- Email: [email protected]
On the magnetic moments of hadrons
Vol. 18(1980)
REPORTS
ON
MA THE M.4 TICAL
ON THE MAGNETIC
No. 1
PHYSICS
MOMENTS
OF HADRONS
L. P. HUGHSTON* and M. SHEPPARD** The Mathematical (Receiced
A new method for calculating based on twistor theory.
Institute, Oxford, England August 22. 1978)
the ratios of hadronic
magnetic moments
is presented,
I. Introduction The purpose of this article is to present some studies towards the understanding of hadronic magnetic moments. Our approach is novel inasmuch as we shall be working within the framework of twistor theory. In units of nuclear magnetons’ the magnetic moments of the proton and the neutron have been measured, respectively, to be pp = 2.7928456(11) and pu, = - 1.91304211(88). Note that the ratio of these numbers is approximately - 3/2. It has been known for some time now that using arguments of a group theoretical nature2 it is possible to derive the value - 3/2 for ,uJ,u,,. In what follows we shall show how these arguments can be carried over, in a relatively straightforward way, to twistor theory. Plan ofthe paper. This section will conclude with a brief review of notation and some background material. In Section II a center mass twistor is introduced for massive multi-twistor systems. It is a simple skew-symmetric twistor representing a point on the complex world-line of the system. We construct the corresponding quantum operator, using standard twistor quantization procedures, and discuss some of its properties. In Section III we introduce a hadronic magnetic moment operator, and we prove a result useful in the evaluation of its matrix elements. The center of mass operator plays a central role in the discussion. We finish in Section IV with a calculation of the protonneutron magnetic moment ratio.
of
*Fellow of Lincoln College, Oxford. **Junior Research Fellow, Wolfson College, Oxford, and S.E.R.C. Research Fellow. ‘One nuclear magneton = eh/2m,c = 3.1524515(53) x lo-‘* MeV gauss-‘. The value quoted here for p,, is given by Cohen and Taylor [2] and the value for p(. is given by Greene et ul. [3]. ‘See, for example, Beg et al. [l]. 1531
54
L. P. HUGHSTON
and M. SHEPPARD
Notation. The standard conventions for twistors and two-component spinors will be used here3. Furthermore, triplets of twistors will be denoted 27 withj = 1,2,3. The conjugate twistor operators -d/LIZ; will be denoted 2:. Note that we have the commutation relations
[Z$
z^;] = s;sy.
(1.1)
Cohomological considerations enter only minimally into the arguments which follow later in this paper, and thus we find it convenient to speak of twistor functions $(ZT) rather than elements of cohomology groups. As usual we put 24 = (of, rcAlj)for the spinor parts of Zy. We say that Z; is restricted to the (finite) space-time point xp ifof = rXA-%,?j. We shall write pXZy = for ZT thus restricted.
(iXAA’nA,j,X,4*j)
(1.2)
Similarly, we shall write
for the twistor function example, the identity
$(Z;)
restricted
crP*twy)
to the spacetime
point s’. We have, for
= PxL4’w;),
where VAA,= a/ax**‘, and where PAAp= TT ApjziJa is the
momentum
(1.4) operator
(with
fi6
= - a/&& Baryon states. Hadron states in general can be represented in terms of holomorphic functions of three twistors. These functions are required to be in suitable eigenstates of mass, spin, baryon number, electric charge, hypercharge, isospin, and the two SU(3) Casimir operators. Holomorphic twistor differential operator expressions can be formulated for these various observables. The connection between the twistor function description of a particle state and the conventional spinor wave-function representation of the state is established by means of a twistor contour integral formula 4. Incorporated in the contour integral formula is a certain combination of spinors, this spinor coefficient structure being uniquely determined by the state of the baryon: thus, the characteristics of the baryon state are coded, in a dual way, into the spinor coefficient structure (see Sheppard [ 11J).
‘See Penrose [S] and [7]. 4Twistor contour integral formulae of the general sort used here are discussed in, for example, Penrose [7], [8], Hughston [4] and Penrose [9].
MAGNETIC
MOMENTS
OF HADRONS
55
Let us denote the three ?r-spinors nA,j by uA,, dA., and sA,, respectively. The expressions for the spinor fields associated with the proton and the neutron can then be written explicitly as follows: Y;‘(x)
=
Kp
i
uA’uB’dgq&,(Z;)An,
(1.5)
. Y;‘(x)=
dA’dB’uB+q,bn(Z;)An,
K,
(1.6)
I where-$,(Zy) and $,,(Zy) are twistor functions appropriate to a proton state and a neutron state, respectively. Here Arc is the natural projective 5-form on the space of the n-spinors: An =
EA’P’EB’Q’EC’R’
d
x,.idn,,jdq
,d+,,dnp
L.d71R.,,.Ei.ikt:“‘.“,
(1.7)
where cijk is the totally skew-symmetric tensor; wedges have been suppressed. Note that p,$( 24) can be regarded as a function of xAA’ and ~~~~The An integration eliminates the 7cAsjdependence, leaving behind the spinor field as a function of xAA’ alone. The normalization factors ICY and K, are designed to ensure that if the twistor functions $( ZT) correspond to normalized states then the spinor fields YUA’(x) emerge automatically with the proper normalization. In the forthcoming discussion we shall be interested in the case for which our nucleons are in a definite spin state with respect to some axis, i.e., either with s, = l/2 (spin up) or with s, = - l/2 (spin down). In this connection it is useful to introduce a constant spinor dyad (OA!,Lo,) normalized so that oA, lA’ = 1. We say that the spinor field ‘PA’(x) is in an eigenstate of spin with s, = l/2 if for all xAA’ we have Yy,, tA’ = 0. Similarly, we have s, = - l/2 if YA,oA’ = 0 over all of space’. If $,(Zq) represents a proton in an s, = l/2 state, then the only relevant spinor coefficient structure6 for the contour integral formula is uOuOd’ - u’u’d’. Note, in particular, that for an s, = l/2 proton state the coefficient u’u’d’ - u’u’d’leads to a vanishing integral. An s, = l/2 proton state is, in fact, determined by the following two conditions: first, that u”uod’ and - uOu’dO are the only two coefficient structures that yield a non-trivial integral; second, that they yield the same value for the integral. For an s, = l/2 neutron the relevant coefficient structure is d”dou’ - d’d’u”. For a A + state with s, = l/2 the structure is 6 - I12(u’u’d’ + 2u0u’do) and for a A0 state with s, = l/2 the relevant structure is given by 6-1i2(d0d0~’ + 2d”d’ u’). 5The spinor dyad {o”., lA..} is related to the direction of the spin-axis zAA’by the fomula o (A, (at) (P”“) is the expectation value of the momentum operator for the state under consideration. We require that {o”,, tA,} be chosen such that PA’ is real. we write cl/duo 6Here we use the notation u” = 0”‘~~’ and similarly u1 = tA’uA,,etc., Furthermore, =.-- lAla/au,*, etc. = z$ where (P”‘*)
56 II.
L. P. HUGHSTON
and M. SHEPPARD
The center of mass twistor
In this section we shall consider some facts which are valid not only for systems of three twistors, but in fact for massive systems composed of any number (two or more) of twistor constituents. Associated with any set of twistors Z; is a skew-symmetric twistor Rza defined by .
R@
=2,,,-2
zfz$ J
,@ik
3
(2.1)
where h?jk is given by7 A.ik = zj Zk~olO aP.
(2.2)
Inasmuch as I@ is skew we can easily verify that RzP is indeed skew. Furthermore, the fact that the squared mass of the system Zg is given by yg = M,
where M, is the complex satisfies the normalization
conjugate condition
&jij,
using
(2.3)
of M’j, it is straightforward
to check that Rap
RzBI aB = 2 *
(2.4)
Next we note that RkP is simple, that is to say, it satisfies R[apR’ila = 0. We can verify the relations
(2.5)
(2.5) by noting that (2.1) can be rewritten R”b = 2m-2~
where Yi is defined according
A
Y@A 3
in the form (2.6)
to Y5 =
zy ,j,.
(2.7)
Equation (2.6) shows that RaP is the skew product of a pair of twistors of valence one, this being the necessary and sufficient condition that (2.5) is satisfied. Since RmB is a simple skew-symmetric twistor satisfying (2.4) it corresponds to a point in complex Minkowski space. This point lies on the complex center ofmass of the Zy system. The remaining points of the complex center of mass are obtained by ‘The infinity twistor I”# is defined by:
p
See Penrose [7] for further discussion.
=
.
MAGNETIC
MOMENTS
57
OF HADRONS
propagating away from the point determined by R”” in the direction of the total momentum of the system8. Although we will not make use of the fact here, it is perhaps worth noting for future reference that the condition that RzP should lie in the forward tube is that the vector WAA’
=
Yiy,,.,
be timelike and future-pointing. To see that this is the case, observe that by varying %A in the expression Y: LA we obtain the various points which lie on the line determined by R”@ in CP,. Thus, if R”fl is to correspond to a forward tube point then Yi AAmust lie in the top half of CP, for all values of IbA.In other words, we must have WAA.
l%A/iA’> 0
(2.9)
for all LA; this, however, is just the condition that W,J, be timelike and future-pointing. Now we apply the twistor quantization’ procedure zj, + Zj, to the twistor R”O. Concerning the operator I? thereby obtained, the following result can be stated: PROPOSITION
1. Ij’$(Z~)
is in a mass eigenstate
with mass eigenvalue
m, then we
have the identity P,@DIJ/(z;) where XaB is the simple skew-symmetric
(2.10)
= XaBp,$(q>, twistor representing
the Minkowski
space point
XAA’
Proof:
First of all, note that if X& represents second of its indices. then we have the relation y”
= xi,
the rc-part of XzB with respect to the
XPB’.
(2.11)
Furthermore, if all n of the twistors Zy are restricted to the point xAA’ then Zj” must be of the form X”‘nBIj for some nASj;-in other words, we have the relation px
The proof of (2.10) then proceeds
z; = X”B’7tB.j.
(2.12)
as follows:
*It is perhaps worth noting, as is discussed in Hughston [4], that the remaining points on the complex world-line of the system can be generated by applying the transformation Z; -+ Z; + I”@AjkZ>, where Ajt is a complex skew-symmetric tensor. It suffices, in fact, to choose Ajt = i Mjk, where 1 is a parameter. ‘References for twistor quantization include Penrose [6] - [9] and Woodhouse [lo].
58
L. P. HUGHSTON
p,P~Ic/(Zp)
and M. SHEPPARD
= 2m-2p,ZqZP LPk$(Zj”) = 2m-2XaA’XBB’pxnATjnBpk tijj”$(Zg) =m - 2 X$ x@Q,
&A’B’7rA~j7TB~k iwlj( z,;,
= mp2 X@p, M2$(Zg),
(2.13)
where M2 = M, h?j is the mass-squared operator. Using the fact that $(Zg) is in a mass eigenstate we see that (2.10) then follows immediately lo. n III. The magnetic moment operator
The hadronic magnetic moment operator k is defined by /i = oap Map + ~7”~M;I*p,
(3.1)
where we have put MS@= ,c;ZpiP)?Zj
Y
(3.2)
M$ = - p&jkZj’k,*(b.f;,.
(3.3)
and
In equation (3.3) the operator R,*, is the Hermitian conjugate of l?,
so that we have (3.4)
Note, incidentally, that Mzp is the Hermitian conjugate of MaB, and that the minus sign appearing in equation (3.3) is a consequence of the fact that a convenient arrangement of indices has been made in the expression. The constant p will be discussed later. The matrix 4 is the standard SU(3) electric charge matrix, defined by
(3.5) Observe that the charge matrix is tracefree and Hermitian. The constant twistor cols appearing in (3.1) is required to have the properties fsafi = C/JW a,flP’ = 0:
(3.6)
Thus, CS,@ is necessarily of the form (3.7)
where c”~’ IS ’ a symmetric spinor. The twistor a”@is the complex conjugate of crap. “The classical analogue of Proposition
1 is, incidentally,
also valid. We have: p,R”B = x”B.
59
MAGNETICMOMENTSOFHADRONS
The operator @is evidently a rather complicated object, and accordingly we require a fair amount of additional apparatus in order that we should have at our disposal an effective procedure for evaluating its matrix elements. We shall now proceed to establish a result that, as will be demonstrated in Section IV, is very useful in this connection. Let us define a pair of new operators r and Z by writing r = psj7rA,k 0::
(;@i
(3.8)
and f = P&iO$ a;l?i, where, as is customary,
(3.9)
we have put izj, = --alaw,!,
bA’j = -
apnA,j
for the spinor parts of the operator z^’“,. The operators magnetic moment operator i in the following way:
z and 7 are related
(3.10) to the
PROPOSITION 2.LetI)( 24) be a twistorfunction appropriate to a particular hadronic eigenstute; and let C denote the spinor coefficient structure appropriate, within a contour integral formula, to that state. Furthermore, let us put C= l7l7, where Il consists entirely of 7cAljcoefficients, and ficonsists entirely of ?r’Ajcoefficients. Then thefollowing formula is valid: (3.11) Proof: The reasoning involved here simplifies considerably in the event that Z involves nAPjcoefficients alone, i.e., in the case that Z= n Let us, then, consider that possibility first, and generalize later to the case of arbitrary 2: As it turns out, a somewhat stronger relation holds in the case Z = Nathan equation (3.11). Writing T= ozs MaB
(3.12)
we can deduce the following relation: p,Tti with no contour
= rp,$,
(3.13)
integral having been taken. Here is the proof of the validity of (3.13):
60
L. P. HUGHSTON
=
and M. SHEPPARD
/.4&j” “A’B’pxXA’kR~’ q,lr/
= p&l EjkQ,p Z$?)~ zi, $.
(3.17) (3.18)
Note that in the passage from (3.14) to (3.15) we have used the following useful lemma, due to G.A.J. Sparling: CV’p,$ = (iXA“Q$i
+ p,dYA’)II/.
(3.19)
Equation (3.19) can be established, without much difficulty, by using the differential chain rule. In going from (3.15)to (3.16)we have used the main content of Proposition 1, taken in the form p&,,‘$
= - ixB~cpX$,
(3.20)
where the spinor operator A,.’ is defined by (3.21) In going from (3.16) to (3.17) we again use Proposition 1, this time in the form PXLB~$
= &.4~WPxti.
(3.22)
It is perhaps worth noting that in (3.22) the pX operator is superfluous, the relation being valid even without its presence. The passage (3.17) to (3.18) is straightforward, a simple consequence of equation (3.7). Given the validity of (3.13), we can integrate so as to obtain (3.23) Equation (3.23) remains valid, moreover, if we include fi coefficients in the formula: l7p,fiT$An
=
$
i
L’rp,fi$An.
(3.24)
The proof of (3.24) proceeds in two steps. First we note that the slightly modified formula Ilp,Tfi$An
i
=
I
l7zp,&A1c
(3.25)
is certainly valid, since it can be obtained from (3.23) with the simple substitution ti + fi$. (We must note in making this substitution that if Ic/is in a mass eigenstate then so is fill/.)
MAGNETIC
MOMENTS
61
OF HADRONS
Thus, the validity of (3.24) hinges on showing that
np,[fi,T]l)A72 = 0.
(3.26)
To demonstrate (3.26) we observe that fiis an operator of a certain degree in GA’,say n. Since t+Gis in a definite quantum state, and the degree of fi is coordinated with II/, it follows that if [ fi, T] does not contain a term of degree II then (3.26) holds. A short calculation will show that this is indeed the case. So as to bypass some rather complicated formulae, we shall illustrate the matter explicitly only for the case n = 1, leaving the more general case to the reader. Now, from equation (3.16) we see that T= p$ ~~‘~‘(n~&.~$;
- n,.,&&.)
(3.27)
where ii/ Putting
= 2m- 27c,~,o~A~iiAs.
l? = 2; and using the commutation
(3.28)
relations
[Cl$, j2\] = 8;8;,
(3.29)
we then derive without much difficulty the following expression for the commutator which we are interested: [i?;, T] = -2mp2pEj
~A’B’7CA,k~ng,,~;;tAt;t$.
in
(3.30)
The term appearing in (3.30) is manifestly an expression of degree n = 3 in $2. It is true generally, in fact, that if fi is of degree II, then the commutator [ii, T] will be of degree n + 2. Thus (3.26) is valid, and consequently we have succeeded in proving (3.24). Having proven relation (3.24) we must now consider the complementary relation lIlp,fifiiAx Y! where ?is the Hermitian
conjugate
IIpxi%+bAx,
=
(3.31)
i of T:
T= O”fi M* 0’
(3.32)
Inasmuch as i = T+ T we see that (3.31) and (3.24) taken in conjunction clearly imply statement (3.11). As a helpful abbreviation let us define the integral operator Sz, by In, =
Adlp,fi.
(3.33)
I Using this notation, sz,@
the proof of (3.31) proceeds
= -Q&Ei,CWjii~~~Z^;)~
as follows: (3.34)
62
L. P. HUGHSTON
and M. SHEPPARD
(3.35)
(3.36) (3.37) (3.38) (3.39) In the derivation above, the passage from (3.34) to (3.35) is simply definition (3.4); and the subsequent expansion into spinor parts, shown in (3.36), is quite routine. We go from (3.36) to (3.37) making use of reasons similar to those involved in showing the validity of (3.26). One sees that if fi is of degree II, then the expression beneath the integral sign in the first term of (3.36) is of degree n + 2; accordingly we see that term must vanish when the integral is taken. In the second term of(3.36) we apply the identity integral sign in the first term of (3.36) is of degree II + 2; accordingly we see that that term must vanish when the integral is taken. In the second term of (3.36) we apply the 212A’r EA’;r ;tS zz M2E AB s A B
(3.40)
identity in order to pass to (3.37), and then use the fact that $ is in a mass eigenstate in order to obtain (3.38). With an adjustment of indices and an accompanying sign change, we apply definition (3.9) in order to obtain (3.39). n IV. The proton-neutron magnetic moment ratio The magnetic moment expectation value
associated
with a hadron
C~lciltil>
state rc/ is defined to be the
(4.1)
of the magnetic moment operator /.?. Unfortunately, we do not know yet how to calculate from theoretical principles the value of the constant p that appears in the definition of the operator I;. We can, however, take the ratios of expressions of the form (4.1) and these will be independent of the value of the constant .LL. We can, for example, look at the ratios <$,IArc/,>l(~“lA&A
(4.2)
where $, is a proton state in a definite state of s, = l/2, and where +,, is a neutron state in a definite state of s, = l/2. Now ,LI$,) is not a pure proton eigenstate. The operator b knocks $P out of its eigenstate into a mixed state. In fact, we have, as will be shown shortly,
MAGNETIC
&f$J
MOMENTS
= C+f$J)
OF HADRONS
63 (4.3)
+ /@ItiLl+ >
where1 t,Gd+ ) is a A + eigenstate, with s, = l/2, and where c(and p are certain numerical coefficients. Since I$, ) and I t,Gd+ ) correspond to distinct types of particles, they are orthogonal: (4.4) +iqh+ > = 0. As a consequence
of (4.4) we see that
ot&lL4~,> = PZ-
(4.5)
Similarly, we have
iw”>
= PYl$n) + P.lsl$B >
(4.6)
where 1I,$~o) is a A0 eigenstate with s, = l/2, where 1~and 6 are numerical coefficients. And thus we have
WlLiI$“)
= PY,
(4.7)
showing that the ratio pp/pL, is given by a/y. What remains to be shown is a method for calculating the coefficients a, 8, y, and 6. For this purpose we invoke Proposition 2. With the aid of Proposition 2 we can write px Ct,QAn
p, C/it,GAn = I
(4.8)
I
where c’ is a modified spinor coefficient structure which can be easily expressed, according to a procedure which will be described below, in terms of Z7, fi, z, and 7. The modified coefficient structure c’ can be uniquely written in the form c’=
(4.9)
a coefficient structure incompatible with $, and where 5 and q are factors. Since I”is incompatible with Ic/it follows from (4.8) that we
px C$An.
pX C@tjAn = 5 I Therefore
<$ represents
(4.10)
i
the part of @II/which lies in the direction
bIti> = (Iti> + 1lvo>
of $, i.e., we have (4.11)
where
($lvP) = 03 the value of q being determined
(up to phase) by the condition
(4.12) that ICJI) be normalized.
64
L. P. HUGHSTON
and M. SHEPPARD
Let us see how Z’ in (4.8) is determined. First of all notice that in (3.11) we can integrate by parts in the first term so as to transfer the action of r from p, Z?$ to 17. Let us denote -r IZ (the minus sign having come from the integration by parts) by the symbol l7’. Furthermore, notice that I7 being of degree n in 7?5 in (3.11) and t being of degree one means that the degree of fiiappears to be too large for that term to survive. If, however, n > 1, then one of the Zi operators in fican act on the IX; spinor in z”(see equation (3.9)) leaving behind -S:$. (If n > 1 this can happen in several different ways and a collection of terms is obtained.) The net effect is to leave behind a modified operator ft in the second term of (3.1 l), this operator being of degree n, as desired. Thus we have C’= nfi+
ni?P
(4.13)
in equation (4.8). Now we shall consider in detail the problem of the proton-neutron moment ratio. Since we have Z= l7 for the proton and the neutron, the problem somewhat. For a proton with s, = l/2 we have, recalling Section I,
magnetic simplifies
ZIP = uOuOdl - uOuldO. Now we wish to calculate UP, where fib = - r UP. An expansion the operator z gives (cf. equation (3.8)) the following formula: -P -17 = $(@
after putting
of
(4.15)
(quark occupation number) operators
u^” = uOa/&JO,
9O =
into components
- u^‘) - *@o - 5’) - $($” _ ii),
where we have defined the homogeneity
a0 =
(4.14)
Gi =
da/ad,
dOa/adO,31 = did/ad’, s”a/asO, 9 = slalasl,
(4.16) (4.17) (4.18)
OA.B.= oCA.I~,). We find, applying (4.15) to (4.14), that we have jC1’7b = $u”uodl
Some elementary
+ +u”uldo.
(4.19)
algebra lets us rewrite (4.19) in the form (4.20)
where Il, + is the coefficient structure appropriate = -!nA+ $
(u”uod’
for a A + state with s, = l/2, namely:
+ 2u”u’do).
(4.21)
MAGNETIC
MOMENTS
OF HADRONS
65
Inspection of (4.20) and comparison with (4.3) shows that c( = 1. Proceeding on to the neutron, we note from Section I that Lln = dOdOu’ - d”d’uo. Applying
(4.22)
- z we obtain F-‘l&
which can be rewritten
= -$dOdou’
_ $d”&O,
(4.23)
as (4.24)
where fl,o is the spinor coefficient structure appropriate n,o = -f-(d”douL Pj
to a if0 state with s, = l/2, i.e.:
+ 2d0d’zt0).
(4.25)
Inspection of (4.2) together with (4.6) shows that y = - 2/3. Now we can calculate the ratio a/y, and we observe the desired value - 312 for the ratio of the magnetic mo~zent qfthe proton to the magnetic moment qf the neutron. Remarks. The calculation above can be obviously modified so as to yield the ratios of the magnetic moments of other low-lying baryons. It can also be modified so as to enable one to calculate the electromagnetic transition rates for certain neutral mesons, and for this the full content of Proposition 2 is required. We shall discuss the case for neutral mesons elsewhere (see Sheppard
Lll)).
The authors would like to express thanks to B. D. Bramson, C. J. Isham, R. Penrose. G. A. J. Sparling, Z. Perjes, and A. S. Popovich for useful discussions.
REFERENCES [l] Beg, M., B. W. Lee, and A. Pais: Plzys. Rev. lefts 13(1964), 514. [2] Cohen. E. R., and B. H. Taylor: J. Phys. Chern. Rej: Data 2( 1973), 663. [3] Greene, G. L., N. F. Ramsey, W. Mampe. J. M. Pendlebury, K. Smith, W. D. Dress, P. D. Miller, and P. Perrin: Physics fefts 71B(1977), 297. [4] Hughston, L. P.; A particle ci~~~cu~ion scheme based on the theory qf twisters, D. Phil. Thesis, Oxford 1976. [S] Penrose, R.: 7’hestructure qfspacetime, in Battelle Rencontres, C. M. Dewitt and J. A. Wheeler, editors; Benjamin, 1968.
66
L. P. HUGHSTON
and M. SHEPPARD
[6] Penrose, R.: Iut. J. Theor, Phys. I (1968), 61. [7] -: fioistor theory, ifs aims and achievements, in Quantum Gravity, C. J. Isham, R. Penrose, and D. W. Sciama, editors; Oxford University Press, 1975. [8] -: Twisters and particles, in Quantum Theory and the Structure of Time and Space, L. Castell, M. Drieschner.
C.F. von Weizacker, editors; C. Hanser Verlag, 1975.
[9] -: Rep. Math. Phys. 12(1977). 65. [IO] Woodhouse, N. M. J. in Group 7heoretical Methods in Physics, Springer-Verlag, 1976. [l l] Sheppard, M. C.: Aspects of twistor particle theory, D. Phil. Thesis, Oxford 1979.
REPORTS
ON
MA THE M.4 TICAL
ON THE MAGNETIC
No. 1
PHYSICS
MOMENTS
OF HADRONS
L. P. HUGHSTON* and M. SHEPPARD** The Mathematical (Receiced
A new method for calculating based on twistor theory.
Institute, Oxford, England August 22. 1978)
the ratios of hadronic
magnetic moments
is presented,
I. Introduction The purpose of this article is to present some studies towards the understanding of hadronic magnetic moments. Our approach is novel inasmuch as we shall be working within the framework of twistor theory. In units of nuclear magnetons’ the magnetic moments of the proton and the neutron have been measured, respectively, to be pp = 2.7928456(11) and pu, = - 1.91304211(88). Note that the ratio of these numbers is approximately - 3/2. It has been known for some time now that using arguments of a group theoretical nature2 it is possible to derive the value - 3/2 for ,uJ,u,,. In what follows we shall show how these arguments can be carried over, in a relatively straightforward way, to twistor theory. Plan ofthe paper. This section will conclude with a brief review of notation and some background material. In Section II a center mass twistor is introduced for massive multi-twistor systems. It is a simple skew-symmetric twistor representing a point on the complex world-line of the system. We construct the corresponding quantum operator, using standard twistor quantization procedures, and discuss some of its properties. In Section III we introduce a hadronic magnetic moment operator, and we prove a result useful in the evaluation of its matrix elements. The center of mass operator plays a central role in the discussion. We finish in Section IV with a calculation of the protonneutron magnetic moment ratio.
of
*Fellow of Lincoln College, Oxford. **Junior Research Fellow, Wolfson College, Oxford, and S.E.R.C. Research Fellow. ‘One nuclear magneton = eh/2m,c = 3.1524515(53) x lo-‘* MeV gauss-‘. The value quoted here for p,, is given by Cohen and Taylor [2] and the value for p(. is given by Greene et ul. [3]. ‘See, for example, Beg et al. [l]. 1531
54
L. P. HUGHSTON
and M. SHEPPARD
Notation. The standard conventions for twistors and two-component spinors will be used here3. Furthermore, triplets of twistors will be denoted 27 withj = 1,2,3. The conjugate twistor operators -d/LIZ; will be denoted 2:. Note that we have the commutation relations
[Z$
z^;] = s;sy.
(1.1)
Cohomological considerations enter only minimally into the arguments which follow later in this paper, and thus we find it convenient to speak of twistor functions $(ZT) rather than elements of cohomology groups. As usual we put 24 = (of, rcAlj)for the spinor parts of Zy. We say that Z; is restricted to the (finite) space-time point xp ifof = rXA-%,?j. We shall write pXZy = for ZT thus restricted.
(iXAA’nA,j,X,4*j)
(1.2)
Similarly, we shall write
for the twistor function example, the identity
$(Z;)
restricted
crP*twy)
to the spacetime
point s’. We have, for
= PxL4’w;),
where VAA,= a/ax**‘, and where PAAp= TT ApjziJa is the
momentum
(1.4) operator
(with
fi6
= - a/&& Baryon states. Hadron states in general can be represented in terms of holomorphic functions of three twistors. These functions are required to be in suitable eigenstates of mass, spin, baryon number, electric charge, hypercharge, isospin, and the two SU(3) Casimir operators. Holomorphic twistor differential operator expressions can be formulated for these various observables. The connection between the twistor function description of a particle state and the conventional spinor wave-function representation of the state is established by means of a twistor contour integral formula 4. Incorporated in the contour integral formula is a certain combination of spinors, this spinor coefficient structure being uniquely determined by the state of the baryon: thus, the characteristics of the baryon state are coded, in a dual way, into the spinor coefficient structure (see Sheppard [ 11J).
‘See Penrose [S] and [7]. 4Twistor contour integral formulae of the general sort used here are discussed in, for example, Penrose [7], [8], Hughston [4] and Penrose [9].
MAGNETIC
MOMENTS
OF HADRONS
55
Let us denote the three ?r-spinors nA,j by uA,, dA., and sA,, respectively. The expressions for the spinor fields associated with the proton and the neutron can then be written explicitly as follows: Y;‘(x)
=
Kp
i
uA’uB’dgq&,(Z;)An,
(1.5)
. Y;‘(x)=
dA’dB’uB+q,bn(Z;)An,
K,
(1.6)
I where-$,(Zy) and $,,(Zy) are twistor functions appropriate to a proton state and a neutron state, respectively. Here Arc is the natural projective 5-form on the space of the n-spinors: An =
EA’P’EB’Q’EC’R’
d
x,.idn,,jdq
,d+,,dnp
L.d71R.,,.Ei.ikt:“‘.“,
(1.7)
where cijk is the totally skew-symmetric tensor; wedges have been suppressed. Note that p,$( 24) can be regarded as a function of xAA’ and ~~~~The An integration eliminates the 7cAsjdependence, leaving behind the spinor field as a function of xAA’ alone. The normalization factors ICY and K, are designed to ensure that if the twistor functions $( ZT) correspond to normalized states then the spinor fields YUA’(x) emerge automatically with the proper normalization. In the forthcoming discussion we shall be interested in the case for which our nucleons are in a definite spin state with respect to some axis, i.e., either with s, = l/2 (spin up) or with s, = - l/2 (spin down). In this connection it is useful to introduce a constant spinor dyad (OA!,Lo,) normalized so that oA, lA’ = 1. We say that the spinor field ‘PA’(x) is in an eigenstate of spin with s, = l/2 if for all xAA’ we have Yy,, tA’ = 0. Similarly, we have s, = - l/2 if YA,oA’ = 0 over all of space’. If $,(Zq) represents a proton in an s, = l/2 state, then the only relevant spinor coefficient structure6 for the contour integral formula is uOuOd’ - u’u’d’. Note, in particular, that for an s, = l/2 proton state the coefficient u’u’d’ - u’u’d’leads to a vanishing integral. An s, = l/2 proton state is, in fact, determined by the following two conditions: first, that u”uod’ and - uOu’dO are the only two coefficient structures that yield a non-trivial integral; second, that they yield the same value for the integral. For an s, = l/2 neutron the relevant coefficient structure is d”dou’ - d’d’u”. For a A + state with s, = l/2 the structure is 6 - I12(u’u’d’ + 2u0u’do) and for a A0 state with s, = l/2 the relevant structure is given by 6-1i2(d0d0~’ + 2d”d’ u’). 5The spinor dyad {o”., lA..} is related to the direction of the spin-axis zAA’by the fomula o (A, (at) (P”“) is the expectation value of the momentum operator for the state under consideration. We require that {o”,, tA,} be chosen such that PA’ is real. we write cl/duo 6Here we use the notation u” = 0”‘~~’ and similarly u1 = tA’uA,,etc., Furthermore, =.-- lAla/au,*, etc. = z$ where (P”‘*)
56 II.
L. P. HUGHSTON
and M. SHEPPARD
The center of mass twistor
In this section we shall consider some facts which are valid not only for systems of three twistors, but in fact for massive systems composed of any number (two or more) of twistor constituents. Associated with any set of twistors Z; is a skew-symmetric twistor Rza defined by .
R@
=2,,,-2
zfz$ J
,@ik
3
(2.1)
where h?jk is given by7 A.ik = zj Zk~olO aP.
(2.2)
Inasmuch as I@ is skew we can easily verify that RzP is indeed skew. Furthermore, the fact that the squared mass of the system Zg is given by yg = M,
where M, is the complex satisfies the normalization
conjugate condition
&jij,
using
(2.3)
of M’j, it is straightforward
to check that Rap
RzBI aB = 2 *
(2.4)
Next we note that RkP is simple, that is to say, it satisfies R[apR’ila = 0. We can verify the relations
(2.5)
(2.5) by noting that (2.1) can be rewritten R”b = 2m-2~
where Yi is defined according
A
Y@A 3
in the form (2.6)
to Y5 =
zy ,j,.
(2.7)
Equation (2.6) shows that RaP is the skew product of a pair of twistors of valence one, this being the necessary and sufficient condition that (2.5) is satisfied. Since RmB is a simple skew-symmetric twistor satisfying (2.4) it corresponds to a point in complex Minkowski space. This point lies on the complex center ofmass of the Zy system. The remaining points of the complex center of mass are obtained by ‘The infinity twistor I”# is defined by:
p
See Penrose [7] for further discussion.
=
.
MAGNETIC
MOMENTS
57
OF HADRONS
propagating away from the point determined by R”” in the direction of the total momentum of the system8. Although we will not make use of the fact here, it is perhaps worth noting for future reference that the condition that RzP should lie in the forward tube is that the vector WAA’
=
Yiy,,.,
be timelike and future-pointing. To see that this is the case, observe that by varying %A in the expression Y: LA we obtain the various points which lie on the line determined by R”@ in CP,. Thus, if R”fl is to correspond to a forward tube point then Yi AAmust lie in the top half of CP, for all values of IbA.In other words, we must have WAA.
l%A/iA’> 0
(2.9)
for all LA; this, however, is just the condition that W,J, be timelike and future-pointing. Now we apply the twistor quantization’ procedure zj, + Zj, to the twistor R”O. Concerning the operator I? thereby obtained, the following result can be stated: PROPOSITION
1. Ij’$(Z~)
is in a mass eigenstate
with mass eigenvalue
m, then we
have the identity P,@DIJ/(z;) where XaB is the simple skew-symmetric
(2.10)
= XaBp,$(q>, twistor representing
the Minkowski
space point
XAA’
Proof:
First of all, note that if X& represents second of its indices. then we have the relation y”
= xi,
the rc-part of XzB with respect to the
XPB’.
(2.11)
Furthermore, if all n of the twistors Zy are restricted to the point xAA’ then Zj” must be of the form X”‘nBIj for some nASj;-in other words, we have the relation px
The proof of (2.10) then proceeds
z; = X”B’7tB.j.
(2.12)
as follows:
*It is perhaps worth noting, as is discussed in Hughston [4], that the remaining points on the complex world-line of the system can be generated by applying the transformation Z; -+ Z; + I”@AjkZ>, where Ajt is a complex skew-symmetric tensor. It suffices, in fact, to choose Ajt = i Mjk, where 1 is a parameter. ‘References for twistor quantization include Penrose [6] - [9] and Woodhouse [lo].
58
L. P. HUGHSTON
p,P~Ic/(Zp)
and M. SHEPPARD
= 2m-2p,ZqZP LPk$(Zj”) = 2m-2XaA’XBB’pxnATjnBpk tijj”$(Zg) =m - 2 X$ x@Q,
&A’B’7rA~j7TB~k iwlj( z,;,
= mp2 X@p, M2$(Zg),
(2.13)
where M2 = M, h?j is the mass-squared operator. Using the fact that $(Zg) is in a mass eigenstate we see that (2.10) then follows immediately lo. n III. The magnetic moment operator
The hadronic magnetic moment operator k is defined by /i = oap Map + ~7”~M;I*p,
(3.1)
where we have put MS@= ,c;ZpiP)?Zj
Y
(3.2)
M$ = - p&jkZj’k,*(b.f;,.
(3.3)
and
In equation (3.3) the operator R,*, is the Hermitian conjugate of l?,
so that we have (3.4)
Note, incidentally, that Mzp is the Hermitian conjugate of MaB, and that the minus sign appearing in equation (3.3) is a consequence of the fact that a convenient arrangement of indices has been made in the expression. The constant p will be discussed later. The matrix 4 is the standard SU(3) electric charge matrix, defined by
(3.5) Observe that the charge matrix is tracefree and Hermitian. The constant twistor cols appearing in (3.1) is required to have the properties fsafi = C/JW a,flP’ = 0:
(3.6)
Thus, CS,@ is necessarily of the form (3.7)
where c”~’ IS ’ a symmetric spinor. The twistor a”@is the complex conjugate of crap. “The classical analogue of Proposition
1 is, incidentally,
also valid. We have: p,R”B = x”B.
59
MAGNETICMOMENTSOFHADRONS
The operator @is evidently a rather complicated object, and accordingly we require a fair amount of additional apparatus in order that we should have at our disposal an effective procedure for evaluating its matrix elements. We shall now proceed to establish a result that, as will be demonstrated in Section IV, is very useful in this connection. Let us define a pair of new operators r and Z by writing r = psj7rA,k 0::
(;@i
(3.8)
and f = P&iO$ a;l?i, where, as is customary,
(3.9)
we have put izj, = --alaw,!,
bA’j = -
apnA,j
for the spinor parts of the operator z^’“,. The operators magnetic moment operator i in the following way:
z and 7 are related
(3.10) to the
PROPOSITION 2.LetI)( 24) be a twistorfunction appropriate to a particular hadronic eigenstute; and let C denote the spinor coefficient structure appropriate, within a contour integral formula, to that state. Furthermore, let us put C= l7l7, where Il consists entirely of 7cAljcoefficients, and ficonsists entirely of ?r’Ajcoefficients. Then thefollowing formula is valid: (3.11) Proof: The reasoning involved here simplifies considerably in the event that Z involves nAPjcoefficients alone, i.e., in the case that Z= n Let us, then, consider that possibility first, and generalize later to the case of arbitrary 2: As it turns out, a somewhat stronger relation holds in the case Z = Nathan equation (3.11). Writing T= ozs MaB
(3.12)
we can deduce the following relation: p,Tti with no contour
= rp,$,
(3.13)
integral having been taken. Here is the proof of the validity of (3.13):
60
L. P. HUGHSTON
=
and M. SHEPPARD
/.4&j” “A’B’pxXA’kR~’ q,lr/
= p&l EjkQ,p Z$?)~ zi, $.
(3.17) (3.18)
Note that in the passage from (3.14) to (3.15) we have used the following useful lemma, due to G.A.J. Sparling: CV’p,$ = (iXA“Q$i
+ p,dYA’)II/.
(3.19)
Equation (3.19) can be established, without much difficulty, by using the differential chain rule. In going from (3.15)to (3.16)we have used the main content of Proposition 1, taken in the form p&,,‘$
= - ixB~cpX$,
(3.20)
where the spinor operator A,.’ is defined by (3.21) In going from (3.16) to (3.17) we again use Proposition 1, this time in the form PXLB~$
= &.4~WPxti.
(3.22)
It is perhaps worth noting that in (3.22) the pX operator is superfluous, the relation being valid even without its presence. The passage (3.17) to (3.18) is straightforward, a simple consequence of equation (3.7). Given the validity of (3.13), we can integrate so as to obtain (3.23) Equation (3.23) remains valid, moreover, if we include fi coefficients in the formula: l7p,fiT$An
=
$
i
L’rp,fi$An.
(3.24)
The proof of (3.24) proceeds in two steps. First we note that the slightly modified formula Ilp,Tfi$An
i
=
I
l7zp,&A1c
(3.25)
is certainly valid, since it can be obtained from (3.23) with the simple substitution ti + fi$. (We must note in making this substitution that if Ic/is in a mass eigenstate then so is fill/.)
MAGNETIC
MOMENTS
61
OF HADRONS
Thus, the validity of (3.24) hinges on showing that
np,[fi,T]l)A72 = 0.
(3.26)
To demonstrate (3.26) we observe that fiis an operator of a certain degree in GA’,say n. Since t+Gis in a definite quantum state, and the degree of fi is coordinated with II/, it follows that if [ fi, T] does not contain a term of degree II then (3.26) holds. A short calculation will show that this is indeed the case. So as to bypass some rather complicated formulae, we shall illustrate the matter explicitly only for the case n = 1, leaving the more general case to the reader. Now, from equation (3.16) we see that T= p$ ~~‘~‘(n~&.~$;
- n,.,&&.)
(3.27)
where ii/ Putting
= 2m- 27c,~,o~A~iiAs.
l? = 2; and using the commutation
(3.28)
relations
[Cl$, j2\] = 8;8;,
(3.29)
we then derive without much difficulty the following expression for the commutator which we are interested: [i?;, T] = -2mp2pEj
~A’B’7CA,k~ng,,~;;tAt;t$.
in
(3.30)
The term appearing in (3.30) is manifestly an expression of degree n = 3 in $2. It is true generally, in fact, that if fi is of degree II, then the commutator [ii, T] will be of degree n + 2. Thus (3.26) is valid, and consequently we have succeeded in proving (3.24). Having proven relation (3.24) we must now consider the complementary relation lIlp,fifiiAx Y! where ?is the Hermitian
conjugate
IIpxi%+bAx,
=
(3.31)
i of T:
T= O”fi M* 0’
(3.32)
Inasmuch as i = T+ T we see that (3.31) and (3.24) taken in conjunction clearly imply statement (3.11). As a helpful abbreviation let us define the integral operator Sz, by In, =
Adlp,fi.
(3.33)
I Using this notation, sz,@
the proof of (3.31) proceeds
= -Q&Ei,CWjii~~~Z^;)~
as follows: (3.34)
62
L. P. HUGHSTON
and M. SHEPPARD
(3.35)
(3.36) (3.37) (3.38) (3.39) In the derivation above, the passage from (3.34) to (3.35) is simply definition (3.4); and the subsequent expansion into spinor parts, shown in (3.36), is quite routine. We go from (3.36) to (3.37) making use of reasons similar to those involved in showing the validity of (3.26). One sees that if fi is of degree II, then the expression beneath the integral sign in the first term of (3.36) is of degree n + 2; accordingly we see that term must vanish when the integral is taken. In the second term of(3.36) we apply the identity integral sign in the first term of (3.36) is of degree II + 2; accordingly we see that that term must vanish when the integral is taken. In the second term of (3.36) we apply the 212A’r EA’;r ;tS zz M2E AB s A B
(3.40)
identity in order to pass to (3.37), and then use the fact that $ is in a mass eigenstate in order to obtain (3.38). With an adjustment of indices and an accompanying sign change, we apply definition (3.9) in order to obtain (3.39). n IV. The proton-neutron magnetic moment ratio The magnetic moment expectation value
associated
with a hadron
C~lciltil>
state rc/ is defined to be the
(4.1)
of the magnetic moment operator /.?. Unfortunately, we do not know yet how to calculate from theoretical principles the value of the constant p that appears in the definition of the operator I;. We can, however, take the ratios of expressions of the form (4.1) and these will be independent of the value of the constant .LL. We can, for example, look at the ratios <$,IArc/,>l(~“lA&A
(4.2)
where $, is a proton state in a definite state of s, = l/2, and where +,, is a neutron state in a definite state of s, = l/2. Now ,LI$,) is not a pure proton eigenstate. The operator b knocks $P out of its eigenstate into a mixed state. In fact, we have, as will be shown shortly,
MAGNETIC
&f$J
MOMENTS
= C+f$J)
OF HADRONS
63 (4.3)
+ /@ItiLl+ >
where1 t,Gd+ ) is a A + eigenstate, with s, = l/2, and where c(and p are certain numerical coefficients. Since I$, ) and I t,Gd+ ) correspond to distinct types of particles, they are orthogonal: (4.4) +iqh+ > = 0. As a consequence
of (4.4) we see that
ot&lL4~,> = PZ-
(4.5)
Similarly, we have
iw”>
= PYl$n) + P.lsl$B >
(4.6)
where 1I,$~o) is a A0 eigenstate with s, = l/2, where 1~and 6 are numerical coefficients. And thus we have
WlLiI$“)
= PY,
(4.7)
showing that the ratio pp/pL, is given by a/y. What remains to be shown is a method for calculating the coefficients a, 8, y, and 6. For this purpose we invoke Proposition 2. With the aid of Proposition 2 we can write px Ct,QAn
p, C/it,GAn = I
(4.8)
I
where c’ is a modified spinor coefficient structure which can be easily expressed, according to a procedure which will be described below, in terms of Z7, fi, z, and 7. The modified coefficient structure c’ can be uniquely written in the form c’=
(4.9)
a coefficient structure incompatible with $, and where 5 and q are factors. Since I”is incompatible with Ic/it follows from (4.8) that we
px C$An.
pX C@tjAn = 5 I Therefore
<$ represents
(4.10)
i
the part of @II/which lies in the direction
bIti> = (Iti> + 1lvo>
of $, i.e., we have (4.11)
where
($lvP) = 03 the value of q being determined
(up to phase) by the condition
(4.12) that ICJI) be normalized.
64
L. P. HUGHSTON
and M. SHEPPARD
Let us see how Z’ in (4.8) is determined. First of all notice that in (3.11) we can integrate by parts in the first term so as to transfer the action of r from p, Z?$ to 17. Let us denote -r IZ (the minus sign having come from the integration by parts) by the symbol l7’. Furthermore, notice that I7 being of degree n in 7?5 in (3.11) and t being of degree one means that the degree of fiiappears to be too large for that term to survive. If, however, n > 1, then one of the Zi operators in fican act on the IX; spinor in z”(see equation (3.9)) leaving behind -S:$. (If n > 1 this can happen in several different ways and a collection of terms is obtained.) The net effect is to leave behind a modified operator ft in the second term of (3.1 l), this operator being of degree n, as desired. Thus we have C’= nfi+
ni?P
(4.13)
in equation (4.8). Now we shall consider in detail the problem of the proton-neutron moment ratio. Since we have Z= l7 for the proton and the neutron, the problem somewhat. For a proton with s, = l/2 we have, recalling Section I,
magnetic simplifies
ZIP = uOuOdl - uOuldO. Now we wish to calculate UP, where fib = - r UP. An expansion the operator z gives (cf. equation (3.8)) the following formula: -P -17 = $(@
after putting
of
(4.15)
(quark occupation number) operators
u^” = uOa/&JO,
9O =
into components
- u^‘) - *@o - 5’) - $($” _ ii),
where we have defined the homogeneity
a0 =
(4.14)
Gi =
da/ad,
dOa/adO,31 = did/ad’, s”a/asO, 9 = slalasl,
(4.16) (4.17) (4.18)
OA.B.= oCA.I~,). We find, applying (4.15) to (4.14), that we have jC1’7b = $u”uodl
Some elementary
+ +u”uldo.
(4.19)
algebra lets us rewrite (4.19) in the form (4.20)
where Il, + is the coefficient structure appropriate = -!nA+ $
(u”uod’
for a A + state with s, = l/2, namely:
+ 2u”u’do).
(4.21)
MAGNETIC
MOMENTS
OF HADRONS
65
Inspection of (4.20) and comparison with (4.3) shows that c( = 1. Proceeding on to the neutron, we note from Section I that Lln = dOdOu’ - d”d’uo. Applying
(4.22)
- z we obtain F-‘l&
which can be rewritten
= -$dOdou’
_ $d”&O,
(4.23)
as (4.24)
where fl,o is the spinor coefficient structure appropriate n,o = -f-(d”douL Pj
to a if0 state with s, = l/2, i.e.:
+ 2d0d’zt0).
(4.25)
Inspection of (4.2) together with (4.6) shows that y = - 2/3. Now we can calculate the ratio a/y, and we observe the desired value - 312 for the ratio of the magnetic mo~zent qfthe proton to the magnetic moment qf the neutron. Remarks. The calculation above can be obviously modified so as to yield the ratios of the magnetic moments of other low-lying baryons. It can also be modified so as to enable one to calculate the electromagnetic transition rates for certain neutral mesons, and for this the full content of Proposition 2 is required. We shall discuss the case for neutral mesons elsewhere (see Sheppard
Lll)).
The authors would like to express thanks to B. D. Bramson, C. J. Isham, R. Penrose. G. A. J. Sparling, Z. Perjes, and A. S. Popovich for useful discussions.
REFERENCES [l] Beg, M., B. W. Lee, and A. Pais: Plzys. Rev. lefts 13(1964), 514. [2] Cohen. E. R., and B. H. Taylor: J. Phys. Chern. Rej: Data 2( 1973), 663. [3] Greene, G. L., N. F. Ramsey, W. Mampe. J. M. Pendlebury, K. Smith, W. D. Dress, P. D. Miller, and P. Perrin: Physics fefts 71B(1977), 297. [4] Hughston, L. P.; A particle ci~~~cu~ion scheme based on the theory qf twisters, D. Phil. Thesis, Oxford 1976. [S] Penrose, R.: 7’hestructure qfspacetime, in Battelle Rencontres, C. M. Dewitt and J. A. Wheeler, editors; Benjamin, 1968.
66
L. P. HUGHSTON
and M. SHEPPARD
[6] Penrose, R.: Iut. J. Theor, Phys. I (1968), 61. [7] -: fioistor theory, ifs aims and achievements, in Quantum Gravity, C. J. Isham, R. Penrose, and D. W. Sciama, editors; Oxford University Press, 1975. [8] -: Twisters and particles, in Quantum Theory and the Structure of Time and Space, L. Castell, M. Drieschner.
C.F. von Weizacker, editors; C. Hanser Verlag, 1975.
[9] -: Rep. Math. Phys. 12(1977). 65. [IO] Woodhouse, N. M. J. in Group 7heoretical Methods in Physics, Springer-Verlag, 1976. [l l] Sheppard, M. C.: Aspects of twistor particle theory, D. Phil. Thesis, Oxford 1979.