Pandey, V P

Rajput, B S

IACS Institutional Repository

Indian J. r a y s .  U A  {* ). 101-169 (1998)IJP A— an tntematipnal journalB S Rajput and V P Pandey Department of Physics, Kuraaun University,Nainital-263 001, IndiaReceived 24 September 1997, accepted 25 November 1997Abstract : Supersymmetrized SchrOdinger equation for Fermion-Dyon system bps been obtained by dimensional reduction of supersymmetrized four-dimensional harmonic oscillator and it has been interpreted as an ensemble of two Schrtkjinger and one Pauli's equation each describing the motion of an electrically charged particle in the field of a Dyon with different magnetic charges.Keywords : Schrfidinger equation, supersymmetry, Fermion-Dyon system PAC$ Nos, : 11.30.Pb, 03.65.Ge 11. IntroductionMonopoles and dyons became the intrinsic parts of all current grand unified theories [1] with the enormous potential importance in connection with their role in catalyzing proton dacay [2,3], the quark confinement problem of QCD [4,5] and RCD [6,9] and CP-violation in terms of non-zero vacuum angle [10], The dyon-fermion dynamics has been worked out by various authors [2 , 11-13] and it has been shown that the nature of dyons is strongly perturbed by fermionic sector which couples with them. In our recent paper [13], we have undertaken the study of dyon-fermion bound states and showed that in dyon-fermion system the fermion moves on a cone with its apex at the dyon and axis along its angular momentum. It has also been shown that the exact solution of Dirac equation for such a system is not possible due to the presence of terms vanishing more rapidly than r* in the potential of the system.Keeping in view the symmetry of Schrddinger equation for a fermion in the field of monopole [14,15] and the difficulties faced [16] in supersymmetrizing the Pauli’s equation© 1998IACS162 B S Rajput and V P Pandeyof a fermion in the field of dyon, in the present paper, we try to obtain the supersymmetrized solution of Schrddinger equation of fermion in dyonic field by the dimensional reduction of supersymmetrized generalized four-dimensional Harmonic oscillator that we can derive the modifications in our earlier results of eigen values and eigen functions of fermion-dyon system as the result of supersymmetrization. The supersymmetrized Schrddinger equation for this system has been interpreted as describing the quantum dynamics of a supermultplet of two spin- 0  and one spin-1 / 2  particles in a Coulomb field. It has also been interpreted as an ensemble of two Schrddinger and one Pauli’s equation each describing the motion of an electrically charged particle in the field of a dyon with different magnetic charges.2. Dimensional reduction of four-dimensional harmonic oscillator to fermion- dyon systemLet us consider the motion of a fermion of mass (m = l/2) and charge in the field of a dyon carrying generalized charge9 = *2 “ ^2  'in terms of electric and magnetic charges e 2 and g 2 respectively. Its Schrddinger equation may be written as [13]HD\if=Ew (2 .1 )where (2 .2 )with II l*K = exev (2.3)P = e , g 2.In eq. (2.3), V is the potential of the field of dyon. Rescaling by a(r = cor), eq. (2.1) may be written as follows :h '"1 = —D a r (2.4)with H™  =(rv* - r  + P*/ r ) , for a  = 4 - Eand H f  =(rv* + r  + P*/ r ) , for a  = VE, (2.5)where < II 3^ 1 (2 .6 )Equation (2.4) may be written in the following specific form :» i T V= [ r ( p k + f]v^=  -§ v - (2.7)Supersymmetrized Schrtidinger equation etc 163The corresponding angular momentum operator may be written as follows :~ e f k )  + P 'rr  (2-8)with L20t) = 1(1 + /) + e f g l  = / '( f ' + l). (2.9)Solution of eq. (2.7) has been obtained in the following form in our earlier paper 113]:u(r )(2.10)where Yf g {, m, (9, 0) are the spherical harmonics for a fermion-dyon system and u{r) is given as follows in terms of confluent hyper-geometric functions:u(r) = (ar ) (J exp ( - a r ) f |a  -  2a, 2arwith a =  -i +[( / '+ I/2)2 - ( f , * , ) 1]1 1 /2(2. i n(2.11a)The corresponding bound stale energy of fermion-dyon system has been obtained in the Allowing form [13] :( e ,e , ) 2r (r + i ) - e2s 2 + i  + i +n(2.12)where n is an integer and we have chosen fermionic mass m = 1 /2, Equation (2.4) is equivalent to the Schrodinger equation for the four-dimensional oscillator [141- -  K - H y  = — y  a (2-13)wilh the constraintX y  = -  i ~  y  = P y  0(0(2.14)where H= - r " 1 JL 2 A 1+ —iT" 1 9_ ( . a 9 ) I 9* ' sin 0 -t— + — -- - - - - —r 2 d y r 9y r 2 sin 0 90 V 06 ) sin2 0 <902 _1sin2 03 2  3 23 + 2cos0 3d(02 dipdco(2.15)Solution of eq. (2.14) may be written asy  = y{r, e, 0, (o) = e*°> (y, 0, 0), (2.16)where to is the angular velocity in the domain (0, 4/i). The condition for single-valuedness oi V/ requires that P should be quantized in half integral units :d 1P  =  e l * 2  =164 B S Rajput and V P ParideyThe system described by eqs. (2.13) and (2! 14) possesses SO(4, 2) spectrum symmetry. The angular momentum operator corresponding to Hamiltonian of eq. (2.15) operates through the following relationL t W =  exp [/Po]!,. (JT)Vf. (2.17)wherer -  8.[}k) = e - p J — h h .  ' rsin2 6(2.18)showing that the projected SU(2) generators have a P-dependent contribution. Choosing: * 2 e91 K'?r / 3* + ' f )or in spherical coordinatesVr = 0, V, -  0, Vq = - g 2 cos9. \as the potential for a monopole of strength g2, this L of eq. (2.18) may reaiily be recasted in the form given by eq. (2.8) for fermion-dyon system. Then Hamiltoriian of eq. (2.13) for four-dimensional harmonic oscillator reduces to that of eq. (2.7) for the motion of a fermion through the field of a dyon. This reduction takes place under the projectionk.R+  x 5 3 -> R+ x S 2 (2.19)imposing the mapping(Z, ,Z 2 ) ^ ( r , Z , / Z 2 ),z i = V^T+ 'V^T’ Z2 = V 'T +such that z a Z o = | Z , |2 + | z 2 |2 = r > 0.(2.20)(2.21)(2.22)Taking a 2-sphere of radius r and projecting the point r —  of C -> 5 2 through north pole, we get_LL -  i 2Z 2 ‘  r ~ r 3or r. = ff.(Za ) = Z a a'abZh,which is Kustaanheirtio-Stiefel (KS)  transformation with 0) as standard Pauli matrices. This projection defines 4 principal fibre bundleM ~  R+ x S 3with 1/(1) as the structural group.Supersymmetrized Schr&dinger equation etc 1653. Supersymmetrized Fermion-Dyon systemIn order to supersymmetrize the system described by eq. (2.7), let Us start with the supersymmetrization of generalized four-dimensional harmonic oscillator described by cq. (2.13) and then perform the dimensional reduction through eqs. (2.19) and (2.20). To meet this end, let us choose the superchargesThis H has a n = 2 conformal supersymmetry. In addition to the operators Q, Q* and H, let us also construct the following operators :(3.1)(3.2)where W is a real function of Zan Z and r\ satisfies the following Clifford algebra :K .n » }  = o.{r]^, 77J } = 2<5fl^  fora, 6=1,2.(3.3)Choosing W(Z, Z) = A In (Zfl Za ), (3.4)where A e R, the domain of Clifford algebra of eq. (3.3), we get/ / =  ^ { e . e +}(3.5)where (3.6)<ind S ‘ = j (3.7)and(3.8)72A(2HO166 B S Rajput and V P PandeyThese generators satisfy the Osp (2.1).structure relation and remain invariant under the SU(2) action generated by the operatorsOperator C, given by eq. (3.6), commutes with all the Osp (2,1) generators and also with J. As such, the full invariant algebra of this problem is Osp (2,1) ® SU(2) ® U(l). Within this algebra, the Hamiltonian of eq. (3.5) for supersymmetried four-dimensional harmonic oscillator may be generalized in the form of the following operatorBut the generators Q, Qf, S and S+ do not commute with x  and hence the supersymmetry of R will be lost under the projection involving this constraint. Thus, we modify this constraint to the following equivalent conditionwhere the operator C, given by eq. (3.6) commutes with every element of Osp (2,1) ® SU(2) algebra and it will affect the projection without breaking any symmetry of the four-dimensional system. Condition (3.12) may be understood as the supersymmetrized version of the constraint (2.14). Then eq. (2.16) may be obtained in the following supersymmetrized formwhere j ( is SU(2) generator given by eq. (3.9) and j ) n) is given by the following supersymmetrized form of eq. (2.8):(3.9)R = H + K = H + ZaZ a. (3.10)The constraint (2.14) may then be written in the following formshowing that every component of \j/ transforms according to the same U(l) represenWionC\jf = 2 P\jf9 (3.12)(3.13)where (3.14)Then eq. (2.17) is supersymmetrized into the following form(3.15)with * k mP k ’ [ P - j 2 ) v ? (3.17)and V ® as the potential of a Dirac monopole of unit strength. In eqr (3.16) £. is the spin matrix given by eq. (3.7) with' l l  = ^ « [ 7 J +«74]. *12 “  + *72]• (3.18)Choosing chiral basisr = \-io> f  }  5,4 - ( ?  o randwe getandy 5 — _ y I y 2 yv - ( i  .:)•4 o ’ 0S' =0 00 l / 2 a ' (3.19)which is fully reducible form showing that the system under consideration comprises two spin-0 and one spin-1/2 particles. Then equationD-  K -Rw = — w areduces to the following supersymmetrized version of eq. (2.7)R {K) w = — w, awhere /?<*> = r[Pi -  ( P - 1 / 2 X ) V ° ] 2  + r( X - P ) 2 -  PX + 1/4X 2  2Xr'S‘r r 2Rescaling by a, eq. (3.20) may be written into the form given by eq. (2.1) where h d  = [ p , - ( P - 1 / 2 X ) V , D]J - K / \ x \[ ( X - P ) 2  - P X  + 1/4X2] 2 Xx‘S ‘(3.20)(3.21)WJ w 3(3.22)The spectrum of this hamiltonian may be obtained in the form of following supersymmetric generalization of eq. (2 .1 2 )E =e } e 2+ + 1/4(9- A ) 2  -  1/4(1 - * ) «  + 1 / 2  + n]]= En . J , a ,  j ’ (3.23)where x  stands for chirality, j{j + 1) gives the eigen values of the operator J2  with J given by eq. (3.16) and a  denotes the eigen values of the operator A given byA = i [ e fS +] + «[fi+.S]. (3.24)The eigen functions corresponding to these energy eigen values (3.23) satisfy the following equations:t iD\j,m,a,x,n) = E\j,m,a,x.n),J2\j, m, a,x,«) = ;'(j +1)|j, m, a,x> n),Jj\j, m, a,x> n) = m\j,m,a,x,n),A\j, m, a, X, n) = a\j, m, a, x,n),Y51/'. m, a, x, n) = x\j< a, x, n). (3.25)The eigen states |j, m, a, x>n) obviously belong to a representation of Osp (2,1) ® SU(2 ) Setting A= P, the two lower components of eq. (3.22) becomee}g2x. aHp = ( p - e , V ) 2 - clc2H W3(3.26)which is the Hamiltomian of Pauli’s equation for the spin -.1/2 particle of charge e\ (and mass = 1 /2 ) in the field of dyon carrying electric and magnetic charges e2 and g2 respectively. For this case, eq. (3.23) reduces ton'J'a 4 [V>0' + l ) - « f * 2  + l /4  + 1/2(1 -  a) + «]]which is Pauli’s generalization of eq. (2.12).For H D given by eq. (3.22), the supersymmetric equation for the fermion moving in the field of a dyon may be written as follows :H Dyf = E y , (3.28)where eigen values L  are given by eq. (3.23) and the corresponding eigen functions satisfy eq. (3.25).LM ^' tjrnmciriicu ocnroauiger equation etc 1694, ConclusionThe eq. (3.28) can be interpreted as describing the quantum dynamics of a supermultiplet of two spin-0 and one spin-1/2 particles in a Coloumb field. It can also be looked at as an ensemble of two Schrddinger and one Pauli equation each describing the motion of a particle with electric charge-e2  in the field of dyon with electric charge e} and with magnetic charges respectively equal to (elg2 - l / 2 ) / e ,  , (e,g 2  + l /2) / e ,  andg2 . Taking e] as the electric charge of the point particle sitting at the origin, we can fix the electric charge of the supermultiplet to be e2. From the coupling to the potential V P , we see that the spin- 0  particles must be assigned magnetic charges equal to g2 ± 1 /2 ^ while the spin-1 / 2  particle will have its magnetic charge equal to g2.Referencesf l] CP Dokos and T N Tomaras Phys. Rev. D212940 (1980)[2] C G Callan (Jr) Phys. Rev. D26 2058 (1982); D25 2141 (1982)131 V A Rubakov JETP. Lett. 33 645 (1981); Nucl. Phys. B203 311 (1982)[41 S Mandelstam Phys. Rep. C23 245 ( 1976); Phys. Rev. D19 249 (1979)151 G 'l Hooft Nucl. Phys. B138 1 (1978)|6J B S Rajput, J M S Rana and H C Chandola Pros Theor. Phys. 82 153 (1989); Can. J. Phys. 69 1441 (1991)|7| B S Rajput, H C Chandola and S Sah ll. Nuovo Cim. 106A 509 (1993)[8| B S Rajput, H C Chandola and J M S Rana Phys. Rev. D43 3550 (1991)|9| B S Rajput and J M S Rana Int J. Theor. Phys. 32 357 (1993)[101 E Witten Phys. Lett. B86 283 (1979)| l l ]  B S Rajput, H C Chandola and R Bist ll. Nuovo. Cim. 104A 697 (1991)112] B S Rajput, V P Pandey and H C Chandola ll Nuovo Cim. 102A 1507 (1990), Can J. Phys. 67 1002 (1989)113] B S Rajput and V P Pandey ll. Nuovo Cim. 11 IB 275 (1996)[ 14] E D'Hoker and L Vinet Nucl. Phys B260 79 (1985)[15] S Durand, J M Lina and L Vinet Lett. Math. Phys. 17 289 (1989)[16] E D'Hoker and L Vinet Comm. Math. Phys. 97 341 (1985); Phys. Rev. Lett 55 1043 (1985)

Supersymmetrized Schrodinger Equation for Fermion-Dyon System

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Supersymmetrized Schrodinger Equation for Fermion-Dyon System

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