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SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 56B (1997) 142-147
Generalized Killing equations for spinning spaces and the role of Killing-Yano Tensors Mihai Visinescua* aDepartment Romania
of Theoretical
Physics, Institute of Atomic Physics, P.O.Box MG-6, Magurele, Bucharest,
The generalized Killing equations for the configuration space of spinning particles (spinning space) are analysed. Solutions of these equations are expressed in terms of Killing-Yano tensors. The constants of motion can be seen as extensions of those from the scalar case or new ones depending on the Grassmann-valued spin variables. The general results are applied to the case of the four-dimensional Euclidean Taub-NUT spinning space.
1. SPINNING EQUATIONS
SPACES
AND
KILLING
Spinning particles, such as Dirac fermions, can be described by pseudo-classical mechanics models involving anticommuting c-numbers for the spin-degrees of freedom. The configuration space of spinning particles (spinning space) is an extension of an ordinary Riemannian manifold, parametrized by local coordinates {zfi}, to a graded manifold parametrized by local coordinates {zp, $+}, with the first set of variables being Grassmann-even (commuting) and the second set Grassmann-odd (anticommuting) [l-3]. The equation of motion of a spinning particle on a geodesic is derived from the lagrangean:
c = ;g&)sY
W” + ;g&$~~Dr.
The corresponding given by: H =
world-line hamiltonian
~g’YIIaII,
(1) is
R 44
where the parentheses denote full symmetrization over the indices enclosed. In general the symmetries of a spinning-particle model can be divided into two classes. First, there are four independent generic symmetries which exist in any theory [l-3]: _ 1. Proper-time hamiltonian
translations
2. Supersymmetry charge
generated
by the super-
(6)
3. Chiral symmetry charge r, = $&
by the
generated
,... &jfil
. . .?p,
by the chiral
(7)
= 0.
(3) If we expand J(z) II, $) in a power series in the canonical momentum
4. Dual supersymmetry supercharge
Q* = i{r*, Qo} *E-mail
generated
H (2),
Qo =&M‘,
(2)
where II,, = g&Y is the covariant momentum. For any constant of motion J(z, I&$), the bracket with H vanishes {HJ}
then the bracket {H,J} vanishes for arbitrary II, if and only if the components of J’ satisfy the generalized Killing equations [l] :
address:
mvisinOtheorl.ifa.ro
0920-5632/97/S] 7.00 0 1997 Elsevier Science B.V. All rights reserved. PII: SO920-5632(97)003 19-8
generated by the dual
M. Visinescu/Nuclear
Physics B (Proc. Suppl.) 56B (1997) 142-147
where d is the dimension of space-time. The second kind of conserved quantities, called non-generic, depend on the explicit form of In the recent literature the metric g&z). there are exhibited the constants of motion in the Schwarzschild [4], Taub-NUT [5-71, KerrNewman [3] spinning spaces. In what follows we shall deal with the nongenetic constants of motion in connection with the Killing equations (5) looking for the general features of the solutions. In general the constants of motion can be seen as extensions of the constants from the scalar case or new ones depending on the Grassmann-valued spin variables {$P}. Let us assume that the number of terms in the series (4) is finite. That means that, for a given n, (n+l) ,7@, ..+,+, vanishes and the last non-trivial equation from the system of Killing equations (5) becomes homogeneous :
glected 7 the components ,7’ “!. Pm (772< n) will Pl. receive a non-trivial spin contribution. Therefore the quantity (11) is no more conserved and the actual constant of motion is
(12) in which Jir!.P,,, with m < n has a non-trivial spin-dependent expression. We shall illustrate the above construction with a few examples. Since for n = 0 eq. (10) is trivial, we shall consider the next case, namely n = 1. In this case eq. (10) is satisfied by a Killing vector R,: R(w)
= 0.
(13)
Introducing this Killing vector in the RHS of the generalized Killing equation (5) for n = 0 one obtains for the Killing scalar [7] $0) = & 2
In order to solve the system of coupled differential equations (5) one starts with a J$:!..,n solution of the homogeneous equation (9). This solution is introduced in the right-hand side (RHS) of the generalized Killing equation (5) for J$.$!_, and the iteration is carried on to n = 0. In fact, for the bosonic sector, neglecting the Grassmann variables {v+!+}, all the generalized Killing equations (5) are homogeneous and decoupled. The first equation shows that J(O) is a trivial constant, the next one is the equation for the Killing vectors and so on. In general, the homogeneous equation (9), for a given n, in which all spin degrees of freedom are neglected, defines a Killing tensor of valence n
. . . r.P
(11)
is a first integral of the geodesic equation [8]. For the spinning particles, even if one starts with a Killing tensor of valence n, solution of eq. (10) in which all spin degrees of freedom are ne-
+‘$”
lPP1
(14)
where the square bracket denotes antisymmetrization with norm one. A more involved example is given by a Killing tensor Kpy satisfying equation (10) for n = 2: &,,;A\) = 0.
(15)
Unfortunately it is not possible to find a closed, analytic expression for the spin corrections to the quantity (11) in terms of the components of the Killing tensor K,,” and its derivatives. But assuming that the Killing tensor Kpy can be written as a symmetrized product of two Killing-Yano tensors, the construction of the conserved quantity (12) is feasible. We remind that a tensor is called a Killing-Yano tensor of valence fPl...Pr r [9] if it is totally antisymmetric and satisfies the equation fw.../&l(Pp;x)
J- = J-j;!..,, IP
143
=
0.
(16)
For the generality, let us assume that the Killing tensor can be written as a symmetrized product of two different Killing-Yano tensors K$” = ;(f/,f;”
+ f~J.j‘“)
(17)
144
M. VisinescdNuclear Physics B (proc. Suppl.) S6B (1997) 142-147
where ff” is a Killing Yano tensor of type i and the Killing tensor has two additional indices i, j to emphasize the fact that it is formed from two different Killing-Yano tensors (i # j). Introducing the Killing tensor (17) in the RHS of eq. (5) for n = 1 we get a spin contribution to the Killing vector [3]: I(f)’ ‘3
;$‘$“(f~~Dyfj’x
=
+;f:pcjh7,
+ f;j”Dvf/‘i”x
+ fj%hJp)
(18)
Combining eq. (22) and (23) with the aid of eqs. (12) we get a constant of motion which is peculiar to the spinning case: J = f&W”
( RxnX + ;Rp;c]““*lir”) .
(24)
In fact this constant of motion is not completely new and it can be expressed in terms of the quantities (14) and (21). Another $-dependent solution of eq. (9) for n = 1 can be generated from a Killing-Yano tensor of valence T:
and using this quantity in the RHS of eq. (5) for n = 0 we get for the Killing scalar
3;(t) = fp,&&...@ Pv2 . **vr.
J(o) ij
Again introducing this quantity in the RHS of eq. (5) for n = 0 we get for J(O) :
=
-
@
W’I’(R,v~, f,!‘,fy
,.
J-w
=
~(-l)‘+‘fIr~...P*:~*+‘] .p
where the tensor ci,,“~ is ci/.Wx= -2fi[vx;&
(20)
Higher orders of the generalized equations (5) can be treated similarly, but the corresponding expressions are quite involved. In what follows we shall analyze the homogeneous equation (9) looking for solutions depending on the Grassmann variables {@‘}. Even the lowest order equation with n = 0 has a non-trivial solution [7,10] $0) = ;f@“$“$Y
(21)
where f,,” is a Killing-Yano tensor covariantly constant. Moreover J(O) is a separately conserved quantity. Going to the next equation (9) with n = 1, a natural solution is: J7(‘ cc) = R,fd%”
(22)
where R, is a Killing vector and again fxc is a Killing-Yano tensor covariantly constant. Introducing this solution in the RHS of the eq. (5) with n = 0, after some calculations, we get for $0) [lo]:
(25)
. . . $P’Fi’
(26)
and the constant of motion corresponding to these solutions of the Killing equations is [lo]:
Qf =
fPl...JIyP
. . . ?p
+ ~(-l)‘+‘f~,,~~;~~+~] *?y . . . ?p+‘.
(27)
Therefore the existence of a Killing-Yano tensor of valence T is equivalent to the existence of a supersymmetry for the spinning space with supercharge Qr which anticommutes with Qe. A similar result was obtained in ref. [ll] in which it is discussed the role of the generalized Killing-Yano tensors, with the framework extended to include electromagnetic interactions. 2.
TAUB-NUT
SPINNING
SPACE
In a special choice of coordinates the euclidean Taub-NUT metric takes the form ds”
=
V(T) (dr” + r”uY2 + r2 sin’&&“) +16m”V-‘(r)(dx
withV(r)=l+ tors DA = R$&,,
a.p
+ cos8d~)~
(28)
There are four Killing vec-
A = 0,...,3
(29)
145
M. Visinescu/Nuclear Physics B (Proc. Suppl.) 56B (1997) 142-147
Kepler-type
where D1
=
d -sinlpBO--coscpcot6-+--
Dz
=
coscpd,
D3
=
-
Do
=
G’
8
-sincp
coscp d sin0 dx a(P sincp d c0teav sin8 dx
a
a+__
d d (30)
corresponding to the invariance of the metric (28) under spatial rotations (A = 1,2,3) and x translations (A = 0). In the purely bosonic case these invariances would correspond to conservation of angular momentum and “relative electric charge” [12-151:
q =
lJ: T’
(31)
16m” V(T) (2 + cost’+)
(32)
where p’= V-~(T)? is the “mechanical momentum” which is only part of the momentum canonically conjugate to r’. On the other hand in the Taub-NUT geometry there are known to exist four Killing-Yano tensors [13]. The first three Killing-Yano tensors fi are covariantly constants (with vanishing field strength) fi
=
8m(dx + cosddp) A dxi --E&l
=
i [axj’+
(-&-4mE)
F]
(36)
where the conserved energy E, from eq. (2), is
&J
;&xc+
problem
+ $)dxj
A C&k,
i = 1,2,3.
(33)
E
= =
[i’+(-$,‘].
iv-‘(T)
(37)
The components Kipv involved with the Runge vector (36) are Killing tensors and they can be expressed as symmetrized products of the KillingYano tensors fi (33) and fy (34) as in eq. (17) 17,131. Starting with these results from the bosonic sector of the Taub-NUT space one can proceed with the spin contributions. Using eq. (27) we can construct from the Killing-Yano tensors (33) and (34) the supercharges Qi and Qy. The supercharges Qi together with QO from eq. (6) realize the N=4 supersymmetry algebra [7]: {QA,QB}
=
-S~~ABH,
A,B =0,...,3
(38)
making manifest the link between the existence of the Killing-Yano tensors and the hyper-KShler geometry of the Taub-NUT manifold. The generalized Killing equation (5) for n = 0 shows that with each Killing vector R$ (29) there is an associated Killing scalar BA: BA =
i ~RA[,,;v]$~$“.
(39)
The fourth Killing-Yano tensor is fy
=
8m(dx + cosedp) A dr +4r(r + 2m)(l + $-)
sin 8d@A &
(34)
Therefore the total angular momentum and “relative electric charge” become in the spinning case [5,7] J’ =
z+;,
(40)
Bo +q
(41)
and having only one non-vanishing component of the field strength
Jo
f YrfQ
where J’= (Ji, Jz, Js) and B’= (BI, Ba, Bs). We mention that the above constants of motion are superinvariant:
=
2(1 +
2-J
rsin8.
(35)
In the Taub-NUT case there is a conserved vector analogous to the Runge-Lenz vector of the
=
{JA,Qo}=O
,
A=0 ,...,
3.
(42)
146
M. Visinescu/Nuclear Physics B (Proc. Suppl.) 56B (1997) 142-147
In order to obtain the spin dependent contribution to the Killing tensor zp, we shall use eq. (18) in conjunction with eqs. (33) and (34) for the Killing-Yano_tensors fi and fy to get a spin dependent part S, to the Killing vector:
These components of the spin are separately conserved an: can be combined with the angular momentum J to define a new improved form of the angular momentum 1i = Ji - Si with the property that it preserve the algebra
eTz, = 2, + Sz
{IiT Ij)
(43)
where & are the standard Killing vectors (29). There is no contribution quadratic in the spin [6] since the quantity (19) vanishes taking into account the fact that the complex structures fi are covariantly constant and the self-duality of the Taub-NUT geometry [7]. Thus only the $Jdependent part of the Killing vectors ,!?,, contributes to the Runge-Lenz vector for the spinning space E = $”
* iYS” + s$ *P.
(44)
In terms of the supercharges Qi and Qy , the components of the Runge-Lenz vector /? are given bY [71 Ri = f {QY, Qi}
,
The non-vanishing some algebra):
i= 1,2,3.
(45)
Poisson brackets are (after
Jk, {Ji, Jjl = Eijk {Ji,Kj) = Eijkxk, {&,Kj)
=
a(A
(46) (47) -2E)
CijkJk
(48)
similarly to the results from the bosonic sector 1131. In what follows we shall consider the conserved quantities peculiar to the spinning case. Taking into account the existence of the Killing-Yano covariantly constants tensors fi (33), three constants of motion can be obtained using the prescription (2 1) : ,
i = 1,2,3
(49)
which realize an SO(3) Lie-algebra similar to that of the angular momentum (46): {Si,
sj} =eijkSk.
(50)
(51)
and that it commutes with the SO(3) algebra generated by the spin Si {Ii, Sj} = 0.
(52)
Let us note also the following Dirac brackets of Si with supercharges {Si,Qo}
=
-%,
{Si, Qj)
=
1 -(&j&e + 2
QjkQk)-
(53)
We can combine these two SO(3) algebras (50), (51) to obtain the generators of a conserved SO(4) symmetry among the constants of motion, a standard basis for which is spanned by Mi* = 1i f Si [7]. We should like to remark that there is no spin component like in equation (49) to be used for a improved “relative electric charge” Jo. The reason is that the fourth KillingYano tensor fy (34) is not covariantly constant. Of course, we can add to JO a combination of the three separately conserved quantities Si but this is not a natural “improved relative electric charge”. Finally, let us consider’s solution of the homegeneous eq. (9) for n = 1 of the type (22). Using the Killing vectors (30) and the Killing-Yano tensors (33) we can form the combinations: J-‘l! Ajp
=
RApfjXc$‘V, A=0 ,..., 3
,
j=l,2,3.
(54)
After some algebra we get the new constants of motion of the form (24): JAj
si = ~fi.“vv
= EijkIk
RAB~’
=
fj~o’d”$”
=
-4iSj JA
)
A=0 ,...,
3
(
,
i- iR~[~;p]$“@
j=l,2,3.
>
(55)
Strictly Speaking, the C0nStaIIt.S JAj are not completely new, being expressed in terms of the
M. Visinescu/Nuclear Physics B (Proc. Suppl.) 568 (1997) 142-147
constants JA (42) and Sj (49). However, the combinations (55) arise in a natural way as solutions of the generalized Killing equations and appear only in the spinning case. Moreover, we can form a sort of Runge-Lenz vector involving only Grassmann components: Li = ‘CijkSjJk , i,j,k = 1,2,3 (56) m with the commutation relations like in eqs. (47), (48):
Jj}
=
CijkLk,
{Li, Lj}
=
(36
{Li,
(57) s”> $eijkJk.
(58)
Note also the following Dirac brackets of Li with supercharges:
{Li,Qo}
= -&fijkQjJky
{b,Qj)
= &(Gjk&Oh - bjQk& +QiMj).
3. CONCLUDING
(59)
(60) REMARKS
The aim of this paper was to point out the important role of the Killing-Yano tensors to generate solutions of the generalized Killing equations. This aspect is closely connected with the fact that the Killing-Yano tensors can be understood as objects generating non-genetic supersymmetries [3]. For the first orders of eqs. (5) and (9) (n I: 2), which are usually encountered in theories of interest, we presented the complete form of the solutions. With some ability, it is possible to investigate the higher orders of the generalized Killing equations, but it seems that one cannot go much far with simple, transparent expressions. The extension of these results for the motion of spinning particles in spaces with torsion and/or in the presence of an electromagnetic field will be discussed elsewhere. Acknowledgements
The author wishes to thank the organizers of the Ahrenshoop Symposium, Buckow 1996, for their kind invitation and financial support that made possible his participation in this very enjoyable event.
147
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