Gauge Theory of Gravitation A Unified Formulation of Poincare and (Anti}De Sitter Gauge Theories


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1 711 Progress of Theoretical Physics, Vol. 80, No.4, October 1988 Gauge Theory of Gravitation A Unified Formulation of Poincare and (Anti}De Sitter Gauge Theories Takeshi SHIRAFUJI and Masahumi SUZUKI Physics Department, Saitama University, Urawa 338 (Received May 28, 1988) We formulate the (anti)de Sitter gauge theory by requiring the ';scale invariance besides the (anti)de Sitter gauge invariance. Owing to the former invariance, the resultant theory does not depend on the dimensional constant which specifies the constant curvature of the initial (anti)de Sitter spacetime. A metric and an affine connection can be defined on spacetime in a ';scale invariant manner. The spacetime then becomes a RiemannCartan space, and its curvature and torsion are just the two ';scale invariant parts of the (anti)de Sitter gauge field strength. The Poincare gauge theory is reproduced in a special gauge. We also discuss the (anti)de Sitter gauge useful for studying global aspects of asymptotically nonflat spacetimes. 1. Introduction Gauge theory of gravity was first formulated by Utiyama as the gauge theory of the Lorentz group/) then by Nakano as that of the translation group,2) and finally by Kibble as that of the Poincare group (Poincare gauge theory or "PGT" for brevity):3) Gauging the Poincare group introduces two kinds of gauge fields, the translation gauge field and the Lorentz gauge field. The underlying spacetime is a Riemann Cart an space with a metric and a nonsymmetric affine connection, and it is characterized by the curvature and the torsion which are identified with the Lorentz and the translation gauge field strengths, respectively. This theory of gravity is based on the gravitational Lagrangian given by a sum of linear and quadratic invariants of curvature and torsion, and provides a general framework to describe the gravitational interaction in microscopic world. 4 )7) The PGT is constructed by starting from Minkowski spacetime where special relativistic law is valid. This seems quite natural because special relativity has been confirmed to the highest energies available at present. As is well known, however, exact solutions of the PGT are not in general asymptotically flat, but asymptotically approach spacetime with constant curvature (namely, de Sitter or antide Sitter spacetime).8) Accordingly, we could reasonably anticipate that we would be led to the PGT even if we started from de Sitter (or antide Sitter) spacetime instead of Minkowski spacetime. The purpose of this paper'is to show that this is indeed the case. The de Sitter gauge theory (or "dsgt") is formulated by gauging the isometry group 0(4,1) of de Sitter spacetime, and the antide Sitter gauge theory (or "AdSGT") is obtained by gauging the isometry group 0(3,2) of antide Sitter spacetime. 9 ):18) We shall denote by R({ })=121]/.Q2 the constant curvature of the initial spacetime of which the isometry group is gauged. Here.Q is a positive constant with dimension of length, and 1]= + 1 for the dsgt and 1]= 1 for the AdSGT. Then the dimensional
2 712 T. Shirajuji and M. Suzuki constant Q enters the resultant theory,and we denote it by dsgt[q] (or AdSGT[Q]) when it is necessary to specify the constant curvature of the initial spacetime. In the dsgt and AdSGT formulated in the past, the Q was considered to be a fundamental constant of the theory, and two arguments were put forward about the transition of dsgt (or AdSGT) to PGT: One is to take the limit Q_HlO,9),10) and the other is based on the symmetry breaking of the gauge group from 0(4,1) (or 0(3,2» to 0(3, 1).12H8) In this paper we take a different standpoint: We do not take Q as a fundamental constant, but construct the dsgt (and AdSGT) by requiring it to be independent of the value of Q. The resultant theory is then shown to coincide with the PGT. In order to do this, we introduce a transformation which defines a correspondence between dsgt[q] and dsgt[q'=aq] (or between AdSGT[Q] and AdSGT[Q'=AQ]) with A being a positive constant parameter. Under this transformation, the fivedimensional vector field.;a(x), which is. a generalization of the fivedimensional coordinate function specifying the initial de Sitter (or antide Sitter) spacetime as a hypersurface in a fivedimensional flat space, is scaled as,;ra(x)= A.;a(x), and the gauge field A ab vex) is changed so that the spacetime metric be kept unchanged: Furthermore, this transformation does not involve any coordinate change. We shall refer to this transformation as the ';scale transformation. The matter fields, such as the spinor fields representing quarks and leptons, are assumed to be left unchanged under ';scale transformations. It is then shown that the spinor wave equation is also ';scale invariant. We shall require the equivalence between dsgt[q] and dsgt[q'] (or AdSGT[Q] and AdSGT[Q']): More specifically, we shall postulate that the gravitational action be invariant under both gauge and ';scale transformations. It is then found that the gravitational Lagrangian is made of the invariants formed of the spacetime metric, the curvature and the torsion. The basic properties of (anti)de Sitter spacetime are briefly reviewed in 2 in order to fix our notations:*) In particular, the spinor wave equation is discussed as the typical example of the field equations of matter fields. The isometry group of (anti)de Sitter spacetime is gauged in 3, and the geometric structure of underlying spacetime is studied in 4. The ';scale transformation is introduced in 5, and the (anti)de Sitter gauge field strength is decomposed into ';scale invariant parts. The extension of ';scale transformations is attempted in 6 in order to cover positiondependent parameters A(x), and the gravitation~il Lagrangian is constructed by assuming its invariance under ';scale as well as gauge transformations. Various special gauge choices are discussed in 7, and conclusions are summarized in the final section. An Appendix is devoted to derive the spin or wave equation in (anti)de Sitter spacetime. 2. The (anti)de Sitter spacetime The (anti)de Sitter spacetime is a space with constant curvature, and it can be embedded in a fivedimensional flat space as a hypersurface by19) *) We use the abbreviation "(anti)de Sitter" in those statements which apply both to de Sitter and antide Sitter cases.
3 Gauge Theory of Gravitation 713 (Z l) where e(a=:=0~3, 5) are fivedimensional cartesian coordinates and 7]ab denotes the fiat metric, 7]ab=diag(l, +1, +1, +1, 7]55=7]). Here Q is a positive constant determining the constant curvature, and we put 7]= + 1( 1) for de Sitter (antide Sitter) spacetime. The quadratic form (Z 1) is kept invariant under (anti)de Sitter transformations whose infinitesimal expressions read (Z Z) where ({Jab = ({J[ab] are infinitesimal, ten parameters. These transformations generate the isometry group 0(4,1) (0(3,2» of de Sitter (antide Sitter) spacetime. 20 ) Given a coordinate patch {x"}, the metric tensor g"li is derived from the fivedimensional coordinates ~a(x) as with The RiemannChristoffel curvature tensor is then given by By virtue of (Z 1), the sa", of (Z 4) satisfies the orthogonality relation, (Z 3) (Z 4) (Z 5), (Z 6) Throughout this paper fivedimensional latin indices are raised or lowered with the metic 7]ab or 7]ab. As for greek indices referred to the coordinate patch {x"}, the metric tensor, g"li or its inverse g"li, is used to lower or raise them. Then we have in particular, (Z 7) Let us consider a fourcomponent spinor field describing spinliz particles in (anti)de Sitter spacetime: It is changed under infinitesimal (anti)de Sitter transfor, mations of (Z Z) as ' (2 8) where sab are the (anti)de Sitter generators defined by (Z 9) with r a being 4 x 4 matrices which satisfy {ra, rb} =  Z7]ab. (Z 10)
4 714 T. Shirafuji and M. Suzuki for 0(4,1), for 0(3,2). (211) Here the Minkowski metric is given by 7]km=diag( 1, + 1, + 1, + 1). As is shewn in the Appendix, the spin or wave equation is derived from the Lagrangian density LM(O)= j g [(i/2) call (rav,}ol 7t)O)ra) +m ] with = t yo for 0(4,1), or (212a) x [(i/12) (SabV )OL 17)o)sab) + (im/24)c b v r a ] with =i trsyofor 0(3,2), where Vv(O) is defined by V v(o) = [all+(i/2)(~acb v ecall)sab/(~ ~)], (212b) (213) and totally antisymmetric symbols cabcde and C VAPI1 are normalized as C0123S= + 1 and c0 123 = 1, respectively. The spinor wave equation takes the form, (214) both in de Sitter and antide Sitter spacetimes. It is also shown in the Appendix that the wave equation (214) is reduced to Dirac's spinor wave equation by a unitary transformation. 2 1) 3_ Gauging the (anti)de Sitter group Let us generalize the relation (24) and the Lagrangian density for the spinor field given by (212a, b) so that they be invariant under (anti)de Sitter gauge transformations with wab(x) being infinitesimal arbitrary functions.. This is achieved by introducing the gauge field Aabv=A[abl v which transforms as (31) and then by replacing the ordinary derivative ap in (24) and (212a, b) by the covariant derivative D p Namely, we assume that the metric tensor is given by the relation (23) with cap now being defined to be (32) Equation (3 I) ensures that the cap transforms as a vector under (anti)de Sitter gauge transformations. The Lagrangian density for the spin or field is obtained from (212q) and (212b) for dsgt and AdSGT, respectively, by replacing the ordinary derivative av by the covariant derivative, Dv =[av+(i/2)a ab vsab]. (33) If we define the derivative fjll by
5 Gauge Theory of Gravitation Z ~a bv_~b av Dv=Dv+ 2 ~.~ Sab, (3 4) the Lagrangian density then reads for dsgt, and (3 5) for AdSGT. Incidentally we notice that we shall meet the derivative jjv again in 5, where we discuss ~scale transformations: The spinor wave equation derived from the Lagrangian density will be given there. The fivedimensional vector field e is supposed to satisfy the condition (2 1), and hence the orthogonality relation (2 6) is still valid. Since the gl'v and a I' are both derived from ~a and A ab v, the latter two fields are the basic dynamical variables. The gauge field behaves like a vector under general coordinate transformations, while it is not a tensor for (anti)de Sitter gauge transformations. We then define the field strength by (3 7) which transforms like a secondrank anti symmetric tensor. The commutator of Dv is then [DI', Dv]ua=F.abI'VUb (3 8) for any vector field ua. Throughout this paper we understand that Dv denotes the covariant derivative for latin indices and that it is ordinary differentiation for greek ones; namely, we put, for example, 4. Geometric interpretations A vector under general coordinate transformations will be referred to as a world vector, while a vector for (anti)de Sitter gauge transformations will be called a local vector. Given a world vector field VI', we can define a local vector field by va = al' VI', which is orthogonal to the field ~a. Conversely, for any local vector field ua, we can form a world vector field UI'= ai'u a, which can be solved to give ua= ai'ui' if and only if U a is orthogonal to ~a. For a world vector field VI', we define the covariant derivative [7 v by (4 1) with va= ai'vi'. This is assumed to hold for any world vector field, and therefore the affine connection coefficients are given by
6 716 T. Shirafuji and M. Suzuki (4 2) from which follows the metric condition, (4 3) The gauge field A ab v (or the covariant derivative Du) defines parallel transport of local vectors: Given a curve C in spacetime connecting two points P and Q, then a local vector ua(p) at P can be parallel transported along C to the point Q, defining a parallel transported vector field on C, U"a(x(il», which satisfies DU/la = du/la +Aa U b dx v =0 Dil  dil bv /I dil. (4 4) Here il is an arbitrary parameter specifying the curve. The relation (4 1) means that a world vector V" at P is to be parallel transported as follows: One first represents the vector as a local vector va=cap VI' at P, then parallel transports va along C to Q, and finally expresses the resultant local vector at Q, V/la(Q), as a world vector, V//(Q)=ca"(Q) V//a(Q). The connection coefficients rp'u of (4 2) are not symmetric with respect to f.1. and v: So the underlying spacetime is RiemannCartan space characterized by curvature and torsion defined respectively by and The commutator of [1 u is then (4 5) (4 6) (4 7) for any world vector field V!. Computing the lefthand side of (4 7) with the help of (3 2), (3 8) and (4 1), we obtain The torsion tensor, on the other hand, is related to the field strength by (4 8) (4 9) as is seen from (3 8) and (4 6). We shall now study in more detail the geometrical meanings of the gauging procedure explained in the previous section. First of all we notice that the gauging of (anti)de Sitter invariance implicitly assumes the existence of a flinfbein field E(x) ={Ea{x); a=0~3, 5} by referring to which local vectors and spinors are defined: The scalar product g(, ) is also supposed to be defined with the property, g(ea, Eb)= T}ab. For a local (anti)de Sitter transformation, (4 10)
7 Gauge Theory of Gravitation 717 (4'11) a local vector field Ua(x), for example, is changed like (4 '12) From the flinfbein at a spacetime point P,*) we can construct a fivedimensional fiat space q; p attached to spacetime at P: Each vector formed of E(P) is considered as a position vector in q; p with the initial point being located at P. The scalar product g(, ) defines an invariant distance in q; P. If we put the basic relation between c a,., and g,.,)) implies Accordingly we can identify {E,.,; J.l=O~3} (4'13) (4'14) with the coordinate basis of the tangent Minkowski space T p : The fourdimensional subspace of q; p which is spanned by {E,.,} and which involves the point P is then identified with Tp Let us next define the origin Op of q; p by requiring that o;p = e E a, and represent points of q; p by their coordinates (xa) with respect to this origin. The hypersurface satisfying X X = 7JQ2( = ~.~) is an (anti)de Sitter space in q; P, which is denoted as SP. The space Tp is tangent to Sp, and hence the latter is also tangent to spacetime at P (see Fig. l(a». Now consider another spacetime point Q near P with the coordinates x"'+dx"'. When the flinfbein E(Q) at Q is parallel transported to P, it becomes a flinfbein at P, E,/P) = {E"a(P)}, which is related to E(P) by (4'15) with A~a)) being the (anti)de Sitter gauge field. This relation defines how to parallel transport local vectors from Q to P: For a local vector U(Q)= Ua(Q)Ea(Q), the transported vector is '\ I, (4 '16) (a) (b) Fig.1. The five dimensional fiat space CUP. (a) The origin Op, the tangent Minkowski space Tp and the (anti )de Sitter space SP. (b) The nearby point Q represented on SP. *) Hereafter in this section we shall denote by P a spacetime point with the coordinates (x P ).
8 718 T. Shirafuji and M. Suzuki where the last equation defines U//a(P). Using (4 15) in (4 16) gives (4 17) which is also obtainable from (4 4). So the rule of parallel transport implied by (4 15) agrees with the previous definition by (4 4). The nearby two spacetime points P and Q are connected by a tangent vector at P, (4 18) which represents the position of Q on Tp: The point Q can also be said to lie on Sp, because the space Sp is tangent to Tp at P. The fundamental relation postulated in ~ 3, (4 19) then means that the vector OpQ in V p coincides with t;//(p), the parallel transported vector from Q to P (see Fig. l(b)). We can therefore define the following onetoone mapping from V Q onto V p : (1) The origin OQ is mapped to the origin Op, and (2) a point of V Q with fivedimensional coordinates {Xa(Q)} is mapped to the point of V p with the coordinates {x//a(p)}. Here X//a(P) are related to Xa(Q) by (4 17). Owing to (4 19), the point Q of V Q is mapped correctly to the point of V p represented by (4 18). Also, the (anti)de Sitter space SQ of V Q is mapped onto Sp, but the tangent Minkowski space TQ at Q is not mapped onto T p Using this mapping successively, loops in spacetime from P to P can be developed on Sp.*),22) Given a loop C from P to P, (4 20) its development on Sp means the curve on Sp defined by {t;//a(x; A); 0::;;,1::;; I}, where t;//a(x; A) denotes the vector obtained by parallel transporting t;a(x(a)) along C from X(A) to x(a=l). In particular, the round trip of t;a(x) at P along the loop C determines the location of the initial point J5 of the development. For an infinitesimal loop, the dislocation vector PP = Llt;a EaCP) is a tangent vector with its components given by (4 21) This result agrees with that for the development associated with the affine connection of (4 2):23) This is quite reasonable because there is no distinction between the two spaces Sp and Tp in the infinitesimal neighborhood of P. *) This is to be compared with the development associated with an affine connection, in which case loops in spacetime from P to P are developed on the tangent Minkowski space T p
9 Gauge Theory of Gravitation ~scale transformations So far we have been assuming that the local vector field ~a(x) satisfies the condition ~. ~= 7JQ2, where Q2 stands for the magnitude of the constant curvature of the original spacetime from which we started. Once the (anti)de Sitter invariance is gauged, however, spacetime is curved differently from the original spacetime, and the constant curvature of the original spacetime is not a directly observable quantity. So it seems that there does not exist any reason to stick to a particular value of Q. Instead, it will be much more appealing to suppose that physically observable effects are invariant under ~scale transformations, (5 1) where A is a dimensionless constant parameter. In order to find the transformation rule of the gauge field under (5 1), we shall require that the cap is ~scale invariant. According to (2 3), this will ensure that the metric tensor is also ~scale invariant. It follows from (3 2) that this requirement for cap is satisfied if the gauge field is changed as (5 2) which we shall assume henceforth. Using (5 2) in (4 2), we see with the help of (2 6) that the affine connection coefficients r;v are also invariant under ~scale transformations: Thus, both the curvature and the torsion of spacetime are ~scale invariant. Equation (5 2) implies that the gauge field strength of (3 7) transforms as F ab pv ~ Fabpv + (ll/a)(~a FbCpv  ~b Facp.v)~c/ (~.~) from which we observe that Tapv and R ab pv defined respectively by and (5 3) (5 4) R ab pv= F ab pv+ (~a FbCpv  e F ac pv)~ci (~.~) + (c a pc b v c b pc a v) / (~.~) (55) are the ~scale invariant parts of the gauge field strength: It follows from (4 8) and (4 9) that (5 6) Inspection of (5 2) also shows that the field Aabv defined by Aab v= A ab v+(~acbv~ ecav)/(~.~) (5 7) does not change under ~scale transformations. Since both ~a and c b v transform as a vector under (anti)de Sitter gauge transformations, the field Aabv behaves in the same manner as the gauge field. The derivative operator jjv introduced by (34) is just the covariant derivative with respect to this modified gauge field, and hence is
10 720 T. Shirajuji and M. Suzuki ~scale invariant. Equation (5 7) gives DJJ~a=o and (4 2) implies (5 8) (5 9) This latter equation means that the modified covariant derivative DJJ is commutable with the covariant derivative {1 JJ for world tensors: Namely, the relation (5 10) is valid jor any local vector field U a. This condition is stronger than that for DJJ given by (4 1), and it allows us to have (5 11) We assume that the spinor field remains unchanged under ~scale transformations. Then, the Lagrangian density (3 5) for dsgt (or (3 6) for AdSGT) is ~scale invariant, and it can be rewritten by means of (5 10) as follows: for dsgt, or (5 12) L M = cabcdecjjap(j CCACdpC e,,{(if 6) fsab(d JJ + v JJ/2)rj; + {im/24)c b JJfrarj;] (5 13) for AdSGT, where we have ignored totalderivative terms, and the VJJ denotes the vector part of the torsion tensor, VJJ= T.~JJ. The spinor wave equation is then given by for dsgt, or (5 14) cabcdecjjap(j CCACdpC e (J[sab(DJJ + v JJ/2) + (m/ 4)c b JJra]rj; =0 (5 15) for AdSGT. It should be noted that (5 15) can be rewritten into the same form as (5 14): This is shown in the same manner as the spinor wave equation in antide Sitter spacetime (see the Appendix). 6. The action integral The basic dynamical variables of the present theory are the ~a, the gauge field AabJJ and other fields collectively called matter fields.*l The field ~a is constrained by the condition ~ ~=constant, however, and only four of its five components are independent. This constraint can be accommodated with the help of Lagrange's multiplier method. But we do not take that way here, and instead we shall extend the framework so that ~. ~ can be any positiondependent function. *l We do not take jfabv of (5 7) as the basic entity, because ~a and jfabv are not independent of each other, being constrained by (5 S).
11 Gauge Theory of Gravitation 721 We still suppose that a fivedimensional flat space CUp is attached to each spacetime point P; the CUp contains the tangent space Tp and (antime Sitter space Sp whose constant curvature is determined by (.; ';)1. Since the value of.;.; now varies from point to point, the fundamental relation (419) should be modified. We shall postulate this time that the direction vector OpQ in CUp is equal to the renormalized one, [';';(P)/';';(Q)yI2~iP),instead of ~ip) itself. Then the vector PQ in CUp is given by with (61) (62) This is the desired relation which replaces (419). The orthogonality relation (26) is still valid, and the metric tensor is expressed by (23) as before. The covariant derivative of world vectors, [7 v, is still defined by (4 I), and the affine connection coefficients are given by (42) with sav now being expressed by (62). The curvature tensor is related to the gauge field strength by (48), while the expression for the torsion tensor (49) has an additional term given by (1/2)(o,H)vo/'al') lnl.; ';1. The ';~cale transformation (5 I) can also be extended to positiondependent parameters A(x). The gauge field is assumed to transform like (52) as before. Then the sa v of (62) is ';scale invariant. The transformation rule of the gauge field strength (53) now has an extra term, [1/A2(.;.;)][a~(.;asbvesav)_(J.l~v)]. As regards the ';scale invariant parts, Tal'V and Rabl'lJ, the former is given by (54) with an additional term, (1/2)(sal'a V  Saval')lnl'; ';1, while the latter by (55) without alteration. We also notice that these ';scale invariant parts still coincide with the torsion and the curvature of spacetime, respectively. N ow we are ready to construct the gravitational action, 4 IG= jd x"/g LG. (63) We shall require that the Lagrange function LG be (i) linear or quadratic in the first derivatives of the gauge field A ab v, (io invariant under (anti)de Sitter gauge transformations, (iii) invariant under ';scale transformations with arbitrary A(x), and (iv) invariant under general coordinate transformations. The requirement (io together with (iii) indicates that the LG should involve the gauge field only through the ';scale invariant parts of the gauge field strength, T a I'V and R ab I'V. Since these ';scale invariant parts are just the torsion and the curvature of spacetime, we find in view of (i) that the gravitational Lagrange function is made of linear or quadratic invariants of the torsion and the curvature of spacetime. This is quite the same as that in Poincare gauge theory. The most general gravitational Lagrange function satisfying the four require
12 722 T. Shirajuji and M. Suzuki ments (i)~(iv) thus consists of ten terms, a linear invariant of the curvature, three quadratic invariants of the torsion, six quadratic invariants of the curvature*) and finally a possible cosmologica'l term. The matter part of the Lagrange function is also assumed to satisfy requirements (i)~(iv). The Lagrange function for the spin or field, for example, meets these requirements, because it is constructed in terms of the covariant derivative with respect to Jiabll which is,;scale invariant. Finally, a short remark is in order about the field,;a. It follows from (ii) and (iii) that the field equation for e is automatically satisfied when other fields (A ab 11 and matter fields) obey their field equations. Accordingly, the field,;a is not a true dynamical variable, but an auxiliary field which serves to make the present theory manifestly (anti)de Sitter gauge invariant. This is also the reason why we do not include the e in requirement (0. (0 The Poincare gauge 7. Gauge conditions Let us consider the gauge in which the,;a satisfies**). e=finite functions (k=o~3), (7 1) where.q is a finite constant. Here 8 is an infinitesimally small parameter, and 0(8) means a term of the order of 8: We shall ignore all the terms of O( 8 n ) with n ~ 1. Then the gauge condition (7 1) is preserved under with (7 2) (7 3) where E is an infinitesimal parameter. The transformation law of e can be rewritten as with (7 4) (7 5) Thus, the fourvector field e is subject to internal Poincare gauge transformations in this gauge. So we shall refer to this gauge as the Poincare gauge. From the orthogonality relation (2 6), it follows that E 5 11 should be 0(8), and hence that the field E a 11 reduces to the tetrad field in this gauge: Namely, *) Only five quadratic invariants of the curvature are independent when action principle is applied. 24 ) **) This gauge is achieved by two steps: First, choose the gauge with e=.q+ 0(8 2 ) and e= 0(8), and then make the ';scale transformation.;a f. (1/8).;a.
13 Gauge Theory of Gravitation 723 for a=k, for a=5. (7 6) Since alllni ~. ~I = O( ( 2 ), all the additional terms proportional to alllni ~. ~I that appeared in 6 can be disregarded: For example, (6'2) reduces to (3 2). Consistency of (3 2) with (7 6) requires that A~511=O(0),. and therefore we have the following expression for the tetrad field: where Dlle and A k II are respectively defined by Dlle=alle+A~mll~m, A\=lim[A~511~5]. 8~O (7'7) (7'8) (7 9) (7 '10) According to (7'7), the field A~511 tends to zero in the limit, o~o: This field gives nontrivial contributions, however, when it appears in the renormalized form of A \. The basic fields in this gauge are thus e, Akmll and A k ll. (The fifth component of ~a is an infinitely large constant, and hence it is not a dynamical variable in this gauge.) The allowed gauge transformations are the internal Poincare gauge transformations of (7'4), under which the fields Akmll and A\ transform as (7'1l) (7'12) respectively. The A km ll is therefore identified with the Lorentz gauge field, and the A kll is interpreted as the gauge field for internal translations described by the param. eters ck(x): These two fields just constitute the Poincare gauge field of the Poincare gauge theory as formulated by Kawai. 7 ) As for matter fields collectively denoted by q, the transformation rule for Poincare gauge transformations reads (7'13) because the w\term is O(co) and can be ignored. Therefore, matter fields undergo only local Lorentz transformations, and do not change under internal translations. (ii) The restricted Poincare gauge Take a more, special gauge satisfying eo=:o y which can be put into e=o, by a suitable ~scale transformation. Equation (3'2) now gives (7~14) (7'15)
14 724 T. Shirajuji and M. Suzuki e\'=.qa~5v, (7 16) and therefore the (anti)de Sitter gauge field just consists of the tetrad fielq and the Lorentz gauge field in this gauge. Internal translations are not allowed any longer, and the ordinary Poincare gauge theory is recovered. So we shall refer to this gauge as the restricted Poincare gauge. (iii) The (anti)de Sitter gauge For an asymptotically flat spacetime, it is convenient to choose the Poincare gauge. Kawai has proposed a nontrivial boundary condition for the e(x), namely, to take asymptotically, (7'17) where Xk are cartesian coordinates of the asymptotic Minkowski spacetime. 25 ) It is not yet clear whether (7 '17) is gaugeequivalent to the trivial boundary condition of c;k=o at infinity.. For Baeklertype solutions which asymptotically approach (anti)de Sitter spacetime, however, it would be more convenient to choose a gauge in which the field e(x) asymptotically tends to the fivedimensional coordinates c;(o)a(x) of the asymptotic (anti)de Sitter spacetime: More precisely, take the fivedimensional metric 7Jab as the same as that of the asymptotic (anti)de Sitter spacetime, and assume that ';:'c;=const (=';:(O).c;(O») everywhere, asymptotically. (7 18) The gauge field then vanishes asymptotically. We shall name this gauge the (anti) de Sitter gauge. We expect this gauge to be a convenient framework to study mass and spin of the systems described by those exact solutions. 26 ) 8. Conclusions We have formulated the (anti)de Sitter gauge theory in such a manner that it does not depend on the dimensional constant.q which specifies the constant curvature of the initial (anti)de Sitter spacetime. The basic dynamical variables are the fivedimensional vector field e, the gauge field A ab v and matter fields. Here matter fields mean those fields which exist before gauging procedure is applied to the isometry group of the initial spacetime of constant curvature. The underlying spacetime is a RiemannCartan space which possesses a metric and a nonsymmetric affine connection, both of which are defined by covariant derivatives of the field,;:a.' The,;:scale transformation plays a crucial role in the present approach. This transformation produces change in c;a and A ab v: In particular, the e is subject to a scale change like,;:a f4 A(x)e with the parameter A(x) being an arbitrary positive function. However, no change is produced in the metric, the affine connection and matter fields. We have made the basic postulate that physically meaningful content of the
15 Gauge Theory of Gravitation 725 (anti)de Sitter gauge theory be ';scale invariant in' addition to being (anti)de Sitter gauge invariant. The gravitational Lagrange function is then a scalar formed of the metric tensor, the curvature tensor and the torsion tensor, since only these three are ';scale invariant among those tensors constructed from the basic dynamical variables,.;a and A ab ll, and their first derivatives_ Consequently, the most general gravitational Lagrange function is the same as that in the Poincare gauge theory. In the (anti)de Sitter gauge theory presented here, the curvature and the torsion tensors are just the two ';scale invariant parts of the gauge field strength. Thus, the ';scale invariant structure of the (anti)de Sitter gauge theory coincides with that of RiemannCartan space. This is to be compared with the situation in the Poincare gauge theory, where the curvature and the torsion tensors are directly related to the Lorentz and the translation gauge field strengths, respectively. As for matter fields, we have discussed the spinor field as the typical example. In this case the Lagrange function is ';scale invariant because the covariant derivative acting on the spinor field is ';scale invariant. Three gauge choices have been illustrated as examples: (1) The Poincare gauge which reproduces Kawai's approach to the Poincare gauge theory, (2) the restricted Poincare gauge which reproduces the ordinary Poincare gauge theory and (3) the (anti)de Sitter gauge which is appropriate when spacetime asymptotically approaches (anti)de Sitter space. It seems interesting to study the asymptotically nonflat exact solutions of the Poincare gauge theory in the light of the present theory. In particular, we expect that the (anti)de Sitter gauge will be a convenient tool to investigate energy, momentum and angular momentum of the systems described by those exact solutions. Appendix  The Spinor Wave Equation in (anti)de Sitter SPacetime*) The spin or wave equation in (anti)de Sitter spacetime has been discussed by many people since Dirac's pioneering Vl:'ork. 21 ),27) It seems, however, that little attention has been paid on the relation to the tetrad formalism on curved spacetime. We shall therefore start from the tetrad formalism, and establish its relation to Dirac's spin or wave equation. Let ekll(x) (k=o~3) be a tetrad (or a vierbein) field in (anti)de Sitter spacetime, and introduce a spinor field rp<e)(x) by referring to it. This spin or field undergoes spin or transformations for local Lorentz transformations of the tetrad field: For infinitesimal ones given by (A'I) the rp<e) transforms like (A'2) where (f)km( = (f)[km]) are infinitesimal arbitrary functions antisymmetric in k and m, and Skm are the Lorentz generators, Skm=(ij4)[y\ ym] with yk being 4x4 Dirac *) We use the same notations and conventions as in 2.
16 726 T. Shirajuji and M. Suzuki matrices. The Dirac equation for (e) reads (iek U yk17v'e)+m) (e)=o, (A 3) which follows from the Lagrangian density LM(O)= j  g [(if2)ek U (e)(yk17 v'el 1t)e)yk) +m (e) (e)]. Here m is the mass, [7 U (e) is defined by [7 U (e) = au + (if 2)ilkmuSkm (A 4) (A 5) with ilkmu( =il[km]u) being the Ricci rotation coefficients formed of the tetrad field, and finally the adjoint (e) is (e)= (e)t yo. Now, since (anti)de Sitter spacetime is a hypersurface satisfying e~a=r;q2, we can define a flinfbeiri field e={ea,}={ea,b} (a'=0~3, 5) by for a'=k=0~3, for a'=5, which satisfies the orthonormality relation, (A 6) (A 7) We can regard the spinor field also as an 0(4,1) (or 0(3,2)) spin or field referred to this flinfbein fields:*) 0(4,1) (or 0(3,2)) spinors are denoted as instead of (e). For (anti)de Sitter transformations of the flinfbein field given by**) with A a'c' A b'd' r;c'd'= r;a'b', the spin or field transforms like ~ U(A), where U(A) satisfies (A S) (A 9) (A 10) Here we understand that the primed latin indices, a', b' etc., are raised or lowered with the flat metric r;a'b' or r;a'b'. For infinitesimal transformations Aa'b,=r;a'b'+Wa'b.' with Wa'b'= W[a'b'], U(A) is given by U(A)=l + (i/2)wa'b,sa'b'. (A n) It follows from (A n) that U(A) is pseudounitary, satisfying UtyO=yOU 1 Ut y5 yo= y5 yo U 1 So we define the adjoint by for 0(4,1), for 0(3,2). (A 12) *) The 0(4,1) and 0(3,2) correspond to de Sitter and antide Sitter spacetimes, respectively. **) These transformations can be local with Aa'b' being positiondependent.
17 Gauge Theory of Gravitation 727 so that it transforms as if) f4 if)u(jl)1. for 0(4,1), for 0(3,2), (A B) (A 14) The covariant derivative (A 5) can be rewritten as a five dimensional form, rlv=av+(i/2)lia'b'vs a 'b', (A 15) where the generalized Ricci rotation coefficients Lia'b'v are given by  {Llkmv Lla'b'V= 0 for (a', b')=(k, m) (k, m=0~3), for (a', b')=(5, k) or (k, 5). (A 16) The Lagrangian density of (A4) can be expressed by u~ing rp, its adjoint if) of (AB) and their covariant derivatives as follows: for 0(4,1) or LM(O)=J g ($a,/q)[icb'v if)(sa'b'rlvvvsa'b')rp+ imif)r,a' rp] for 0(3, 2), where and for a'=k, for a'=5 (A17a) (A17b) (A 19) The Lagrangian density (A 17) is invariant under (anti )de Sitter transformations (A8) and (A9), if the generalized Ricci rotation coefficients are changed like since the derivatives rl vrp and if)rl then transform as (A 20) rlvrp f4 U(A)(rlvrp), if)rlv f4 (if)vv)u(a)1, (A21) apd since $a'.and ca'v are changed as (A 22)
18 728 T. Shirajuji and M. Suzuki fivedimensional flat space, in which the (anti)de Sitter spacetime is embedded, and consider the local (anti)de Sitter transformation which brings the flinfbein e of (A 6) to the global frame e(o}:*} (A 23) Since ea(o}b=oa b, the transformation coefficients Aac' are just the inverse of the components of the flinfbein ec,b: Namely, it follows from (A 6) and (A 7) that k v A C', c' _ e vea a  e a{ r;t;a/q for c'=k for c'=5 (A 24) with 71= Using (A 24) in (A 20), we get the following expression for the generalized Ricci rotation coefficients with respect to the global frame: Lf~OJv=(t;acbv  t;bcav) / (t;. t;), (A 25) where cav are given by (2 4). All the quantities are now expressed by referring to the global frame. In particular, the Lagrangian density of the spinor field given by (A 17) reads LM(O}= )  g [(i/2)cav (f(ra[7 )O)  V v(o} ra) + m(f ] for 0(4,1), or LM(O}=) g (t;a/q)[icbv(f(sab[7 )0>_ 17v(O}sab) + im(fra ] for 0(3,2). Here [7 v(o} stands for the derivative operator [7 v(o}=av+(i/2)lf~ojvsab, (A 26a) (A 26b) (A 27) which uses the generalized Ricci rotation coefficients with respect to the global frame given by (A 25). By means of the relations, with cabcdecfj.vplfcb fj.ccvcdpce (j= 2471)  g (t;a/q), cabcdecfj.vplfccvcdpce (j~ 671)  g (t;acbv t;bcav)/q cabcde= + 1 for (abcde)=(01235), Cp.vPlf= + 1 for (PlIPo)=(0123). The Lagrangian density of (A. 26b) can be rewritten as (A 28a) (A 28b) (A 29). (A 30) *} Note that the five vectors of the global frame are labeled by an unprimed latin index. This is to be contrasted with a fiinfbein like e={ea,}={ea,b} of (A'6) which has two kinds of indices, a primed imd an. unprimed latin indices: An unprimed latin index represents components of a fivevector with respect to the global frame, while a primed one numbers the five vectors of a fiinfbein.
19 Gauge Theory of Gravitation 729 for 0(3,2), which does not involve e/q. The spin or wave equation can be derived from the Lagrangian density, and it reads for 0(4,1), and (A 31) [2~aCbllsab/7 )0)+ m~ara] =o or equivalently (A 32a) (A 32b) for 0(3, 2). Use of (A 28a, b) in (A 32b) immediately gives (A 32a), and (A 32a) can be put in the same form as (A 31) because (~ara) and (Cbllrb) anticommute with each other due to the orthogonality relation (2 6). Accordingly, the spinor wave equation is given by (A 31) in both de Sitter and antide Sitter space times. Equation (A 31) can be rewritten as with the help ofthe relation (~afa)2= ~.~. D=(i/2/~ ~)rarb(~aa/ae ~ba/ae) =(~ara//~ ~)Cbllrball, and the new spinor fields (±) by the unitary transformation, Then they satisfy which are Dirac's spinor equations in (anti)de Sitter spacetime. References (A 33) Let us introduce the operator D by (A 34) (A 35) (A 36) 1) R. Utiyama, Phys. Rev. 101 (1956), ) T. Nakano, in SOryilshi no Honshitsu, ed. M. Taketani (Iwanami, Tokyo, 1963), p. 161 (in Japanese). K. Hayashi and T. Nakano, Prog. Theor. Phys. 38 (1967), 49l. 3) T. W. B. Kibble, J. Math. Phys. 2 (1961), ) K. Hayashi, Prog. Theor. Phys. 39 (1968), 494. K. Hayashi and A. Bregman, Ann. of Phys. 75 (1973), 562. K. Hayashi and T. Shirafuji, Prog. Theor. Phys. 64 (1980), 866, 883; 65 (1981), ) F. W. Hehl, P. von der Heyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. 48 (1976), 393. F. W. Hehl, in Proceedings of the 6th Course of International School of Cosmology and Gravitation on "Spin, Torsion and Super gravity" ed. P. G. Bergmann and V. Sabbata (Plenum, New York, 1980), p. 5. Earlier references are cited in these review papers. 6) E. M. Mielke, Geometrodynamics of Gauge Fields  On the Geometry of Yang Mills and Gravitational Gauge Fields (Akademie Verlag, Berlin, 1987). 7) T. Kawai, Gen. ReI. Grav. 18 (1986), 995; 19 (1987), 1285E; Prog. Theor. Phys. 76 (1986), ) P. Baekler, Phys. Lett. 99B (1981), 329..
20 730 T. Shirajuji and M. Suzuki C. H. Lee, Phys. Lett. 130B (1983), 257. P. Baekler and F. W. Hehl, in From SU(3) to Gravity Festschrift in Honor of Yuval Ne'eman ed. E. Gotsman and G. Tauber (Cambridge U.P., Cambridge, 1985), p ) T. Kawai and H. Yoshida, Prog. Theor. Phys. 62 (1979), 266; 64 (1980), ) A. Inomata and M. Trinkala, Phys. Rev. D19 (1979), ) P. Fre, Nuovo Cim. 53B (1979), ) S. W. MacDowell and F. Mansouri, Phys. Rev. Lett. 38 (1977), ) P. K. Townsend, Phys. Rev. D15 (1977), ) K. S. Stelle and P. C. West, Phys. Rev. D21 (1980), ) J. P. Hsu, Phys. Rev. Lett. 42 (1979), 934; Nuovo Cim. 61B (1981), ) T.. Fukuyama, Z. Phys. C10 (1981), 9; Ann. of Phys. 157 (1984), 321. T. Fukuyama and K. Kamimura, Nuovo Cim. 74B (1983), ) A. A. Tseytlin, Phys. Rev. ])26 (1982), ) H. Pagels, Phys. Rev. D29 (1984), ) See, for example, S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space Time (Cambridge U.P., Cambridge, 1973). 20) For general discussions of the (anti )de Sitter group, see, for example, F. GUrsey, in Group Theoretical Concepts and Methods in Elementary Particle Physics, ed. F. GUrsey (Gordon. Breach, New York, 1962), p ) P. A. M. Dirac, Ann. of Math. 36 (1935), ) See, for example, S. Kobayashi and K. Nomizu, Foundations of Differential Geometry I (John Wiley, New York, 1963). 23) See Eq. (3 14) of K. Hayashi and T. Shirafuji, Prog. Theor. Phys. 58 (1977), ) K. Hayashi and T. Shirafuji, Prog. Theor. Phys. 65 (1981), ) T. Kawai, Prog. Theor. Phys. 79 (1988), 920. T. Fukui, T. Kawai and H. Saitoh, Prog. Theor. Phys. 79 (1988), ) For an attempt using the ordinary Poincare gauge theory, see P. Baekler, R. Hecht, F. W. Hehl and T. Shirafuji, Prog. Theor. Phys. 78 (1987), ) See Ref. 20) for a review. Many references are cited also in W. Drechsler, Fortsch. der Phys. 23 (1975), 607; W. Drechsler and R. Sasaki, Nuovo Cim. 46A (1978), 527.
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