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Gravity as Yang-Mills gauge theory
Nuclear Physics B262 (1985) 144-158 (~) North-Holland Publishing C o m p a n y
GRAVITY AS Y A N G - M I L L S G A U G E T H E O R Y H. D E H N E N and F. G H A B O U S S I
Fakultiit fiir Physik der Universit~it Konstanz, D-7750 Konstanz, Postfach 5560, Fed. Rep. Germany Received 5 March 1985 We propose a Yang-Mills field theory of gravity based on a unitary phase-gauge-invariance of the lagrangian where the gauge transformations are those of the S U ( 2 ) × U(1) symmetry of the 2-spinors. In the classical limit this microscopic theory results in Einstein's metrical theory of gravity, where we restrict ourselves in a first step to its linearized version.
I. Introduction
In Einstein's general relativity the gravitational interaction is described by the metric field of space-time based on the equivalence of inertial and (passive) gravitational mass (equivalence principle). According to Einstein's field equations the source of the metric field (gravitational field) is the energy-momentum stress tensor of matter, which is given ultimately because of the quantum structure of matter by the expectation values of the energy-momentum stress operators. Thus Einstein's description of gravity is a purely macroscopic one and on this level in best accordance with the known experimental data at least within the first-order approximation*. Accordingly the classical metric gravitational field represents the background of space-time on which all other physics happens; it reacts back on the metric field according to Einstein's field equations. But here the first problem arises. The back-reaction of a single quantum process of matter on the classical metric field cannot be considered consistently in Einstein's theory [1]. Furthermore on the purely microscopic level there exist neither a theory nor experiments of gravitational interactions up to the present day. However there is some evidence that, for the gravitational interaction of elementary particles the metrical description of gravity may not be the suitable one. First of all, all attempts at quantization of the metric field have been without success. Secondly, only in the case of a classical particle does the mass drop out of the equation of motion for gravitation as a consequence of the equivalence principle, so that all masses are moved in the same way by gravity and the gravitational field * So long as the solar mass quadruple moment is unknown, the perihelion shift of Mercury is not a precise test with regard to the second-order approximation. 144
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can be described by a geometrical "guiding field". But in contrast to this, the motion of a quantum particle state is not independent of the particle mass even in the presence of a classical gravitational field [2]. In spite of the validity of the equivalence principle the particle mass does not drop out of the Schr6dinger equation. Thus there is no universal guidance of the quantum processes by gravity [3] and consequently there is no hint of a necessary geometrization of the gravitational interaction on the quantum level. Finally, Einstein's metrical theory of gravitation has a completely alternative structure to the successful gauge theories of electroweak and strong interactions. Therefore a unification of all interactions needs a modification of the standard description of gravity. The possibilities discussed until now mainly consist of supergravity [4] and modern K a l u z a - K l e i n theories [5], where in both cases the metric is still retained as a fundamental structure of gravitation on the microscopic level. But these models have other problems, especially for instance, the required existence of too m a n y exotic elementary particles. For this reason, in this p a p e r we want to point out the possibility of the construction of a unitary gauge theory of gravity on the microscopic level in accordance with the gauge theories of electroweak and strong interactions. Then the gravitational interaction would also be produced by the exchange of the usual vector gauge bosons on the background of the Minkowski space-time of special relativity. Thus on the quantum level, gravitation would not be described by a metrical tensor field, but by gauge vector fields*. Only in a certain classical limit, i.e. by cooperation of many vector bosons, can Einstein's metrical description of classical gravity result in at least its tested linearized version. According to this line the non-euclidean metric of space-time is not a fundamental, but more an effective field produced by combination of gauge vector fields like a condensation phenomenon. It has only a classical meaning describing the classical gravitational field alone. We emphasize that we are pursuing here the idea that quantum or microscopic physics has priority and follows directly from a few very general first principles, whereas all macroscopic physics must be deduced from its classical limit.
2. The group theoretical concept Following the above mentioned line the first aim of this p a p e r is a quantum theoretical description of the gravitational interaction between elementary particles using a usual gauge field theory based on a phase-gauge-invariance of the lagrangian where, with respect to the interpretation of the quantum theory, only unitary gauge transformations are allowed. Furthermore, because gravity couples to all particles, a gauge transformation is needed which results from the transformation of the * After preparing this paper we were informed that another attempt in this direction had been made [6] but with very different assumptions, e.g. a euclidean metric and a non-Yang-Mills lagrangian.
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intrinsic spinor structure of each particle and not from a transformation between different particles of any multiplet. Finally, gravity is connected with the concept of "mass". However this concept is still a classical one and should therefore be avoided in a quantum approach to gravity. In accordance with the modern gauge theoretical treatment of interactions one has to start with massless particles and the "mass" appears subsequently by a dynamical process, namely, as a consequence of spontaneous symmetry breaking. As we shall see later in sect. 4 this is also the only satisfactory way of applying Einstein's tensor theory in the classical limit starting from a microscopic vector gauge theory of gravity. On doing so, the empirical equivalence between the inertial and gravitational mass of particles (which is the basis of general relativity) may be found theoretically on the microscopic level. However in the following we do not want to discuss the mass problem in detail; that would be outside the bounds of this paper. We consider only the gravitational interaction of massless particles as a first step. Thus we start from the general transformation group GL(2, C) of the 2-spinors which we take as the most general symmetry group. However, all generators of this group can be constructed by complex linear combinations of the generators of SU(2) x U(1), i.e. of the unit and Pauli matrices [7]. Thus instead of a gauge theory for the complete group GL(2, C), we propose to gauge only the S U ( 2 ) x U(1) symmetry by which the basic quantum theoretical requirement of the unitarity of the phase-gauge-transformations is automatically fulfilled. This idea is supported by the mathematical fact that one can reduce the group GL(2, C) to its maximal compact subgroup U(2) by contraction [8]. However, it is well-known that the group SL(2, C) is the covering group of the Lorentz group. From this point of view we indeed gauge the Lorentz group in the sense that we gauge its "basic" transformations [7], namely those of SU(2). Although we start from the 2-spinor calculus for massless particles we perform our investigations mainly in the 4-spinor representation, i.e. in the 4 x 4 representation of SU(2) x U(1) with respect to a later consideration of particle masses. Furthermore we restrict ourselves to a treatment of the theory on the level of the first quantization. However in the case of the compact gauge group SU(2) × U(1) there are no problems with quantization; the theory is renormalizable in the usual way and it gives no ghost-fields. According to our gauge group there exist 4 gauge vector fields and it is possible 'to introduce with respect to the reducibility of the 4 x 4 representation of SU(2) x U(1), 2 or even 3 different gauge-coupling constants. However because all gauge bosons mediate the same interaction between all spin-½ particles, i.e. gravity, we give all these coupling constants the same value*. Under this presumption we first construct as follows a gauge theory of the SU(2) x U(1) symmetry for gravity on the Minkowski space-time background and then we show that in the classical limit a metric description of gravity for the expectation values results, where the * A d e e p e r u n d e r s t a n d i n g o f this ansatz can be e x p e c t e d from a unification o f all i n t e r a c t i o n s at s o m e later date.
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m e t r i c fulfils E i n s t e i n ' s field e q u a t i o n s . H e r e we restrict ourselves for s i m p l i c i t y to the l i n e a r i z e d theory. 3. The l a g r a n g i a n of microscopic gravity W e define the t r a n s f o r m a t i o n m a t r i c e s o f the g r o u p S U ( 2 ) x U ( 1 ) r e p r e s e n t a t i o n by*
in the 4 × 4
U = e iA°(x~)'° with
(3.1)
(o oo)
I cr "r = ~ 0
(3.1a)
(or° is the u n i t - m a t r i x a n d or 1, cr2, tr 3 are the Pauli matrices). The c o m m u t a t o r relations for the g e n e r a t o r s z a are: [ ~.b, re] = ie bCza .
(3.1b)
T h e n the s p i n o r s t a t e - f u n c t i o n ~0 a n d the D i r a c matrices 7 ~' t r a n s f o r m as: 6 ' = Uq,,
~,'"= U v " U - ' ,
(3.2)
with the i n v a r i a n t r e l a t i o n : y(" y~} = *7"~
(3.2a)
(r/~'~ = rh,~ = d i a g ( - 1 , 1, 1, 1) M i n k o w s k i - m e t r i c ) . A c c o r d i n g l y the c o v a r i a n t s p i n o r derivative r e a d s : D . ~ = ( 0 . + igto.)qJ, ( g is the u n i f o r m g a u g e - c o u p l i n g c o n s t a n t ) with the S U ( 2 ) x U ( 1 ) defined by w, = w~,j a ,
(3.3) g a u g e fields to.. (3.3a)
w h i c h t r a n s f o r m u n d e r g a u g e t r a n s f o r m a t i o n as"* , 1 i w , = U w , U - + - UI~U -1 . g
(3.4)
F u r t h e r m o r e it f o l l o w s f r o m (3.2) that the y " are o n l y c o v a r i a n t l y c o n s t a n t , i.e. D,~T" =- Y~'l~ + ig[w~, y " ] = 0 ,
(3.5)
which, t o g e t h e r with the c o n d i t i o n (3.2a), is the d e t e r m i n a t i o n e q u a t i o n for y " in a d d i t i o n to the field e q u a t i o n s (3.11) a n d (3.12). N o w we define the g a u g e - f i e l d strength o f m i c r o s c o p i c gravity in the usual w a y b y the c o m m u t a t o r o f the c o v a r i a n t derivative given in (3.3): 1 V,~ ~ z-- [ D , , D~] = w~l. - o9,1~ + ig[t%, to~]. tg
(3.6)
* In the following Einstein's sum convention is used. Latin indices a, b, etc. run from 0 to 3 and are related to the basis of the group algebra. Greek indices run from 0 to 3 and are related to the space-time coordinates. ** The symbol I/z means the usual partial derivative with respect to the coordinate x*'.
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For the components of Fp`, with respect to r ~ it follows immediately from (3.3a) and (3.1b): Fp`~, = w~alp` - w p`,l ~ - geab~co p`btO~c ,
(3.7)
satisfying the Bianchi identities F~CP`~I* l + gF'abCFb[p`d'OX lc
=
0.
(3.8)
In the case of the p u r e gauge field
i g ~ u u "1 , g
(3.9a)
Fp`~, - 0.
(3.9b)
COp, ~ O~p` = ~
it holds: Fp`~=- 0,
With the principle of minimal coupling the gauge-invariant Lagrange density of gauge gravity for a massless particle takes the very simple form: h l~P.v ab 3 f = 167rk" p`~a--b ~7 -~hiq~yp`D~,@ + h.c. .
(3.10)
Here we have employed as group-space metric the Minkowski matrix rl ab, because we need (with respect to the classical limit of this microscopic theory) the U(1) gauge potential to be time-like and that of SU(2) to be space-like*. Furthermore we have taken the opportunity to introduce besides the gauge-coupling constant g a second coupling constant k between the two gauge-invariant parts of the Lagrange density. Both coupling constants must be determined by experiment or by the transition to the macroscopic classical limit, where k comes out to be proportional to the newtonian gravitational constant (7, whereas g does not possess a classical limiting value and has only the meaning of a microphysical "heavy charge". However, both coupling constants are necessary in the following and cannot be substituted by one another. The field equations following from the action principle belonging to (3.10) are given by Op`F p`~" + ge"bCF~"wp`,. = 2 ¢rkg~{ 7 ~, ra}qJ
(3.11)
,y~(Op` + igcop`)@ = 0.
(3.12)
and, with (3.5), by
Finally the canonical energy-momentum tensor of the whole system takes the form with respect to (3.12): TltV
1. - v h vaa Rv~ =~th(~by ~0tp`-qTip`y"~0)-~--~k(coo~,tp`F - z lr~ctfl~?a - ~ --o,0o.o.j.
(3.13)
* Evidently the action corresponding to (3.10) is bounded from below after a Wick rotation of time, according to which not only the space-time metric but also the group-space metric goes over from a pseudo-euclidean to a euclidean structure.
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It has the property (with the use of the field equations): = ~ghOy t%cr ~D,O + h.c.,
(3.13a)
where o-~,~= ½[y,, y~] is the spin operator and the right-hand side represents a 4-force density between the gauge field and spin of the particle*. Evidently in our microscopic theory of gravitation there exists 4 gauge-vector fields, to,o, which couple to 4 "heavy" gauge currents of matter (@field) as is usual in Yang-Miils field theories: j~'~ = ½gt~{~/~, r"}qJ.
(3.14)
Due to the fact that the U(1) gauge current j , o is proportional to the time-like probability current density and is therefore correlated with the 4-momentum density of the particle field, we expect that the field equations (3.11), (3.12) are indeed those for gravity. This supposition will be confirmed subsequently by investigation of the classical limit of this microscopic theory.
4. The classical limit
We can be sure that the theory proposed in sect. 3 is a theory of gravity, if in the classical limit Einstein's metrical theory of gravitation results. For this it must be shown firstly, that according to Ehrenfest's theorem the expectation values of the 4-momentum of the particle field (qJ-fieid) fulfils the usual equations of motion for classical gravity. This means in detail, that from the equations of motion the classical gravitational force must be read off; it must be reducible to geometrical connection coefficients (macroscopic field strengths), because classical gravity is a geometrical theory. Secondly, it must be shown that the connection coefficients are metrical and thirdly, that they fulfil Einstein's field equations with the energy-momentum tensor of the ~b-field (matter field) as source. In doing this we restrict ourselves for simplicity to the linearized Einstein theory as the only empirically confirmed part of the complete theory.
4.1. THE EQUATION OF MOTION
We start from the result (3.13a), neglecting in the classical limit the interaction between the gauge field and spin of the particle. Then we have** Tg°lo+ Tfl j = 0.
(4.1)
* The right-hand side of (3.13a) results from the fact that the Dirac matrices within the lagrangian (3.10) depend explicitly on the space-time coordinates according to (3.5). Choosing the gauge-invariant canonical energy-momentum tensor instead of (3.13) the right side of (3.13a) vanishes identically, which implies (4.1) automatically. ** Latin indices j, k, etc. run from 1 to 3 and are related to the space-like coordinates only.
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After insertion of (3.13) into (4.1) and integration over the 3-dimensional hypersurface, t = const, we obtain: ~9° ½ih(0Y°Ol" - 0['T°0) d3x = ~ k
(w,~l,,F
-
)lo d3x
(Fo Fa3)l. d3x ,
(4.2)
where boundary integrals concerning the pure 0-field are dropped out with respect to the normalization condition for the wave function f 0*0 d3x = 1. Considering (3.13) the canonical energy-momentum tensor of the matter field is given by: T,~(0)
l. - ~01~,= ~th(Oy
~I,,Y~0) •
(4.3)
According to this the integrand on the left-hand side of (4.2) is identical with the 4-momentum density T,°(O) of the particle field. From now on, with the use of the field eqs. (3.11), eq. (4.2) takes the form (neglecting boundary integrals of the 0-field):
Oof [ T.°(O) ltl -41rk
I
½figO(y°to~ + w . y ° ) 0 ] d3x
1 a/3 a 3 [(Fo,oF3c~a )13-~(F~ F~3)I~,] d x.
(4.4)
Here the bracket on the left-hand side represents the gauge-invariant canonical 4-momentum density of the particle field obtained from (4.3) by replacing the ordinary derivatives with the covariant ones. However, the second term on the left-hand side of (4.4) is of the order of the coupling constant g compared to the first term and therefore can be neglected within a first-order approximation with respect to gravitational interaction. Then one obtains with the help of the Bianchi identities (3.8) and the field equations (3.11) the following equation of motion for the expectation value of the 4-momentum of the 0-field:
Oof T.°(O)d3x=h f F.~aj ~a d3x.
(4.5)
Here on the right-hand side
K.=-hfF.~.j"~ d3x=fF.~a~hgO{y, r~}O 1
--ct
d3x
(4.6)
represents the classical gravitational 4-force as a product of the gauge-field strengths and the gauge currents of matter.
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In view of a classical geometrical interpretation the integrand in (4.6) must be reduced to a product between classical non-algebra-valued connection coefficients and the energy-momentum tensor of the matter field. For this reason we write:
-- a , ra}~b d3x K , = I F~,~bt~a bl5hg~b{y
(4.6a)
and introduce as "correspondence condition" between the quantum and the classical theory of gravity the following normalization of the gauge-vector fields:
co ~to b = ~ b + eab(x, ) ,
(4.7)
leabl "~ 1
(4.7a)
where we choose
in view of a weak field approximation to the theory*. According to this, in the case of the pure gauge-fields (3.9a) where F~,~ -= 0, one has to take, in (4.7), e~ =- 0 which is always possible. Then (small) deviations from the pure gauge-field case produce (small) values for F,~a and e~b different from zero, so that the normalization condition (4.7) has no restrictive meaning. Accordingly the U(1) gauge field CO,o is time-like, whereas the three SU(2) gauge fields are space-like with respect to the group-space metric introduced in (3.10). Only in this way can the pseudo-riemannian structure of the metric be guaranteed later. Consequently, in the case of the linearized theory the Kronecker symbol in (4.6a) can be substituted by the left-hand side of (4.7). One obtains immediately with the use of (3.3a): gp.
=
bl u f F~btov ~hgO{3, , to~}0 d3x.
(4.8)
By this procedure we have obtained the product in (4.6a) separated into two factors, which are both (pseudo-) scalars with respect to the gauge transformations and therefore can be interpreted classically: the first scalar in (4.8) corresponds to the connection coefficients, the second one should be related to the energy-momentum tensor of the matter field. For an investigation of this last relation we eliminate the ~o~ field in the anticommutator of (4.8) with the help of the field equation (3.12). Then we get, using (3.2a):
l hgd~{ 3,~, to v} ~b = ~ih ( d~3,~b I~ - d~t~3/~b),
(4.9)
where the terms proportional to the spin operator cr~'~ interacting with F~,~b are neglected with regard to the classical limit. Comparison with (4.3) shows that the right-hand side of (4.9) is identical with the energy-momentum tensor T ~ ( O ) of the matter field. We mention explicitly that this essential result, which connects the * If we avoid in (3.3) the explicit appearance of a gauge-coupling constant, this must be introduced at the latest in (4.7), see also (4.16).
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gauge currents with the energy-momentum tensor and which makes possible at last the transition to Einstein's theory, is valid only in the absence of the mass term in the field equation (3.12). Hence with the use of (4.9) eq. (4.8) takes the form:
K, = f F~.~aoJ~T~"(O)d3x,
(4.10)
which has now the structure of the usual gravitational force. We note that as a consequence of relation (4.9) the gauge-coupling constant g no longer appears explicitly in (4.10). For the full-linearized classical limit of the equation of motion one has to use in (4.10) the energy-momentum tensor T"~(0) in the lowest WKB approximation of the free particle field. One finds from (4.3) (see appendix): Tt"~](¢) = 0 ,
T,," (0) = 0 ,
(4.11)
according to which T""(O) is symmetric and traceless. The first property results from neglecting the spin of the particles and the last one is a direct consequence of the masslessness of the particles and valid only in that case. Applying this result to the relation (4.10) we find for the equation of motion (4.5):
ao f T"°(*)d3x= f F:(,~oJ:)T~'~(*)d3x,
(4.12)
which we have now to compare with the corresponding one according to general relativity: 0o
f
T"°(@) d 3 x - -
f F',~.T~'(O)
d3x-
F .aT
(0) d3x-
(4.13)
The last term on the right-hand side of (4.13) vanishes because of F"~. = 0 in the case of a traceless energy-momentum tensor (see (4.11)) as source of the classical gravitational field (cf. eq. (4.32)). Then we find by comparison of (4.12) and (4.13) the following correlation between the geometrical connection coefficients F " . ~ and the microscopic gauge-field strengths F.~.: F " ~ = -Fa(~to~).
(4.14)
Thus we have arrived at a geometrical description of the gravitational force in the classical limit of our microscopic gauge theory without using any principle like the equivalence principle. 4.2. THE M E T R I C
In order to find out the metrical structure of our connection coefficients we reform the right-hand side of eq. (4.14) by inserting the definition (3.7) and obtain: 1
IzA
a
a
a
1
a
a
p.
- (to.~to ~)fA] -5(to ~1~+ tO~l~)~oa .
(4.15)
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Here the bracket has indeed the structure of a l i n e a r i z e d Christoffel symbol with the metric defined by g~,v --- to~oto~ •
(4.16)
Then the connection (4.15) is a m e t r i c one after choosing the gauge-fixing condition: to(~l~> = 0
(4.17)
as a fully relevant constraint for the classical metrical limit of our theory, which is realizable in the lowest W K B limit of eq. (3.11). Obviously, the microscopic theory possesses more structures than appear in the classical metrical limit. For the metrical limit the solutions of the microscopic theory are restricted to those gauge fields which are at least in the WKB limit Killing vector fields with respect to the Minkowski metric. Simultaneously the gauge transformations are b o u n d e d to those, which allow (4.17) to become invariant, i.e. in the case of linearized gauge transformations (see (4.21)) only gauge functions ;ta are allowed satisfying bc
hal~,h~+ gea hbl(~w~>c = 0.
(4.17a)
Then the covariant derivative of the metric (4.16) vanishes with respect to the connections defined by (4.15) and (4.17): g~,~Lx - F ~ , x g ~
- F~g~,~
=0.
(4.18)
In the case of p u r e gauge fields (3.9a) with F,~, --- 0 the connection coefficients (4.14) and (4.15) vanish identically and the metric (4.16) is reduced to the Minkowski metric r/,~, where in the normalization condition (4.7) e, b - 0 is valid. In general by decomposition of the metric (4.16) according to g~,~=rl~,~+h~,,,
Ih.~l~l
(4.19)
one finds with the use of (4.7) the following relations within the linearized theory: hl~ v = F,abC~o~bOjav , vt e~ b = to ~b toan~,~.
(4.20)
Herewith the normalization condition (4.7) takes the form of the usual tetrad condition in general relativity to .btovagt'Lv = t~ ba ,
(4.20a)
where g~'~ = 7/~'~- h ~'~ with g~'~g~A= 6 ~ *. Of course, the metric (4.16) is not a gauge-invariant quantity (pseudo-scalar). From this it follows that from a gauge transformation of the gauge fields a coordinate transformation of the metric field is induced. Here we restrict ourselves within the • N o t e t h a t in contrast to the r e m a i n i n g d e s i g n a t i o n in this paper, here g " ~ inverse to g~,~.
"0~%l~°g~o, b u t g ~
is
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linearized theory to infinitesimal transformations. Then from (3.1), (3.3a) and (3.4) the gauge-field transformation is as follows (IA~I¢ 1): 1
(4.21)
toga = tolza --g l~al~ -- 8abCabtot~c •
Herewith we obtain according to (4.16) the " n e w " metric: g~,,, . =. "q~,~ . + .h j , , =.
w . ~,,,to ~ ,
(4.22)
where in view of (4.19) h~,~=h~-
a .. to~Aal~_ 1gt°~Aal
(4.22a)
However, in the case of a transition from g~,~ to g~,~ by the infinitesimal coordinate transformation x " = x t' + sC'(x ~) (with Ix l) there exists the well-known transformation law for the deviations from the Minkowski metric: h~,~ = ht,~ - ~:t,l~- ~:~lt, •
(4.23)
Comparison of (4.22a) and (4.23) yields immediately with respect to (4.17) the following connection between infinitesimal gauge and coordinate transformations: 1
= - to A~. g
(4.24)
However, only those gauge transformations are allowed which do not disturb the gauge fixing condition (4.17), see (4.17a). Accordingly, the (infinitesimal) coordinate transformations are limited. As we see later, the gauge-fixing condition (4.17) implies (on the level of the field equations) the de Donder gauge for the metric*, to which the coordinate transformations are bounded, see (4.29).
4,3. THE F I E L D E Q U A T I O N S
After definition of the metric (4.16) through the gauge potentials it remains to be shown that the metric or its connection coefficients satisfy Einstein's linearized field equations. These must follow as the classical limit of the gauge-field equations (3.11). Therefore one first has to remove their algebra-valuedness. For this we multiply eqs. (3.11 ) by the gauge fields toga and obtain with the use of (3.3a) and (3.7): /x
a
o,. ( Fa.,to ,.) -
/.t
a
/.L
a
= 2~rlig~{ %,.
,o,.}~,
(4.25)
The right-hand side is with respect to the basic relation (4.9) already proportional to the energy-momentum tensor of the matter field; terms containing the spin tensor * The usual de Donder gauge for the metric (4.19) reads ht,~p~-½hl~ = 0 (h = h~'~) and corresponds to the Lorentz gauge in electrodynamics, see ref. [9].
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or~'" are neglected with regard to the classical limit. On the left-hand side of (4.25) the second and third term can be neglected in view of the linearization. Thus we get: ~' " =47r -k a,,(F~vw~) h T~(qJ),
(4.26)
which is reduced with respect to (4.11) to its symmetric and traceless part: o
a,,(F~(~w~)) =47r
T~,(q,)
(4.27)
with a. ( F ~ o J ) - 0 bt
va
__
(4.28)
.
Obviously, also in the classical limit of the field equations - as in the classical limit of the equations of motion - the gauge-coupling constant g no longer appears explicitly, so that it cannot be determined by comparison with the classical Einstein theory. Using (3.7) the last equation (4.28) implies: (~%~1~- °°~at~)w ~a = 0.
(4.28a)
On the other hand from the metrical condition (4.17) it follows that (w~,alv+ w~al~,)w~a = 0.
(4.28b)
Then by addition and subtraction of (4.28a) and (4.28b) we obtain with respect to (4.16), (4.17) and (4.19): h~[~ = 0,
hi, = 0 ,
(h = h ~ f ) ,
(4.29)
whereby eq. (4.28) is fulfilled. Consequently the gauge-fixing condition (4.17) induces the special de Donder gauge (4.29) of the metric for the case of traceless field equations. The remaining equation (4.27) results from using (4.14) (remembering also (4.16) and (4.17)) in: k 0 ~ , F ~ = - 4 ~ r - ~ T~,~(~b).
(4.30)
These results can now be compared with Einstein's field equation for the traceless energy-momentum tensor in its linearized version (c = 1): O. F " ~
- O,~F~'.~ = - 8 7 r G T . ~ ,
(4.31)
where with regard to (4.19)
F~'~ = ½hl~.
(4.31a)
Taking into account explicitly the tracelessness of the Ricci tensor R~,~ (as a consequence of the tracelessness of the energy-momentum tensor) in the linearized
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form and considering additionally the usual de Donder gauge, we obtain in view of (4.31a) and in agreement with (4.29): h~r~l~ = 0,
hl~l~ = 0 0 h[~ = 0 ~ F ~
= 0.
(4.32)
Then eqs. (4.30) and (4.31) are identical by choosing the coupling constant k as follows: k = 2hG.
(4.33)
Herewith Einstein's linearized field equations are arrived at as the classical limit of our microscopic gauge field equations. Finally we note that, if the gauge fields to, a become massive as a consequence of a spontaneous symmetry breaking, the transition to Einstein's theory performed above would lead immediately to a cosmological constant A, proportional to the square of the gauge-field masses.
5. Conclusions
As already mentioned in sect. 2 the second quantization of our theory in its delivered form is straightforward. However there are some other points for discussion as well as remarks concerning an extension of the theory. At first we note that, according to the condition (4.7) and the definition (4.16) the gauge fields t%a are dimensionless. Then, in view of (3.3) the gauge-coupling constant g, which has the meaning of a microscopic heavy charge (see (3.14)) and which disappears in the classical limit, has the dimension of a reciprocal length. This may be the Planck length ~ and consequently g should be proportional to the Planck mass v ~ / G (in accordance with the accepted strength of the gravitational interaction, 10 '9 GeV*), rather than a new universal constant. However its true value remains undetermined within our theoretical approach and can be fixed only experimentally. However, here exists the difficulty of any microscopic theory of gravity, namely impossibility of a direct confrontation with experiments in the near future. Thus for constructing such theories there remains only the foundation on the most general and well-established first principles, like the postulate of unitary compact gauge symmetries and, as indirect proof the transition to the tested macroscopic limit. In this respect we want to point to the fact that the transition from our microscopic vector field theory of gravity to Einstein's macroscopic tensor field theory with the energy-momentum tensor of matter as source is not trivial, because the energymomentum tensor must be constructed from the currents of matter according to the vector field theory. But this procedure is possible, as we have seen in eq. (4.9), only in the case of massless particles. Therefore, the masslessness is a very essential presumption in our model and "mass" can be introduced only dynamically by * Note that in this case the total coupling constant in (3.11) would be with respect to (4,37) the square root of the gravitational constant, i.e. kg = v/-hG, in accordance with the situation in electrodynamics.
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spontaneous symmetry breaking. We hope that in this way - as already suggested in sect. 2 - the equivalence principle, which has not been used within our approach, can be deduced. By all means, the classical concept of the geometrization of gravity as a consequence of the equivalence principle follows in our theory from the microscopic "minimal gauge principle". In the present p a p e r the gravitational interaction has been investigated only between (massless) fermions. However, as already stated in sect. 2, gravity couples to all fields, i.e. also to bosonic ones. Following the line of this p a p e r the construction of the gravitational interaction concerning bosons should be possible by a grand unification of all interactions using higher U ( N ) symmetry groups acting on a higher-dimensional spin-isospin space*. Finally the question arises whether it is possible to even reach Einstein's non-linear theory. So far as we know, the answer to this question can only be given after the application of an iterative approximation method with all its difficulties. However it cannot be excluded that the limit in question can only be arrived at after a spontaneous symmetry breaking solving simultaneously the mass problem.
Appendix With the WKB ansatz = ~ aM(x~')(--ih) n e (i/~)s~x~)
(A.1)
n=O
one obtains in the limiting case h - ~ 0 the following leading expressions for the canonical energy-momentum tensor (4.3): T"~(t~) = -tioy~ao SI"
(A.2)
and for the field equation (3.12) in the case of free particles: y~aoSi, = 0.
(A.3)
The only non-trivial solution of (A.3) is (disregarding an irrelevant factor): ~oy~ao ~ S Ir ,
SI~SI~ = 0,
(A.4)
which means that the probability current is light-like in the lowest WKB limit, as is to be expected in the case of massless free particles. Then the energy-momentum tensor (A.2) for the free matter field reads in its classical limit: T'"(t#)~SI'S
I~ ,
T,/" (~b) -- 0.
(A.5)
Note added in proof We are aware of the fact, that the exact condition (4.17) is very strong, because it cannot be fulfilled in general by a gauging alone (the trace of (4.17) however is * In p r e p a r a t i o n .
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achievable generally), so that it restricts also the set of admissible solutions for c o n s t r u c t i o n of the macroscopic classical limit. O n the other h a n d we r e m e m b e r that our theory is still i n c o m p l e t e because of the masslessness of the particles, in c o n s e q u e n c e of which the set of solutions is also restricted already on the microscopic level. There are some hints that that these two facts are related to o n e a n o t h e r a n d that therefore a final j u d g e m e n t of the m e a n i n g of the c o n d i t i o n (4.17) will be possible o n l y in c o n n e c t i o n with solving the mass problem, e.g. with the help of a Higgs field. Thus, in the present form of the theory even a n a p p r o x i m a t e fulfilment of the c o n d i t i o n (4.17) is sufficient.
References [1] Don N. Page and C.D. Geilker, Phys. Rev. Lett. 47 (1981) 979 [2] A.W. Overhauser and R. Collela, Phys. Rev. Lett. 32 (1974) 1237; R. Collela, A.W. Overhauser and S.A. Werner, Phys. Rev. Lett. 34 (1975) 1472 [3] L. Rosenfeld, Quantentheorie und Gravitation in Entstehung, Entwicklung und Perspektiven der Einsteinschen Gravitationstheorie, ed. by J. Treder (Akademie-Verlag, Berlin, 1966); H. H6nl, Phys. Bl~itter37(2) (1981) 26 [4] P. van Nieuwenhuizen, Phys. Reports 68 (1981) 189 [5] A. Salam and J. Strathdee, Ann. Phys. (NY) 141 (1982) 316; E. Witten, Nucl. Phys. B186 (1981) 412 [6] H.R. Pagels, Phys. Rev. D29 (1984) 1960 17] B.G. Wybourne, Classical groups for physicists (Wiley, NY 1974); E. Merzbacher, Quantum mechanics, 2nd ed. (Wiley, NY 1970), p. 271 [8] C. Nash and S. Sen, Topology and geometry for physicists (Academic Press, London, 1983) chap. 7.6 [9] L.D. Landau and E.M. Lifschitz, Klassische Feldtheorie (Akademie Verl., Berlin, 1964) § 101
GRAVITY AS Y A N G - M I L L S G A U G E T H E O R Y H. D E H N E N and F. G H A B O U S S I
Fakultiit fiir Physik der Universit~it Konstanz, D-7750 Konstanz, Postfach 5560, Fed. Rep. Germany Received 5 March 1985 We propose a Yang-Mills field theory of gravity based on a unitary phase-gauge-invariance of the lagrangian where the gauge transformations are those of the S U ( 2 ) × U(1) symmetry of the 2-spinors. In the classical limit this microscopic theory results in Einstein's metrical theory of gravity, where we restrict ourselves in a first step to its linearized version.
I. Introduction
In Einstein's general relativity the gravitational interaction is described by the metric field of space-time based on the equivalence of inertial and (passive) gravitational mass (equivalence principle). According to Einstein's field equations the source of the metric field (gravitational field) is the energy-momentum stress tensor of matter, which is given ultimately because of the quantum structure of matter by the expectation values of the energy-momentum stress operators. Thus Einstein's description of gravity is a purely macroscopic one and on this level in best accordance with the known experimental data at least within the first-order approximation*. Accordingly the classical metric gravitational field represents the background of space-time on which all other physics happens; it reacts back on the metric field according to Einstein's field equations. But here the first problem arises. The back-reaction of a single quantum process of matter on the classical metric field cannot be considered consistently in Einstein's theory [1]. Furthermore on the purely microscopic level there exist neither a theory nor experiments of gravitational interactions up to the present day. However there is some evidence that, for the gravitational interaction of elementary particles the metrical description of gravity may not be the suitable one. First of all, all attempts at quantization of the metric field have been without success. Secondly, only in the case of a classical particle does the mass drop out of the equation of motion for gravitation as a consequence of the equivalence principle, so that all masses are moved in the same way by gravity and the gravitational field * So long as the solar mass quadruple moment is unknown, the perihelion shift of Mercury is not a precise test with regard to the second-order approximation. 144
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can be described by a geometrical "guiding field". But in contrast to this, the motion of a quantum particle state is not independent of the particle mass even in the presence of a classical gravitational field [2]. In spite of the validity of the equivalence principle the particle mass does not drop out of the Schr6dinger equation. Thus there is no universal guidance of the quantum processes by gravity [3] and consequently there is no hint of a necessary geometrization of the gravitational interaction on the quantum level. Finally, Einstein's metrical theory of gravitation has a completely alternative structure to the successful gauge theories of electroweak and strong interactions. Therefore a unification of all interactions needs a modification of the standard description of gravity. The possibilities discussed until now mainly consist of supergravity [4] and modern K a l u z a - K l e i n theories [5], where in both cases the metric is still retained as a fundamental structure of gravitation on the microscopic level. But these models have other problems, especially for instance, the required existence of too m a n y exotic elementary particles. For this reason, in this p a p e r we want to point out the possibility of the construction of a unitary gauge theory of gravity on the microscopic level in accordance with the gauge theories of electroweak and strong interactions. Then the gravitational interaction would also be produced by the exchange of the usual vector gauge bosons on the background of the Minkowski space-time of special relativity. Thus on the quantum level, gravitation would not be described by a metrical tensor field, but by gauge vector fields*. Only in a certain classical limit, i.e. by cooperation of many vector bosons, can Einstein's metrical description of classical gravity result in at least its tested linearized version. According to this line the non-euclidean metric of space-time is not a fundamental, but more an effective field produced by combination of gauge vector fields like a condensation phenomenon. It has only a classical meaning describing the classical gravitational field alone. We emphasize that we are pursuing here the idea that quantum or microscopic physics has priority and follows directly from a few very general first principles, whereas all macroscopic physics must be deduced from its classical limit.
2. The group theoretical concept Following the above mentioned line the first aim of this p a p e r is a quantum theoretical description of the gravitational interaction between elementary particles using a usual gauge field theory based on a phase-gauge-invariance of the lagrangian where, with respect to the interpretation of the quantum theory, only unitary gauge transformations are allowed. Furthermore, because gravity couples to all particles, a gauge transformation is needed which results from the transformation of the * After preparing this paper we were informed that another attempt in this direction had been made [6] but with very different assumptions, e.g. a euclidean metric and a non-Yang-Mills lagrangian.
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intrinsic spinor structure of each particle and not from a transformation between different particles of any multiplet. Finally, gravity is connected with the concept of "mass". However this concept is still a classical one and should therefore be avoided in a quantum approach to gravity. In accordance with the modern gauge theoretical treatment of interactions one has to start with massless particles and the "mass" appears subsequently by a dynamical process, namely, as a consequence of spontaneous symmetry breaking. As we shall see later in sect. 4 this is also the only satisfactory way of applying Einstein's tensor theory in the classical limit starting from a microscopic vector gauge theory of gravity. On doing so, the empirical equivalence between the inertial and gravitational mass of particles (which is the basis of general relativity) may be found theoretically on the microscopic level. However in the following we do not want to discuss the mass problem in detail; that would be outside the bounds of this paper. We consider only the gravitational interaction of massless particles as a first step. Thus we start from the general transformation group GL(2, C) of the 2-spinors which we take as the most general symmetry group. However, all generators of this group can be constructed by complex linear combinations of the generators of SU(2) x U(1), i.e. of the unit and Pauli matrices [7]. Thus instead of a gauge theory for the complete group GL(2, C), we propose to gauge only the S U ( 2 ) x U(1) symmetry by which the basic quantum theoretical requirement of the unitarity of the phase-gauge-transformations is automatically fulfilled. This idea is supported by the mathematical fact that one can reduce the group GL(2, C) to its maximal compact subgroup U(2) by contraction [8]. However, it is well-known that the group SL(2, C) is the covering group of the Lorentz group. From this point of view we indeed gauge the Lorentz group in the sense that we gauge its "basic" transformations [7], namely those of SU(2). Although we start from the 2-spinor calculus for massless particles we perform our investigations mainly in the 4-spinor representation, i.e. in the 4 x 4 representation of SU(2) x U(1) with respect to a later consideration of particle masses. Furthermore we restrict ourselves to a treatment of the theory on the level of the first quantization. However in the case of the compact gauge group SU(2) × U(1) there are no problems with quantization; the theory is renormalizable in the usual way and it gives no ghost-fields. According to our gauge group there exist 4 gauge vector fields and it is possible 'to introduce with respect to the reducibility of the 4 x 4 representation of SU(2) x U(1), 2 or even 3 different gauge-coupling constants. However because all gauge bosons mediate the same interaction between all spin-½ particles, i.e. gravity, we give all these coupling constants the same value*. Under this presumption we first construct as follows a gauge theory of the SU(2) x U(1) symmetry for gravity on the Minkowski space-time background and then we show that in the classical limit a metric description of gravity for the expectation values results, where the * A d e e p e r u n d e r s t a n d i n g o f this ansatz can be e x p e c t e d from a unification o f all i n t e r a c t i o n s at s o m e later date.
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147
m e t r i c fulfils E i n s t e i n ' s field e q u a t i o n s . H e r e we restrict ourselves for s i m p l i c i t y to the l i n e a r i z e d theory. 3. The l a g r a n g i a n of microscopic gravity W e define the t r a n s f o r m a t i o n m a t r i c e s o f the g r o u p S U ( 2 ) x U ( 1 ) r e p r e s e n t a t i o n by*
in the 4 × 4
U = e iA°(x~)'° with
(3.1)
(o oo)
I cr "r = ~ 0
(3.1a)
(or° is the u n i t - m a t r i x a n d or 1, cr2, tr 3 are the Pauli matrices). The c o m m u t a t o r relations for the g e n e r a t o r s z a are: [ ~.b, re] = ie bCza .
(3.1b)
T h e n the s p i n o r s t a t e - f u n c t i o n ~0 a n d the D i r a c matrices 7 ~' t r a n s f o r m as: 6 ' = Uq,,
~,'"= U v " U - ' ,
(3.2)
with the i n v a r i a n t r e l a t i o n : y(" y~} = *7"~
(3.2a)
(r/~'~ = rh,~ = d i a g ( - 1 , 1, 1, 1) M i n k o w s k i - m e t r i c ) . A c c o r d i n g l y the c o v a r i a n t s p i n o r derivative r e a d s : D . ~ = ( 0 . + igto.)qJ, ( g is the u n i f o r m g a u g e - c o u p l i n g c o n s t a n t ) with the S U ( 2 ) x U ( 1 ) defined by w, = w~,j a ,
(3.3) g a u g e fields to.. (3.3a)
w h i c h t r a n s f o r m u n d e r g a u g e t r a n s f o r m a t i o n as"* , 1 i w , = U w , U - + - UI~U -1 . g
(3.4)
F u r t h e r m o r e it f o l l o w s f r o m (3.2) that the y " are o n l y c o v a r i a n t l y c o n s t a n t , i.e. D,~T" =- Y~'l~ + ig[w~, y " ] = 0 ,
(3.5)
which, t o g e t h e r with the c o n d i t i o n (3.2a), is the d e t e r m i n a t i o n e q u a t i o n for y " in a d d i t i o n to the field e q u a t i o n s (3.11) a n d (3.12). N o w we define the g a u g e - f i e l d strength o f m i c r o s c o p i c gravity in the usual w a y b y the c o m m u t a t o r o f the c o v a r i a n t derivative given in (3.3): 1 V,~ ~ z-- [ D , , D~] = w~l. - o9,1~ + ig[t%, to~]. tg
(3.6)
* In the following Einstein's sum convention is used. Latin indices a, b, etc. run from 0 to 3 and are related to the basis of the group algebra. Greek indices run from 0 to 3 and are related to the space-time coordinates. ** The symbol I/z means the usual partial derivative with respect to the coordinate x*'.
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For the components of Fp`, with respect to r ~ it follows immediately from (3.3a) and (3.1b): Fp`~, = w~alp` - w p`,l ~ - geab~co p`btO~c ,
(3.7)
satisfying the Bianchi identities F~CP`~I* l + gF'abCFb[p`d'OX lc
=
0.
(3.8)
In the case of the p u r e gauge field
i g ~ u u "1 , g
(3.9a)
Fp`~, - 0.
(3.9b)
COp, ~ O~p` = ~
it holds: Fp`~=- 0,
With the principle of minimal coupling the gauge-invariant Lagrange density of gauge gravity for a massless particle takes the very simple form: h l~P.v ab 3 f = 167rk" p`~a--b ~7 -~hiq~yp`D~,@ + h.c. .
(3.10)
Here we have employed as group-space metric the Minkowski matrix rl ab, because we need (with respect to the classical limit of this microscopic theory) the U(1) gauge potential to be time-like and that of SU(2) to be space-like*. Furthermore we have taken the opportunity to introduce besides the gauge-coupling constant g a second coupling constant k between the two gauge-invariant parts of the Lagrange density. Both coupling constants must be determined by experiment or by the transition to the macroscopic classical limit, where k comes out to be proportional to the newtonian gravitational constant (7, whereas g does not possess a classical limiting value and has only the meaning of a microphysical "heavy charge". However, both coupling constants are necessary in the following and cannot be substituted by one another. The field equations following from the action principle belonging to (3.10) are given by Op`F p`~" + ge"bCF~"wp`,. = 2 ¢rkg~{ 7 ~, ra}qJ
(3.11)
,y~(Op` + igcop`)@ = 0.
(3.12)
and, with (3.5), by
Finally the canonical energy-momentum tensor of the whole system takes the form with respect to (3.12): TltV
1. - v h vaa Rv~ =~th(~by ~0tp`-qTip`y"~0)-~--~k(coo~,tp`F - z lr~ctfl~?a - ~ --o,0o.o.j.
(3.13)
* Evidently the action corresponding to (3.10) is bounded from below after a Wick rotation of time, according to which not only the space-time metric but also the group-space metric goes over from a pseudo-euclidean to a euclidean structure.
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It has the property (with the use of the field equations): = ~ghOy t%cr ~D,O + h.c.,
(3.13a)
where o-~,~= ½[y,, y~] is the spin operator and the right-hand side represents a 4-force density between the gauge field and spin of the particle*. Evidently in our microscopic theory of gravitation there exists 4 gauge-vector fields, to,o, which couple to 4 "heavy" gauge currents of matter (@field) as is usual in Yang-Miils field theories: j~'~ = ½gt~{~/~, r"}qJ.
(3.14)
Due to the fact that the U(1) gauge current j , o is proportional to the time-like probability current density and is therefore correlated with the 4-momentum density of the particle field, we expect that the field equations (3.11), (3.12) are indeed those for gravity. This supposition will be confirmed subsequently by investigation of the classical limit of this microscopic theory.
4. The classical limit
We can be sure that the theory proposed in sect. 3 is a theory of gravity, if in the classical limit Einstein's metrical theory of gravitation results. For this it must be shown firstly, that according to Ehrenfest's theorem the expectation values of the 4-momentum of the particle field (qJ-fieid) fulfils the usual equations of motion for classical gravity. This means in detail, that from the equations of motion the classical gravitational force must be read off; it must be reducible to geometrical connection coefficients (macroscopic field strengths), because classical gravity is a geometrical theory. Secondly, it must be shown that the connection coefficients are metrical and thirdly, that they fulfil Einstein's field equations with the energy-momentum tensor of the ~b-field (matter field) as source. In doing this we restrict ourselves for simplicity to the linearized Einstein theory as the only empirically confirmed part of the complete theory.
4.1. THE EQUATION OF MOTION
We start from the result (3.13a), neglecting in the classical limit the interaction between the gauge field and spin of the particle. Then we have** Tg°lo+ Tfl j = 0.
(4.1)
* The right-hand side of (3.13a) results from the fact that the Dirac matrices within the lagrangian (3.10) depend explicitly on the space-time coordinates according to (3.5). Choosing the gauge-invariant canonical energy-momentum tensor instead of (3.13) the right side of (3.13a) vanishes identically, which implies (4.1) automatically. ** Latin indices j, k, etc. run from 1 to 3 and are related to the space-like coordinates only.
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H. Dehnen, F. Ghaboussi / Gauge-field theory of gravity
After insertion of (3.13) into (4.1) and integration over the 3-dimensional hypersurface, t = const, we obtain: ~9° ½ih(0Y°Ol" - 0['T°0) d3x = ~ k
(w,~l,,F
-
)lo d3x
(Fo Fa3)l. d3x ,
(4.2)
where boundary integrals concerning the pure 0-field are dropped out with respect to the normalization condition for the wave function f 0*0 d3x = 1. Considering (3.13) the canonical energy-momentum tensor of the matter field is given by: T,~(0)
l. - ~01~,= ~th(Oy
~I,,Y~0) •
(4.3)
According to this the integrand on the left-hand side of (4.2) is identical with the 4-momentum density T,°(O) of the particle field. From now on, with the use of the field eqs. (3.11), eq. (4.2) takes the form (neglecting boundary integrals of the 0-field):
Oof [ T.°(O) ltl -41rk
I
½figO(y°to~ + w . y ° ) 0 ] d3x
1 a/3 a 3 [(Fo,oF3c~a )13-~(F~ F~3)I~,] d x.
(4.4)
Here the bracket on the left-hand side represents the gauge-invariant canonical 4-momentum density of the particle field obtained from (4.3) by replacing the ordinary derivatives with the covariant ones. However, the second term on the left-hand side of (4.4) is of the order of the coupling constant g compared to the first term and therefore can be neglected within a first-order approximation with respect to gravitational interaction. Then one obtains with the help of the Bianchi identities (3.8) and the field equations (3.11) the following equation of motion for the expectation value of the 4-momentum of the 0-field:
Oof T.°(O)d3x=h f F.~aj ~a d3x.
(4.5)
Here on the right-hand side
K.=-hfF.~.j"~ d3x=fF.~a~hgO{y, r~}O 1
--ct
d3x
(4.6)
represents the classical gravitational 4-force as a product of the gauge-field strengths and the gauge currents of matter.
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In view of a classical geometrical interpretation the integrand in (4.6) must be reduced to a product between classical non-algebra-valued connection coefficients and the energy-momentum tensor of the matter field. For this reason we write:
-- a , ra}~b d3x K , = I F~,~bt~a bl5hg~b{y
(4.6a)
and introduce as "correspondence condition" between the quantum and the classical theory of gravity the following normalization of the gauge-vector fields:
co ~to b = ~ b + eab(x, ) ,
(4.7)
leabl "~ 1
(4.7a)
where we choose
in view of a weak field approximation to the theory*. According to this, in the case of the pure gauge-fields (3.9a) where F~,~ -= 0, one has to take, in (4.7), e~ =- 0 which is always possible. Then (small) deviations from the pure gauge-field case produce (small) values for F,~a and e~b different from zero, so that the normalization condition (4.7) has no restrictive meaning. Accordingly the U(1) gauge field CO,o is time-like, whereas the three SU(2) gauge fields are space-like with respect to the group-space metric introduced in (3.10). Only in this way can the pseudo-riemannian structure of the metric be guaranteed later. Consequently, in the case of the linearized theory the Kronecker symbol in (4.6a) can be substituted by the left-hand side of (4.7). One obtains immediately with the use of (3.3a): gp.
=
bl u f F~btov ~hgO{3, , to~}0 d3x.
(4.8)
By this procedure we have obtained the product in (4.6a) separated into two factors, which are both (pseudo-) scalars with respect to the gauge transformations and therefore can be interpreted classically: the first scalar in (4.8) corresponds to the connection coefficients, the second one should be related to the energy-momentum tensor of the matter field. For an investigation of this last relation we eliminate the ~o~ field in the anticommutator of (4.8) with the help of the field equation (3.12). Then we get, using (3.2a):
l hgd~{ 3,~, to v} ~b = ~ih ( d~3,~b I~ - d~t~3/~b),
(4.9)
where the terms proportional to the spin operator cr~'~ interacting with F~,~b are neglected with regard to the classical limit. Comparison with (4.3) shows that the right-hand side of (4.9) is identical with the energy-momentum tensor T ~ ( O ) of the matter field. We mention explicitly that this essential result, which connects the * If we avoid in (3.3) the explicit appearance of a gauge-coupling constant, this must be introduced at the latest in (4.7), see also (4.16).
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gauge currents with the energy-momentum tensor and which makes possible at last the transition to Einstein's theory, is valid only in the absence of the mass term in the field equation (3.12). Hence with the use of (4.9) eq. (4.8) takes the form:
K, = f F~.~aoJ~T~"(O)d3x,
(4.10)
which has now the structure of the usual gravitational force. We note that as a consequence of relation (4.9) the gauge-coupling constant g no longer appears explicitly in (4.10). For the full-linearized classical limit of the equation of motion one has to use in (4.10) the energy-momentum tensor T"~(0) in the lowest WKB approximation of the free particle field. One finds from (4.3) (see appendix): Tt"~](¢) = 0 ,
T,," (0) = 0 ,
(4.11)
according to which T""(O) is symmetric and traceless. The first property results from neglecting the spin of the particles and the last one is a direct consequence of the masslessness of the particles and valid only in that case. Applying this result to the relation (4.10) we find for the equation of motion (4.5):
ao f T"°(*)d3x= f F:(,~oJ:)T~'~(*)d3x,
(4.12)
which we have now to compare with the corresponding one according to general relativity: 0o
f
T"°(@) d 3 x - -
f F',~.T~'(O)
d3x-
F .aT
(0) d3x-
(4.13)
The last term on the right-hand side of (4.13) vanishes because of F"~. = 0 in the case of a traceless energy-momentum tensor (see (4.11)) as source of the classical gravitational field (cf. eq. (4.32)). Then we find by comparison of (4.12) and (4.13) the following correlation between the geometrical connection coefficients F " . ~ and the microscopic gauge-field strengths F.~.: F " ~ = -Fa(~to~).
(4.14)
Thus we have arrived at a geometrical description of the gravitational force in the classical limit of our microscopic gauge theory without using any principle like the equivalence principle. 4.2. THE M E T R I C
In order to find out the metrical structure of our connection coefficients we reform the right-hand side of eq. (4.14) by inserting the definition (3.7) and obtain: 1
IzA
a
a
a
1
a
a
p.
- (to.~to ~)fA] -5(to ~1~+ tO~l~)~oa .
(4.15)
H. Dehnen, F. Ghaboussi / Gauge-field theory of gravity
153
Here the bracket has indeed the structure of a l i n e a r i z e d Christoffel symbol with the metric defined by g~,v --- to~oto~ •
(4.16)
Then the connection (4.15) is a m e t r i c one after choosing the gauge-fixing condition: to(~l~> = 0
(4.17)
as a fully relevant constraint for the classical metrical limit of our theory, which is realizable in the lowest W K B limit of eq. (3.11). Obviously, the microscopic theory possesses more structures than appear in the classical metrical limit. For the metrical limit the solutions of the microscopic theory are restricted to those gauge fields which are at least in the WKB limit Killing vector fields with respect to the Minkowski metric. Simultaneously the gauge transformations are b o u n d e d to those, which allow (4.17) to become invariant, i.e. in the case of linearized gauge transformations (see (4.21)) only gauge functions ;ta are allowed satisfying bc
hal~,h~+ gea hbl(~w~>c = 0.
(4.17a)
Then the covariant derivative of the metric (4.16) vanishes with respect to the connections defined by (4.15) and (4.17): g~,~Lx - F ~ , x g ~
- F~g~,~
=0.
(4.18)
In the case of p u r e gauge fields (3.9a) with F,~, --- 0 the connection coefficients (4.14) and (4.15) vanish identically and the metric (4.16) is reduced to the Minkowski metric r/,~, where in the normalization condition (4.7) e, b - 0 is valid. In general by decomposition of the metric (4.16) according to g~,~=rl~,~+h~,,,
Ih.~l~l
(4.19)
one finds with the use of (4.7) the following relations within the linearized theory: hl~ v = F,abC~o~bOjav , vt e~ b = to ~b toan~,~.
(4.20)
Herewith the normalization condition (4.7) takes the form of the usual tetrad condition in general relativity to .btovagt'Lv = t~ ba ,
(4.20a)
where g~'~ = 7/~'~- h ~'~ with g~'~g~A= 6 ~ *. Of course, the metric (4.16) is not a gauge-invariant quantity (pseudo-scalar). From this it follows that from a gauge transformation of the gauge fields a coordinate transformation of the metric field is induced. Here we restrict ourselves within the • N o t e t h a t in contrast to the r e m a i n i n g d e s i g n a t i o n in this paper, here g " ~ inverse to g~,~.
"0~%l~°g~o, b u t g ~
is
154
14. Dehnen, F.. Ghaboussi / Gauge-field theory of gravity
linearized theory to infinitesimal transformations. Then from (3.1), (3.3a) and (3.4) the gauge-field transformation is as follows (IA~I¢ 1): 1
(4.21)
toga = tolza --g l~al~ -- 8abCabtot~c •
Herewith we obtain according to (4.16) the " n e w " metric: g~,,, . =. "q~,~ . + .h j , , =.
w . ~,,,to ~ ,
(4.22)
where in view of (4.19) h~,~=h~-
a .. to~Aal~_ 1gt°~Aal
(4.22a)
However, in the case of a transition from g~,~ to g~,~ by the infinitesimal coordinate transformation x " = x t' + sC'(x ~) (with Ix l) there exists the well-known transformation law for the deviations from the Minkowski metric: h~,~ = ht,~ - ~:t,l~- ~:~lt, •
(4.23)
Comparison of (4.22a) and (4.23) yields immediately with respect to (4.17) the following connection between infinitesimal gauge and coordinate transformations: 1
= - to A~. g
(4.24)
However, only those gauge transformations are allowed which do not disturb the gauge fixing condition (4.17), see (4.17a). Accordingly, the (infinitesimal) coordinate transformations are limited. As we see later, the gauge-fixing condition (4.17) implies (on the level of the field equations) the de Donder gauge for the metric*, to which the coordinate transformations are bounded, see (4.29).
4,3. THE F I E L D E Q U A T I O N S
After definition of the metric (4.16) through the gauge potentials it remains to be shown that the metric or its connection coefficients satisfy Einstein's linearized field equations. These must follow as the classical limit of the gauge-field equations (3.11). Therefore one first has to remove their algebra-valuedness. For this we multiply eqs. (3.11 ) by the gauge fields toga and obtain with the use of (3.3a) and (3.7): /x
a
o,. ( Fa.,to ,.) -
/.t
a
/.L
a
= 2~rlig~{ %,.
,o,.}~,
(4.25)
The right-hand side is with respect to the basic relation (4.9) already proportional to the energy-momentum tensor of the matter field; terms containing the spin tensor * The usual de Donder gauge for the metric (4.19) reads ht,~p~-½hl~ = 0 (h = h~'~) and corresponds to the Lorentz gauge in electrodynamics, see ref. [9].
14. Dehnen, F. Ghaboussi / Gauge-field theory of gravity
155
or~'" are neglected with regard to the classical limit. On the left-hand side of (4.25) the second and third term can be neglected in view of the linearization. Thus we get: ~' " =47r -k a,,(F~vw~) h T~(qJ),
(4.26)
which is reduced with respect to (4.11) to its symmetric and traceless part: o
a,,(F~(~w~)) =47r
T~,(q,)
(4.27)
with a. ( F ~ o J ) - 0 bt
va
__
(4.28)
.
Obviously, also in the classical limit of the field equations - as in the classical limit of the equations of motion - the gauge-coupling constant g no longer appears explicitly, so that it cannot be determined by comparison with the classical Einstein theory. Using (3.7) the last equation (4.28) implies: (~%~1~- °°~at~)w ~a = 0.
(4.28a)
On the other hand from the metrical condition (4.17) it follows that (w~,alv+ w~al~,)w~a = 0.
(4.28b)
Then by addition and subtraction of (4.28a) and (4.28b) we obtain with respect to (4.16), (4.17) and (4.19): h~[~ = 0,
hi, = 0 ,
(h = h ~ f ) ,
(4.29)
whereby eq. (4.28) is fulfilled. Consequently the gauge-fixing condition (4.17) induces the special de Donder gauge (4.29) of the metric for the case of traceless field equations. The remaining equation (4.27) results from using (4.14) (remembering also (4.16) and (4.17)) in: k 0 ~ , F ~ = - 4 ~ r - ~ T~,~(~b).
(4.30)
These results can now be compared with Einstein's field equation for the traceless energy-momentum tensor in its linearized version (c = 1): O. F " ~
- O,~F~'.~ = - 8 7 r G T . ~ ,
(4.31)
where with regard to (4.19)
F~'~ = ½hl~.
(4.31a)
Taking into account explicitly the tracelessness of the Ricci tensor R~,~ (as a consequence of the tracelessness of the energy-momentum tensor) in the linearized
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H. Dehnen, F. Ghaboussi / Gauge-field theory of gravity
form and considering additionally the usual de Donder gauge, we obtain in view of (4.31a) and in agreement with (4.29): h~r~l~ = 0,
hl~l~ = 0 0 h[~ = 0 ~ F ~
= 0.
(4.32)
Then eqs. (4.30) and (4.31) are identical by choosing the coupling constant k as follows: k = 2hG.
(4.33)
Herewith Einstein's linearized field equations are arrived at as the classical limit of our microscopic gauge field equations. Finally we note that, if the gauge fields to, a become massive as a consequence of a spontaneous symmetry breaking, the transition to Einstein's theory performed above would lead immediately to a cosmological constant A, proportional to the square of the gauge-field masses.
5. Conclusions
As already mentioned in sect. 2 the second quantization of our theory in its delivered form is straightforward. However there are some other points for discussion as well as remarks concerning an extension of the theory. At first we note that, according to the condition (4.7) and the definition (4.16) the gauge fields t%a are dimensionless. Then, in view of (3.3) the gauge-coupling constant g, which has the meaning of a microscopic heavy charge (see (3.14)) and which disappears in the classical limit, has the dimension of a reciprocal length. This may be the Planck length ~ and consequently g should be proportional to the Planck mass v ~ / G (in accordance with the accepted strength of the gravitational interaction, 10 '9 GeV*), rather than a new universal constant. However its true value remains undetermined within our theoretical approach and can be fixed only experimentally. However, here exists the difficulty of any microscopic theory of gravity, namely impossibility of a direct confrontation with experiments in the near future. Thus for constructing such theories there remains only the foundation on the most general and well-established first principles, like the postulate of unitary compact gauge symmetries and, as indirect proof the transition to the tested macroscopic limit. In this respect we want to point to the fact that the transition from our microscopic vector field theory of gravity to Einstein's macroscopic tensor field theory with the energy-momentum tensor of matter as source is not trivial, because the energymomentum tensor must be constructed from the currents of matter according to the vector field theory. But this procedure is possible, as we have seen in eq. (4.9), only in the case of massless particles. Therefore, the masslessness is a very essential presumption in our model and "mass" can be introduced only dynamically by * Note that in this case the total coupling constant in (3.11) would be with respect to (4,37) the square root of the gravitational constant, i.e. kg = v/-hG, in accordance with the situation in electrodynamics.
H. Dehnen, F. Ghaboussi / Gauge-field theory of gravity
157
spontaneous symmetry breaking. We hope that in this way - as already suggested in sect. 2 - the equivalence principle, which has not been used within our approach, can be deduced. By all means, the classical concept of the geometrization of gravity as a consequence of the equivalence principle follows in our theory from the microscopic "minimal gauge principle". In the present p a p e r the gravitational interaction has been investigated only between (massless) fermions. However, as already stated in sect. 2, gravity couples to all fields, i.e. also to bosonic ones. Following the line of this p a p e r the construction of the gravitational interaction concerning bosons should be possible by a grand unification of all interactions using higher U ( N ) symmetry groups acting on a higher-dimensional spin-isospin space*. Finally the question arises whether it is possible to even reach Einstein's non-linear theory. So far as we know, the answer to this question can only be given after the application of an iterative approximation method with all its difficulties. However it cannot be excluded that the limit in question can only be arrived at after a spontaneous symmetry breaking solving simultaneously the mass problem.
Appendix With the WKB ansatz = ~ aM(x~')(--ih) n e (i/~)s~x~)
(A.1)
n=O
one obtains in the limiting case h - ~ 0 the following leading expressions for the canonical energy-momentum tensor (4.3): T"~(t~) = -tioy~ao SI"
(A.2)
and for the field equation (3.12) in the case of free particles: y~aoSi, = 0.
(A.3)
The only non-trivial solution of (A.3) is (disregarding an irrelevant factor): ~oy~ao ~ S Ir ,
SI~SI~ = 0,
(A.4)
which means that the probability current is light-like in the lowest WKB limit, as is to be expected in the case of massless free particles. Then the energy-momentum tensor (A.2) for the free matter field reads in its classical limit: T'"(t#)~SI'S
I~ ,
T,/" (~b) -- 0.
(A.5)
Note added in proof We are aware of the fact, that the exact condition (4.17) is very strong, because it cannot be fulfilled in general by a gauging alone (the trace of (4.17) however is * In p r e p a r a t i o n .
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H. Dehnen, F. Ghaboussi / Gauge-field theory of gravity
achievable generally), so that it restricts also the set of admissible solutions for c o n s t r u c t i o n of the macroscopic classical limit. O n the other h a n d we r e m e m b e r that our theory is still i n c o m p l e t e because of the masslessness of the particles, in c o n s e q u e n c e of which the set of solutions is also restricted already on the microscopic level. There are some hints that that these two facts are related to o n e a n o t h e r a n d that therefore a final j u d g e m e n t of the m e a n i n g of the c o n d i t i o n (4.17) will be possible o n l y in c o n n e c t i o n with solving the mass problem, e.g. with the help of a Higgs field. Thus, in the present form of the theory even a n a p p r o x i m a t e fulfilment of the c o n d i t i o n (4.17) is sufficient.
References [1] Don N. Page and C.D. Geilker, Phys. Rev. Lett. 47 (1981) 979 [2] A.W. Overhauser and R. Collela, Phys. Rev. Lett. 32 (1974) 1237; R. Collela, A.W. Overhauser and S.A. Werner, Phys. Rev. Lett. 34 (1975) 1472 [3] L. Rosenfeld, Quantentheorie und Gravitation in Entstehung, Entwicklung und Perspektiven der Einsteinschen Gravitationstheorie, ed. by J. Treder (Akademie-Verlag, Berlin, 1966); H. H6nl, Phys. Bl~itter37(2) (1981) 26 [4] P. van Nieuwenhuizen, Phys. Reports 68 (1981) 189 [5] A. Salam and J. Strathdee, Ann. Phys. (NY) 141 (1982) 316; E. Witten, Nucl. Phys. B186 (1981) 412 [6] H.R. Pagels, Phys. Rev. D29 (1984) 1960 17] B.G. Wybourne, Classical groups for physicists (Wiley, NY 1974); E. Merzbacher, Quantum mechanics, 2nd ed. (Wiley, NY 1970), p. 271 [8] C. Nash and S. Sen, Topology and geometry for physicists (Academic Press, London, 1983) chap. 7.6 [9] L.D. Landau and E.M. Lifschitz, Klassische Feldtheorie (Akademie Verl., Berlin, 1964) § 101