the spectral asymmetry arises from an infinite n u m b e r o f localized states. To calculate the spin requires more care since it turns out to depend on the wavefunctions. The pure fermionic (Noether) contribution to the angular mom e n t u m (from one flavor) is
jr = f dZr g/t( _ieijriOJ+ ½o.3)~,
(17)
while the interaction term contributes Jim = ~ dZr ~u-i~gj~,m=½~
d2r[~t,~leij r'Aj ,
(18)
since the relevant component of the interaction part of the energy-momentum tensor is O~t = -j°A ( Note that only the sum J=Jf+J~,, is gauge invariant. For the angular m o m e n t u m to be even under charge conjugation ½ [gt*, ~,] must be the induced charge defined by the spectral asymmetry. Since, however, both the induced charge and the vector potential is known for the solution (6) we can evaluate Ji.~ explicitly to get [ 20 ] Ji,t=
f dZreijrB(r)e
jl 01
~-TB(r)=-
q2
n
(19)
This contribution does depend on the wavefunctions as is easily seen by considering the same quantity for the flux-string potential (12), where the result is twice 478
24 November 1988
that of (19). To calculate the remaining piece, Jr, for an arbitrary flux we need the explicit fermion spectrum. We have not been able to do this in general, but we argue that J r = q2/n so the net contribution from the fermions is zero. For integer units o f flux and no electrostatic potential the whole contribution comes from the zero modes. Including the contribution o f the 0th mode (which is not localized), we get J r = (½n)(½~)[½+ 3+...+½(2¢--l)]=~nO2=q2/n. We must include the last zero mode for consistency with our result for the induced charge. Although we have not shown that J~+Ji= 0 for general values of the flux, we conjecture that it should indeed he true, since there is no reason to believe that the spin should be nonanalytic in the flux. Also note that the interesting cases of one and two flavors (corresponding to two and one unit o f flux respectively) is covered by our demonstration. We are also consistent with the result o f Paranjape and Brown who considered the A ° = 0 case [ 2 1 - 2 4 ] ~3 Finally we discuss our results in comparison with earlier work. Model (4) was considered by Pisarski and Rao in ref. [ 13 ]. They derive the interaction ( I 1 ), but conclude that the charge qe excitations obey 0-statistics with 0 = 4rcqZ/n, so we do not agree with their conclusion (their result is correct for cyons, as discussed above). The spin o f the excitations was calculated by Schonfeld in ref. [ 12 ] who obtained zero. He, however, calculated a different quantity, namely the angular m o m e n t u m constructed from the gauge noninvariant and nonsymmetric Noether stress tensor. There is no technical discrepancy between our results, but J as defined in (9) satisfies the generalized spin statistics connection. Hagen considered a model with a topological mass but no F ~ term [20]. This case has no propagating gauge field, and corresponds to the e 2 ~ limit o f model 4. Again there is a long range current-charge interaction which gives rise to a change o f statistics. We can see no reason why our conclusions about spin and statistics should change in the large e 2 limit. We ~3 We stress that all the time we calculate the gauge invariant angular momentum, corresponding to the definition of Wilczek [ 16] and Brown [24] rather than the Noether angular momentum considered by Jackiw and Redlich [25 ], and Paranjape [21 ]. Both definitions are perfectly good, but the gauge invariant notion should be used for discussing the generalized spin statistics connection.
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PHYSICS LETTERS B
t h u s agree w i t h H a g e n ' s e v a l u a t i o n o f the spin o f particles, b u t e x p e c t the statistics to be c h a n g e d so t h a t the g e n e r a l i z e d spin statistics r e l a t i o n is satisfied j u s t as in m o d e l ( 4 ) . In this c o n n e c t i o n we s h o u l d also m e n t i o n a v e r y r e c e n t p a p e r by P o l y a k o v w h o argues t h a t t h e c h a r g e d e x c i t a t i o n s o f a n o n l i n e a r s i g m a m o d e l c o u p l e d to a gauge field w i t h a C h e r n S i m o n t e r m ( b u t again w i t h o u t a n F ~ t e r m ) bec o m e f e r m i o n s for # = 1/87r 2 [26 ]. T o t h e e x t e n t t h a t the e x c i t a t i o n s o f t h e n o n l i n e a r s i g m a m o d e l can be c o n s i d e r e d as n o n r e l a t i v i s t i c , we differ f r o m Polya k o v ' s c o n c l u s i o n in t h a t we n e e d # = 1/27r 2 for the fermion boson transmutation. W h i l e w r i t i n g up this work, we l e a r n e d t h a t A.S. G o l d h a b e r a n d R. M a c k e n z i e w e r e c o m p l e t i n g an article o n c y o n s t h a t agrees w i t h a n d c o m p l e m e n t s o u r conclusions.
Acknowledgement We t h a n k K. J o h n s o n , S. K i v e l s o n , X . G . W e n , A. Zee, a n d A.S. G o l d h a b e r for s t i m u l a t i n g a n d useful discussions.
References
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[2] F. Wilczek, Phys. Rev. Lett. 49 (1982) 957. [3] D.P. Arovas, R. Schrieffer, F. Wilczek and A. Zee, Nucl. Phys. B 251 (1985) 117. [4] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. [5] D.P. Arovas, R. Schrieffer and F. Wilczek, Phys. Rev. Lett. 53 (1984) 722. [6] P.B. Wiegman, Phys. Rev. Len. 60 (1988) 821. [7] Y.S. Wu, Phys. Rev. Lett. 52 (1984) 2103. [8] F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250. [9] D. Thouless and Y.S. Wu, Phys. Rev. B 31 (1985) 1191. [ 10] W. Siegel, Nucl. Phys. B 156 (1979) 135. [ l l ] S . Deser, R. Jackiw and S. Templeton, Ann. Phys. 140 (1982) 372. [ 12 ] J.F. Schonfeld, Nucl. Phys. B 185 ( 1981 ) 157. [ 13] R.D, Pisarski and S. Rao, Phys. Rev. D 32 ( 1985 ) 2081. [ 14] A.N. Redlich, Phys. Rev. D 29 (1984) 2366. [ 15 ] K. Johnson, Proc. 1984 APS meeting (Santa Fe), (World Scientific, Singapore, 1985 ). [ 16 ] F. Wilczek, Phys. Rev. Lett. 48 ( 1982 ) 1144. [ 17] A.S. Goldhaher, Phys. Rev. Lett. 49 (1982) 905. [ 18] R. Jackiw, Phys. Rev. D 29 (1984) 2375. [ 19 ] A.J. Niemi and G.W. Semenoff, Phys. Rev. Lett 51 ( 1983 ) 2077. [20] C.R. Hagen, Phys. Rev. D 31 (1985) 2135. [21 ] M.B. Paranjape, Phys. Rev. Lett. 55 (1985) 2390. [22] M.B. Paranjape, Phys. Rev. D 36 (1987) 3766. [23] M.B. Paranjape, Phys. Rev. Lett. 57 (1986) 500. [24] L.S. Brown, Phys. Rev. Lett. 57 (1986) 499. [25] R. Jackiw and A.N. Redlich, Phys. Rev. Lett. 50 (1983) 555. [26] A.M. Polyakov, Mod. Phys. Lett. 3 (1988) 325.
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