- Email: [email protected]
Spin and statistics of CP1 skyrmions
Volume 146B, number 5
18 October 1984
PHYSICS LETTERS
SPIN AND STATISTICS OF CP1 SKYRMIONS A.M. DIN Institute of Theoretical Physics, University of lmsanne,
BSP Dorigny, CH-1015 Luusanne, Switzerland
and W.J. ZAKRZEWSKI ’ Ct?RN,Geneva, Switzerland Received 8 June 1984
We investigate the properties of n-skyrmion systems in the (2 + 1)-dimensional CP’ model with an additional Hopf term in the action. For a general time dependence of the skyrmion trajectories, we derive a simple expression for the action in terms of a complex plane line Integral. In the special case of skyrmion rotations and exchanges this expression reduces to closed contour integrals showing the unusual spin and statistics of the skyrmions.
The skyrmion picture [I] of particles interpreted as solitions in non-linear sigma models has turned out to be very useful for understanding the phenomenology of low energy particle interactions. The quantum properties of skyrmions are intimately related to the topological structure of the theory and to extract these properties in an explicit way it is illustrative to consider the case of sigma models in (2 + 1) dimensions [2]. The O(3) model [3] has been studied in some detail for this purpose but we fmd it more convenient to study instead the (in fact essentially equivalent) CP1 model [4] in (2 + 1) dimensions. The action of this model is taken to be S = Jdt
d2x [D,z Iy*z t (0/27r)A,Jr],
(1)
where z = (z 1, z2) is a complex vector fulfilling /zIy = 1 and the covariant derivative Dfiz = a Mz Z * a,z z makes the gauge invariance under z -+eiolz manifest. The e-term appearing in S is given in terms of the conserved topological current
r On leave of absence from Department of Mathematical Sciences, University of Durham, Durham, England.
0370-2693/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Jfi = -(i/2n)#rhD
YzDAz
(2)
and the “gauge” potential A, =iZa,z,
(3)
which obviously in a local function of the basic field, in contradiction to the O(3) model where the relationship is non-local. The B-term is nothing but the Hopf term which is related formally to the Chem-Simon term of gauge theory [5] since from (2) and (3) it follows that A,Jp
= (-1/2n)A,Wh+lA
= (- 1/4n)~@A,,
Fvh .
Solitons are time independent solutions of the equations of motion corresponding to S. But for such SOlutions the &term is identically zero (because of the time-derivative) and the solitons are just the instantons (or anti instantons) of the CP1 model in two euclidean dimensions. The existence of these topological solutions is predicted from the homotopy n,(CPl) = Z and the homotopy class (winding number) is given explicitly in terms of the topological charge Q = Jd2x
Jo =; [d2x(]D+zi2
- lD_z12),
(4) 341
Volume
146B, number
5
PHYSICS
where we have introduced derivatives with respect to the complex variables x, = x I f ix2. Explicitly one can represent the most general ninstanton solution (fulfilling D-z = 0) in the form z = f/If1 with f = [(x+ - al).**(x+
- an), Xx+ - bl)+**(x+ - bn)3 (5)
where ai # bj for all i and j. In the dilute instanton gas limit where in the complex plane each ai is close to bi and far from the others the n-instanton looks like a system of anyons [6] (bound states of magnetic flux tubes and charged particles) and can be treated approximately as a classical system of particles [2,7]. The correspondence is however somewhat deceptive since the “substructure” given by the a-parameters (the so-called instanton quarks in the language of ref. [S] and the b-parameters (the instanton anti-quarks) implies a further degree of freedom which is crucial for the understanding of the quantum properties of the system. We here consider the n-skyrmion configuration without making any approximations and allow the quark and antiquark parameters a and b as well as the scale X to vary arbitrarily as a function of time t belonging to a finite interval say [0, T] . In case when the t = 0 and t = T configurations are identical the skyrmion defines a map S3 + CP1 - S2 and since n3(S2) = Z we know that in each topological charge (Q) sector the skyrmion manifold is multiply connected. This fact is basically responsible for the spin and statistics properties of skyrrnions [9]. To be completely general we will however first consider a situation where ai( b,(t), h(t) vary arbitrarily in the internal [0, T] (but still subject to ai f bj and h f 0). In the evaluation of the action S = Jr dt 1 d2x f? the kinetic part of L? will (adiabatically) give a contribution ET + O(T-l), where E = n is the energy of the n-skyrmion system, which just corresponds to the usual quantum mechanical time development. The behaviour of the Hopf term H=;dt
jd2xd”“~~a,za,~‘a,z
(6)
0
is more interesting 342
and we will now show how to cal-
1984
culate it explicitly. The basic problem is to evaluate directly the x-integration. In the case of the topological charge (4) the integrand Jo is quickly seen to be a total derivative and so it reduces to a line integral. The integrand h = epVhZ * a,z a,? - a,,z in (6) is not obviously a total divergence but it nevertheless turns out to be so after some algebraic manipulations. To see this we write in general z = f/If1 with f = (fl(x+), fz(x+)) and the integrand becomes h =
Wifi4)[@,fla+f2 - a,f,a+f,)
x (fla+f2 -f,a+f,)+ - c.c.1 .
(7)
The identity a+(fl/GrfP)
= Ifl-4(fla+f2
shows that h can be rewritten h = m[(a,fla+f2
-f2a+fl)
(8)
as
- atf2a+fl).Fllf2~fi2i
-cd, (9)
which is precisely a total derivative expression allowing H to be represented as a line integral: - a,f2a+fi)
H=$
f1
x f21f12-C.C.
1 .
(10)
The x+ contour consists of a circle at infinity and encloses all the singularities of the integrand. We now insert the explicit form (5) for fl and f2 and find the integrand to be
-C;i cg (x+ - ijai)(X+ - bi>
where the dots refer to time derivatives. From the first term there is a non-vanishing contribution at the poles x, = bj and from the second term there is a contribution at infinity. The final result for H is:
H=;;dt(-?&+-+n; -c.c.).
j-d2xA,Jp
0
=&fdt
18 October
LETTERS
(12)
i 0 This expression involves line integrals in the complex plane of the relative vectors bj - ai and the scale A. For the case of the initial skyrmion configuration at t = 0 coinciding with the final configuration at t = T,
Volume
146B, number
PHYSICS
5
i.e. the set {~:ai(7’)}issome permutation of the set {ai(O and similarly for b (while for h we must have h(0) = h(T)), the r integral reduces to a closed contour integral and the value of H is therefore always an integer multiple of 2 n. To be specific, let us investigate the case of a one-skyrmion configuration f = (x+ - a, X(x+ - b)) an d consider a 27r rotation a(t) = a exp(it), b(t) = b exp(it), t E [0,2x]. Then 2n HZ_; r dt(iti)=2n, (13) 0
since the relative vector b(t) - a(t) just makes one revolution around the origin. In general the value of H thus depends on how many times the closed contours of the relative position vectors of the quarks and antiquarks revolve around the origin. As a further example we may consider a two skyrmion configuration f= ((x+ - al) X (x+ - a& h(x+ - br)(x+ - b2)) where we exchange al and a2.The answer for H is now -+_
a1
a1 -b1 a1 -I--------
a2 - bl
a2
at - b,+b?-C’C’ aI I
a2
L
.
22
(14) i
I
18 October 1984
LETTERS
If we thus perform a 2n-rotation and an exchange as done above, this factor is just exp(ie), which for the choice of the parameter 0 = 0 becomes +l , i.e. the skyrmion is quantized as a boson, and for 0 = rr it becomes -1, i.e. quantisation as a fermion. This result is independent of the orientation of the transformation path but the result of the exchange operation could come out to be +l instead of -1 by choosing a different exchange path. When taking 0 < 0 < rr, to somehow interpolate between bosons and fermions, the paths and their orientation can be chosen so as to obtain any factor exp(S), where I is integer. In ref. [2], it was noted that there was an arbitrariness in the assignment of a factor exp(i0) or exp(-i0) to the exchange operation, but as we see it, this assignment problem appears to be more generic. Another problem is that the Hopf term considered in ref. [2] to describe the classical particle correspondence with the skymrion field theory does not quite agree with our general expression (12) even if the bound-state degree of freedom is frozen. The above suggests that the consistency and possible implications of interpolating quantum statistics of skyrmions warrants further study. These problems will be discussed elsewhere.
.
The terms [al/(al - bl)] and [a2/(a2 - bl)] combine to give a single closed contour integral and similarly for the terms [iI/(ul - b2)] and [;~/(a2 - b2)] The value of H now depends on whether b 1 and b2 lie within the closed contour traced out by al and ~2, and if so, how many times the contour revolves. In the case, say, when only bl lies inside the contour which revolves once, we find H = 2n. This value is the same as for the rotation considered in (13) and is an indication that, as anticipated, the exchange operation and the 2rr-rotataion belong to the same connectivity (or homotopy) class, which is but a manifestation of the connection between spin and statistics [9]. However the evaluation of H following an exchange quarks (or antiquarks) was seen to depend crucially on how this exchange was performed. The effect of the Hopf term corresponding to a certain transformation on the parameters of an n skyrmion wave function is simply, as it can be seen from eq. (l), to multiply with a factor exp(SH/2rr).
References [l] T.H.R. Skyme,
Proc. R. Sot. London
Ser A 247 (1958)
260. [2] F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) F. Wilczek and A. Zee, UC Santa Barbara ITP-84-25
preprint
2250; NSF-
(1984).
[3] A.A. Belavin and A.M. Polyakov,
JETP Lett.
22 (1975)
245. [4] H. Eichenherr, E. Cremmer
Nucl. Phys. B146 (1978)
215;
and J. Scherk,
Phys. Lett. 74B (1978)
341;
V. Golo and A. Perelomov,
Phys. Lett. 79B (1978)
112.
[5] S. Deser, R. Jackiw (1982)
and S. Templeton,
Phys. Rev. Lett. 48
975.
[6] F. Wilczek, Phys. Rev. Lett. 49 (1982) [7] Y.S. Wu, University
of Washington
957.
preprint
40048-
Q7P4 (1984). [S] V. Fateev, (1979)
1. Frolov
and A. Schwartz,
Nucl. Phys. B154
1.
[9] D. Finkelstein
and J. Rubinstein,
J. Math. Phys. 9 (1968)
1762.
343
18 October 1984
PHYSICS LETTERS
SPIN AND STATISTICS OF CP1 SKYRMIONS A.M. DIN Institute of Theoretical Physics, University of lmsanne,
BSP Dorigny, CH-1015 Luusanne, Switzerland
and W.J. ZAKRZEWSKI ’ Ct?RN,Geneva, Switzerland Received 8 June 1984
We investigate the properties of n-skyrmion systems in the (2 + 1)-dimensional CP’ model with an additional Hopf term in the action. For a general time dependence of the skyrmion trajectories, we derive a simple expression for the action in terms of a complex plane line Integral. In the special case of skyrmion rotations and exchanges this expression reduces to closed contour integrals showing the unusual spin and statistics of the skyrmions.
The skyrmion picture [I] of particles interpreted as solitions in non-linear sigma models has turned out to be very useful for understanding the phenomenology of low energy particle interactions. The quantum properties of skyrmions are intimately related to the topological structure of the theory and to extract these properties in an explicit way it is illustrative to consider the case of sigma models in (2 + 1) dimensions [2]. The O(3) model [3] has been studied in some detail for this purpose but we fmd it more convenient to study instead the (in fact essentially equivalent) CP1 model [4] in (2 + 1) dimensions. The action of this model is taken to be S = Jdt
d2x [D,z Iy*z t (0/27r)A,Jr],
(1)
where z = (z 1, z2) is a complex vector fulfilling /zIy = 1 and the covariant derivative Dfiz = a Mz Z * a,z z makes the gauge invariance under z -+eiolz manifest. The e-term appearing in S is given in terms of the conserved topological current
r On leave of absence from Department of Mathematical Sciences, University of Durham, Durham, England.
0370-2693/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Jfi = -(i/2n)#rhD
YzDAz
(2)
and the “gauge” potential A, =iZa,z,
(3)
which obviously in a local function of the basic field, in contradiction to the O(3) model where the relationship is non-local. The B-term is nothing but the Hopf term which is related formally to the Chem-Simon term of gauge theory [5] since from (2) and (3) it follows that A,Jp
= (-1/2n)A,Wh+lA
= (- 1/4n)~@A,,
Fvh .
Solitons are time independent solutions of the equations of motion corresponding to S. But for such SOlutions the &term is identically zero (because of the time-derivative) and the solitons are just the instantons (or anti instantons) of the CP1 model in two euclidean dimensions. The existence of these topological solutions is predicted from the homotopy n,(CPl) = Z and the homotopy class (winding number) is given explicitly in terms of the topological charge Q = Jd2x
Jo =; [d2x(]D+zi2
- lD_z12),
(4) 341
Volume
146B, number
5
PHYSICS
where we have introduced derivatives with respect to the complex variables x, = x I f ix2. Explicitly one can represent the most general ninstanton solution (fulfilling D-z = 0) in the form z = f/If1 with f = [(x+ - al).**(x+
- an), Xx+ - bl)+**(x+ - bn)3 (5)
where ai # bj for all i and j. In the dilute instanton gas limit where in the complex plane each ai is close to bi and far from the others the n-instanton looks like a system of anyons [6] (bound states of magnetic flux tubes and charged particles) and can be treated approximately as a classical system of particles [2,7]. The correspondence is however somewhat deceptive since the “substructure” given by the a-parameters (the so-called instanton quarks in the language of ref. [S] and the b-parameters (the instanton anti-quarks) implies a further degree of freedom which is crucial for the understanding of the quantum properties of the system. We here consider the n-skyrmion configuration without making any approximations and allow the quark and antiquark parameters a and b as well as the scale X to vary arbitrarily as a function of time t belonging to a finite interval say [0, T] . In case when the t = 0 and t = T configurations are identical the skyrmion defines a map S3 + CP1 - S2 and since n3(S2) = Z we know that in each topological charge (Q) sector the skyrmion manifold is multiply connected. This fact is basically responsible for the spin and statistics properties of skyrrnions [9]. To be completely general we will however first consider a situation where ai( b,(t), h(t) vary arbitrarily in the internal [0, T] (but still subject to ai f bj and h f 0). In the evaluation of the action S = Jr dt 1 d2x f? the kinetic part of L? will (adiabatically) give a contribution ET + O(T-l), where E = n is the energy of the n-skyrmion system, which just corresponds to the usual quantum mechanical time development. The behaviour of the Hopf term H=;dt
jd2xd”“~~a,za,~‘a,z
(6)
0
is more interesting 342
and we will now show how to cal-
1984
culate it explicitly. The basic problem is to evaluate directly the x-integration. In the case of the topological charge (4) the integrand Jo is quickly seen to be a total derivative and so it reduces to a line integral. The integrand h = epVhZ * a,z a,? - a,,z in (6) is not obviously a total divergence but it nevertheless turns out to be so after some algebraic manipulations. To see this we write in general z = f/If1 with f = (fl(x+), fz(x+)) and the integrand becomes h =
Wifi4)[@,fla+f2 - a,f,a+f,)
x (fla+f2 -f,a+f,)+ - c.c.1 .
(7)
The identity a+(fl/GrfP)
= Ifl-4(fla+f2
shows that h can be rewritten h = m[(a,fla+f2
-f2a+fl)
(8)
as
- atf2a+fl).Fllf2~fi2i
-cd, (9)
which is precisely a total derivative expression allowing H to be represented as a line integral: - a,f2a+fi)
H=$
f1
x f21f12-C.C.
1 .
(10)
The x+ contour consists of a circle at infinity and encloses all the singularities of the integrand. We now insert the explicit form (5) for fl and f2 and find the integrand to be
-C;i cg (x+ - ijai)(X+ - bi>
where the dots refer to time derivatives. From the first term there is a non-vanishing contribution at the poles x, = bj and from the second term there is a contribution at infinity. The final result for H is:
H=;;dt(-?&+-+n; -c.c.).
j-d2xA,Jp
0
=&fdt
18 October
LETTERS
(12)
i 0 This expression involves line integrals in the complex plane of the relative vectors bj - ai and the scale A. For the case of the initial skyrmion configuration at t = 0 coinciding with the final configuration at t = T,
Volume
146B, number
PHYSICS
5
i.e. the set {~:ai(7’)}issome permutation of the set {ai(O and similarly for b (while for h we must have h(0) = h(T)), the r integral reduces to a closed contour integral and the value of H is therefore always an integer multiple of 2 n. To be specific, let us investigate the case of a one-skyrmion configuration f = (x+ - a, X(x+ - b)) an d consider a 27r rotation a(t) = a exp(it), b(t) = b exp(it), t E [0,2x]. Then 2n HZ_; r dt(iti)=2n, (13) 0
since the relative vector b(t) - a(t) just makes one revolution around the origin. In general the value of H thus depends on how many times the closed contours of the relative position vectors of the quarks and antiquarks revolve around the origin. As a further example we may consider a two skyrmion configuration f= ((x+ - al) X (x+ - a& h(x+ - br)(x+ - b2)) where we exchange al and a2.The answer for H is now -+_
a1
a1 -b1 a1 -I--------
a2 - bl
a2
at - b,+b?-C’C’ aI I
a2
L
.
22
(14) i
I
18 October 1984
LETTERS
If we thus perform a 2n-rotation and an exchange as done above, this factor is just exp(ie), which for the choice of the parameter 0 = 0 becomes +l , i.e. the skyrmion is quantized as a boson, and for 0 = rr it becomes -1, i.e. quantisation as a fermion. This result is independent of the orientation of the transformation path but the result of the exchange operation could come out to be +l instead of -1 by choosing a different exchange path. When taking 0 < 0 < rr, to somehow interpolate between bosons and fermions, the paths and their orientation can be chosen so as to obtain any factor exp(S), where I is integer. In ref. [2], it was noted that there was an arbitrariness in the assignment of a factor exp(i0) or exp(-i0) to the exchange operation, but as we see it, this assignment problem appears to be more generic. Another problem is that the Hopf term considered in ref. [2] to describe the classical particle correspondence with the skymrion field theory does not quite agree with our general expression (12) even if the bound-state degree of freedom is frozen. The above suggests that the consistency and possible implications of interpolating quantum statistics of skyrmions warrants further study. These problems will be discussed elsewhere.
.
The terms [al/(al - bl)] and [a2/(a2 - bl)] combine to give a single closed contour integral and similarly for the terms [iI/(ul - b2)] and [;~/(a2 - b2)] The value of H now depends on whether b 1 and b2 lie within the closed contour traced out by al and ~2, and if so, how many times the contour revolves. In the case, say, when only bl lies inside the contour which revolves once, we find H = 2n. This value is the same as for the rotation considered in (13) and is an indication that, as anticipated, the exchange operation and the 2rr-rotataion belong to the same connectivity (or homotopy) class, which is but a manifestation of the connection between spin and statistics [9]. However the evaluation of H following an exchange quarks (or antiquarks) was seen to depend crucially on how this exchange was performed. The effect of the Hopf term corresponding to a certain transformation on the parameters of an n skyrmion wave function is simply, as it can be seen from eq. (l), to multiply with a factor exp(SH/2rr).
References [l] T.H.R. Skyme,
Proc. R. Sot. London
Ser A 247 (1958)
260. [2] F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) F. Wilczek and A. Zee, UC Santa Barbara ITP-84-25
preprint
2250; NSF-
(1984).
[3] A.A. Belavin and A.M. Polyakov,
JETP Lett.
22 (1975)
245. [4] H. Eichenherr, E. Cremmer
Nucl. Phys. B146 (1978)
215;
and J. Scherk,
Phys. Lett. 74B (1978)
341;
V. Golo and A. Perelomov,
Phys. Lett. 79B (1978)
112.
[5] S. Deser, R. Jackiw (1982)
and S. Templeton,
Phys. Rev. Lett. 48
975.
[6] F. Wilczek, Phys. Rev. Lett. 49 (1982) [7] Y.S. Wu, University
of Washington
957.
preprint
40048-
Q7P4 (1984). [S] V. Fateev, (1979)
1. Frolov
and A. Schwartz,
Nucl. Phys. B154
1.
[9] D. Finkelstein
and J. Rubinstein,
J. Math. Phys. 9 (1968)
1762.
343