UWThPh-81-3
NOTES ON GAUGE THEORY AND GRAVITATION*
R.P. Wallner
Institut für Theoretische Physik Universität Wien
Abstract
In order to investigate whether A. Einstein's general relativity theory (CRT) fits into the general scheme of a gauge theory, first the concept of a (classical) gauge theory is outlined in an introductionary spacetime approach. Having thus fixed the notation and the main properties of gauge fields, CRT is examined to find out what the gauge potentials and the corresponding gauge group might be. In this way, we are led to the possibility of interpreting GRT as a gauge theory of the 4-dimensional translation group T(4) = (1^,+)» and where the gauge potentials are in- corporated in a T(4)-invariant way via orthonormal anholonomic basis 1-forms. To include also the spin aspect we just indicate a natural exten- sion of GRT by gauging also the Lorentzgroup, whereby a Riemann-Cartan • spacetime (U.-spacetime) comes into play. As usual in our papers, the calculus of exterior forms is used throughout. As, however, our notation has got up for itself a little bit during the course of time, a short overview is given in Sec. 2.
+) Work supported by "Fonds zur Förderung der wissenschaftlichen Forschung in Österreich", project no. 4291. 1. Introduction
1.1 What is a gauge theory? Nowadays, classical (= nonquantized) gauge theory is formulated in terms of principal fiber bundels with connection and associated vectorbundles [l]. But the use of these modern concepts, although highly desired, sometimes thrusts aside the origin and the under- lying idea of what is called a "gauge procedure" or "gauge principle". Therefore, in Sec. 3, we revive the idea of introducing gauge fields (potentials) as (exterior) compensating fields in order to regain in- variance of any field theory under their symmetry group after the group action has become spacetime dependent ("local gauge transformation", the transformation of a field under a symmetry group with spacetime dependent parameters [2]). In this way, gauge fields are closely linked to the in- variance properties of the corresponding Lagrangian, and as a consequence, are incorporated in certain identities arising from that invariance. All this together amounts in a list of certain properties gauge fields have to obey and by which it is summarized what we want to call a (classical) "gauge theory" (see end of Sec. 3).
1.2 What has gauge theory to do with gravitation? The best theory of gravitation still available is Einstein's general relativity theory (CRT), no doubt. It is well-tested enough, simple enough (compared to the rich- ness of its consequences) and beautiful enough (in particular because of its geometric-mathematical structure) to let them both feel enthusiastic, the pragmatician and the esthetical idealist. Nevertheless, there are still some essential problems in this theory: (1) CRT contains singularities, which cause some problems in understanding the physical consequences of high densities of matter as well as the role of a singularity on a manifold at all (which are contrary con- cepts by definition). (2) CRT still resists quantization already in the linearized case, not to speak of the full theory because of its nonlinear character, (3) CRT still resists unification with other physical theories, in parti- cular because of the problems mentioned in (1) and (2)„ To overcome these defects, one is well advised to look at other physical theories, which, to some extend, submit a unification and quantization (including renormalization) scheme, arising from what we called before a "gauge theory" [3]. According to this one expects that, once CRT is re- considered and reformulated as a gauge theory similar to the theories of weak, electromagnetic and strong forces, gravity will be mollified to fit into the general scheme, thus leading to a unified theory of all known physical forces. But all these hopes and expectations are, however, very speculative at the moment and as far as I can see', there is no immediate evidence for solving all problems merely by formulating gravity as an (perhaps extended) gauge theoretical version of CRT. Therefore I want to stress another aspect which is very close at hand: As mentioned above, (classical) gauge theory 'has experienced a considerable amount of geometrization the last decade, where the notion of fiberbundles has taken place in it. Nowadays, gauge theory is as imbued with geometrical ideas as CRT, so it is simply desirable to combine these theories.
1.3 How do we proceed? Several attempts were made'to formulate any gravity theory as a gauge theory of a certain group, but most of them were guided by what one wants and not what one already has, thus overlooking sometimes the best theory of gravitation already available: CRT. Here, we try a different approach. Starting in Sec. 3 with a simple introduction into the gauge idea in general and ending up with the main properties of gauge fields, we go on and look at CRT, analyse the basic variables and investi- gate whether they may be interpreted as gauge fields and what the correspond- ing gauge group might be (Sec. 4). For that purpose, the formulation of gauge theory at the same footing as CRT (over spacetime - no bundle view- point) happens to be the most appropriate one. As it turns out, we shall interpret CRT as a gauge theory of the 4-dimensional translation group T(4) = (1^,+), where general coordinate transformations are interpreted as local (= spacetime dependent) trans- lations. To include also the spin aspect, we just line out the further gauge procedure for the Lorentzgroup, thus arriving at a Riemann-Cartan (UL-) spacetime as underlying manifold (Sec. 5). Finally, the main results and aspects are summarized in Sec. 6. 1.4 What is assumed? In general, we assume spacetime M4 to be a 4-dimen-
CO sional connected paracompact C -manifold, endowed with an indefinite metric of signature 1 (hyperbolic manifold, nomenclature follows [4]). For physical applications, matter fields are represented by vector (spinor-) valued p-forms
2. Exterior Forms
2.1 Let M11 be a n-dimensional manifold and E (Mn) := E (Mn,3R) the module , p co p of (scalar valued) exterior p-forms over E (Mn ) := C (Mn ;3R) , the set of smooth mappings from H to the real numbers 3R. The direct sum of E (M ), p = O,l,...,n, is then widened to a (graded) algebra by the componentwise extended exterior product
A: Ep x Eq ->• E p+q V p,q 6 [O,n] , p + q —< n i... .i which is defined here in the following construction of a p-basis {e }, i, = 1,... ,n V k, of E out of a 1-basis {e1}, i = l,...,n, of E,: K p 1 i,...! i. i [i, i ] e P := e A ... A eP := p! e ®...©ep (2.1)
where ® denotes tensor product and square brackets antisyranetrization as usual. Thus any p-form <|> G E can be written as i . . .i = e P = (2 2) • * *lJi, . • • • i-L l ' *iJ - . * . »i JL *[L**>4****i i Ll-J ' 1 l P1 l P ' l P where vertical bars demand summation over i, < i. < . . . < i . From (2.1/2) 12 p it follows the well known commutation law
4, A i|» = (-)- ty A 4> if <}> € E , ijj 6 E . (2'.3)
Recall, the exterior product of a p-form $ and a function f is simply f A ij> := f«(f>. Nevertheless, the wedge sign will be used sometimes in this case cither. Differentiation is represented by the exterior derivative d: E •»• E v p G [0,n-l], which is uniquely, defined by the properties
(1) d is 2R- linear
(2) d((j) A ij;) = d
(3) d o d = O (2.4)
(4) df = ordinary differential of f € E o (5) d is local (domain of
2.2 If the manifold M admits a pseudo-Riemannian metric g, we use the latter to define a scalar product in E (m,3R) V m G M , simply defined by
e. := g.. e3 6 E . (2.6)
We call it "dual" because it is the unique 1-form basis corresponding to the dual vector basis {£.} (e (£.} =6 .) by e. = g(5.,*) and therefore
i' i Note; Basis and dual basis with respect to g are both 1-forms but the latter is closely linked to (and represents therefore) the metric g (or vice versa) . There is a natural extension of introduced by the inner product (or
contraction) i: E,1 x E p '-»• E p-1. (E- 1. := 0) , i(a)6 := i(a,d>), with the de- fining properties (a 6 E ):
(1) i(ct)(
(2) i(a)f =0 V f 6 E (2.7) o (3) i(a)e = a (a = a e. = a.e ) .
Exterior derivative and contraction define the Lie derivative Ä: E, x E ->• - l P -»-E / £(a/!|>) =: &(a)(j>, where
£(a) := i(a)d. + di(a) V a 6 E . (2.8)
Note; Ä(a)<)> H JKcOd), where a is the corresponding vector field to ex £ E * ' (i.e. ex =: g(a, •) ) .
2.3 Let M be orientable, i.e. there% exists a continuous nonvanishing c 6 E (Mn) . We define the metric volume element by
:= /(-)S g e'"'" e E (2.9)
where g = det(g. .) = det(
i1"'in in 11 "e P := i(e P) ... i(e )e . (2.1O)
Just as e provides a volume element for M , any p-form <£> defines a p- dimensional volume element and hence may be integrated over a p-dimensional compact submanifold N of Mn, the corresponding integral is denoted by / V may be interpreted as p-linear mappings of © T M into V, V m £ M (usual notation, m see [4]). Thus V-valued p-forms are thought as (row) vectors in V with TR-valued p-forms as components with respect to a basis of V: E (Mn,V)a
Examples. ({E } basis of V) f\ n n (1) V = m. a 6 EP (M ,V'=:]R) = : Ep , di 6 E q(M ,V) =: Eq © V . Define A:E x (E © V) -* E ©V p q p+q •^ by a A (j> := a A if) © E A (2) V = V and V endowed with the structure of a Lie algebra,
n n
n D A 6 Ep (M ,fl) =: EP ® a, d, 6 Eq (M ,V) =: Eq © V . Define A : (Ep © Q) x (Eq © V) •*• Ep+ q © V a A by A A cp := A A X ad > = A A
6(L A 10 = 6L A K + L A 6K = 6* A [3L/3$ A K + (-)pr /. A because of (2.3). Thus we have
IT- (L A 10 = If A K + (-)pr L A |T (2.11) dtp d(fi d for any L G E , cj> £ E . The use and utility of this notation will become obvious in the next section. 2.6 Lagrangians. Let M1* = spacetirae. An action integral for the Lagrangian formulation of any classical field theory, leading to field equations of at most second order, is of the type I (A) = / L( where «T :=|t :=1^- (-)P d^j- (2.13) ofjj 3 under tlie action of a certain T.ie group G. As is well known, these symmetry properties may be deduced from the symmetry properties of (2.11), that is, the form of the field equations "-T = o is left unchanged in A if öl = O in A (2.14) (<5 = infinitesimal action of G) . Condition (2.14) may be weakened to 6L = exact functional of 64* / but we ignore that delicacy for simplicity. Note: As Examples. For $ and i. (1) ij> = x 6 E (M1* z U, K4) chart mapping d (2) <(» £ E (M1*, K) scalar field (massive, neutral) L(dx,(j),d(j)) = [d (3) i|i £ E (M4, S) Dirac- field; (y = y dx^, y Dirac matrices) >,d (4) iji 6 E (M4,S) Rarita-Schwinger (spin 3/2)-field L(dx,ü,di(0 = — (ib A YcY A dip - du A YrY A ti) - — ijJ A YcY A Y A ifj 2*> 5 2 »> 3. The Gauge Idea 3.1 Global gauge invariance. Consider a I-agrangian 4-form of type L = = L((j>,d(j)) which is invariant under the infinitesimal action 6 of a Lie group G, i.e. 5L = O , L = L((()/d<(i) , (3.1) where the vector (spinor-, ..) valued p-forms $ provide a representation of G (see examples above). Carrying out the variation (3.1) explicitly immediately leads to the identity (use 5d = do) - 6 d«j =0 if "T = 0 , «j := 6 Let X , a= 1,..., q := dim G denote the generators of G in the ij>-repre- 3. sentation and e , a = l,...,q, constant infinitesimal group parameters, then 6 ("global" action of' G because ea constant) and (3.2) becomes - X (j) A ::T H d»j (3.5) 3. Si t v;hich yields q (= dim G) weakly conserved currents d?:ja-0, «Ja!,Xa+A|^ (3.6) in the case ::T = O. 3.2 Local gauge invariance. Suppose now t = e (x) V a, i.e. e spacetime rfependent ("local" action of G, G is now called "gauged"), then (3.3/4) merely leads to the trivial case ead"j + dea A ::j = O because of the arbitrariness of E , de V a. To circumvent this awkward situation, we introduce q new auxiliary variables A , a = 1,...,q := dim G to compensate the q undesired terms ^ de .To this end, start again with another Lagrangian L1 = L'( 6L' = S = Ea x * A ::T' + ea d"j' + dea A ::j' + OAa A — = 0 3Aa where the primed expressions have an analogous meaning as above. As the new fields A are introduced to cancel the inhomogeneous terms de A :'j , we try the simplest of all possibilities ("minimal procedure"), namely «j- := X A A 1^-5 ^ ' (3.9) a a aa<)> »a and = - dea (3.10) whereby a symmetry condition similar to (3.5) is obtained. Note that by (3.1O), the Aa have to be 1-forms V a, the kind of their coupling to the (p-field is, however, restricted by-(3.9) only. To trace this particular coupling, we observe that "j^ has to be in- variant under the group action V a-(i.e. 6::j ' = 0 V a) , as d<|> (resp. a 3L'/3cW)) transforms contragrediently to A (resp. 3L'/to. ), as far as the inhomogeneous parts are concerned:'. a a 6d To proceed/ define AA4>:=A AX(p (by which A is interpreted as Lie- cl algebra valued 1-form, see Sec. 2.4), Example (3): A G E (M4,g)) thus getting X <(> = 3(A A 4>)/3Aa and therefore 31' _ 9(AA 4.) 31' 31.' _ 3L' . 3Aa ~ 3Aa 3d The last identity simply says that dcj> and A A $ have to appear in L' only through the combination 11 D 6D(|> = E X D£ . (3.14) 3, Performing the variation of A A <}> under the group action 6 6 (A A ^ (f ,Da . = structure constants of G) and using (3.13) we see, however, that the simple transformation property of the A in (3.1O) will not suffice in general to guarantee (3.14), but has to be replaced by 6Aa : = - dea - f\ Ab C =: - Dea (3.15) be E which meets (3.1O) in the case of an abelian group. The q (- dim G) com- pensating l~forms A obeying the local gauge transformation (3.15) under "their'1 gauge group G are called (G-) gauge potentials. Because of (3.15) , the gauge potentials may be transformed to zero by local gauge transformations, at least locally. But then, locally where A = 0 V a, L1 ( (3.16) as the coupling of the gauge potentials to the - X <}> A »T = d»j - f°, Ab A "j =: D"j (3.17) aT a ba Jc a where the primes are omitted according to (3.16). Remarks (1) Consider E (M4,g) ^ A = Aa & X , where {X } = basis of Q, a = 1,...,q. a a , a a a a a C Then <5A = 6A ® Xa + A & 6Xa and therefor_ e (6A) = OA - f .b eA E if 6 corresponds to G. In this more precise notation of (3.15), we see that the choice (3.10) was not really wrong as long as we were not prepared to look at the A as forming the components of one object, the Lie-algebra valued 1-form A E E (M4,g). The need for this was introduced by (3.14) as a consequence of invariance of (3.9) under G. But the latter simply implies that ::j in addition are components of " a a Lie-algebra valued 3-form "j. £ E (M^g) (not to be confused with ::j in (3.2/3)), transforming like an "isoveccor", i.e. 6::j = <5;:ja ® X + . a + 5:ja ® 6X = e :-'ja fc ®X <$=>• (6::j)a = f\ E ;:jc (6"ja = O as J a J oa c J be J J above; the group indices are rised and lowered by the invariant group metric of G). Thus (3.17) is merely a component version of a G- covariant identity of the corresponding isovectors :Jj £ E-d^g), J D::j € E (M1*,g), etc. Note tliat in the case of an abelian group (<$A) = 6A and (3.9/10) is sufficient to start the gauge procedure. We will use this fact in Sec. 4 to gauge the translation group T(4). (2) Although supererogatory, we want to give another argument for (3.12). Use (3.9/10) to get P a a (-) 6 Comparison with yields P a 6 Read off the coefficients of 6$ and 6D$ according to Sec. 2.5 and obtain 3i.'/8d (lex indicates that $ in D " ^ J. * ' *\ -3 .1 — 'N J. (3.18) do) ddcp •. o (p ex which reveals- the G-invariance-of the Euler-Lagrange expression !CT', as expected« ; (3) Observe that the "minimal coupling" was a result of the "minimal pro- cedure" to obtain local gauge invariance, i.e. to obtain de A "j' + * . cl + 6Aa A 8/-'/9Aa = o, in (3.8) by (3.9/1O). In this sense the term "minimal" is legitimate, because one can think of attaining local gauge invariance by other exterior fields than gauge potentials, which may result in a more complicated kind of coupling. However, if the compensating fields are to be gauge potentials, i.e. the "minimal procedure" is performed, than there is no other coupling than "minimal coupling". 3.3 Lagrangians for the gauge fields. The gauge fields Aa demand, inter- preted as physical (exterior) fields, an action principle for their own. According to the general scheme for Lagrangians leading to field equations of at most second order, we assume'L = L (e,A ,dA ), where any basis o o 1 lt lt {e } 6 E^M^/IR) (i = 0,1,2,3, or equivalently, e 6 E]L(M ,3R ))i& "ited explicitly in L , as they may, in general, provide a representation of the gauge group (e„g. orthonormal basis for the Lorentzgroup), but not necessarily as gauge fields. Recall that any 1-form basis will be needed to construct a 4-form L , e.g. by "-operation. Of course, L has to be invariant under the corresponding local gauge group, a constraint which restricts the various possibilities of L con- siderably. To see this, perform the same variation procedure as in the case of the iji-fields 14 3L . 3L ' 3L ÖL = 6e A -r-2" + 6A A —- + ödAS A —— H 0 , (3.19) a a ex 3A 3dA use FA in (3.15) and -Se := E (Y e)' , where Y are the generators of G in a a the e-representation , to get the following identities !i°3ii&A^A:!! . (3.20) 3L 3L 3L p = f\ (Ab A-—- + dAb A ——) (3.21) ex where (A A A)a := fa A A A°. By (3.2O) we see that the gauge potentials have to appear in L only via the combination Fa := dAa + 4- (A A A)a (3.22) the (G-) gauge field strengths, a = 1,..., q = dim G, which transform homogeneously under G, i.e. like an isovector (6F) = fu e F . The use of (3.20) in (3.21) finally yields - *b J where 3L 31 8L MJ 3Aa 3dAa 3dAa are the Euler-Lagrange expressions' for the (free) gauge potentials (com- pare (3.18)), D"J is defined analogously to (3.17) and ::t. * := 3L /3e3 i c • (o)j o ex turn out to be the gauge invariant- canonical energy/momentum currents (see [7]; 3/3ei means that the e to construct A = A ,e G E are held fixed, l ex i. l that is, the 1-forms A are treated as independent variables, and not their components A . 6 E ) . i o Note that in the case Y = 0 V a, the right hand side of (3.23) vanishes ' a identically. Because of f , = f, this implies 3I/3F ^ F or ::F and L has to be squared in the field strengths. A standard example for this is the "Yang-Mills "-like version L... = — Fa A XF . However, in general iM 2. a 15 - (Y £ O) , there is no need for the free Lagrangian to be "Yang-Mills"- a like, that means, to be squared in the field strengths. Example : Take G -> SO (3,1), AS -* ulj = u j , pa -> nlj = n lj := dulj + •f to1 Aw., {e1} orthonormal basis, Y ->• Y. . = Y,- . .-, , where (Ye) = = 6. . e and indices are moved by ri = diag (-1,1,1,1). The analogue of ij m (3.23) is i**.. H 8[1 A '«t(o)j] i (3.25) where — "J. . = — 6L /ou1-' is the usual spin current of the gauge fields, 2 3.J 2 )0 and (3.25) relates the SO (3, l)-covariant divergence of J. . to the anti- symmetric part of the canonical energy/momentum current "t. . in the well known way. The identities (3.25), however, do not restrict L to the simple "Yang-Mills "-like version ^ y ft1"3 A "ti. ., but allov; a variety of Lagrangians: ,(°) . I n^ A "e o * = 2 " A eij := 2 13.. .- <2) i A e. A «(O A ek) L(3) := 4 nlj A ek A «{{J- A e.) (3.26) 0 A • IK j A A .» A «,^ A . , Note in particular that there is also a Lagrangian.linear in the gauge field strengths, L(o). 16 3.4 Gauge theory defined. After having defined gauge potentials, gauge field strengths and gauge groups, we are now ready to summarize their main properties, thus declaring also what we want to call a (classical) gauge theory. In general, we are dealing with two kinds of objects. On the one hand, w<2 have certain V-valued p-forms over space time, F = dA + —(A A A) , the gauge field strengths, in order to preserve local gauge invariance of L , see Sec. 3.3. (e) The corresponding Euler-Lagrange expressions ::J := S L /6A have to a o obey certain identities by reason of local gauge invariance of L again, see (3.23). (f) The coupling of the gauge potentials to the G-fields (j> is performed minimally, L1 ( To combine G-fields and gauge potentials within one action principle, we• first introduce a coupling constant K by A =: icA1, wherefore D L(e,^,d^>,A,dA) = iM 4. Gravitation 4.1 CRT as a gauge theory. As mentioned in the introduction, we try to interpret Einstein's general relativity theory (CRT) as a gauge theory. To this (and, we have to look for gauge potentials in the theory, which obey, by construction, the conditions (a) - (g) of Sec. 3.4. In searching these potentials we first have to reformulate conventional (pure) CRT in terms of 1-forms according to the fact that gauge potentials have to be 1-forms. Secondly, these 1-forms have to be interpreted as forming one Lie-algebra-valued 1-form, whereby the corresponding gauge group comes into play. After having found both, the gauge fields and the gauge group, we shall proceed asking whether and in which way, this gauge theory may be extended in order to overcome, at some time or other, the problems of conventional CRT. 4.2 CRT in terms of exterior forms. Looking at the usual Lagrangian density in CRT LE = j v^g"R (4.1) (usual terminology [6], Sir times gravitational constant =: velocity of light =: 1) is rather deplorable because there are no 1-forms at the first glance. Everything is made up of tensorial functions g (tensorial O-form, (g ) 6 E (v'SüR'* ^K1*), V1* = Riemannian spacetime) and their derivatives only (R = R(g,r,3P), T = F(g,8g)). But if we pass to the Lagrangian 4-form / = i /^ R a4x = i Ag R dx° A dx1 A dx2 A dx3 (4.2) b Z 2 h we see that there are 1-forms indeed, dx^, which we can use to define V V V e y := gu v e , 'e := dx . (4.3) Using the equations of Cartan v de u + /i y A e v = O • (4.4a) a*? + 1V A Jia = RV (4.4b) ]i a y p 19 (ft the Levi-Civita connection 1-forms with respect to the basis {e }, P V its components being the usual Christoffel symbols /t = T e ; R = v at - p ya y = R i ie , the Riemannian curvature 2- forms, its components forming 'the Riemann-christoffel tensor), we are able to rewrite (4.2) like this L =RP A «a V . (4.5) p E 2 v p As L is obviously invariant with respect to mere change of the basis/ we prefer an orthonormal, anholonomic basis {e1}, ( V g = e ® e = (4.6) e1 = n.... e1 © ej (e =: h e , h the usual vierbein-fieIds and g = h h. ). The )J V pv y 3.V Cartan equations for that basis now read de1 + A1. A ej = 0 ' (4.7a) d^1. + ti-. A ^. = R1. (4.7b) 3^3] where n. ^ :- n A.1 = -t because of metricity. The Lagrangian (4.2/5) JC now becomes •J . (4.8) and thus the Einstein current "G (:JGT = G ::e , G = G = Einstein tensor with respect to the basis {e }) gets := 2" Rij A "^^ ~ ""r + Xt~ (4'9) where 20 (4.10) corresponds to the so called "super-field strength" and «"• t := - — A.* . . Ah (A/... AA "e + Au. AA «-e \) /,(4.11i ii\) 2 13 m ra to one of the pseudo-currents/ representing the canonical energy/momentum current of ehe gravitational field (see [5] for further details) . Because of (4.8) - (4.11) it looks as if there were two kinds of 1- forms in the theory, e and H. But solving (4.7a) with respect to h. . yields, after some ' - 3 calculations 'Llj = ~ [Ke^de1 - ife^Jde^ + e. A ife1) i(ej)dek] (4.12) 2 i 1C v:hich exhibits that the six n, 3 are completely determined by the four basis 1-forms e . But if this is the case, why not formulate Einstein's theory in terms of e and de only? Insert (4.12) into (4.7b) and (4.8) to find Lp = L (e,de), in particular L (e,de) = - d(e. A "de1) - \- de1 A a"3 -A "(de. A e.) + E i 2. 3 i (4.13) + -j de1 A e. A "(dej A e.) 4 i D ]- and the Einstein current :{G" turns out to be the Euler-Lagrange expression for the fields e : «•k L_E E, E .,Jt ,..Jc , . ... "G := - — = •; — := T — + d — — = -t + d"IT (4.14) 6eK , . 6e, 3e. 3de, k k k k with :CIF, 5:t as in (4.9/1O/11), but now in terms of e and de: 8L _k E i1 k l k i1 -IT := -r— = - e A "(de. A e ) + -^ e A -(de A e.) (4.15) ode i 2 i 9L 3L "tKk := -—E = i(eKk )L^ - i(ekK)de. A —E - (4.16) oe, E i 9de. k i 21 and where Lfe equals L but the exact term d(e. A ::de ) discarded. This E E i reformulation (4.13) - (4.16) of Einstein's general relativity in terms of the 1-forms e (and their exterior derivatives Je ) does not only simplify practical calculations considerably (no connection-, no curvature components are to be calculated) , but in addition reveals that CRT fits into the general scheme of our Lagrangian formalism, L = L(e,de), by which various field theoretical aspects are reflected in a more distinct way, e.g. the canonical energy/momentum character of ::t in (4.16). However, as we want to look at CRT as a gauge theory and the only variables we are left with are 1-forms e , it looks as if we had found the desired gauge potentials. 4.3 Where are the gauge potentials? Before investigating whether the orthonormal basis 1-forms e may be interpreted as gauge potentials, i r~ there is a comment on the Levi-Civita connection forms X. , in place, which are frequently viewed to be gauge potentials of the homogeneous Lorentzgroup . And indeed, under local Lorentzrotations of the basis, infinitesimally written as 6 e1 = ^.(x) ej . ij - , e\ = (4.17) 2TfOc u ~J e K. the n. transform as Loren tz gauge potentials 6 = - de - * e - e =: - DE (4.18) iO u jC K. , _ ._. a ij a ij , _a _mn (compare (3.15)r using A ->• A. , e -> e , and f -> r . . = . . DC 1 3^^* = n [6?? 6^" + önp 6™^]) . But as the six I13 are completely determined by four basis 1- forms e1, they are not independent in the sense of con- dition (a), Sec. 3.4. Therefore, they will not suffice to act as compensat- ing fields and are thus ruled out as gauge potentials of the Lorentzgroup, at least in strict sense we are dealing with. Now we are left again with the orthonormal 1-forms e , which are obviously not gauge potentials of -the Lorentzgroup, but rather Lorentz- fields (SO(3, 1) -fields) according to (4.17). So let us try another trans- formation group frequently used in CRT, the (local) coordinate transfor- mation group, its action infinitesimally written as 22 6coord * = E(X) ' P = °'1'2'3 (4.19) for any coordinate functions x . Interpreting (4.19) as expressing the action of a gauged group, the anholonomic l-forms e are obviously not suited to serve as "compensating fields" for that group, as, by construc- tion, they do not transform under (4.19) at all: 1 <5 coord, e = o , i = 0,1,2,3 . (4.2O) Thus it is not astonishing that the basis l-forms e1 do not fit into the general scheme (a) - (g) , whereby they are ruled out again as gauge potentials corresponding to (4.19). This result is, at the first glance, rather disappointing since (4.19) looks very much like the local ("gauged") u version of infinitesimal translations in Minkowskian spacetime (5x = e^ = const V u) , and any interpretation' of CRT as a gauge theory of the (4- dimensional) translation group T (4) is well suited to explain the pre- ference of the mass/energy aspect in this theory. Thus some authors insist on having formulated a " trans lational gauge theory" with use of a certain kind of orthonormal basis as "gauge potentials" of that group (see e.g. [8] and further references therein) . However, as the underlying idea of gauging the translation group T (4) seems to be very reasonable, we try to follow that line, thus interpreting (4.19) as expressing the local action of T(4) . 4.4 Gauging the translation group. To find out the gauge potentials, we perform the gauge procedure outlined in Sec. 3.2. Thus start with ordinary 4 field theory over Minkowskian spacetime M* , based on L = L(x,dx,d>rd as usual (global translations, e not to be confused with (2.9)). Note that 6x = e follows our usual terminology, but as x is a chart mapping, 23 <5x =erelates translations to the spacetime diffeomorphism group (the same goes for the gauged case). Considering invariance of L(x,dx, 91 6L ~= 6x A + 5dx A (4.22) 9dx ex (recall that we treat x and dx as B4-valued O-forms and 1-forms, resp., quite analogous to the ik = O (4.23) 3x ex i.e. the coordinates must not appear explicitly in the Lagrangian to guarantee global translation-invariance. Now, if we replace e by £ = e(x) (local, or "gauged" translations), the symmetry <5L = 0 does not hold, except in the trivial case 3L/3x| ex = 8L/3dx = O because 6L = e A + de A iL. (4.24) 3x ••>dx ' ex According to the general gauge idea, we introduce 4 exterior fields c 6 E , y = 0,1,2,3 (or combined in compact notation c £ E (M^/H4), IR1' = Lie-algebra of T(4) ) , to compensate the inhomogeneous terms in (4.24), i.e. take L' = L1 (x,dx,4>,d 3L' 6L' = 6x A + 6dx A !4~ + oc A — = 0 (4.25) 3dx 8c In order to restate the original symmetry condition (4.23), we perform the "minimal procedure" analogously to (3.9)ff, that is 31.' _ 3L' (4.26) 8dx ~ 3c As T(4) is abelian, we know from the general considerations in Sec. 3.2 that (4.26/27) is sufficient to perform the gauge procedure, see remark (1). Furthermore, (4.27) reveals that c actually corresponds to a gauge potential of a 4-diraensional abelian group, as is the case with T (4). Now, (4.26) simply says that c has to appear in L1 through the com- bination e := dx + c only, whereas (4.27) reveals that this combination is invariant under gauged translations, i.e. 6e = O , e := dx + c =: Dx (4.28) where Dx := dx + c A x = dx + c^ A X x = dx + c, as 6x = e^X x = £v"/ ,,x = B . V y xV according to (3.4) and (3.13), that is, D corresponds to a T(4)-covariant exterior derivative. Note that D (4.29) in order to meet L in the case c = O ("minimal coupling" of the gauge potentials c to the chart mapping x, as the latter are assumed to be the only fields the translation group acts on, see (4.21)). Remark . According to (3.17) one might have expected some (weak) conserva- tion laws corresponding to the symmetry 6 L = 0. As we are dealing with translations, the presumably weakly conserved "charge "-currents 8l./3c are expected to correspond to the energy/momentum current ::t. And indeed, this is the case as ::t := 3/-/9e = 3L/3dx = 3L/8c. However, the correspond- ing weak conservation law d"t = O (or rather a generalized version of it, as the coordinate basis dx has to be replaced by an anholonomic one) is obtained in a different way, using, the identity £(e.)l = di(e.)L V i, which yields 25 . A ::T + (-}p i(e.) 4.5 Lagrangians for the free T(4)-gauge potentials. Analogously to Soc. 3-3 we assume L = L (e/c,dc), where e is already taken as the local o o translation invariant orthonormal basis of (4.28). Then C :--= de = de (4.31) by (4.28), C being the T(4)-field strength, analogously to (3.22) but the A A A-term discarded because T(4) is abelian. But then 6JL = O simply gives 3jL /3c| = O, that is, the T(4)-gauge potentials have to appear in L only through the local translational invariant e and de. There are several independent possibilities for L , e.g. L(Q) -iVAKe. o 2 i , (1) 1 , i . ... L = — de A "de. o 2 i (4.32) L(2) = 4" äe1 A e. A ::(dej A e.) 0^1 3 L(3) =4- de1 A Remark. Of course, L^ is not the only possible choice one can make. How- ever, any choice has to lead to a gravity theory comparable with experi- 26 raents. By this criterion, for example, L is ruled out as the correspond- ing field equations do not admit a -"Newtonian-limit" version of the theory. On the other hand LQ(2 ) vanishes identically in the case of spherical symmetry (where de ^ e ) and is thus not suited as a Lagrangian for its own. LQ finally, which is used by Hehl et al. ([8], [9]) leads to a gravitation theory undistinguishable from CRT up to the 5th order of post- Newtonian approximation, as shown e.g. in [ 1O] . Einstein's Lagrangian, however, has one essential property - it is, in addition, invariant under local (space time dependent) Lorentz rotations of the basis 1-forms e , the corresponding (symmetric) Einstein current ::G thus has to be coupled to a symmetric matter energy/momentum current, as one expects within a trans- 'lational gauge theory, where the spin aspect is left unbased. 4.6 A natural extension; U -theories. The last point in the preceding re- mark already suggests a .natural extension of the T(4)-gauge theoretical version of CRT, namely, to include also the spin aspect by gauging the homogeneous Lorentzgroup. To this end, start again with a T(4)-gauge theory based on (£2 = essentially the gravitational constant, corresponding to the squared translational coupling constant) Ue,de, and perform the "minimal procedure" to gauge the Lorentzgroup (viewing e and L(e,de,4>,d(f>,io,du>) = L (e,4>,D where D Remark on teleparallelism. Any decoupling of the Lorentzgroup by K •*• O (and therefore fi1-3 -> 0 V i,j) ought to lead back to the T(4)-gauge theory based on (4.33), that is, in the case of L = L%, to CRT ("translation- T v . limit"). However, the corresponding condition ti = O V i,j restricts the underlying spacetime now more than the one we started with, i.e. space- time of a translational gauge theory has to admit a teleparallel basis (the one with respect to which to = O globally) and is then called Weitzen- bock-(Ttt-) spacetime, in contrast to the more general Riemannian space- time V4 of GRT„ In principle, this point of view is acceptable, once the T(4)-invariant basis in (4.28) is interpreted as a teleparallel basis (modulo global Lorentz rotations) and the field equations of CRT are accompanied by constraints for the. Riemannian curvature R1. according to i ' 3 the teleparallelism conditions n . = O. 28 5. Conclusion In carrying over the concept of .a gauge theory (outlined in Sec. 3) to Einstein's general relativity theory (CRT), we are led to the possibility of interpreting GRT as a (pure) gauge theory of the 4-diraensional trans- lation group T(4), where the corresponding gauge potentials c are incor- porated in a T(4)-invariant way via orthonormal anholonomic basis- 1-forras (Sec, 4). This seems to be very reasonable, as CRT stresses in particular the mass/energy aspect of physics.'To include also the spin aspect on an equal footing, we just indicate in Sec. 4.6, how CRT may be extended by performing in addition the gauge procedure for the Lorentzgroup, and by which a Riemann-Cartan manifold as-underlying spacetime is suggested. For obvious reasons, the concept of a gauge theory was outlined at the same level as CRT - over spacetime. But as in modern gauge theory the notion of fiber bundles is extensively used, there are some remarks to be made, to link both approaches. For the sake of briefness, the terminology is taken for granted, otherwise see [l]. In the fiber bundle formulation of a gauge theory, the Lie-groups we are dealing with are contained as typical fibers of principal fiber bundles (P »M1* ,7f;G) over spacetime M1*/ where P denotes the bundle space and ir: P -> Ml( a projection. In that picture a gauge potential is viewed as a % certain Lie-algebra-valued 1-form over P, A E E (P,g), called the connection of the principal fiber bundle. The usual "matter-fields", as counterparts of the gauge potentials, are V-valued, horizontal and equivariant p-forms 'u over P, A £ E (P,V), where V is a reoresentation space of G. As G is the p typical fiber of (P,Mlf ,ii;G) and as the matter fields are equi variant, any action of G on v is uniquely linked with vertical automorphisms in P, called "gauge transformations". Now, the gauge potentials A 6 E (M1*,g), G-fields <}> G E (M1*,V) and gauge transformations we were dealing with in Sec. 3 are connected with the fiber bundle objects via (local) sections P .^ ..'b . cf P (i.e. s: MH 2 U •*• P, 7i o s = id ), by A : = s-A and For example, in the case of the Lorentzgroup, the corresponding principal fiber bundle is known to be the bundle of orthonormal bases and any local section corresponds to a local choice of a basis. As the ..'v» gauge potential u = s"co depends on sections, it depends on the ba^is chosen. And the same goes for the gauge potential c for the translation group T(4) = (H^.+J with the argument reversed. As by (4.27), the T(4)- gauge potentials depend on the choice of coordinates, the corresponding principle fiber bundle will be a "coordinate bundle", where local sections correspond to a local choice of coordinates (i.e. P equals U x 3R1* locally, because G = T(4) = (Kl+, + ), but the image of any section has to be diffeomorphic to an open neighbourhood in 3R4, in order to provide a "coordinate section"). Further details will be traced in [7]. Acknowledgements I would like to thank Prof. W. Thirring for several comments during the earlier stage of this work. I am also indebted to members of the "Institut für Physik und Astrophysik" of the Max Planck Institute in Munich, Garching, where part of this paper was given as a seminar talk and where numerous questions, in particular by Prof. J. Ehlers and Doz. E.G. Schmidt, caused me to clarify some details more rigorously. It is also a great pleasure to thank Mrs. F. 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