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Canonical formulation of the spherically symmetric einstein-Yang-Mills-Higgs system for a general gauge group
ANNALS
OF PHYSICS
108,
79-98 (1977)
Canonical Formulation Einstein-Yang-Mills-Higgs
of the Spherically Symmetric System for a General Gauge Group* P. CORDERO+
International
Centre for Theoretical Physics, Trieste, Italy Received October 15, 1976
The dynamics of the spherically symmetric system of gravitation interacting with scalar and Yang-Mills fields is presented in the context of the canonical formalism. The gauge group considered is a general (compact and semisimple) N parameter group. The scalar (Higgs) field transforms according to an unspecified M-dimensional orthogonal representation of the gauge group. The canonical formalism is based on Dirac’s techniques for dealing with constrained hamiltonian systems. First the condition that the scalar and Yang-Mills fields and their conjugate momenta be spherically symmetric up to a gauge is formulated and solved for global gauge transformations, finding, in a general gauge, the explicit angular dependence of the fields and conjugate momenta. It is shown that if the gauge group does not admit a subgroup (locally) isomorphic to the rotation group, then the dynamical variables can only be manifestly spherically symmetric. If the opposite is the case, then the number of allowed degrees of freedom is connected to the angular momentum content of the adjoint representation of the gauge group. Once the suitable variables with explicit angular dependence have been obtained, a reduced action is derived by integrating away the angular coordinates. The canonical formulation of the problem is now based on dynamical variables depending only on an arbitrary radial coordinate r and an arbitrary time coordinate 1. Besides the gravitational variables, the formalism now contains two pairs of N-vector variables, (R, xa), (8, xo), corresponding to the allowed Yang-Mills degrees of freedom and one pair of M-vector variables, @I, x,,), associated with the original scalar field. The reduced Hamiltonian is invariant under a group of r-dependent gauge transformations such that R plays the role of the gauge field (transforming in the typically inhomogeneous way) and in terms of which the gauge covariant derivatives of 0 and h naturally appear. No derivatives of R appear in the Hamiltonian and the gauge freedom allows us to define a gauge in which R is zero. Also the r and t coordinates are fixed in a way consistent with the equations of motion. Some nontrivial static solutions are found. One of these solutions is given in closed form; it is singular and corresponds to a generalization of the singular solution found in the literature with different degrees of generality and the geometry is described by the Reissner-Nordstrom metric. The other solution is defined through its asymptotic behavior. It generalizes to curved space the finite energy solution discussed by Julia and Zee in flat space.
* To be submitted for publication. t On leave of absence from Departamento Universidad de Chile, Santiago, Chile.
de Fisica, Facultad de Ciencias Fisicas y Matemtiticas,
79 Copyright All rights
Cl 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSlY
0003-49
I6
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P. CORDER0
1. INTRODUCTION At the classical level the existence of particle-like solutions found by ‘t Hooft [l] and Polyakov [2] has particularly enhanced the interest in classical non-Abelian gauge theories. Many generalizations of such solutions have subsequently been found both in flat and curved space-time [3-l I], most of which, when explicitly developed, refer only to the case when the gauge group is isomorphic to the rotation group W(3). In this paper we generalize results of a previous paper [8] considering the spherically symmetric Einstein-Yang-Mills-Higgs system associated with a general gauge group. We begin by studying a systematic way of reducing the variables of the problem to variables with no angular dependence and show how many degrees of freedom in the spherically symmetric problem are present in the formalism before any gauge or coordinate fixation is performed. In the main part of the paper we present a canonical formulation of the dynamics of the spherically symmetric problem of interacting scalar, Yang-Mills, and gravitational fields. The scalar field is considered to be a Higgs field in the sense that it provides mass to some of the gauge fields through the mechanism of spontaneous symmetry breaking. The analysis of the dynamics is an application of Dirac’s techniques for constrained Hamiltonian systems [12, 131 to the gauge invariantly coupled system of scalar, Yang-Mills, and gravitational fields to a Hamiltonian “reduced” by the requirement of spherical symmetry (reduced in the sense that the Hamiltonian generates only the motions consistent with spherical symmetry). In this reduced Hamiltonian formalism the dynamical variables also “reduce”; they depend only on an arbitrary coordinate r and an arbitrary time coordinate t. By means of Dirac’s canonical method we are able to present the whole dynamical problem in a clear and orderly way from a general point of view. There is a clear distinction between the constraints and the actual equations of motion, and the coordinate and gauge fixation is performed in the characteristic way developed by Dirac. The question of quantization is, of course, at the back of our mind when using the canonical formulation. The concept of spherically symmetric (scalar or Yang-Mills) fields used in this paper [8] refers to fields whose angular dependence is fixed by the condition that the effect of a rotation be compensated by a gauge transformation. The fields that we consider, then, are not necessarily manifestly spherically symmetric (MSS) but only spherically symmetric up to a gauge (SSUG). (Compare the present definition with that used in Refs. [14, 151.) In Section 2 we study the implications on the form of the scalar and Yang-Mills fields (and their conjugate momenta) when the condition of being spherically symmetric up to a gauge is imposed. The consequent equations are integrated to find the explicit and most general angular dependence of the dynamical variables. Our treatment, therefore, is different in still another way from the treatments of most of the papers quoted in that we derive the most general angular dependence of the
EINSTEIN-YANG-MILLS-HIGGS
SYSTEM
81
fields instead of feeding in as an ansatz a particular form for the solutions (the Wu-Yang ansatz [16] is used in the case when the gauge group is SO(3)) and without discussing its generality and the problem of gauge choice. also in Section 2 it is shown that it is necessary that the gauge group admit a subgroup jsomorphic to the rotation group in order that SSUG solutions can exist: otherwise the dynamical variables can only be MSS. It is not difficult to imagine a (nonrigorous) picture of why this has to be so: The effect of rotations can only be Furthermore, the number of degrees of freedom compensated by (gauge) “rotations.” is connected to the angular momentum content of the adjoint representation of the gauge group. In Section 3 the angular coordinates are integrated away in the expression of the action integral to set forth a reduced action principle. Each of the terms appearing in the action integrand (and the Hamiltonian) is written explicitly in terms of the reduced variables depending only on r and t. These variables are described as vectors to avoid using too many indices. The Yang-Mills degrees of freedom are described by the vectors R and 0 belonging to the N-dimensional space of the adjoint representation of the gauge group and by their conjugate momenta xR and K~. R is associated with the r-component of the Yang-Mills field, W, , and 0 is associated with WO . The reduced variables associated with the degreesof freedom allowed to the scalar field are the couple of conjugate variables (h, xh) belonging to a generic M-dimensional representation space corresponding to the orthogonal representation under which the scalar field transforms. Jn Section 4 the residual gauge freedom allowed by the reduced action is presented and it is observed that R behaves like the compensating field (gauge field), transforming in the typically inhomogeneous way, in terms of which gauge covariant derivatives of 0 and h naturally arise. No derivatives of the R field appear in the Hamiltonian, and the gauge group is shown to have as many r-dependent parameters as R has components, thus allowing the choice of a gauge in which R is zero. The remaining coordinates (r and t) are also fixed. From the equations of motion, the constraints and the coordinate and gauge fixations, we derive, in Section 5, equations for the remaining variables valid for the static case. Two nontrivial solutions of these equations are presented. The first is a generalization of the well-known singular solution found by many authors in flat or curved space-time. The geometry corresponds to the Reissner-Nordstrom metric. the Yang-Mills field has a form which is an N-dimensional generalization of the Wu-Yang ansatz and the Higgs field takes an essentially constant value (corresponding to a minimum of its self-potential) apart from a trivial angular dependencethat gives it topological properties asthe asymptotic form of the scalar field usedby ‘t Hooft [J J, The other solution is a generalization of the finite energy solution found by .JuJia and Zee [3] in flat space-time. Finally. there are three appendixes. Appendix A defines the basic notation. the indices, and dynamical variables and it also gives a sketchy presentation of the Hamiltonian formulation of the problem of interacting Yang-Mills and scalar fields with gravitation. In Appendix B we give some useful properties of the vectors of the adjoint representation space, the cross product defined between them and some
82
P. CORDER0
matrices acting on them. The equations of motion and constraints for the spherically symmetric case are given in Appendix C.
2. SPHERICAL SYMMETRY
Since we are going to study the spherically symmetric motions of the system we must first of all study the general form that the gravitational, Yang-Mills, and scalar fields must have in agreement with the space symmetry required, In the following we give separately the angular dependence allowed for the space metric gij , the space components of the Yang-Mills fields WA<(A = 1, 2,..., N; i = r, 0, v) and the scalar field @ as a consequence of the requirement of spherical symmetry. As will be seen in Appendix A there is no need to study the time components g,, and WA, since in the Hamiltonian method [12, 13, 171 they play the role of arbitrary functions, not the role of dynamical variables. (a) The most general spherically symmetric gii is known to have the form
gij
=
diag[+(r.t), $A(F,~),e2A(rst)sin2e].
(2.1)
where the functions t.~and X of Y and t are arbitrary functions that describe the allowed freedom of the gravitational field. The coordinates 8 and y are the standard polar angles, while Yis an arbitrary radial coordinate. The conjugate momenta&j is a contravariant density of rank two which has as its most general form &j =
&ag[$r&fi
8, $n,e-2A(sin e)-11. 7~~(r,t) will be the conjugate
sin 8, &7-,e-2Asin
(2.2)
As the notation already suggests, ~~(r, t) and momenta of p and h, respectively. (b) We now require that the gauge vector fields _WA = WAiai, (A = l,..., N; i = r, 8, 9) and their conjugate momenta TA = fl,& be spherically symmetric. _WAmust satisfy the requisite that the effect of any space rotation may be compensated by a gauge transformation; that is, _WA is required to be spherically symmetric up to a gauge. For practical reasons we restrict the application of this concept to global gauge transformations. The gauge group T must be compact if we want to guarantee the positive defi-* nitiveness of the Yang-Mills field kinetic energy term in the Hamiltonian. Further, for simplicity, we take T to be semisimple. Under these conditions we can always choose the metric of the group, in terms of the structure constants CABCof T, to be gAB
=
-&CD~cCc~~
=
8AB,
9
(2.3)
and therefore we need not worry about the height of the group indices. In the following we also disregard the possibility of having many different gauge coupling constants
EINSTEIN-YANG-MILLS-HIGGS
SYSTEM
83
as would happen if group T were expressible as a direct product of smaller groups. (All the simplifying restrictions made in this paragraph can be lifted without great difficulty.) The condition of spherical symmetry of the gauge fields and their momenta is obtained equating the effect of a small rotation (generated by L, ? a 1, 2, 3) with the effect of a small global gauge transformation Sd&
) ?y”] = c”,,sA”~‘?
(2.4a)
= -z;
(2.4b)
or
&vc.
and g,, ) g--(*iQq
= -CABCNfg-(“y,
(2.h)
= -z;
(2.5b)
cg-(l’I)Ir”.
The minus signs are chosen for convenience and Nz represents the ratio between the infinitesimal gauge parameters 6/lB of the gauge transformation needed to compensate the infinitesimal rotation around the a-axis and the angle of rotation, NE = -(SAB/Sa”).
(2.6)
If we were considering local gauge transformations then (2.4) would have an extra term (l/f) SLIA,j &jai , where f is the gauge self-coupling constant. The N iV matrices Z,, are defined in (2.4) as linear combinations of the matrices C, , (2.7)
(CBYC = CA,, that define the adjoint representation
of the gauge group algebra, Z, = N,BC, .
The infinitesimal
generators
of rotation
(2.X)
which we use are
L, = sin CJI(~/ZI) + cos p cot f!Xa/+),
(2.9a)
L1 = cos rp(i?/i%) - sin q cot &a!@,),
(2.9b)
& = q&p,
(2 9c)
and they satisfy
Ll 2Lb1= %x&r . It is straightforward that
to prove from the Jacobi identity
ET 3Zbl = %bC&
(2.10) satisfied by L, , L, . and _WA (2.11)
84
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CORDER0
which shows that the three matrices Z, close an SO(3) algebra. But since this algebra is a subalgebra of T, then either the gauge group admits a subgroup isomorphic to SO(3) or the 2, are identically zero. The latter would correspond to MSS gauge fields and we shall not consider it any further. If the 2, do not vanish, on the other hand, then it is possible to parametrize Tin accordance with SO(3) so that N, = 8;
and
N;‘=
0
for
B’ = 4,..., N.
(2.12)
To write Eqs. (2.4) explicitly it is convenient to use a vector notation for the WAi. Wi means an N-vector of components WAi. In this notation Eqs. (2.4) become wi,, = -z,wi, Wi,* = -(Z, wi,,
(2.13) cos ‘p + Z, sin rp) W’ - SzWa cot 9,
(2.14)
cos y - Z, sin 9) Wi + SgW@- 8: sin-” OW”] tg 8.
= -[(Z,
(2.15)
It is not difficult to integrate the system (2.13)-(2.15) after noticing that many relations such as e-z8mZ2eZ3a= Z, cos 9) + Z, sin y
(2.16)
hold (they all come from (2.11)). The most general solution of the above system of equatiohs is (written for the covariant Wi) Wi = ~Fz~“Czz”[R(r, where the N-vector-valued
t), 0(r, t) - N, , (-Z,0
- NJ sin 01,
(2.17a)
functions R, 0 are only restricted by
Z,R = 0
and
(Z&2 0 = -0.
(2.17b)
The presence of the vectors N, and N, (defined in (2.6)) and (2.17a) proves to be very convenient later on. A similar solution is found for xi, d=fe
-ZSme-Z~e[xR sin 8, &7cosin 19,-&Z3xo],
(2.18a)
with ZgcR = 0
and
(Z,)2 X@ = -7re.
(2.18b)
where 7~ = ZJr, t) and xg = x(r, t) are arbitrary functions of r and t that will play the role of conjugate momenta to R and 0. We want to know how many scalar (dynamical) variables enter into the SSUG field (2.17). In the standard angular momentum terminology R is an m = 0 vector while 0 is an arbitrary (real) linear combination of the (complex) eigenvectors of Z, corresponding to the eigenvalues fi. If the representation of SO(3) defined by
EINSTEIN-YANG-MILLS-HIGGS
SYSTEM
85
the Z, decomposes into p0 representations with I = 0, p representations with integer I # 0 and p’ representations with half integer 1then (ij Z, has p0 +p linearly independent eigenvectors Sli (k = l,..., p. i p) with eigenvalue zero, and (ii) 2,2 has 2p linearly independent eigenvectors with eigenvalues ( &i)” := ~~~1. These we choose to be real vectors X, and Y, such that z,x, Z,Y,
= YI, ) = -x, )
k = I,2 . ...) p,
Therefore the vectors R and 0 have as their most general expression R =
c
(2.19a)
ali(r, t) S, ,
k=l
(2.19b)
--W = f Mr, 9 & -- xdr, r>Yd
(2.19~)
h=l
We see that p0 + 3p arbitrary scalar real functions appear in the general solution of Eq. (2.4). The p’ half integer I representations obviously play no role in this respect. In the particular case of T = SO(3) since p. = 0, p -= 1, p’ =I=0 we are back to the case of three arbitrary functions that were called a, 4, and x in Ref. [8]. If T = W(3) then either p,, = 1, p = 1, p’ = 2 if the rotations are compensated by the group generated by A,, A,, A,, and therefore four functions will appear in the general solution or, if the subalgebra SO(3) of SU(3) is obtained from A, , A,, and A, then 17” = 0, p == 2, and p’ = 0 and there are six arbitrary functions in the general form of Wj (see Ref. 191). (c) Next we impose on the scalar field @ the condition of being spherically symmetric up to a global gauge transformation. Without loss of generality we assume that * is real and that it transforms according to an M-dimensional orthogonal representation of T (the Q representation for short). In analogy with (2.4) and (2.5) we demand that &$br = - TBrdNf@’
(IT A = 1, 2,..., M)
= -J:o@A,
(2.2&i) (2.20b)
and L u
-
r ?r@
=
-
E
-JL
(2.2la)
TBrANfg-(1/2)n;;j
&4/2)4
.
(2.2lb)
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P. CORDER0
The M x M matrices J, are linear combinations of the matrices TB representing the generators of the gauge group in the 0 representation and they must satisfy the angular momentum algebra
[J, , &I = +Jc .
(2.22)
The general solution of (2.21) and (2.22) is
ip = e-JWJ@h(u, t),
(2.23a)
where J&r,
t) = 0
(2.23b)
and 7rQ = e -JSqe-Jeex,(r, t) sin 8,
(2.24a)
&%(r, t> = 0,
(2.24b)
where
h(r, t) and xh(r, t) being M-vector-valued functions. The general form of h and xh is again (like R) a linear combination of the m = 0 linearly independent vectors existing in the representation of SO(3) induced by the Q representation of the gauge group. Equations (2.1) (2.17), (2.18), (2.23) and (2.24) are the main results of this section. To some extent they can be generalized to the case when the SO(3) symmetry requirement is replaced by a requirement of D-symmetry, D being one of the many nontrivial isometry groups that (curved) 3-space can admit. The dependence on the coordinates on each minimal invariant variety (~9and y in the case of SO(3)) can be made explicit and eliminated from the dynamical problem [18].
3. THE HAMILTONIAN OF THE SPHERICALLY SYMMETRIC MOTIONS
The analysis of the dynamics of the Einstein-Yang-Mills-Higgs system is greatly simplified by restricting the general Hamiltonian formalism summarized in Appendix A, to the fields with spherical symmetry found in Section 2. The generators X1, Zi, and G defined in Appendix A take the form (3.1)
(3.2a) (3.2b) (3.3)
EINSTEIN-YANG-MILLS-HIGGS
87
SYSTEM
The equations of motion can then be obtained from the extremization action
of the restricted
(3.4)
They are given in Appendix C. The variations are not all independent howeversince (2.17b), (2.18b), (2.23b), and (2.24b) must be preserved in time. To avoid introducing extra Lagrange multiplier terms (such as D . Z,R, K . 2,x, , etc.) we have to take care that the equations of motion leave the variables R, ?$ , 8, no, and h, xh in the subspace defined by the equations mentioned above. We can proceed in this way because the extra Lagrange multipliers turn out to be zero, or, in other words, one can check explicitly that Eqs. (C14)-(C19) are satisfied in the three static solutions found in Section 5. Equations (C14)-(C19) turn out to be conditions on the Lagrange multipliers N, N’, and W, the only nontrivial one being (3.7) below. In writing the reduced action we have assumed that the angular dependence in the integrand of the full action given by (A9) was a common factor sin 0 integrated away (J sin 8 & dp = 47~) and for this we have assumed that N and NT are functions of r and f only, while W, has been assumed to have the form W,
=
e-Z3”e‘eZ”8W(r,
t)
(3.5)
to cancel exactly the angular dependence in the last term of the jntegrand defining S. These assumptions represent no loss of generality as was discussed in Ref. [8]. Furthermore, the explicit form of 6 given below is such that z&
= 0,
(3.6)
and, to guarantee that Z,R = 0 is preserved in time we have to set z,w
= 0.
(3.7)
Next we give the derived explicit expression corresponding to each one of the reduced generators split into the different contributions coming from the gravitational, Yang-Mills, and scalar fields. (a) The functions $7 and 2,” are [19] $7 = p-yg4
_ ~rTT,rA + p$g”
_ 2~~~1 + 3x’2 - &-A)),
(b) To obtain sIM
and 2:”
(3.8) (3.9)
rt. is necessary to evaluate BRi first, the result
being .dBr
x
;e-z3’e-Z?e(-N3
_L
0
;,,
Z,e) sin 0,
(3.10a)
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,@ = f e-Z8me-Z2BZ,(8’ + 8 x R) sin 8,
(3. lob)
cc&?,” = ‘r e-z3’J’e-z2e(Qt + Q x R).
(3. IOC)
where L!.@is defined in Appendix A and the cross product between N vectors is defined in terms of the structure constants of the gauge group, (8 x R)A = CAB@RC
(3.11)
(see Appendix B). We notice that the expressions in round brackets in (3.1Ob), (3.10~) look like a gauge covariant derivative of 0, where the role of the compensating field is played by R. Inserting (2.17), (2.1 S), and (3.10) into (A3) and (A4), one finds
1 + 2f2@ u-2A(Q 2,‘”
x
Z,Q - N2)2 + 2e-“(0’
+ 0 x R)2},
(3.13)
= X@* (6’ + Q x RI,
G“YM = --rr,~R--(~ox~+Z3~O~Zge)-~;(.
In the derivations following: (i) (ii) (iii) 7c, ’ 8 = 0, (iv)
(3.12)
(3.14)
necessary to get the above results frequent use is made of the
The 2, matrices are antisymmetric and traceless, their exponentiation yields orthogonal matrices, vectors belonging to different eigenspaces of 2, are orthogonal, the results of Appendix
e.g.,
B.
(c) Finally, to obtain the functions SI,“b &! , and @ we need to know that the gauge covariant derivative Vi+ (defined in (A8)) of the scalar field is VT@ = ~+‘~e-~@[h - RATAh],
(3.15a)
V,@ = -e-J~pe-J2e@ATAh,
(3.15b)
V,O = e-Jaqe-J2e(Z3@)A T,h sin 6.
(3.15c)
Again in (3.15a) we can see what will turn out to be the gauge covariant derivative of h with R again appearing as the gauge field.
EINSTEIN-YANG-MILLS-HIGGS
89
SYSTEM
Jn the above use has been made of the identity te-
.%De-Z,e)BD TBrA(e-J3.3me-J’p*z _ (e-.‘3ae-.y-,
T,“,
I
The results of Section 2 together with (3.15) yield
.TP’p =
Xh.
(h’ - R”T,h),
(3.18)
Jn Eq. (3.19) boldface type has been used for the N x N matrix TA becausethat is the factor that carries the N vector index A, i.e.,
In Eqs. (‘3.15)~(3.18), on the other hand, boldface type has been used to indicate M vectors. It was said in (3.2) that H8 = 0. This is proved by first obtaining that se = nh . OATAh and then showing that the A4 vector OATAh belongs to the eigenspaceof (R,)” associated with the eigenvalue -1, while nIL satisfiesRp,, -- 0.
4. GAUGE FREEDOM AND GAUGE
FIXATION
Now that we have the explicit form of the terms that appear in the definition of the action (3.4) it is possible to check that the reduced action 3 is invariant under the gauge group defined by (4.Ja) (4.1 b) @cA
+
,-frAc~@$+$rAG,
(4. Ic)
?r;c,
-+
e-f'ACAT~CBef~c~,
(4.ld)
hde-fr4T.4h,
rc,, -+
e-fr.4TA,h,
(4.le)
(4.lf)
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P. CORDER0
where rA is an arbitrary r dependent N-vectorial function satisfying 2$(r)
= 0.
(4.2)
This local gauge group is characterized by p0 + p arbitrary functions TA(r), p. and p being the numbers defined in the discussion following Eq. (2.18) and it is obviously a subgroup of the original local gauge group. In Ref. [8], for example, where pa = 0 and p = 1 there appeared just one function (called y(r)) characterizing the residual gauge freedom. The gauge freedom we have then equals the number of independent components of R and it is natural to choose the gauge in which R is absent, i.e., being in a general gauge we can go to the gauge R = 0 by first solving the equation for I’ (coming from (4. la)), @-““C”)’
efi-Ac.oj
=
-e-frAC.qBCBefrACAe
(4.3)
The formal solution of this equation, defined up to a constant, can be found by iteration, the first terms of the series being (4.4) Given this r we can now transform to our gauge R=O
(gauge fixation).
(4.5)
A convenient coordinate fixation is eA = r
(space coordinate fixation),
(4.6)
n-@= 0
(time coordinate fixation).
(4.7)
These conditions must be preserved in time and they must fix the value of the Lagrange multipliers W, NV, N. In fact, to guarantee that (4.6) is valid for all time we require that h = 0 which implies, together with (4.7), that NT = 0 (see Eq. (C7)). Requiring that +P = 0 and that J? = 0 implies (see Eqs. (C9) and (Cl)), N[2e-“(1 - 2rp’) + 2e” -
&fe2rp2eu(Q
x Z,O
+f zr-zeun~ + if 2e--rrci _ N3)2+ f-2em“@‘2 + $r-2em~~~
f @2e-ub’2- r2euU - +e”[(@ . T/z)” + (Z,Q - T/z)~] - 4(Nre-p)’ = 0
(4.8)
and W’ -j- fNefir2nR
= 0.
WY
EINSTEIN-YANG-MILLS-HlGGS
91
SYSTEM
The last two equations can indeed be used together with the rest of the equations of motion to determine W and N. For a careful discussion of gauge transformations and gauge choice see Ref. [20].
5. STATIC SOLUTIONS The equations of motion for our variables p, 7~, ? h, v,, , R, xR , 0, x0 , h, xh are obtained without any difficulty by evaluating the Poisson brackets of every one of these variables with the total reduced Hamiltonians
They can be found in Appendix C. To these equations one must add the constraints flL = 0, Z& = 0, and e = 0, also given in Appendix C. We have already mentioned at the end of Section 4 how the equations x =~ 0. T, = 0, and tZ = 0 determine the functions IV’, N, and W. The equations of motion ensure that the constraints 3; = 0, =z$ = 0, and e = 0 are preserved in time and therefore that there are no new constraints in the formalism. The first two of these constraints themselves can be used to obtain p and 7~~in terms of the matter fields and it can be shown that the equations for 7jn and fi become identities. In the present section we look for static solutions. A solution is static if all gauge and coordinate transformation invariant quantities do not change in time. Tn our case it means that all quantities left after requiring that Eqs. (4.5)-(4.7) be satisfied, must be time independent. It is possible to reduce the problem to the following set of equations for 0, h, W, p, and N l
(Ne-I@) -- T - ‘fz
z,o x (0 x z&I - N3) + g
[eATAh - T/z - (Z@)A
T,h
w >: (0 ;c W)
. (Z,T)/r] = 0,
(5.2)
(Nr2e-41’)’ + Ne@[(8ATA)2 h + ((Zs@)A TA)2h - 2r2eU$
h] -.fWATAxn
:
0.
(5.3) r2W’ r Ne-u __ - (Q x W) x 0 - (Z,(0 c Ne” I
x W)) Y. 2.p
+ .f2r2WAT.J?
* Th = 0,
(5.4) ’ From the gauge and coordinate
conditions
and the equations of Appendix
C written for the
staticcasewe get R = 0, VA = 0, xW = 0, N’ = 0 E quation(Cl) is usedto eliminateTCR, Eq. (C2) is usedto eliminateXQ and (C3) to eliminate xh (see (5.7)). Now Eq. (5.2) follows from (C5); Eq. (5.3)from (C6); Eq. (5.4)from (C13);Eq. (5.5)from (Cl]) and Eq. (5.6) from (C9).
92
P.
2ty.i + eaer 1 C
(Q 2N2 x Y2
_ (Q x -W - %I2 _ (@*TA@’ + ((Z,Q)* 4f 9% 4
_ f ‘r2(wAi-,&)z -pu(hq e2p
1 +
(Q
X
Wz
r2U2
-fg+-,=
4N2
2rp’ -
CORDER0
_
(Q
X
f;Q&-
Ns)’
_
T,h)2
1,
(5.5)
(@*T~h)~+y@)*
TAhJ
2N2
L
+ f2rz(WATAh)z
-- r2U(h2)
4N2
2
+ r2W’2
1
W2
r2h’2
w-v-
4
2e”(re-uN)’ +
N
= , ‘(5.6)
where x, in (5.3) is given by xh = (fr2eu/N)
WATAh.
(5.7)
(i) The trivial solution to these equations corresponds to no fields except for the gravitational one, 8 = N2, h = 0, W = 0. The geometry is that of the Schwarzschild solution. (ii) The second simplest solution is 8 = 0,
h = h, = const,
xh = 0.
(5.8)
In fact, (5.3) shows that (au/ah’)
/h-h,
=
0,
i.e., ho is a field value which minimizes the symmetry breaking potential, (5.4) and the vanishing of xh it follows that r2W’/NeP
= -q
= const,
(5.9)
while from (5.10)
which implies W’ = -q
[’ (Ne”/p2) dp.
Inserting the last result in (5.7) it is seen that
q*T,/&f
= (qATAh,)r = 0,
(5.12)
showing that there is a connection between the orientation of the charge q in W space and the orientation of the “vacuum expectation value” of the scalar field h, . This orientation can be fixed making use of the surviving gauge freedom with constant gauge functions mentioned after (4.3). See, however, Ref. [20]. Next we substitute the previous results in Eq. (5.5), obtaining e-2u = 1 - (2m/r) + ((g” + q2)/4r2), where m is an integration
(5.13)
constant and g2 = I/f”.
(5.14)
EINSTEIN-YANG-MILLS-HIGGS
93
SYSTEM
Had we not assumed in Appendix A that U vanishes at its minimum would be an extra term proportional to r2 on the right-hand side of the well-known effect of the presence of a cosmological term in the makes the geometry not asymptotically flat). With or without such Eq. (5.6) gives N --_ e-u and henceforth the solution is (without the extra cosmological
= e-z3”e-z2*
(5.15)
term)
9 == e-J3me-Jq,, , w,
(h = &) there (5.13). This is theory (which an extra term
(5.16a)
[!! )
r
{js” -_. ~- 1 _ z!!!. + i r
g2 + q2
7 r2(dP + sin2 0 dv2),
(5.16~)
where q and h, are constrained to satisfy (5.12). This spherically symmetric singular solution does not have surviving massive components of the gauge field W, . In fact. it exists even without introducing a scalar field at all [8]. (iii) The finite energy solutions which include massive gauge fields cannot be written explicitly even in the simplest cases, such as those analyzed for example, by ‘t Hooft [1] and by Julia and Zee [3] in flat space-time. It is not difficult to check. however, that the following asymptotic forms are consistent with Eqs. (5.2)-(5.4): (a) As r -+ 0 0 ---f (c-r2 + 1) N, ,
(5.17a) (5.17b)
W + Car) N3, h - @I W ,
(5.17c)
where M, is an I = 1, m = 0 vector in the M-dimensional representation space. i.e., (J1” + Jzz + Ja2) M, = -2M, and J3Ma = 0, while a, b, and c are constant. (b) As r -+ co h --t h, (minimum of U(P)), (5.18:~) W-M (constant), (5.18b) with MAT,,h, == 0 and Q + (epRr) 8,) (5.18~) where .Q* is a matrix of the form %B
= f’~rP~i-
+ QA~QB~ -
S,,SBc
>
(5.19)
PAr = j-TArrhf ,
(5.20a)
QAP G f
(5.20b)
Tm-,rh~. SAC5%2"*f&CMB. (z&AD
(5.20~)
94
P. CORDER0
The vectors II, , M, 8, are constants which must be such that (5.18~) tends to zero to have finite energy.
APPENDIX
A: NOTATION AND GENERAL HAMILTONIAN FORMULATION
We use the indices relativistic indices, space polar indices, space Cartesian indices, indices of the adjoint representation of the gauge group, indices of an arbitrary real and orthogonal representation of the gauge group.
I*, v = 0, r, 8, v i, j = r, 9, fp a,b = 1,2,3 A, B, C ,... = 1, 2,. .., N
r, A, 2 = 1, 2)...) A4
We deal with the Hamiltonian formulation of the gauge invariant interacting system of a real scalar field @jr, a Yang-Mills field WA,“, and the gravitational field described by the metric tensor g,, . The canonical variables are the spatial components of the metric gij , the space components of the gauge potentials WAi, and the scalar fields Qr together with their respective conjugate momenta rTii, rrAi, and 7rr. The Hamiltonian of the system is [13, 17,211 H = s d3x(N.Xl
+ NiZi
+
f WA,G.J,
where the arbitrary functions N and Ni are the combinations the space-time line element in the form ds2 = -N2
dt2 + gii(Ni dt + dY)(Ni
(AlI
of the g,, that define
dt + dxj).
(A9
The WA0 is the arbitrary time component of the gauge field. The generators of normal and tangential deformations (2’ and Si) and of gauge transformations GA are
=
g-1/2(TijTij
+ ~[g-l12,02
-
iT2)
-
g1/2~
+
$g-li2gii(&inAj
+ gl12gijVi@ . V,@ + 2grRJ(*)]
+
w 0,
&&4iWAj)
(A3)
EINSTEIN-YANG-MILLS-HIGGS
95
SYSTEM
all of which vanish weakly [12, 131.The vector density ~47~~ is gAi
=
1 ijkFA, 2E
3k 3
(Ah)
= &cijk(WA,j - WAj*k - fCASCWBiWC~),
gij is the inverse of g,, . The scalar self-coupling U(@) is not specified but to be in accordance with our denomination of Higgs field, it should be thought as being
u(a) = b(@ - ay,
(A74
which is zero at its minimum to avoid introducing an undesired cosmological term. Another possible choice for the potential could be the popular u(a) = b(l - cos j (P2y).
(A%)
The gauge derivative of @ is given by Vi@ = @r,i - f WBiTBr,&DA,
(A@
where the matrices TB were already introduced in (2.20). The action is given by
It is this action that we consider in Section 3 to obtain the reduced action 3. Quantities with one group index A (or one index r) will sometimes be called N vectors (or M vectors) and will be indicated by boldface type, with the corresponding index omitted; e.g., instead of N;;” we write N, (or instead of hr we write h). Both indices “A” and “P can indifferently be written up or down.
APPENDIX
B: VECTORS AND MATRICES
OF THE ADJOINT
REPRESENTATION
In Section 2 we introduced the N-dimensional N, of components N: and the N :f N antisymmetric matrices (-&>Ac
=
N%Anc
.
@I)
(We remind the reader that the group indices are raised and lowered with the metric (2.3J.J Defining the antisymmetric cross product of any two N vectors by (U x V)A = CAB&JBV~
032)
it is immediate that ZJ-J = N, x U.
(B3)
96
P.
CORDER0
The last result together with (2.11) yields Nz x Nb = GA%,
(B4)
which, in turn, implies that the three vectors N, have the same norm, (Nl)2 = (N2)2 = (NJ2.
(B5)
The square of the N, vectors must be understood as C, N;;4N,A . Two other properties which are useful are (uxv)xw+(wxu)xv+(Yxw)xu=o
0%)
and Z,(U
APPENDIX
x V) = (Z,U)
x v + u x (Z,V).
C: THE EQUATIONS IN THE SPHERICALLY
OF MOTION SYMMETRIC
(B7)
AND CONSTRAINTS CASE
The variation of the reduced action ,!? defined by Eq. (3.4) together with (3.8), (3.9) (3.12)-(3.14), (3.17)-(3.19) yields equations of motion for the canonical variables 8, xg , h, rch , which we write in the form they take after considering EL>=u3h,%,R,XR, the coordinate and gauge choices (4.5), (4.6) and (4.7)
lk = f 2Nr-2eunR+ f W’, 6 = if 2Ne-une + NW - f0
(Cl) x
W,
h = Nr2e+nls + N?(h’ - RATAh) -f
. = nR
-2f
w WATAh,
-2Ne-u0’ x 0 + Ne-ur2h’ * Th - Nrz,
(C3) x
0
+ Nrrfi *Th +fW X n,, . IQ = 2f-2Nr-2eu( 8 x Z,O - N2) x Z,S + f-2(Ne--rr0’)’ - Neu[OATAh . Th - (Z@)” T,h * (Z,T) h] + (N’x,)’
. xh = (Ne-V2h’)’
(C4) +
f W x ~0, (C5)
+ Neu[(O . T)2 h + (Z,O . T)2 h]
- 2NeYr2 g
h + (Nrxh)’ -f
WATAnh ,
VW
x = NV,
(C7)
p = -tNr-2 e- %r,, + N*p’ + (Nr)‘,
(C8)
* = N[2e-“(1 =&4
- 2rp’) + 2e” - &f 2r-2eYrri + (f2/4) e+nE
- &ff-2r-2e+u(Q
x Z,Q -
bQ2 + fp2e-“Qr2
+ $r-2ee-u&
+ +r2e-ph’2- r2euU - $eu[(@* Th)2 + (Z,O . Th)2]] - 4(Nre-u)‘,
(C9)
EINSTEIN-YANG-MILLS-HIGGS
+,, = N[--4e-“(1
97
SYSTEM
- 2-p’) +f 2r-2eurr~ + ~~-2r-2e“(0 x 2,8
- N3)2
f. r-2e-@T; - r2e-“h’” f 2r2e”U]
(c-10)
f J[(Nre-l*)’ - (N’r2e-u)‘]. The constraints (3. I). (3.2), and (3.3), respectively, become 2[e-“(1 - 2rp’) - eU]f -+ .f-“[$r-“e”(e
x
hfV2[eu& Z,8
-
N,)2
J- $r2em uni] +
e-“@‘“I
+
$r-‘em
‘ST:
+ $-2e-W2 + &e”[(O - Th)z -t (Z,O . T/7)“] 4 ellr’l! = 0,
(Cl I)
x0 . 0’ + xh * h’ I r-lr,, -= 0,
(C12)
x 0 + Zgc, ‘k: Z,e) + TT,,. Th =: 0. %r J- 4(x, -
(Cl3)
Furthermore we have to require that the constraints (‘2.17b). (2.18b). (2.23b). and (2.24b) be preserved in time, i.e.. z,ft
= 0,
(Cl1)
Z,icR = 0,
lCl5)
z;o =z --0.
(C16)
Z.&J =- - ic, .
1(‘17)
J,h :: 0,
I(‘1 8)
J3+ =z 0.
((“19)
ACKNOWLEDGMENTS I am indebted to Demetrios Christodoulou and Claudio Teitelboim cussions and for much encouragement. I would also like to thank Rafael for their constructive criticism. I also wish to express my gratitude to International Atomic Energy Agency and UNESCO for hospitality at Theoretical Physics. Trieste.
for many enlightening disBenguria and Sergio Hojman Professor Abdus Salam, the the International Centre for
REFERENCES 1. G. 2. A.
‘T HOOFT, Nucl. Phys. B 79 (1974), 276. M. POLYAKOV. JETP Lett. 20 (1974), 194.
3. B. JULIA AND A. ZEE, Phys. Rev. D 11 (1975), 2227. 4. F, A. BAID AND R. 3. RUSSELL, Pfzy~. Rev. D 11 (1975),
2692.
CHO AND P. G. 0. FREUND, Phys. Rev. D 12 (1975), 6. P. B. YASSKIN, Phys. Rev. D 12 (1975), 2212. 7. P. VAN NIEUWENHUIZER, D. WILKINSON, AND M. J. PERRY, 5. Y.
8.
M.
P. CORDERO
AND
C. TEITELBOIM,
Amt.
Physics,
100 (I 976),
1588. Phys. 607.
Rev.
D 13 (1976).
778.
98 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
P. CORDER0
E. C~RRIGAN, D. I. OLIVE, D. B. FAIRLIE, AND J. NUYTS, Nucl. Phys. B 106 (1976), 475. W. J. MARCIANO AND H. PAGELS, P&s. Rev. D 12 (1975), 1093. F. A. Bars AND J. R. PRIMAK, P&w. Rev. D 13 (1976), 819. P. A. M. DIRAC, Canad. J. Math. 3 (1951), 1; “Lectures on Quantum Mechanics,” Belfer Graduate School of Science, Yeshiva University, New York, 1964. A. J. HANSON, T. REGGE, AM) C. TEITELBOIM, “Constrained Hamiltonian Systems,” Accademia Nazionale dei Lincei, Rome, 1975. M. IKEDA AND Y. MIYACHI, Progr. Theoret. Phys. (Kyoto) 27 (1962), 474. H. G. Loos, Nucl. Phys. 72 (1965), 677. T. T. Wu AND C. N. YANG, in “Properties of Matter under Unusual Conditions” (Mark and Fernbach, Eds.), Interscience, New York, 1969. R. ARNOWITT, S. DESER, AND C. W. MISNER, in “Gravitation: An Introduction to Current Research” (L. Witten, Ed.), Wiley, New York, 1962. D. CHRISTODOULOUAND P. CORDERO, in preparation. B. K. BERGER, D. M. CHITRE, V. E. MONCFUEF, AND Y. NUTKU, Phys. Rev. D 5 (1972), 2467. R. BENGLJRIA, P. CORDERO, AND C. TEITELBOIM, Nucl. Phys. B 122 (1977), 61. C. TEITELBOIM, “Lectures on Generalized Hamiltonian Dynamics,” lecture notes, Princeton University, Spring 1976.
OF PHYSICS
108,
79-98 (1977)
Canonical Formulation Einstein-Yang-Mills-Higgs
of the Spherically Symmetric System for a General Gauge Group* P. CORDERO+
International
Centre for Theoretical Physics, Trieste, Italy Received October 15, 1976
The dynamics of the spherically symmetric system of gravitation interacting with scalar and Yang-Mills fields is presented in the context of the canonical formalism. The gauge group considered is a general (compact and semisimple) N parameter group. The scalar (Higgs) field transforms according to an unspecified M-dimensional orthogonal representation of the gauge group. The canonical formalism is based on Dirac’s techniques for dealing with constrained hamiltonian systems. First the condition that the scalar and Yang-Mills fields and their conjugate momenta be spherically symmetric up to a gauge is formulated and solved for global gauge transformations, finding, in a general gauge, the explicit angular dependence of the fields and conjugate momenta. It is shown that if the gauge group does not admit a subgroup (locally) isomorphic to the rotation group, then the dynamical variables can only be manifestly spherically symmetric. If the opposite is the case, then the number of allowed degrees of freedom is connected to the angular momentum content of the adjoint representation of the gauge group. Once the suitable variables with explicit angular dependence have been obtained, a reduced action is derived by integrating away the angular coordinates. The canonical formulation of the problem is now based on dynamical variables depending only on an arbitrary radial coordinate r and an arbitrary time coordinate 1. Besides the gravitational variables, the formalism now contains two pairs of N-vector variables, (R, xa), (8, xo), corresponding to the allowed Yang-Mills degrees of freedom and one pair of M-vector variables, @I, x,,), associated with the original scalar field. The reduced Hamiltonian is invariant under a group of r-dependent gauge transformations such that R plays the role of the gauge field (transforming in the typically inhomogeneous way) and in terms of which the gauge covariant derivatives of 0 and h naturally appear. No derivatives of R appear in the Hamiltonian and the gauge freedom allows us to define a gauge in which R is zero. Also the r and t coordinates are fixed in a way consistent with the equations of motion. Some nontrivial static solutions are found. One of these solutions is given in closed form; it is singular and corresponds to a generalization of the singular solution found in the literature with different degrees of generality and the geometry is described by the Reissner-Nordstrom metric. The other solution is defined through its asymptotic behavior. It generalizes to curved space the finite energy solution discussed by Julia and Zee in flat space.
* To be submitted for publication. t On leave of absence from Departamento Universidad de Chile, Santiago, Chile.
de Fisica, Facultad de Ciencias Fisicas y Matemtiticas,
79 Copyright All rights
Cl 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSlY
0003-49
I6
80
P. CORDER0
1. INTRODUCTION At the classical level the existence of particle-like solutions found by ‘t Hooft [l] and Polyakov [2] has particularly enhanced the interest in classical non-Abelian gauge theories. Many generalizations of such solutions have subsequently been found both in flat and curved space-time [3-l I], most of which, when explicitly developed, refer only to the case when the gauge group is isomorphic to the rotation group W(3). In this paper we generalize results of a previous paper [8] considering the spherically symmetric Einstein-Yang-Mills-Higgs system associated with a general gauge group. We begin by studying a systematic way of reducing the variables of the problem to variables with no angular dependence and show how many degrees of freedom in the spherically symmetric problem are present in the formalism before any gauge or coordinate fixation is performed. In the main part of the paper we present a canonical formulation of the dynamics of the spherically symmetric problem of interacting scalar, Yang-Mills, and gravitational fields. The scalar field is considered to be a Higgs field in the sense that it provides mass to some of the gauge fields through the mechanism of spontaneous symmetry breaking. The analysis of the dynamics is an application of Dirac’s techniques for constrained Hamiltonian systems [12, 131 to the gauge invariantly coupled system of scalar, Yang-Mills, and gravitational fields to a Hamiltonian “reduced” by the requirement of spherical symmetry (reduced in the sense that the Hamiltonian generates only the motions consistent with spherical symmetry). In this reduced Hamiltonian formalism the dynamical variables also “reduce”; they depend only on an arbitrary coordinate r and an arbitrary time coordinate t. By means of Dirac’s canonical method we are able to present the whole dynamical problem in a clear and orderly way from a general point of view. There is a clear distinction between the constraints and the actual equations of motion, and the coordinate and gauge fixation is performed in the characteristic way developed by Dirac. The question of quantization is, of course, at the back of our mind when using the canonical formulation. The concept of spherically symmetric (scalar or Yang-Mills) fields used in this paper [8] refers to fields whose angular dependence is fixed by the condition that the effect of a rotation be compensated by a gauge transformation. The fields that we consider, then, are not necessarily manifestly spherically symmetric (MSS) but only spherically symmetric up to a gauge (SSUG). (Compare the present definition with that used in Refs. [14, 151.) In Section 2 we study the implications on the form of the scalar and Yang-Mills fields (and their conjugate momenta) when the condition of being spherically symmetric up to a gauge is imposed. The consequent equations are integrated to find the explicit and most general angular dependence of the dynamical variables. Our treatment, therefore, is different in still another way from the treatments of most of the papers quoted in that we derive the most general angular dependence of the
EINSTEIN-YANG-MILLS-HIGGS
SYSTEM
81
fields instead of feeding in as an ansatz a particular form for the solutions (the Wu-Yang ansatz [16] is used in the case when the gauge group is SO(3)) and without discussing its generality and the problem of gauge choice. also in Section 2 it is shown that it is necessary that the gauge group admit a subgroup jsomorphic to the rotation group in order that SSUG solutions can exist: otherwise the dynamical variables can only be MSS. It is not difficult to imagine a (nonrigorous) picture of why this has to be so: The effect of rotations can only be Furthermore, the number of degrees of freedom compensated by (gauge) “rotations.” is connected to the angular momentum content of the adjoint representation of the gauge group. In Section 3 the angular coordinates are integrated away in the expression of the action integral to set forth a reduced action principle. Each of the terms appearing in the action integrand (and the Hamiltonian) is written explicitly in terms of the reduced variables depending only on r and t. These variables are described as vectors to avoid using too many indices. The Yang-Mills degrees of freedom are described by the vectors R and 0 belonging to the N-dimensional space of the adjoint representation of the gauge group and by their conjugate momenta xR and K~. R is associated with the r-component of the Yang-Mills field, W, , and 0 is associated with WO . The reduced variables associated with the degreesof freedom allowed to the scalar field are the couple of conjugate variables (h, xh) belonging to a generic M-dimensional representation space corresponding to the orthogonal representation under which the scalar field transforms. Jn Section 4 the residual gauge freedom allowed by the reduced action is presented and it is observed that R behaves like the compensating field (gauge field), transforming in the typically inhomogeneous way, in terms of which gauge covariant derivatives of 0 and h naturally arise. No derivatives of the R field appear in the Hamiltonian, and the gauge group is shown to have as many r-dependent parameters as R has components, thus allowing the choice of a gauge in which R is zero. The remaining coordinates (r and t) are also fixed. From the equations of motion, the constraints and the coordinate and gauge fixations, we derive, in Section 5, equations for the remaining variables valid for the static case. Two nontrivial solutions of these equations are presented. The first is a generalization of the well-known singular solution found by many authors in flat or curved space-time. The geometry corresponds to the Reissner-Nordstrom metric. the Yang-Mills field has a form which is an N-dimensional generalization of the Wu-Yang ansatz and the Higgs field takes an essentially constant value (corresponding to a minimum of its self-potential) apart from a trivial angular dependencethat gives it topological properties asthe asymptotic form of the scalar field usedby ‘t Hooft [J J, The other solution is a generalization of the finite energy solution found by .JuJia and Zee [3] in flat space-time. Finally. there are three appendixes. Appendix A defines the basic notation. the indices, and dynamical variables and it also gives a sketchy presentation of the Hamiltonian formulation of the problem of interacting Yang-Mills and scalar fields with gravitation. In Appendix B we give some useful properties of the vectors of the adjoint representation space, the cross product defined between them and some
82
P. CORDER0
matrices acting on them. The equations of motion and constraints for the spherically symmetric case are given in Appendix C.
2. SPHERICAL SYMMETRY
Since we are going to study the spherically symmetric motions of the system we must first of all study the general form that the gravitational, Yang-Mills, and scalar fields must have in agreement with the space symmetry required, In the following we give separately the angular dependence allowed for the space metric gij , the space components of the Yang-Mills fields WA<(A = 1, 2,..., N; i = r, 0, v) and the scalar field @ as a consequence of the requirement of spherical symmetry. As will be seen in Appendix A there is no need to study the time components g,, and WA, since in the Hamiltonian method [12, 13, 171 they play the role of arbitrary functions, not the role of dynamical variables. (a) The most general spherically symmetric gii is known to have the form
gij
=
diag[+(r.t), $A(F,~),e2A(rst)sin2e].
(2.1)
where the functions t.~and X of Y and t are arbitrary functions that describe the allowed freedom of the gravitational field. The coordinates 8 and y are the standard polar angles, while Yis an arbitrary radial coordinate. The conjugate momenta&j is a contravariant density of rank two which has as its most general form &j =
&ag[$r&fi
8, $n,e-2A(sin e)-11. 7~~(r,t) will be the conjugate
sin 8, &7-,e-2Asin
(2.2)
As the notation already suggests, ~~(r, t) and momenta of p and h, respectively. (b) We now require that the gauge vector fields _WA = WAiai, (A = l,..., N; i = r, 8, 9) and their conjugate momenta TA = fl,& be spherically symmetric. _WAmust satisfy the requisite that the effect of any space rotation may be compensated by a gauge transformation; that is, _WA is required to be spherically symmetric up to a gauge. For practical reasons we restrict the application of this concept to global gauge transformations. The gauge group T must be compact if we want to guarantee the positive defi-* nitiveness of the Yang-Mills field kinetic energy term in the Hamiltonian. Further, for simplicity, we take T to be semisimple. Under these conditions we can always choose the metric of the group, in terms of the structure constants CABCof T, to be gAB
=
-&CD~cCc~~
=
8AB,
9
(2.3)
and therefore we need not worry about the height of the group indices. In the following we also disregard the possibility of having many different gauge coupling constants
EINSTEIN-YANG-MILLS-HIGGS
SYSTEM
83
as would happen if group T were expressible as a direct product of smaller groups. (All the simplifying restrictions made in this paragraph can be lifted without great difficulty.) The condition of spherical symmetry of the gauge fields and their momenta is obtained equating the effect of a small rotation (generated by L, ? a 1, 2, 3) with the effect of a small global gauge transformation Sd&
) ?y”] = c”,,sA”~‘?
(2.4a)
= -z;
(2.4b)
or
&vc.
and g,, ) g--(*iQq
= -CABCNfg-(“y,
(2.h)
= -z;
(2.5b)
cg-(l’I)Ir”.
The minus signs are chosen for convenience and Nz represents the ratio between the infinitesimal gauge parameters 6/lB of the gauge transformation needed to compensate the infinitesimal rotation around the a-axis and the angle of rotation, NE = -(SAB/Sa”).
(2.6)
If we were considering local gauge transformations then (2.4) would have an extra term (l/f) SLIA,j &jai , where f is the gauge self-coupling constant. The N iV matrices Z,, are defined in (2.4) as linear combinations of the matrices C, , (2.7)
(CBYC = CA,, that define the adjoint representation
of the gauge group algebra, Z, = N,BC, .
The infinitesimal
generators
of rotation
(2.X)
which we use are
L, = sin CJI(~/ZI) + cos p cot f!Xa/+),
(2.9a)
L1 = cos rp(i?/i%) - sin q cot &a!@,),
(2.9b)
& = q&p,
(2 9c)
and they satisfy
Ll 2Lb1= %x&r . It is straightforward that
to prove from the Jacobi identity
ET 3Zbl = %bC&
(2.10) satisfied by L, , L, . and _WA (2.11)
84
P.
CORDER0
which shows that the three matrices Z, close an SO(3) algebra. But since this algebra is a subalgebra of T, then either the gauge group admits a subgroup isomorphic to SO(3) or the 2, are identically zero. The latter would correspond to MSS gauge fields and we shall not consider it any further. If the 2, do not vanish, on the other hand, then it is possible to parametrize Tin accordance with SO(3) so that N, = 8;
and
N;‘=
0
for
B’ = 4,..., N.
(2.12)
To write Eqs. (2.4) explicitly it is convenient to use a vector notation for the WAi. Wi means an N-vector of components WAi. In this notation Eqs. (2.4) become wi,, = -z,wi, Wi,* = -(Z, wi,,
(2.13) cos ‘p + Z, sin rp) W’ - SzWa cot 9,
(2.14)
cos y - Z, sin 9) Wi + SgW@- 8: sin-” OW”] tg 8.
= -[(Z,
(2.15)
It is not difficult to integrate the system (2.13)-(2.15) after noticing that many relations such as e-z8mZ2eZ3a= Z, cos 9) + Z, sin y
(2.16)
hold (they all come from (2.11)). The most general solution of the above system of equatiohs is (written for the covariant Wi) Wi = ~Fz~“Czz”[R(r, where the N-vector-valued
t), 0(r, t) - N, , (-Z,0
- NJ sin 01,
(2.17a)
functions R, 0 are only restricted by
Z,R = 0
and
(Z&2 0 = -0.
(2.17b)
The presence of the vectors N, and N, (defined in (2.6)) and (2.17a) proves to be very convenient later on. A similar solution is found for xi, d=fe
-ZSme-Z~e[xR sin 8, &7cosin 19,-&Z3xo],
(2.18a)
with ZgcR = 0
and
(Z,)2 X@ = -7re.
(2.18b)
where 7~ = ZJr, t) and xg = x(r, t) are arbitrary functions of r and t that will play the role of conjugate momenta to R and 0. We want to know how many scalar (dynamical) variables enter into the SSUG field (2.17). In the standard angular momentum terminology R is an m = 0 vector while 0 is an arbitrary (real) linear combination of the (complex) eigenvectors of Z, corresponding to the eigenvalues fi. If the representation of SO(3) defined by
EINSTEIN-YANG-MILLS-HIGGS
SYSTEM
85
the Z, decomposes into p0 representations with I = 0, p representations with integer I # 0 and p’ representations with half integer 1then (ij Z, has p0 +p linearly independent eigenvectors Sli (k = l,..., p. i p) with eigenvalue zero, and (ii) 2,2 has 2p linearly independent eigenvectors with eigenvalues ( &i)” := ~~~1. These we choose to be real vectors X, and Y, such that z,x, Z,Y,
= YI, ) = -x, )
k = I,2 . ...) p,
Therefore the vectors R and 0 have as their most general expression R =
c
(2.19a)
ali(r, t) S, ,
k=l
(2.19b)
--W = f Mr, 9 & -- xdr, r>Yd
(2.19~)
h=l
We see that p0 + 3p arbitrary scalar real functions appear in the general solution of Eq. (2.4). The p’ half integer I representations obviously play no role in this respect. In the particular case of T = SO(3) since p. = 0, p -= 1, p’ =I=0 we are back to the case of three arbitrary functions that were called a, 4, and x in Ref. [8]. If T = W(3) then either p,, = 1, p = 1, p’ = 2 if the rotations are compensated by the group generated by A,, A,, A,, and therefore four functions will appear in the general solution or, if the subalgebra SO(3) of SU(3) is obtained from A, , A,, and A, then 17” = 0, p == 2, and p’ = 0 and there are six arbitrary functions in the general form of Wj (see Ref. 191). (c) Next we impose on the scalar field @ the condition of being spherically symmetric up to a global gauge transformation. Without loss of generality we assume that * is real and that it transforms according to an M-dimensional orthogonal representation of T (the Q representation for short). In analogy with (2.4) and (2.5) we demand that &$br = - TBrdNf@’
(IT A = 1, 2,..., M)
= -J:o@A,
(2.2&i) (2.20b)
and L u
-
r ?r@
=
-
E
-JL
(2.2la)
TBrANfg-(1/2)n;;j
&4/2)4
.
(2.2lb)
86
P. CORDER0
The M x M matrices J, are linear combinations of the matrices TB representing the generators of the gauge group in the 0 representation and they must satisfy the angular momentum algebra
[J, , &I = +Jc .
(2.22)
The general solution of (2.21) and (2.22) is
ip = e-JWJ@h(u, t),
(2.23a)
where J&r,
t) = 0
(2.23b)
and 7rQ = e -JSqe-Jeex,(r, t) sin 8,
(2.24a)
&%(r, t> = 0,
(2.24b)
where
h(r, t) and xh(r, t) being M-vector-valued functions. The general form of h and xh is again (like R) a linear combination of the m = 0 linearly independent vectors existing in the representation of SO(3) induced by the Q representation of the gauge group. Equations (2.1) (2.17), (2.18), (2.23) and (2.24) are the main results of this section. To some extent they can be generalized to the case when the SO(3) symmetry requirement is replaced by a requirement of D-symmetry, D being one of the many nontrivial isometry groups that (curved) 3-space can admit. The dependence on the coordinates on each minimal invariant variety (~9and y in the case of SO(3)) can be made explicit and eliminated from the dynamical problem [18].
3. THE HAMILTONIAN OF THE SPHERICALLY SYMMETRIC MOTIONS
The analysis of the dynamics of the Einstein-Yang-Mills-Higgs system is greatly simplified by restricting the general Hamiltonian formalism summarized in Appendix A, to the fields with spherical symmetry found in Section 2. The generators X1, Zi, and G defined in Appendix A take the form (3.1)
(3.2a) (3.2b) (3.3)
EINSTEIN-YANG-MILLS-HIGGS
87
SYSTEM
The equations of motion can then be obtained from the extremization action
of the restricted
(3.4)
They are given in Appendix C. The variations are not all independent howeversince (2.17b), (2.18b), (2.23b), and (2.24b) must be preserved in time. To avoid introducing extra Lagrange multiplier terms (such as D . Z,R, K . 2,x, , etc.) we have to take care that the equations of motion leave the variables R, ?$ , 8, no, and h, xh in the subspace defined by the equations mentioned above. We can proceed in this way because the extra Lagrange multipliers turn out to be zero, or, in other words, one can check explicitly that Eqs. (C14)-(C19) are satisfied in the three static solutions found in Section 5. Equations (C14)-(C19) turn out to be conditions on the Lagrange multipliers N, N’, and W, the only nontrivial one being (3.7) below. In writing the reduced action we have assumed that the angular dependence in the integrand of the full action given by (A9) was a common factor sin 0 integrated away (J sin 8 & dp = 47~) and for this we have assumed that N and NT are functions of r and f only, while W, has been assumed to have the form W,
=
e-Z3”e‘eZ”8W(r,
t)
(3.5)
to cancel exactly the angular dependence in the last term of the jntegrand defining S. These assumptions represent no loss of generality as was discussed in Ref. [8]. Furthermore, the explicit form of 6 given below is such that z&
= 0,
(3.6)
and, to guarantee that Z,R = 0 is preserved in time we have to set z,w
= 0.
(3.7)
Next we give the derived explicit expression corresponding to each one of the reduced generators split into the different contributions coming from the gravitational, Yang-Mills, and scalar fields. (a) The functions $7 and 2,” are [19] $7 = p-yg4
_ ~rTT,rA + p$g”
_ 2~~~1 + 3x’2 - &-A)),
(b) To obtain sIM
and 2:”
(3.8) (3.9)
rt. is necessary to evaluate BRi first, the result
being .dBr
x
;e-z3’e-Z?e(-N3
_L
0
;,,
Z,e) sin 0,
(3.10a)
88
P.
CORDER0
,@ = f e-Z8me-Z2BZ,(8’ + 8 x R) sin 8,
(3. lob)
cc&?,” = ‘r e-z3’J’e-z2e(Qt + Q x R).
(3. IOC)
where L!.@is defined in Appendix A and the cross product between N vectors is defined in terms of the structure constants of the gauge group, (8 x R)A = CAB@RC
(3.11)
(see Appendix B). We notice that the expressions in round brackets in (3.1Ob), (3.10~) look like a gauge covariant derivative of 0, where the role of the compensating field is played by R. Inserting (2.17), (2.1 S), and (3.10) into (A3) and (A4), one finds
1 + 2f2@ u-2A(Q 2,‘”
x
Z,Q - N2)2 + 2e-“(0’
+ 0 x R)2},
(3.13)
= X@* (6’ + Q x RI,
G“YM = --rr,~R--(~ox~+Z3~O~Zge)-~;(.
In the derivations following: (i) (ii) (iii) 7c, ’ 8 = 0, (iv)
(3.12)
(3.14)
necessary to get the above results frequent use is made of the
The 2, matrices are antisymmetric and traceless, their exponentiation yields orthogonal matrices, vectors belonging to different eigenspaces of 2, are orthogonal, the results of Appendix
e.g.,
B.
(c) Finally, to obtain the functions SI,“b &! , and @ we need to know that the gauge covariant derivative Vi+ (defined in (A8)) of the scalar field is VT@ = ~+‘~e-~@[h - RATAh],
(3.15a)
V,@ = -e-J~pe-J2e@ATAh,
(3.15b)
V,O = e-Jaqe-J2e(Z3@)A T,h sin 6.
(3.15c)
Again in (3.15a) we can see what will turn out to be the gauge covariant derivative of h with R again appearing as the gauge field.
EINSTEIN-YANG-MILLS-HIGGS
89
SYSTEM
Jn the above use has been made of the identity te-
.%De-Z,e)BD TBrA(e-J3.3me-J’p*z _ (e-.‘3ae-.y-,
T,“,
I
The results of Section 2 together with (3.15) yield
.TP’p =
Xh.
(h’ - R”T,h),
(3.18)
Jn Eq. (3.19) boldface type has been used for the N x N matrix TA becausethat is the factor that carries the N vector index A, i.e.,
In Eqs. (‘3.15)~(3.18), on the other hand, boldface type has been used to indicate M vectors. It was said in (3.2) that H8 = 0. This is proved by first obtaining that se = nh . OATAh and then showing that the A4 vector OATAh belongs to the eigenspaceof (R,)” associated with the eigenvalue -1, while nIL satisfiesRp,, -- 0.
4. GAUGE FREEDOM AND GAUGE
FIXATION
Now that we have the explicit form of the terms that appear in the definition of the action (3.4) it is possible to check that the reduced action 3 is invariant under the gauge group defined by (4.Ja) (4.1 b) @cA
+
,-frAc~@$+$rAG,
(4. Ic)
?r;c,
-+
e-f'ACAT~CBef~c~,
(4.ld)
hde-fr4T.4h,
rc,, -+
e-fr.4TA,h,
(4.le)
(4.lf)
90
P. CORDER0
where rA is an arbitrary r dependent N-vectorial function satisfying 2$(r)
= 0.
(4.2)
This local gauge group is characterized by p0 + p arbitrary functions TA(r), p. and p being the numbers defined in the discussion following Eq. (2.18) and it is obviously a subgroup of the original local gauge group. In Ref. [8], for example, where pa = 0 and p = 1 there appeared just one function (called y(r)) characterizing the residual gauge freedom. The gauge freedom we have then equals the number of independent components of R and it is natural to choose the gauge in which R is absent, i.e., being in a general gauge we can go to the gauge R = 0 by first solving the equation for I’ (coming from (4. la)), @-““C”)’
efi-Ac.oj
=
-e-frAC.qBCBefrACAe
(4.3)
The formal solution of this equation, defined up to a constant, can be found by iteration, the first terms of the series being (4.4) Given this r we can now transform to our gauge R=O
(gauge fixation).
(4.5)
A convenient coordinate fixation is eA = r
(space coordinate fixation),
(4.6)
n-@= 0
(time coordinate fixation).
(4.7)
These conditions must be preserved in time and they must fix the value of the Lagrange multipliers W, NV, N. In fact, to guarantee that (4.6) is valid for all time we require that h = 0 which implies, together with (4.7), that NT = 0 (see Eq. (C7)). Requiring that +P = 0 and that J? = 0 implies (see Eqs. (C9) and (Cl)), N[2e-“(1 - 2rp’) + 2e” -
&fe2rp2eu(Q
x Z,O
+f zr-zeun~ + if 2e--rrci _ N3)2+ f-2em“@‘2 + $r-2em~~~
f @2e-ub’2- r2euU - +e”[(@ . T/z)” + (Z,Q - T/z)~] - 4(Nre-p)’ = 0
(4.8)
and W’ -j- fNefir2nR
= 0.
WY
EINSTEIN-YANG-MILLS-HlGGS
91
SYSTEM
The last two equations can indeed be used together with the rest of the equations of motion to determine W and N. For a careful discussion of gauge transformations and gauge choice see Ref. [20].
5. STATIC SOLUTIONS The equations of motion for our variables p, 7~, ? h, v,, , R, xR , 0, x0 , h, xh are obtained without any difficulty by evaluating the Poisson brackets of every one of these variables with the total reduced Hamiltonians
They can be found in Appendix C. To these equations one must add the constraints flL = 0, Z& = 0, and e = 0, also given in Appendix C. We have already mentioned at the end of Section 4 how the equations x =~ 0. T, = 0, and tZ = 0 determine the functions IV’, N, and W. The equations of motion ensure that the constraints 3; = 0, =z$ = 0, and e = 0 are preserved in time and therefore that there are no new constraints in the formalism. The first two of these constraints themselves can be used to obtain p and 7~~in terms of the matter fields and it can be shown that the equations for 7jn and fi become identities. In the present section we look for static solutions. A solution is static if all gauge and coordinate transformation invariant quantities do not change in time. Tn our case it means that all quantities left after requiring that Eqs. (4.5)-(4.7) be satisfied, must be time independent. It is possible to reduce the problem to the following set of equations for 0, h, W, p, and N l
(Ne-I@) -- T - ‘fz
z,o x (0 x z&I - N3) + g
[eATAh - T/z - (Z@)A
T,h
w >: (0 ;c W)
. (Z,T)/r] = 0,
(5.2)
(Nr2e-41’)’ + Ne@[(8ATA)2 h + ((Zs@)A TA)2h - 2r2eU$
h] -.fWATAxn
:
0.
(5.3) r2W’ r Ne-u __ - (Q x W) x 0 - (Z,(0 c Ne” I
x W)) Y. 2.p
+ .f2r2WAT.J?
* Th = 0,
(5.4) ’ From the gauge and coordinate
conditions
and the equations of Appendix
C written for the
staticcasewe get R = 0, VA = 0, xW = 0, N’ = 0 E quation(Cl) is usedto eliminateTCR, Eq. (C2) is usedto eliminateXQ and (C3) to eliminate xh (see (5.7)). Now Eq. (5.2) follows from (C5); Eq. (5.3)from (C6); Eq. (5.4)from (C13);Eq. (5.5)from (Cl]) and Eq. (5.6) from (C9).
92
P.
2ty.i + eaer 1 C
(Q 2N2 x Y2
_ (Q x -W - %I2 _ (@*TA@’ + ((Z,Q)* 4f 9% 4
_ f ‘r2(wAi-,&)z -pu(hq e2p
1 +
(Q
X
Wz
r2U2
-fg+-,=
4N2
2rp’ -
CORDER0
_
(Q
X
f;Q&-
Ns)’
_
T,h)2
1,
(5.5)
(@*T~h)~+y@)*
TAhJ
2N2
L
+ f2rz(WATAh)z
-- r2U(h2)
4N2
2
+ r2W’2
1
W2
r2h’2
w-v-
4
2e”(re-uN)’ +
N
= , ‘(5.6)
where x, in (5.3) is given by xh = (fr2eu/N)
WATAh.
(5.7)
(i) The trivial solution to these equations corresponds to no fields except for the gravitational one, 8 = N2, h = 0, W = 0. The geometry is that of the Schwarzschild solution. (ii) The second simplest solution is 8 = 0,
h = h, = const,
xh = 0.
(5.8)
In fact, (5.3) shows that (au/ah’)
/h-h,
=
0,
i.e., ho is a field value which minimizes the symmetry breaking potential, (5.4) and the vanishing of xh it follows that r2W’/NeP
= -q
= const,
(5.9)
while from (5.10)
which implies W’ = -q
[’ (Ne”/p2) dp.
Inserting the last result in (5.7) it is seen that
q*T,/&f
= (qATAh,)r = 0,
(5.12)
showing that there is a connection between the orientation of the charge q in W space and the orientation of the “vacuum expectation value” of the scalar field h, . This orientation can be fixed making use of the surviving gauge freedom with constant gauge functions mentioned after (4.3). See, however, Ref. [20]. Next we substitute the previous results in Eq. (5.5), obtaining e-2u = 1 - (2m/r) + ((g” + q2)/4r2), where m is an integration
(5.13)
constant and g2 = I/f”.
(5.14)
EINSTEIN-YANG-MILLS-HIGGS
93
SYSTEM
Had we not assumed in Appendix A that U vanishes at its minimum would be an extra term proportional to r2 on the right-hand side of the well-known effect of the presence of a cosmological term in the makes the geometry not asymptotically flat). With or without such Eq. (5.6) gives N --_ e-u and henceforth the solution is (without the extra cosmological
= e-z3”e-z2*
(5.15)
term)
9 == e-J3me-Jq,, , w,
(h = &) there (5.13). This is theory (which an extra term
(5.16a)
[!! )
r
{js” -_. ~- 1 _ z!!!. + i r
g2 + q2
7 r2(dP + sin2 0 dv2),
(5.16~)
where q and h, are constrained to satisfy (5.12). This spherically symmetric singular solution does not have surviving massive components of the gauge field W, . In fact. it exists even without introducing a scalar field at all [8]. (iii) The finite energy solutions which include massive gauge fields cannot be written explicitly even in the simplest cases, such as those analyzed for example, by ‘t Hooft [1] and by Julia and Zee [3] in flat space-time. It is not difficult to check. however, that the following asymptotic forms are consistent with Eqs. (5.2)-(5.4): (a) As r -+ 0 0 ---f (c-r2 + 1) N, ,
(5.17a) (5.17b)
W + Car) N3, h - @I W ,
(5.17c)
where M, is an I = 1, m = 0 vector in the M-dimensional representation space. i.e., (J1” + Jzz + Ja2) M, = -2M, and J3Ma = 0, while a, b, and c are constant. (b) As r -+ co h --t h, (minimum of U(P)), (5.18:~) W-M (constant), (5.18b) with MAT,,h, == 0 and Q + (epRr) 8,) (5.18~) where .Q* is a matrix of the form %B
= f’~rP~i-
+ QA~QB~ -
S,,SBc
>
(5.19)
PAr = j-TArrhf ,
(5.20a)
QAP G f
(5.20b)
Tm-,rh~. SAC5%2"*f&CMB. (z&AD
(5.20~)
94
P. CORDER0
The vectors II, , M, 8, are constants which must be such that (5.18~) tends to zero to have finite energy.
APPENDIX
A: NOTATION AND GENERAL HAMILTONIAN FORMULATION
We use the indices relativistic indices, space polar indices, space Cartesian indices, indices of the adjoint representation of the gauge group, indices of an arbitrary real and orthogonal representation of the gauge group.
I*, v = 0, r, 8, v i, j = r, 9, fp a,b = 1,2,3 A, B, C ,... = 1, 2,. .., N
r, A, 2 = 1, 2)...) A4
We deal with the Hamiltonian formulation of the gauge invariant interacting system of a real scalar field @jr, a Yang-Mills field WA,“, and the gravitational field described by the metric tensor g,, . The canonical variables are the spatial components of the metric gij , the space components of the gauge potentials WAi, and the scalar fields Qr together with their respective conjugate momenta rTii, rrAi, and 7rr. The Hamiltonian of the system is [13, 17,211 H = s d3x(N.Xl
+ NiZi
+
f WA,G.J,
where the arbitrary functions N and Ni are the combinations the space-time line element in the form ds2 = -N2
dt2 + gii(Ni dt + dY)(Ni
(AlI
of the g,, that define
dt + dxj).
(A9
The WA0 is the arbitrary time component of the gauge field. The generators of normal and tangential deformations (2’ and Si) and of gauge transformations GA are
=
g-1/2(TijTij
+ ~[g-l12,02
-
iT2)
-
g1/2~
+
$g-li2gii(&inAj
+ gl12gijVi@ . V,@ + 2grRJ(*)]
+
w 0,
&&4iWAj)
(A3)
EINSTEIN-YANG-MILLS-HIGGS
95
SYSTEM
all of which vanish weakly [12, 131.The vector density ~47~~ is gAi
=
1 ijkFA, 2E
3k 3
(Ah)
= &cijk(WA,j - WAj*k - fCASCWBiWC~),
gij is the inverse of g,, . The scalar self-coupling U(@) is not specified but to be in accordance with our denomination of Higgs field, it should be thought as being
u(a) = b(@ - ay,
(A74
which is zero at its minimum to avoid introducing an undesired cosmological term. Another possible choice for the potential could be the popular u(a) = b(l - cos j (P2y).
(A%)
The gauge derivative of @ is given by Vi@ = @r,i - f WBiTBr,&DA,
(A@
where the matrices TB were already introduced in (2.20). The action is given by
It is this action that we consider in Section 3 to obtain the reduced action 3. Quantities with one group index A (or one index r) will sometimes be called N vectors (or M vectors) and will be indicated by boldface type, with the corresponding index omitted; e.g., instead of N;;” we write N, (or instead of hr we write h). Both indices “A” and “P can indifferently be written up or down.
APPENDIX
B: VECTORS AND MATRICES
OF THE ADJOINT
REPRESENTATION
In Section 2 we introduced the N-dimensional N, of components N: and the N :f N antisymmetric matrices (-&>Ac
=
N%Anc
.
@I)
(We remind the reader that the group indices are raised and lowered with the metric (2.3J.J Defining the antisymmetric cross product of any two N vectors by (U x V)A = CAB&JBV~
032)
it is immediate that ZJ-J = N, x U.
(B3)
96
P.
CORDER0
The last result together with (2.11) yields Nz x Nb = GA%,
(B4)
which, in turn, implies that the three vectors N, have the same norm, (Nl)2 = (N2)2 = (NJ2.
(B5)
The square of the N, vectors must be understood as C, N;;4N,A . Two other properties which are useful are (uxv)xw+(wxu)xv+(Yxw)xu=o
0%)
and Z,(U
APPENDIX
x V) = (Z,U)
x v + u x (Z,V).
C: THE EQUATIONS IN THE SPHERICALLY
OF MOTION SYMMETRIC
(B7)
AND CONSTRAINTS CASE
The variation of the reduced action ,!? defined by Eq. (3.4) together with (3.8), (3.9) (3.12)-(3.14), (3.17)-(3.19) yields equations of motion for the canonical variables 8, xg , h, rch , which we write in the form they take after considering EL>=u3h,%,R,XR, the coordinate and gauge choices (4.5), (4.6) and (4.7)
lk = f 2Nr-2eunR+ f W’, 6 = if 2Ne-une + NW - f0
(Cl) x
W,
h = Nr2e+nls + N?(h’ - RATAh) -f
. = nR
-2f
w WATAh,
-2Ne-u0’ x 0 + Ne-ur2h’ * Th - Nrz,
(C3) x
0
+ Nrrfi *Th +fW X n,, . IQ = 2f-2Nr-2eu( 8 x Z,O - N2) x Z,S + f-2(Ne--rr0’)’ - Neu[OATAh . Th - (Z@)” T,h * (Z,T) h] + (N’x,)’
. xh = (Ne-V2h’)’
(C4) +
f W x ~0, (C5)
+ Neu[(O . T)2 h + (Z,O . T)2 h]
- 2NeYr2 g
h + (Nrxh)’ -f
WATAnh ,
VW
x = NV,
(C7)
p = -tNr-2 e- %r,, + N*p’ + (Nr)‘,
(C8)
* = N[2e-“(1 =&4
- 2rp’) + 2e” - &f 2r-2eYrri + (f2/4) e+nE
- &ff-2r-2e+u(Q
x Z,Q -
bQ2 + fp2e-“Qr2
+ $r-2ee-u&
+ +r2e-ph’2- r2euU - $eu[(@* Th)2 + (Z,O . Th)2]] - 4(Nre-u)‘,
(C9)
EINSTEIN-YANG-MILLS-HIGGS
+,, = N[--4e-“(1
97
SYSTEM
- 2-p’) +f 2r-2eurr~ + ~~-2r-2e“(0 x 2,8
- N3)2
f. r-2e-@T; - r2e-“h’” f 2r2e”U]
(c-10)
f J[(Nre-l*)’ - (N’r2e-u)‘]. The constraints (3. I). (3.2), and (3.3), respectively, become 2[e-“(1 - 2rp’) - eU]f -+ .f-“[$r-“e”(e
x
hfV2[eu& Z,8
-
N,)2
J- $r2em uni] +
e-“@‘“I
+
$r-‘em
‘ST:
+ $-2e-W2 + &e”[(O - Th)z -t (Z,O . T/7)“] 4 ellr’l! = 0,
(Cl I)
x0 . 0’ + xh * h’ I r-lr,, -= 0,
(C12)
x 0 + Zgc, ‘k: Z,e) + TT,,. Th =: 0. %r J- 4(x, -
(Cl3)
Furthermore we have to require that the constraints (‘2.17b). (2.18b). (2.23b). and (2.24b) be preserved in time, i.e.. z,ft
= 0,
(Cl1)
Z,icR = 0,
lCl5)
z;o =z --0.
(C16)
Z.&J =- - ic, .
1(‘17)
J,h :: 0,
I(‘1 8)
J3+ =z 0.
((“19)
ACKNOWLEDGMENTS I am indebted to Demetrios Christodoulou and Claudio Teitelboim cussions and for much encouragement. I would also like to thank Rafael for their constructive criticism. I also wish to express my gratitude to International Atomic Energy Agency and UNESCO for hospitality at Theoretical Physics. Trieste.
for many enlightening disBenguria and Sergio Hojman Professor Abdus Salam, the the International Centre for
REFERENCES 1. G. 2. A.
‘T HOOFT, Nucl. Phys. B 79 (1974), 276. M. POLYAKOV. JETP Lett. 20 (1974), 194.
3. B. JULIA AND A. ZEE, Phys. Rev. D 11 (1975), 2227. 4. F, A. BAID AND R. 3. RUSSELL, Pfzy~. Rev. D 11 (1975),
2692.
CHO AND P. G. 0. FREUND, Phys. Rev. D 12 (1975), 6. P. B. YASSKIN, Phys. Rev. D 12 (1975), 2212. 7. P. VAN NIEUWENHUIZER, D. WILKINSON, AND M. J. PERRY, 5. Y.
8.
M.
P. CORDERO
AND
C. TEITELBOIM,
Amt.
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