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Hidden supersymmetries of particle motion in a Wu-Yang monopole field
18 April 1996
PHYSICS LETTERS B Physics Letters B 373 (1996) 125-129
Hidden supersymmetries of particle motion in a Wu-Yang monopole field A.J. Macfarlane, A.J. Mountain D.A.M.lY?, Cambridge University, Silver Street, Cambridge CB3 9EU: UK
Received 22 January 1996 Editor: PV. Landshoff
Abstract
A theory is studied of the supersymmetric motion of a spin-3 particle with SU(2) colour in a Wu-Yang monopole background, in which H = Qo2relates Hamiltonian and supercharge. ?Lvo additional supercharges which anticommute with Qo are found SO that the theory possesses a hidden N = 2 supersymmetry algebra {Qi , Qj} = 8, i (J’ +constant), i, j = 1,2, where J is the conserved total angular momentum.
1. Introduction In any dynamical theory the discovery of hidden symmetries is always an important matter. If the spectrum of a quantum-mechanical problem reveals unexpected degeneracies one knows very well that there must exist hidden symmetries which account for them [ l-41. The hydrogen atom with its SO(4) invariance [ 1 ] affords a familiar example with hidden symmetries generated by the Lenz-Runge vector. Another well-known example that indicates that the hidden symmetries may be supersymmetries is found in the study of the non-relativistic motion of a particle of spin-4 in the background field of a Dirac monopole [ 51. When the gyromagnetic ratio of the particle is so chosen that the the theory possesses a natural supersymmetry [ 51 then, while there exist degeneracies of the spectrum of the quantum-mechanical theory which are well-understood [5], interpretationof this in terms of hidden supersymmetry represents a newer insight [ 61. Work similar in spirit to this has been performed also in the context of motion in a curved background
[ 51. In the cases of the Kerr-Newmann [ 71 and TaubNUT [8,9] backgrounds, it has been found that, in addition to the natural supercharges that square up to the Hamiltonian of the theory, there are respectively one and four additional supercharges that anticommute with the original ones and have squares which yield different constants of the motion. The discovery of the extra supersymmetries is systematically associated with the existence of so-called Killing-Yano tensors. Recently Tanimoto [ lo] has given a comprehensive discussion of Killing-Yano tensors and related supersymmetries for a general curvBd plus electromagnetic background. We note that this discussion applies also to the case mentioned above of the Dirac monopole which provides [ 51 the simplest non-trivial example of a Killing-Yano tensor. Having seen how extra supersymmetries materialise in the case of supersymmetric dynamics in the background field of the Dirac monopole and for the KerrNewman and Taub-NUT metrics, it seems very likely that comparably interesting findings should emerge from the supersymmetric dynamics of a particle of
0370-2693/96/$12.00 @ 1996 Blsevier Science B.V. All rights reserved I’ll SO370-2693(96)00141-4
126
A.J. Macfarlane,
A.J. Mountain/Physics
spin-i and SU(Z)-colour in the background of suitable solutionsof the source-free Yang-Mills equations. A detailed formulation of this dynamics has been presented elsewhere [ 111, in a paper devoted in part to displaying a complete Lagrangian description (found elusive [ 12-141 when the use of fermionic variables is not envisaged) of Wong’s equations [IS] for the non-relativistic motion of a coloured particle in a background Yang-Mills field. Here we apply the formulation to the special case in which the background field is that of the Wu-Yang monopote [ 161 (see also the monograph [ 171, and also [ 181, for related material and references). We show that the theory allows simple choice of Killing-Yano tensors and gives rise thence to two extra hermitian supercharges Qi, i = 1,2. If the original hermitian supercharge Qo of the theory is related to the Hamiltonian H of the (quantum mechanical version of the) theory by H = Q& then the supercharges Qi obey
{Qo, Qi}=o,
{Qiv Qj}=hjKt
[H, QoI ~0,
[H, Qil ~0.
(1)
Here K differs by a constant from iJ2, where J is the conserved total angular momentum of the particle, including the expected contribution that stems from the background field. The results are qualitatively similar to those found for the Dirac monopole [ 61, except for the notable fact that the present theory has a hidden supersymmetry of N = 2 as opposed to N = 1 type. We intend to return in forthcoming work to matters involving more general Yang-Mills background fields. In discussing the search for hidden supersymmetries in some given dynamical theory with standard supersymmetry, for example of type N = 1, we should contrast our approach with a more familiar one [ 191 in which one seeks conditions under which the N = 1 supersymmetry of the theory can be extended to N = 2 or N = 4. There the additional supersymmetries must not only anticommute with the original one but also must, like the original ones, close on the Hamiltonian. The latter requirement is neither sought after nor present in our work, but the extra supersymmetries we deal with must satisfy the conditions laid down in [ 121 in generality to ensure that the former requirement is satisfied. Section 2 summarises material from previous work [ 11 J that is required for use in the present paper. In
L.etters B 373 (19%) 125-129
Section 3 we give expressions in terms of KillingYano and other tensors for supercharges Q that may anticommute with the generator Qc such that Q,’ = H, and derive conditions upon these tensors that ensure {Qa , &} = 0. In Section 4 we solve these conditions explicitly when the background field is that of WuYang [ 16-l 8 1 monopole, finding two independent solutions Q = Qi,i = 1,2, which themselves anticommute, and conform to ( 1) above. For simplicity, we present our discussion in the language of classical mechanics; the extension of it to the quantum case proceeds in straightforward fashion.
2. Basic formalism We set out from the superfield formalism of [ 1l] involving scalar superfields, which contain the space co-ordinates xi and the Majorana fermions 4i from which spin is made [ 201 according to si = -a
ieijk4j$k
,
(2)
and spinorial superfields which contain auxiliary bosonic variables, to be eliminated in the usual way, and Majorana fermions A, from which the charge vector of .SU(2) -colour is made using (3)
Ja = -i ic,&pAy. The Lagrangian we use [ 111 thus is
L = t (iiii + i&& + IA,-& + igAi,c,p,JpA,ki) + ~&~apyApAyhCbj
(4)
-
Here Aia is the Yang-Mills field, and R’tjaits covariant curl. The provision of the canonical formalism is problem free [ll]. We find pi = ii - g&j, {xi,pj}=Sijv {di,
7 {Aa, #i}=Ov
+j}=-iSij,
{A,, Ag}=-i&/3,
(5)
and the Hamiltonian H = ~(pi+gAi,j,)(pi+gAiaja)
+&&Sk,
(6)
127
A.J. Macfarlane.A.J. Mounrain/PhysicsLettersB 373 (19%) 125-129
where the notations (2) and (3) have been used, and Fija = EijkBka. Also Noether’s theorem yields the supercharge QO= (Pi + g&Ja)(Pi
(6)
.
Using the consequence of (5) :
to proceed is already clear from previous work, from the Dirac monopole case in fact. Some preliminary tests soon make it clear that there two independent candidates for the role assigned above to Q, which we may denote henceforth by Qr and Q2. We are therefore led to define Ql
+
i(-)faFaG +i(_)f$Z$, LI P
%jk ad',
Q2 = -GgiaAa
(7)
where f is the Grassmann parity of F, to calculate Poisson brackets, we can show that Q obeys the Poisson bracket relation {Qo , Qo} = -2iH.
(8)
This is the classical analogue of the more familiar quanta1 relationship QG = H. We can also verify that Qc generates canonically the supersymmetry transformation rules employed in the derivation of (4). We can also use (7) to show that {Si 9
sj}
= Eijksk
7ip} = &a&& 9
{ja
9
(9)
which justifies our viewing of Si and ja as spin and colour or charge variables. One knows [ 111 also that in quantum theory, in which we impose the anticommutation relations we represent 4i (in the Schrodinger picture) in terms of Pauli matrices by ai/&. It follows that we represent Si by ai/ confirming that we are dealing with the case of spin one-half. One can also use (7) to compute the time dependence of any classical variable Y via 4i4j
Y={Y,
+
+j+i
H}.
=
= iifij4j
Sij,
+
aiCijkh4jjk
+
iidija&4jL
1
(11)
9
(12)
where the c-tensor is totally antisymmetric and the dtensor is antisymmetric in its first two indices. The idea here is that we display all the dependence on the canonical variables pi , bi and A,, leaving implicit the dependence on the xi given, within f~ in terms of Ai,, or else, elsewhere, in terms of undetermined scalars and tensors. The distinction between Qr and Q2 lies in the fact that the former is odd in the +i and even in the Aa, whereas the opposite holds for Q2. The tensors fij and gia are Killing-Yano tensors, and the general way in which these feature in the general theory and examples of it leads us to suppose all the dependence on the variables pi is as implied by Eqs. ( 11) and ( 12). We have not displayed either a A - A - C$term in(Il),oraA-A_Atermin(12),becauseittums out that no such contributions survive. We turn first to the calculation of {Qa , Qr} using (7). We require the vanishing of the coefficients of all independent terms that arise, due care being taken of symmetry and antisymmetry properties. Thus the R - i terms and the i - 4 - C#J terms tell us that the Killing-Yano tensor fij is antisymmetric and obeys fki,j-fkj,i+cijk=O.
(13)
We therefore adopt the choices (10)
This allows the derivation of the equations of motion for the theory and their consequences for Si and ja. These are presented in [ 111. 3. Additional supersymmetries The intention is to define supercharges Q such that the Poisson bracket {Qc , e} vanishes, to compute (0, e} and to describe the symmetry algebra that emerges for our background field problem. The way
fij = Eijkxk
and
cijk
=
-2Eijk
.
(14)
No term quartic in the 4 variables can survive because they are three in number and anticommuting. There remains only a term of type A - A - C$- q5, and its coefficient vanishes if fjkejin = fjiFkja -
(15)
We turn next to the case of {Qn , Qz}. As in the previous case one is lead to these conditions for the Poisson bracket to vanish:
128 Digjy
DIidjk)y
A.J. Macjarlane. A.J. Mountain/Physics Letters B 373 (19%) 125-129 +
idijy
=
=
0
(
0.
(16)
Here
(17)
Ji = EijkXjpk + Si + ji ,
Here D denotes the (colour gauge) covariant derivative, as, for example, in Digja = 3igi, + g&epyAiygjp, and the square brackets around the subscripts in (16) imply antisymmetrisation. In Section 4, we turn to the special case of the WuYang monopole. 4. The Wu-Yang monopole The specific choice of background Yang-Mills field that we use in our studies here is the Wu-Yang monopole [ 16-181, a solution of the source-free Yang-Mills equations. This is specified by the potential Ai, and the corresponding covariant field Fiia = &ijkBka: gr2 A;, = &iakXk,
gr4 Bka = Xk .& ,
(18)
where g is a coupling constant and r2 = XiXj* To show that Qt defined by ( 11) and ( 14) has Poisson bracket zero with Qa in the Wu-Yang case, it is necessary only to show that ( 14) and ( 18) satisfy ( 15), which is easy to do. To show that there exists Q2 which does likewise requires selection of a suitable form for the KillingYano tensor gia, computation of dijy using ( 16), and demonstration that (17) is satisfied. With the choice this works out straightforwardly for the &a = &iakXk, Wu-Yang monopole. Thus we have the explicit results QI =
EijkXiij4k
-
ii&ijkh4j‘#k
9
Q2 = eijaXiij& + Sih,( 6;n - xi XII/r2).
(19) (20)
A non-trivialcalculation based on (7) is still needed to show that the two hermitian supercharges just found have a vanishing Poisson bracket, so that we have two new supersymmetries. Above we noted the wellknown relation of QO to the Hamiltonian. Now we ask what conserved quantities arise from the Poisson brackets {Qr , Qt } and {Q;! , Qz}. Two non-trivial calculations based on (7) complete the proof of the notable result {Qi , Q,i) = -2iS,$
J2 .
(21)
(22)
which we identify as the total angular momentum of the system under study with the last term arising as the contribution due to the background field. (See [ 211 for a relevant discussion of conserved quantities.) It is a non-trivial calculation to use the equations of motion [ 111 of the system (specialised to the case of the Wu-Yang monopole) to verify that the total angular momentum (22) is conserved in time, a useful consistency check on our considerations. Our construction of the supersymmetry algebra of Qt and Q2 has led us to a well-known conserved quantity, as for the Dirac monopole, instead of closing upon the Hamiltonian. Thus, instead of finding an extended supersymmetry (in the sense of [ 191) we have found the symmetry algebra (21), all quantities in which have vanishing Poisson brackets with the original supercharge and the Hamiltonian of the problem. The significant feature that goes beyond the picture obtained for the Dirac monopole is that the hidden supersymmetry algebra here is of N = 2 type rather than N = 1.
Acknowledgements This paper describes research supported in part by PPARC. A.J.M. thanks PPARC for the grant which supports his research. References [ 11W. Pauli, Z. Phys. 36 (1926) 336; V. Fock, Z. Phys. 98 (1935) 145; V. Batgmann, Z. Phys. 99 ( 1936) 576. [2] J.M. Jauch and E.L. Hill, Phys. Rev. 57 (1940) 641. [ 31 H. Bacry, H. Ruegg and J.M. Souriau, Commun. Math. Phys. 3 (1966) 323. [4] A.O. Barut, A. Biihm and Y. Ne’eman, Dynamical Groups and Spectrum Generating Algebras, Vols 1 and 2, World Sci., Singapore, 1988. [5] E. D’Hoker and L. Vinet, Phys. Lett. B 137 (1984) 72. [6] E de Jonghe, A.J. Macfarlane, K. Peeters and J.W. van Holten, Phys. Le.tt. B 359 (1995) 114. [7] G.W. Gibbons, R.H. Rietdijk and J.W. van Holten, Nucl. Phys. B 404 ( 1993) 42. [ 81 J.W. van Holten, Phys. Lett. B 342 ( 1995) 47. [9] M. Visinescu, Pmt. ICSMP-95, Dubna. Russia. July 1995, to be published.
A.J. Macfarlane,A.J. Mountain/PhysicsLettersB 373 (19%) 125-129 101 M. Tanimoto, The role of Killing-Yano tensors in the supersymmetric quantum mechanics on a curved manifold, preprint, TIT/I-IBP-277/Cosmo-10; grqc 9501006. II] N. Linden, A.J. Macfarlane and J.W. van Holten, Particle motion in a Yang-Mills field: Wong’s equations and their spin one-half analogues, preprint DAMTP-95/37. NBCHEF /95-049, July 1995, hepth/9512071, Czech. J. Phys., to be published. 12 I SK. Wong, Nuovo Cimento A 65 (1970) 689. I3 I R. Montgomery, I&t. Math. Phys. 8 ( 1984) 59. 141 J.E. Marsden and T. S Ratiu, Introduction to Mechanics & Symmetry, Springer, Berlin, 1994. I5 I Chan Hong-MO, J. Faridani and Tsou Sheung Tsun, RAL Report, RAL-95-027, A Non-Abelian Yang-Mills Analogue of Classical Electromagnetic Duality.
129
[ 161 T.T. Wu and C.N. Yang, Phys. Rev. D 12 (1975) 3843; T.T. Wu and C.N. Yang, in Properties of Matter under Unusual Conditions, H. Mark and S. Ferbach, eds., Interscience, New York ( 1969) 349. ] 171 M. Carmeli, Kh. Huleihil and E. Leibowitz, Gauge Fields, Classification and Equations of Motion, World Sci., Singapore, 1989. [ 181 A. Actor, Rev. Mod. Phys. 51 (1979) 461. [19] R.A. Cole and G. Papadopoulos, Class. Quant. Grav. 7 (1990) 427. [20] J.L. Martin, Proc. Roy. Sot. (London) 251 (1959) 536,543. [21] J.W. van Holten and R.H. Rietdijk, J. Geom. Phys. 11 (1993) 559.
PHYSICS LETTERS B Physics Letters B 373 (1996) 125-129
Hidden supersymmetries of particle motion in a Wu-Yang monopole field A.J. Macfarlane, A.J. Mountain D.A.M.lY?, Cambridge University, Silver Street, Cambridge CB3 9EU: UK
Received 22 January 1996 Editor: PV. Landshoff
Abstract
A theory is studied of the supersymmetric motion of a spin-3 particle with SU(2) colour in a Wu-Yang monopole background, in which H = Qo2relates Hamiltonian and supercharge. ?Lvo additional supercharges which anticommute with Qo are found SO that the theory possesses a hidden N = 2 supersymmetry algebra {Qi , Qj} = 8, i (J’ +constant), i, j = 1,2, where J is the conserved total angular momentum.
1. Introduction In any dynamical theory the discovery of hidden symmetries is always an important matter. If the spectrum of a quantum-mechanical problem reveals unexpected degeneracies one knows very well that there must exist hidden symmetries which account for them [ l-41. The hydrogen atom with its SO(4) invariance [ 1 ] affords a familiar example with hidden symmetries generated by the Lenz-Runge vector. Another well-known example that indicates that the hidden symmetries may be supersymmetries is found in the study of the non-relativistic motion of a particle of spin-4 in the background field of a Dirac monopole [ 51. When the gyromagnetic ratio of the particle is so chosen that the the theory possesses a natural supersymmetry [ 51 then, while there exist degeneracies of the spectrum of the quantum-mechanical theory which are well-understood [5], interpretationof this in terms of hidden supersymmetry represents a newer insight [ 61. Work similar in spirit to this has been performed also in the context of motion in a curved background
[ 51. In the cases of the Kerr-Newmann [ 71 and TaubNUT [8,9] backgrounds, it has been found that, in addition to the natural supercharges that square up to the Hamiltonian of the theory, there are respectively one and four additional supercharges that anticommute with the original ones and have squares which yield different constants of the motion. The discovery of the extra supersymmetries is systematically associated with the existence of so-called Killing-Yano tensors. Recently Tanimoto [ lo] has given a comprehensive discussion of Killing-Yano tensors and related supersymmetries for a general curvBd plus electromagnetic background. We note that this discussion applies also to the case mentioned above of the Dirac monopole which provides [ 51 the simplest non-trivial example of a Killing-Yano tensor. Having seen how extra supersymmetries materialise in the case of supersymmetric dynamics in the background field of the Dirac monopole and for the KerrNewman and Taub-NUT metrics, it seems very likely that comparably interesting findings should emerge from the supersymmetric dynamics of a particle of
0370-2693/96/$12.00 @ 1996 Blsevier Science B.V. All rights reserved I’ll SO370-2693(96)00141-4
126
A.J. Macfarlane,
A.J. Mountain/Physics
spin-i and SU(Z)-colour in the background of suitable solutionsof the source-free Yang-Mills equations. A detailed formulation of this dynamics has been presented elsewhere [ 111, in a paper devoted in part to displaying a complete Lagrangian description (found elusive [ 12-141 when the use of fermionic variables is not envisaged) of Wong’s equations [IS] for the non-relativistic motion of a coloured particle in a background Yang-Mills field. Here we apply the formulation to the special case in which the background field is that of the Wu-Yang monopote [ 161 (see also the monograph [ 171, and also [ 181, for related material and references). We show that the theory allows simple choice of Killing-Yano tensors and gives rise thence to two extra hermitian supercharges Qi, i = 1,2. If the original hermitian supercharge Qo of the theory is related to the Hamiltonian H of the (quantum mechanical version of the) theory by H = Q& then the supercharges Qi obey
{Qo, Qi}=o,
{Qiv Qj}=hjKt
[H, QoI ~0,
[H, Qil ~0.
(1)
Here K differs by a constant from iJ2, where J is the conserved total angular momentum of the particle, including the expected contribution that stems from the background field. The results are qualitatively similar to those found for the Dirac monopole [ 61, except for the notable fact that the present theory has a hidden supersymmetry of N = 2 as opposed to N = 1 type. We intend to return in forthcoming work to matters involving more general Yang-Mills background fields. In discussing the search for hidden supersymmetries in some given dynamical theory with standard supersymmetry, for example of type N = 1, we should contrast our approach with a more familiar one [ 191 in which one seeks conditions under which the N = 1 supersymmetry of the theory can be extended to N = 2 or N = 4. There the additional supersymmetries must not only anticommute with the original one but also must, like the original ones, close on the Hamiltonian. The latter requirement is neither sought after nor present in our work, but the extra supersymmetries we deal with must satisfy the conditions laid down in [ 121 in generality to ensure that the former requirement is satisfied. Section 2 summarises material from previous work [ 11 J that is required for use in the present paper. In
L.etters B 373 (19%) 125-129
Section 3 we give expressions in terms of KillingYano and other tensors for supercharges Q that may anticommute with the generator Qc such that Q,’ = H, and derive conditions upon these tensors that ensure {Qa , &} = 0. In Section 4 we solve these conditions explicitly when the background field is that of WuYang [ 16-l 8 1 monopole, finding two independent solutions Q = Qi,i = 1,2, which themselves anticommute, and conform to ( 1) above. For simplicity, we present our discussion in the language of classical mechanics; the extension of it to the quantum case proceeds in straightforward fashion.
2. Basic formalism We set out from the superfield formalism of [ 1l] involving scalar superfields, which contain the space co-ordinates xi and the Majorana fermions 4i from which spin is made [ 201 according to si = -a
ieijk4j$k
,
(2)
and spinorial superfields which contain auxiliary bosonic variables, to be eliminated in the usual way, and Majorana fermions A, from which the charge vector of .SU(2) -colour is made using (3)
Ja = -i ic,&pAy. The Lagrangian we use [ 111 thus is
L = t (iiii + i&& + IA,-& + igAi,c,p,JpA,ki) + ~&~apyApAyhCbj
(4)
-
Here Aia is the Yang-Mills field, and R’tjaits covariant curl. The provision of the canonical formalism is problem free [ll]. We find pi = ii - g&j, {xi,pj}=Sijv {di,
7 {Aa, #i}=Ov
+j}=-iSij,
{A,, Ag}=-i&/3,
(5)
and the Hamiltonian H = ~(pi+gAi,j,)(pi+gAiaja)
+&&Sk,
(6)
127
A.J. Macfarlane.A.J. Mounrain/PhysicsLettersB 373 (19%) 125-129
where the notations (2) and (3) have been used, and Fija = EijkBka. Also Noether’s theorem yields the supercharge QO= (Pi + g&Ja)(Pi
(6)
.
Using the consequence of (5) :
to proceed is already clear from previous work, from the Dirac monopole case in fact. Some preliminary tests soon make it clear that there two independent candidates for the role assigned above to Q, which we may denote henceforth by Qr and Q2. We are therefore led to define Ql
+
i(-)faFaG +i(_)f$Z$, LI P
%jk ad',
Q2 = -GgiaAa
(7)
where f is the Grassmann parity of F, to calculate Poisson brackets, we can show that Q obeys the Poisson bracket relation {Qo , Qo} = -2iH.
(8)
This is the classical analogue of the more familiar quanta1 relationship QG = H. We can also verify that Qc generates canonically the supersymmetry transformation rules employed in the derivation of (4). We can also use (7) to show that {Si 9
sj}
= Eijksk
7ip} = &a&& 9
{ja
9
(9)
which justifies our viewing of Si and ja as spin and colour or charge variables. One knows [ 111 also that in quantum theory, in which we impose the anticommutation relations we represent 4i (in the Schrodinger picture) in terms of Pauli matrices by ai/&. It follows that we represent Si by ai/ confirming that we are dealing with the case of spin one-half. One can also use (7) to compute the time dependence of any classical variable Y via 4i4j
Y={Y,
+
+j+i
H}.
=
= iifij4j
Sij,
+
aiCijkh4jjk
+
iidija&4jL
1
(11)
9
(12)
where the c-tensor is totally antisymmetric and the dtensor is antisymmetric in its first two indices. The idea here is that we display all the dependence on the canonical variables pi , bi and A,, leaving implicit the dependence on the xi given, within f~ in terms of Ai,, or else, elsewhere, in terms of undetermined scalars and tensors. The distinction between Qr and Q2 lies in the fact that the former is odd in the +i and even in the Aa, whereas the opposite holds for Q2. The tensors fij and gia are Killing-Yano tensors, and the general way in which these feature in the general theory and examples of it leads us to suppose all the dependence on the variables pi is as implied by Eqs. ( 11) and ( 12). We have not displayed either a A - A - C$term in(Il),oraA-A_Atermin(12),becauseittums out that no such contributions survive. We turn first to the calculation of {Qa , Qr} using (7). We require the vanishing of the coefficients of all independent terms that arise, due care being taken of symmetry and antisymmetry properties. Thus the R - i terms and the i - 4 - C#J terms tell us that the Killing-Yano tensor fij is antisymmetric and obeys fki,j-fkj,i+cijk=O.
(13)
We therefore adopt the choices (10)
This allows the derivation of the equations of motion for the theory and their consequences for Si and ja. These are presented in [ 111. 3. Additional supersymmetries The intention is to define supercharges Q such that the Poisson bracket {Qc , e} vanishes, to compute (0, e} and to describe the symmetry algebra that emerges for our background field problem. The way
fij = Eijkxk
and
cijk
=
-2Eijk
.
(14)
No term quartic in the 4 variables can survive because they are three in number and anticommuting. There remains only a term of type A - A - C$- q5, and its coefficient vanishes if fjkejin = fjiFkja -
(15)
We turn next to the case of {Qn , Qz}. As in the previous case one is lead to these conditions for the Poisson bracket to vanish:
128 Digjy
DIidjk)y
A.J. Macjarlane. A.J. Mountain/Physics Letters B 373 (19%) 125-129 +
idijy
=
=
0
(
0.
(16)
Here
(17)
Ji = EijkXjpk + Si + ji ,
Here D denotes the (colour gauge) covariant derivative, as, for example, in Digja = 3igi, + g&epyAiygjp, and the square brackets around the subscripts in (16) imply antisymmetrisation. In Section 4, we turn to the special case of the WuYang monopole. 4. The Wu-Yang monopole The specific choice of background Yang-Mills field that we use in our studies here is the Wu-Yang monopole [ 16-181, a solution of the source-free Yang-Mills equations. This is specified by the potential Ai, and the corresponding covariant field Fiia = &ijkBka: gr2 A;, = &iakXk,
gr4 Bka = Xk .& ,
(18)
where g is a coupling constant and r2 = XiXj* To show that Qt defined by ( 11) and ( 14) has Poisson bracket zero with Qa in the Wu-Yang case, it is necessary only to show that ( 14) and ( 18) satisfy ( 15), which is easy to do. To show that there exists Q2 which does likewise requires selection of a suitable form for the KillingYano tensor gia, computation of dijy using ( 16), and demonstration that (17) is satisfied. With the choice this works out straightforwardly for the &a = &iakXk, Wu-Yang monopole. Thus we have the explicit results QI =
EijkXiij4k
-
ii&ijkh4j‘#k
9
Q2 = eijaXiij& + Sih,( 6;n - xi XII/r2).
(19) (20)
A non-trivialcalculation based on (7) is still needed to show that the two hermitian supercharges just found have a vanishing Poisson bracket, so that we have two new supersymmetries. Above we noted the wellknown relation of QO to the Hamiltonian. Now we ask what conserved quantities arise from the Poisson brackets {Qr , Qt } and {Q;! , Qz}. Two non-trivial calculations based on (7) complete the proof of the notable result {Qi , Q,i) = -2iS,$
J2 .
(21)
(22)
which we identify as the total angular momentum of the system under study with the last term arising as the contribution due to the background field. (See [ 211 for a relevant discussion of conserved quantities.) It is a non-trivial calculation to use the equations of motion [ 111 of the system (specialised to the case of the Wu-Yang monopole) to verify that the total angular momentum (22) is conserved in time, a useful consistency check on our considerations. Our construction of the supersymmetry algebra of Qt and Q2 has led us to a well-known conserved quantity, as for the Dirac monopole, instead of closing upon the Hamiltonian. Thus, instead of finding an extended supersymmetry (in the sense of [ 191) we have found the symmetry algebra (21), all quantities in which have vanishing Poisson brackets with the original supercharge and the Hamiltonian of the problem. The significant feature that goes beyond the picture obtained for the Dirac monopole is that the hidden supersymmetry algebra here is of N = 2 type rather than N = 1.
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