Gravitation gauge group
G. T. TER-KAZARIAN(*)
Byurakan Astrophysical Observatory - Armenia 378433
(ricevuto l’11 Luglio 1996; approvato il 20 Novembre 1996)
Summary. — The suggested theory involves a drastic revision of the role of local internal symmetries in the physical concept of curved geometry. Under the reflection of fields and their dynamics from Minkowski to Riemannian space a standard gauge principle of local internal symmetries has been generalized. A gravitation gauge group is proposed, which is generated by hidden local internal symmetries. In all circumstances, it seemed to be of the greatest importance for the understanding of the physical nature of gravity. The most promising aspect in our approach so far is the fact that the energy-momentum conservation laws of gravitational interacting fields are formulated quite naturally by exploiting all the advantages of auxiliary
shadow fields on flat shadow space. The mechanism developed here enables one to
infer Einstein’s equation of gravitation, but only with a strong difference from Einstein’s theory at the vital point of well-defined energy-momentum tensor of gravitational field and conservation laws. The gravitational interaction as well as the general distortion of the manifold G( 2 .2 .3 ) with hidden group Uloc( 1 ) has been considered.
PACS 04.20 – Classical general relativity.
PACS 04.20.Cv – Fundamental problems and general formalism.
1. – Introduction
In spite of its unrivalled simplicity and beautiful features, Einstein’s classical theory of gravitation clashes with some basic principles of field theory. This state of affairs has not changed much up to the present and the proposed abundant models of gravitation are not able to provide an unartificial and unique recipe for resolving the controversial problems of the energy-momentum conservation laws of gravitational interacting fields, the localization of the energy of gravitation waves and also severe problems involved in quantum gravity. It may seem foolhardy to think of the rules governing such issues in the scope of the theory accounting for gravitation entirely in terms of intricated Riemannian geometry. The difficulties associated with this step are notorious; however, these difficulties are technical. Mainly, they stem from the fact
(*) E-mail address: gagoHhelios.sci.am
that Riemannian geometry, in general, does not admit a group of isometries. So, the Poincaré transformations no longer act as isometries. For example, it is not possible to define energy-momentum as Noether currents related to exact symmetries and so on. On the other hand, the concepts of local internal symmetries and gauge fields [1-3] have become a powerful tool and have been successfully utilized for the study of electroweak and strong interactions. Intensive attempts have been made for constructing a gauge theory of gravitation. But they are complicated enough and not generally acceptable. Since none of the solutions of the problems of determination of the gauge group of gravitation and of the Lagrangian of the gravitational field proposed up to now seems to be wholly convincing, one might gain insight into some of the unknown features of the phenomena of local internal symmetries and gravity by investigating new approaches in the hope of resolving such issues. In order to effect a reconciliation, it is the purpose of the present article to explore a number of fascinating features of generating the gauge group of gravitation by hidden local internal
symmetry. This is the guiding principle framing our discussion. In line with this we
think it is of the geatest importance to generalize the standard gauge principle of local internal symmetries. In the meantime, a second trend emerged as a formalism of reflection of physical fields and their dynamics from Minkowski to Riemannian space. There is an attempt to supply some of the answers to the crucial problems of gravitation in a concise form and to trace some of the major currents of thought under a novel viewpoint. In exploiting all the advantages of field theory in terms of flat space, a particular emphasis will be placed just on the formalism of reflection. This is not a final report on a closed subject, but we hope that the suggested theory will serve as a useful introduction and that it will thereby increase our knowledge on the role of local internal symmetries in the physical concept of curved geometry. Of course, much remains to be done before one can determine whether this approach can ever contribute to the larger goal of gaining new insight into the theory of gravity.
2. – Formulation of the principle
In the standard picture, suppose that a massless gauge field Bl(xf) 4TaBla(xf) with
the values in the Lie algebra of group G is a local form of expression of the connection in principle bundle with a structure group G. The set of matter fields are defined as the sections of vector bundles associated with G by the reflection Ff: M4K E, so that
pFf(xf) 4xf, where xfM4is a space-time point of Minkowski flat space specified by
the index (f). The Ff is a column vector denoting a particular component of field taking
values in the standard fiber Fxfupon xf: p 21(U( f )
) 4U( f )
3 Fxf, where U
( f )is a region of
base of principle bundle upon which an expansion into the direct product p21(U( f )
) 4
U( f )3 G is defined. The various suffixes of Ff are left implicit. The fiber is a Hilbert
vector space on which a linear representation Uf(xf) 4exp[2iTauaf(xf) ] of group G
with structure constants Cabcis given. This space is regarded as a Lie algebra of group
G upon which the Lie algebra acts according to the law of adjoint representation: BDad B FfK [B , Ff]. To facilitate writing, we shall consider, mainly, the most
important fields of spin 0, 1
2 , 1. But the developed method may be readily extended to
the field of arbitrary spin s, since the latter will be treated as a system of 2s fermions of half-integral spin. In order to generalize the standard gauge principle below we proceed with a preliminary discussion.
a) As a starting point we shall assume that under gauge field Bfthe basis vectors
el
f transform into four vector fields
e×m(kB f) 4D× m l (kBf) efl, (2.1) where D×l m
are real-valued matrix functions of Bf, and k is a universal coupling constant
by which the gravitational constant will be expressed
(
see eq. (4.7))
. The double occurrence of dummy indices, as usual, will be taken to denote a summation extended over all their values. The explicit form of D×lm
, which must be defined under concrete physical considerations [4-8] (sect. 5, 6), does not concern us here. However, two of the most common restrictions will be placed upon these functions. As far as all known interactions are studied on the base of commutative geometry, then, first of all, the functions D×l
m
will be diagonalized at given (m , l). As a real-valued function of the Hermitian matrix Bf, each of them is Hermitian too and may be diagonalized by a
proper unitary matrix. Hence Dl m (kBf) 4diag(l m l1, l m l2, l m l3, l m
l4), provided with
eigen-values lm
lias the roots of the polynomial characteristic equation
C(lml) 4det (l m lI 2D× m l ) 40 . (2.2) Thus em(kBf) 4D m l (kBf) efl (2.3)
leads to commutative geometry. Consequently, if en(kBf) denotes the inverse matrix
vector V D V c 0 , then aem, enb 4Dl m
Dnl4 dmn, Dl m
Dmk4 dlk. A bilinear form on the vector
fields of sections t of a tangent bundle of R4
: g× : t 7 t KCQ(R4), namely the metric
(gmn), is a section of conjugate vector bundle S2t * (symmetric part of tensor degree)
with corresponding components in basis em(kBf):
gmn4 aem, enb 4 aen, emb 4gnm, em4 gmnen.
(2.4)
In holonomic basis g× 4gmndxm7 dxn. One now has to impose a second restriction on the
functions Dl m
by placing a stringent condition upon the tensor vml (xf) implying
¯flD m k(kBf) 4¯fk
(v
ml (xf) Dmm(kBf))
, (2.5) where ¯lf4 ¯ ¯xl f , so that the v 4vllshould be a Lorentz scalar function of the trace of
curvature form V of connection Bl with the values in the Lie algebra of group G. That
is v 4v( tr V) 4v( d tr B) , (2.6) provided that
.
/
´
V 4!
l Ek FflkdxflR dxfk, B 4Bfldxfl, Fflk4 ¯ f lBk2 ¯ f kBl2 ig[Bl, Bk] . (2.7)Thereupon a functional v has a null variational derivative dv( tr V) dBl
4 0 at local variations of connection BlK Bl1 dBl, namely, v is invariant under Lorentz (L) and
b) Constructing a diffeomorphism xm(xl
f): M4K R4, the holonomic functions
xm(xl
f) satisfy the defining relation
emc m l4 elf1 xfl(Bf) , (2.8) where xf l(Bf) 4emx m l4 2 1 2em
o xf (¯f kD m l 2 ¯flD m k) dxfk. (2.9)The following notational conventions will be used throughout: cm
l4 ¯flx
m
, where the indices m, n, l, t, s, k stand for variables in R4, whereas l, k, m, n, i, j refer to M4. A
closer examination of eq. (2.9) shows that the covector xfl(Bf) realizes the coordinates
xmby providing the criteria of integration
¯fkc m l4 ¯flc m k (2.10)
and undegeneration V c V c 0 [9, 10]. Due to eq. (2.6) and eq. (2.9), one has the Lorentz scalar gauge-invariant functions
x 4 ael f, xflb 4 v 2 2 2 , S 4 1 4 cm l Dm l 4 1 2
g
1 1 v 4h
. (2.11)So, out of a set of arbitrary curvilinear coordinates in R4 the real-curvilinear
coordinates may be distinguished, which satisfy eq. (2.8) under all possible Lorentz and gauge transformations. There is a single-valued conformity between corresponding tensors with various suffixes on R4 and M4, while each co- or contra-variant index transformed incorporating, respectively, with functions cml or clm. Each transformation
of real-curvilinear coordinates xm
K x8m8 was generated by some Lorentz and gauge transformations ¯x 8m 8 ¯xm 4 c m 8 l 8(B 8f) clm(Bf) Ll 8l . (2.12)
There would then exist preferred systems and groups of transformations of real-curvilinear coordinates in R4. The wider group of transformations of arbitrary curvilinear coordinates in R4would then be of no importance for the field dynamics. If
an inverse function clmmeets the condition
¯cml dxn cG l mncll, (2.13) where Gl
mn is the usual Christoffel symbol in accordance with the metric gmn, then the
curvature of R4does not vanish [11].
In pursuing the original problem further we are led to the principle point of a drastic change of the standard gauge scheme and to assume hereafter that a
single-valued, smooth, double-sided reflection of fields FfK F(Ff): FxfK Fx, namely
Ffl R m(xf), takes place under the local group G, where Ff%Fxf, F % Fx, Fx is the fiber
upon x : p21(U) 4U3F
put forth in illustration of a point at issue. Then FmRd (x) 4cmlRcdmR(Bf) Fl R mf (xf) f (Rc) m R d l R mFl R mf (xf) , (2.14)
where R(Bf) is a reflection matrix. The idea of general gauge principle may be framed
into the requirement of invariance of the physical system of fields F(x) under the
finite local gauge transformations UR4 R 8cUfRc1 of the Lie group of gravitation GR
generated by G , where R 8c4 Rc(B8f) if the gauge field Bf(xf) is transformed under G in
standard form, while the corresponding transformations of fields F(x) and their
covariant derivatives are written as . / ´ F8(x) 4UR(x) F(x) ,
(g
m(x) ˜ mF(x))
8 4 UR(x)(g
m(x) ˜mF(x))
. (2.15)The solution of eq. (2.15) may be readily obtained as the reflection of covariant derivatives gn(x) ˜ nFm R d(x) 4S(Bf) c m lRcdmR(Bf) gkDkFfl R m(xf) , (2.16)
where Dl4 ¯lf2 igBl(xf), S(Bf) is the gauge-invariant Lorentz scalar, ˜mis the covariant
derivative in R4: ˜
m4 ¯m1 Gm, provided with the connection [11] Gm(x) 4 1
2S ab
Van(x) ¯mVbn(x), the Sab are generators of Lorentz group, Vam(x) are the
components of the affine tetrad vectors ea in the used coordinate net xm: V am(x) 4
aem, eab; one has gm(x) ¨ em(x) and gl¨eflfor the fields (s 40, 1); but gm(x) 4Vam(x) ga
for the spinor field
(
s 412)
, where gaare Dirac’s matrices. Since the fields Ff(xf) no
longer hold, the reflected ones F(x) will be regarded as the real physical fields. But a conformity eq. (2.14) and eq. (2.16) enables Ff(xf) to serve as an auxiliary shadow field
on the shadow flat space M4. These notions arise basically from the most important
fact that a Lagrangian L(x) of fields F(x) may be obtained under the reflection from the Lagrangian Lf(xf) of the corresponding shadow fields and vice versa. Certainly,
L(x) is also an invariant under the wider group of arbitrary curvilinear transformations x Kx 8 in R4:
JcL(x) Ninv (GR; x Kx 8)4 Lf(xf) Ninv (L ; G), (2.17)
where Jc4 V c V k 4
(
1 12 Vaelf, xfkb V 1Vaxfl, xfkb V)
1 O2(
see eq. (4.1))
, while the internalgauge symmetry G remains hidden symmetry, since it is screened by the gauge group of gravitation GR. We note that the tetrad ea and basis efl vectors meet the
condition . / ´ ra l(x , xf) 4 aea, elfb 4Vma(x) D m l (kBf) , aea f , ebfb 4 aea, ebb 4dab. (2.18)
So, the Minkowskian metric is written hab
4 diag ( 1 , 21 , 21 , 21 ) 4 aefa, ef
b
b 4 aea, eb
3. – Reflection matrix
One interesting offshoot of the general gauge principle is a formalism of reflection. A straightforward calculation for the fields s 40, 1 gives the explicit unitary reflection matrix R(x , xf) 4Rf(xf) Rg(x) 4exp [2iUf(xf) 2Ug(x) ] , (3.1) provided that Uf(xf) 4g
0 xf Bl(xf) dxfl, Ug(x) 4 0 x [R† f GmRf1 c21¯mc] dxm, c f (c m l) , (3.2)where Ug4 0 for the scalar field and Ug1 Ug†4 0 for the vector field, because of Gm1
Gm†4 0 and ¯m(c21c) 40. The function S(Bf) has the form of eq. (2.11), since S(Bf) 4
1 4R † (cmlDl m ) R 414clmDl m
. The infinitesimal gauge transformation UfB 1 2 iTaufa(xf)
yields
UR4 exp
y
igCabcTa0
xf
ub
fBlcdxfl1 U8g2 Ug
z
,(3.3)
where the infinitesimal transformation B 8la4 Bla1 CabcubfBlc2 1
g¯l
fu f
a was used. For
example, the Lagrangian of the isospinor-scalar shadow field Wfis in the form
Lf(xf) 4JfL(x) 4Jc[ (em˜mW)†(en˜nW) 2m2W
†
W] 4
(3.4)
4 Jc[S(Bf)2(DlWf)†(DlWf) 2m2Wf†Wf] .
The Lagrangian of the isospinor-vector Maxwell’s shadow field arises in a straight-forward manner: Lf(xf) 4JcL(x) 42 1 4JcF † mnFmn4 2 1 4 JcS(Bf) 2(F( f ) mn)†F mn ( f ) , (3.5) provided that Fmn4 ˜mAn2 ˜nAm, Fmn( f )4 (cknDmf2 cmkDnf) Akf, Dmf4 DmlDl, (3.6)
with an additional gauge-violating term
(3.7) LGf(xf) 4JcLG(x) 42 1 2z 21 0 Jc(˜mAm)†˜nAn4 4 21 2 z 21 0 JcS(Bf)2(llmDlAfm)†lknDkAfn, llm4 1 2 (d l m1 vlm) ,
where z210 is a gauge-fixation parameter. Finally, the Lagrangian of the
isospinor-ghost fields is written as
Lghf (xf) 4JcLgh(x) 4Jca(em¯mC)†, en¯nCb 4JcS(Bf)2(DlCf)†DlCf.
Continuing along this line we come to a discussion of the reflection of the spinor field Cf(xf)
(
s 4 1 2)
as a solution of eq. (2.15):.
/
´
C(x) 4R(Bf) Cf(xf) , C – (x) 4C–f(xf) RA†(Bf) , gm(x) ˜ mC(x) 4S(Bf) R(Bf) glDlCf(xf) , (˜mC – (x))
gm(x) 4S(Bf)(D
lC – f(xf))
g l RA†(Bf) , (3.9) where RA4 g0 Rg0, Sab 414[ga, gb], Gm(x) 4 1 4Dm , ab(x) g agb, the Dm , ab(x) are Ricci’s
rotation coefficients. The reflection matrix R is in the form of eq. (3.1), provided that we make the change
Ug(x) 4 1 2
0 x R† f ]gmGmRf, gndxn( , (3.10)the ] , ( is an anticommutator. A calculation now gives
S(Bf) 4 1 8 Kcm l ]RA†gmR , g l( 4 inv , (3.11) where (3.12) K 4RA†R 4RAg†Rg4 4 exp
y
21 2 0 x(
]Rf†GAm†gm, gndxn( Rf1 Rf†]gmGmRf, gndxn()
z
, AGm†4 g0Gm†g0.Taking into account that [Rf, gn] 40 and substituting [12]
G A m †gn 1 gnGm4 2˜mgn4 0 (3.13)
into eq. (3.12), we get
K 41 . (3.14) Since UA† RUR4 RAUf†RA8†R 8 UfR†4 RAUf†RA†RUfR†, (3.15) where RA8†R84RA†R 41, then UA† RUR4 g0UR†g0UR4 1 . (3.16)
The Lagrangian of the isospinor-spinor shadow field may be written as (3.17) Lf(xf) 4JcL(x) 4 4 Jc
m
i 2[C – (x) gm(x) ˜ mC(x) 2(
˜mC – (x))
gm (x) C(x) ] 2mC–(x) C(x)n
4 4 Jcm
S(Bf) i 2[C – fglDlCf2 (DlC – f) glCf] 2mC – fCfn
.In the special case of the curvature tensor Rl
mnt4 0 , eq. (2.8) may be satisfied globally in
M4by putting cml4 D m l 4 V m l 4 ¯xm ¯jl , VDV c 0 , x f l4 0 , (3.18) where jl
are inertial coordinates. So, S 4Jc4 1 , which means that one simply has
constructed local G-gauge theory in M4 both in curvilinear as well as inertial coordinates.
4. – Action principle
Field equations may be inferred from an invariant action
S 4SBf1 SF4
LBf(xf) d 4xf1
kLF(x) d4x .(4.1)
The Lagrangian LBf(xf) of the gauge field Bl(xf) defined on M
4 is invariant under
Lorentz as well as G-gauge groups. But the Lagrangian of the rest of fields F(x) defined on R4is invariant under the gauge group of gravitation GR. Consequently, the
whole action eq. (4.1) is G-gauge invariant, since GR is generated by G. The field
equations follow at once in terms of Euler-Lagrange variations, respectively, in M4 and R4,
.
`
`
/
`
`
´
dfLBf dfBa l 4 Jal4 2 dfLFf dfBa l 4 2¯g mn ¯Bla d(kLF) dgmn 4 4 21 2 ¯gmn ¯Ba l k VDV Dkmd (kLF) dDn k 4 2k 2 ¯gmn ¯Ba l Tmn, dLF dF 4 0 , dLF dF– 4 0 , (4.2)where Tmn is the energy-momentum tensor of the fields F(x). Making use of the
Lagrangian of the corresponding shadow fields Ff(xf), in generalized sense, one may
readily define the energy-momentum conservation laws and also exploit all the advantages of field theory in terms of flat space in order to settle or mitigate the difficulties whenever they arise, including the quantization of gravitation, which does not concern us here. Meanwhile, of course, one is free to carry out an inverse reflection to R4whenever it is needed. To render our discussion here more transparent, below we
clarify the relation between gravitational and coupling constants. So, we consider the theory at the limit of Newton’s non-relativistic law of gravitation in the case of static weak gravitational field given by the Poisson equation [11]
˜2g004 8 pGT00. (4.3) In linear approximation g00(kBf) B112kBka(xf) uka, uka4
g
¯D00(kBf) ¯kBkah
0 4 const , (4.4)then
uka˜2kBka4 4 pGT00.
(4.5)
Since eq. (4.2) must match with eq. (4.5) at the considered limit, then the right-hand sides of both equations should be in the same form:
kuka dfL Bf dfBak 4 kukaJ a kB 2 k2 2 uk a¯g00 ¯(kBk a) T004 k 2 (ukau a k) T00. (4.6)
With this final detail carried out, one gets
G 4 k 2 4 pu2c , u2 cf(ukauak) . (4.7)
At this point we emphasize that Weinberg’s argumentation [13], namely, a prediction of attraction between particle and antiparticle, and repulsion between the same kind of particles, which is valid for pure vector theory, no longer holds in the suggested theory of gravitation. Although Bl(xf) is the vector gauge field, eq. (4.7) just furnishes the
proof that only gravitational attraction exists. One final observation is worth recording. A fascinating opportunity has turned out in case one utilizes the reflected Lagrangian of Einstein’s gravitational field
LBf(xf) 4JcLE(x) 4
JcR
16 pG , (4.8)
where R is a scalar curvature. Hence, one readily gets the field equation
g
Rlm2 1 2d m lRh
Dml ¯Dll ¯Bka 4 28 pGUak(kBf) 428pGTlnDnl ¯Dll ¯Bka , (4.9)which obviously leads to Einstein’s equation. Of course, in these circumstances it is straightforward to choose gmn as the characteristic of the gravitational field without referring to the gauge field Bla(xf). However, in this case the energy-momentum tensor
of the gravitational field is well defined. At this vital point the suggested theory strongly differs from Einstein’s classical theory. In line with this, one should use the real-curvilinear coordinates, in which the gravitational field is well defined in the sense that it cannot be destroyed globally by coordinate transformations. Meanwhile, taking into account the general rules of eq. (2.12), the energy-momentum tensor of the gravitational field may be readily obtained by expressing the energy-momentum tensor of the vector gauge field Bla(xf) in terms of the metric tensor and its derivatives:
(4.10) Tmn4 gmlTnl4 gmlclkcinT( f ) ik 4 4 gmlclkcin
g
¯Bla ¯gm 8 n8 c s i¯sgm 8 n8 ¯(¯tgl 8 t8) ¯(¯f kBla) ¯(JcLE) ¯(¯tgl 8 t8) 2 dkiJcLEh
.At last, we should note that eq. (4.8) is not the simplest one among gauge-invariant Lagrangians. Moreover, it must be the same in all cases including eq. (3.18) too. But in the last case it contravenes the standard gauge theory. There is no need to contemplate such a drastic revision of physics. So, for our part we prefer the G-gauge invariant
Lagrangian in terms of M4, LBf(xf) 42 1 4aF f lk(xf), Flkf (xf)bK, (4.11) where Ff
lk(xf) is in the form of eq. (2.7), a , bKis the Killing undegenerate form on the Lie
algebra of group G for adjoint representation. Certainly, an explicit form of the functions Dl
m
(k af) should be defined.
5. – Gravitation at G 4Uloc( 1 )
The gravitational interaction with the hidden Abelian local group G 4Uloc
( 1 ) 4
SOloc
( 2 ) and one-dimensional trivial algebra g× 4R1 was considered in [4-6], wherein
the explicit form of the transformation function Dl m
is defined by making use of the principle bundle p : E KG(2.3). It is worthwhile to consider the major points of it anew in concise form and make it complete by calculations, which will be adjusted to fit the here outlined theory. We start with a very brief recapitulation of the structure of the flat manifold G( 2 .3 ) 4* R2
7 R34 R135 R 3
2 provided with the basis vectors e 0 (la)4
Ol7 sa, where aOl, Otb 4* dlt4 1 2 dlt, asa, sbb 4dab(l , t 46: a, b41, 2, 3), d is
the Kronecker symbol. A bilinear form on vector fields of sections t of tangent bundle of G( 2 .3 ): g×0
: t 7 t KCQ
(G( 2 .3 )
)
, namely the metric (g0(la)(tb)), is a section of conjugate
vector bundle with components g×0(e(la)0 , e(tb)0 ) in basis (e(la)0 ). The G( 2 .3 ) decomposes
into three-dimensional ordinary (R3
f) and time (Tf3) flat spaces G( 2 .3 ) 4Rf35 Tf3with
signatures sgn (R3
f) 4 (111) and sgn(Tf3) 4 (222). Since all directions in Tf3are
equivalent, then by the notion time one implies the projection of the time coordinate on a fixed arbitrary universal direction in T3
f. By this reduction Tf3K Tf1 the transition
G( 2 .3 ) KM44 Rf35 Tf1 may be performed whenever it is needed. Under massless
gauge field a(la)(hf) associated with Uloc( 1 ) the basis e(la)0 transforms at a point hf
G( 2 .3 ) according to eq. (2.3):
e(la)4 D(la)(tb)e(tb)0 ,
(5.1)
while the matrix D is in the form D 4C7R, where the distortion transformations
O(la)4 C(la)t Ot and s(la)4 R
b
(la)Ob are defined. Thereby C(la)t 4 dtl1 ka(la)* dtl, R is
a matrix of the group SO( 3 ) of all ordinary rotations of the planes, each of which involves two arbitrary basis vectors of R3
l, around the orthogonal axes. The angles of
permissible rotations will be determined from a special constraint imposed upon distortion transformations, namely the sum of distortions of the corresponding basis vectors Oland sb has to be zero at a given l:
aO(la), Otbt c l1 1 2eabg as(lb), sgb as(lb), sbb 4 0 , (5.2)
where eabgis an antisymmetric unit tensor. Thereupon tan u(la)4 2 ka(la), where u(la)
is the particular rotation around the axis sa of Rl3. Inasmuch as the R should be
independent of the sequence of rotation axes, then it implies the mean value R 4
1
6
!
i c j c kR(i jk), where R(i jk)is the matrix of rotations carried out in the sequence (i jk)
distortion of the basis pseudo-vector Ol, but the distortion of sa has followed from
eq. (5.2).
Certainly, the whole theory outlined in sects. 1-4 will then hold provided we simply replace each single index m of variables by the pair (la) and so on. Following the standard rules, next we construct the diffeomorphism h(la)(h
f (tb)
): G( 2 .3 ) KG(23) and introduce the action eq. (4.1) for the fields. In the sequel, a transition from the six-dimensional curved manifold G(23) to four-dimensional Riemannian geometry R4is straightforward by making use of the reduction of the three time components e0 a4
1
k2(e(1a)1 e(2a)) of the basis six-vectors e(la)to a single e0in fixed universal direction.
Actually, since the Lagrangian of fields on R4 is a function of the scalars, namely
A(la)B(la)4 A0 aB0 a1 AaBa, so taking into account that A0 aB0 a4 A0 aae0 a, e0 bb B0 b4
A0ae0, e0b B04 A0B0, one readily may perform the required transition. The gravitation
field equation is written as
¯(tb)f ¯f(tb)a(la)2 ( 1 2 z021) ¯(la)f ¯f(tb)a(tb)4 2 1 2k2g ¯g(tb)(mg) ¯a(la) T(tb)(mg). (5.3)
To render our discussion more transparent, below we consider in detail a solution of the spherical-symmetric static gravitational field a(1a)4 a(2a)4 1
k2a0 a(rf). So, u(1a)4 u(2a)4 2 arctan
(
kk2a0 a)and s(1a)4 s(2a). It is convenient to make use of the
spherical coordinates s(11)4 sr, s(12)4 su, s(13)4 sW. The transition G( 2 .3 ) KM4is
performed by choosing in Tf3 the universal direction along the radius vector: xf0 r4 tf,
xf0 u4 xf0 W4 0 . Then, from eq. (5.1) one gets
e04 D00e00, er4 Drrer0, eu4 eu0, eW4 eW0,
(5.4)
provided D 4C7I with components D004 1 1
k k2a0, D r r4 1 2 k k2a0, a0 fa0 r, where e004 j07 sr, er04 j 7 sr, eu04 j 7 su, eW04 j 7 sW, j04 1 k2(O11 O2) and j 4 1 k2(O12 O2). The coordinates x m
(t , r , u , W) implying xm(xfl): M4K R4 exist in the
whole region p21(U) R4
, ¯xm ¯xl f 4 cml4 1 2(D m l 1 vml Dmm) , (5.5)
where, according to eq. (5.4), one has x0 r
4 t , x0 u4 x0 W4 0 . A straightforward calculation gives the non-vanishing components
.
/
´
c0 04 1 2 D 0 0 , c014 1 2tf¯ f rD00, c114 Drr, c224 c334 0 , v1 14 v224 v334 1 , v104 tf¯frD00. (5.6)Although ¯rc004 G001c00and ¯rc114 G111c11, one has
¯tc014 c00¯frD00cG010c004 c00¯rln D00,
where, according to eq. (5.4), the non-vanishing components of the Christoffel symbol are written as G0014 1 2g 00 ¯rg00, G1004 2 1 2g 11 ¯rg00, G1114 1 2g 11 ¯rg11. So the condition
eq. (2.13) holds, namely the curvature of R4is not zero. The curved space R4 has the
group of motions SO( 3 ) with two-dimensional space-like orbits S2where the standard
coordinates are u and W. The stationary subgroup of SO( 3 ) acts isotropically upon the tangent space at the point of sphere S2of radius r. So, the bundle p : R4
K R2has the fiber S24 p21(x), x R4
with a trivial connection on it, where R2is the quotient-space
R4
OSO( 3 ). Outside of the distribution of matter with total mass M, eq. (5.3) in Feynman gauge is reduced to ˜2
fa04 0 , which has the solution
k k2a04 2 GM rf 4 22 rrg f
(
see eq. (4.7))
. So, the line element is written asds2 4
g
1 2 rg 2 rfh
2 dt2 2g
1 1 rg 2 rfh
2 dr2 2 r2( sin2u dW2 1 du2) , (5.8)provided by eq. (5.5) and eq. (5.6). Finally, for example, the explicit form of the unitary matrix URof gravitation group GRin the case of scalar field reads
UR4 R 8 UfR14 e2i [tf¯ f ruf(rf) 1uf(rf) ], (5.9) where R 4Rf4 e 2 igrg 2 rf tf , g(a 802 a0) 4¯fruf(rf) . (5.10)
The Lagrangian of the charged scalar shadow field eq. (3.4) is in the form
Lf(xf) 4 1 2
k
g
7 8h
2 (DlWf)* DlWf2 m2W *f Wfl
, (5.11)where, according to eq. (2.11) and eq. (5.6), one has JC4
1
2
(
1 1VvlmV
)
412 , S 478 .
6. – Distortion of the flat manifold G( 2 .2 .3 ) at G 4Uloc( 1 )
Following [4-8], the foregoing theory can be readily generalized for the distortion of the 12-dimensional flat manifold G( 2 .2 .3 ) 4* R2
7* R27 R34 Gh( 2 .3 ) 5 Gu( 2 .3 ) 4
!
2l , m 415 Rlm34 Rx35 Tx35 Ru35 Tu3 with corresponding basis vectors e(l , m , a)4
Ol , m7 sa%G( 2 .2 .3 ), e i 0 (la)4 Oi l7 sa %G i( 2 .3 ), where Oi 14 1 k2(O1 , 11 eiO2 , 1), Oi 24 1 k2(O1 , 21 eiO2 , 2), eh4 1 , eu4 21 and aOl , m, Ot , nb 4* dlt* dmn (l , m , t , n 41, 2), asa, sbb 4dab. Thereupon aO
i l, Oi tb 4eidi j* dlt. The massless gauge field of distortion
a(l , m , a)(zf) with the values in the Lie algebra of Uloc( 1 ) is a local form of expression of
the connection in principle bundle p : E KG(2.2.3) with a structure group Uloc( 1 ). The
set of matter fields Ff(zf) are the sections of vector bundles associated with Uloc( 1 ) by
the reflection Ff: G( 2 .2 .3 ) KE, so that pFf(zf) 4zf, where the coordinates z(l , m , a)
exist in the whole region p21(U( f )) G( 2 .2 .3 ).
(l , m , a). So the basis e(tnb)transforms into
e(lma)4 D(lma)(t , n , b)e(t , n , b),
(6.1)
provided D 4C7R, where O(lma)4 C
t , n
(lma)Ot , n, s(lma)4 R
b
(lma)sb. The matrices C
generate the group of distortion transformations of bi-pseudovectors Ot , n: C t , n
(lma)4
dt
ldnm1 ka(l , m , a)* dtl* dnm, but the matrices R are the elements of the group SO( 3 )lmof
ordinary rotations of the planes of the corresponding basis vectors of Rlm3. The special
constraint eq. (5.2) holds for the basis vectors of each R3
lm. Thus, the gauge field a(l , m , a)
is generated by the distortion of bi-pseudovectors Ot , n, while the rotation
transformations R follow due to eq. (5.2) and R implies, as usual, the mean value with respect to the sequence of rotation axes. The angles of permissible rotations are tan u(l , m , a)4 2 ka(l , m , a)(zf). The action eq. (4.1) now reads
S 4Saf1 SF4 (6.2)
4
Laf(zf) dz( 1 , 1 , 1 )R R R dz( 2 , 2 , 3 )
1
kg LF(z) dz( 111 )R R R dz( 223 ),where g is the determinant of the metric tensor on G( 223 ) and z(lma)(z(t , n , b)
): G( 2 .2 .3 ) KG(223) was constructed according to eqs. (2.8)-(2.10), provided that ¯z(lma) ¯z(t , n , b) 4 c (lma) (t , n , b)4 1 2(D (lma) (t , n , b)1 v (r , v , g) (t , n , b)D (lma) (r , v , g)) . (6.3)
Right through the variational calculations one infers the field equations (4.2), while ¯(t , n , b)¯(t , n , b)a(l , m , a)2 ( 1 2 z210 ) ¯(l , m , a)¯(t , n , b)a(t , n , b)4 (6.4) 4 J(l , m , a)4 21 2kg g(tnb)(rvg) ¯a(l , m , a) T(tnb)(rvg), where T(tnb)(rvg)4 2 d(kg LF) kg dg(tnb)(rvg) . (6.5)
The curvature of the manifold G
i( 2 .3 ) K Gi( 23 ) (sect. 5), which leads to
four-dimen-sional Riemannian geometry R4, is a familiar distortion,
a( 1 , 1 , a)4 a( 2 , 1 , a)f 1 k2 ah (1a), a( 1 , 2 , a)4 a( 2 , 2 , a) f 1 k2 ah (2a), (6.6)
whence s( 11 a)4 s( 21 a)fs(1a), s( 12 a)4 s( 22 a)fs(2a). Hence ei (la)4 Oi (la)7 s(la), where
O i (la)4 Oi l1 k k2eiah (la)* d t lO
i t. As a consequence, G( 223 ) 4 Gh( 23 ) 5 Gu( 23 ). The other
important case of inner distortion,
a( 1 , 1 , a)4 2a( 2 , 1 , a)f 1 k2 au (1a), a( 1 , 2 , a)4 2a( 2 , 2 , a)f 1 k2 au (2a), (6.7)
leads to s( 11 a)4 2 s( 21 a)fs(1a), s( 12 a)4 2 s( 22 a)fs(2a). Hence, Oi (la)4 Oi l1
k
k2eiau (la)* d
t
lO
i t, where ei (la)4 Oi (la)7 s(la) is the basis in the inner-distorted
mani-fold G
7. – Discussion and conclusions
If we collect together the results just established we finally arrive at the quite promising, if not entirely satisfactory, proposal of generating the group of gravitation by hidden local internal symmetries. Moreover, under the reflection of shadow fields from Minkowski flat space to Riemannian space, one framed the idea of general gauge principle of local internal symmetries G into the requirement of invariance of the physical system of reflected fields on R4under the Lie group of gravitation G
R of local
gauge transformations generated by G. This yields the invariance under the wider group of arbitrary curvilinear coordinate transformations in R4. While the
energy-momentum conservation laws are well defined. The fascinating prospect emerged for resolving or mitigating the controversial problems of gravitation, including its quanti-zation, by exploiting all the advantages of field theory in terms of flat space. Subsequently, one may carry out an inverse reflection into R4whenever it is needed. It was proved that only gravitational attraction exists. One may easily infer Einstein’s equation of gravitation, but with a strong difference at the vital point of well-defined energy-momentum tensor of gravitational field and conservation laws. Nevertheless, for our part we prefer the standard gauge-invariant Lagrangian eq. (4.11), while the functions eq. (2.3) ought to be defined. We considered the gravitational interaction with hidden Abelian group G 4Uloc( 1 ) with the base G( 2 .3 ) of principle bundle as well as
the distortion of the manifold G( 2 .2 .3 ), which yields both the curvature and inner distortion of space and time. However, we believe we have made good headway by presenting a reasonable framework whereby one will be able to verify the basic ideas and illustrate the main features of drastic generalization of the standard gauge principle of local internal symmetries in order to gain new insight into the physical nature of gravity.
* * *
It is a pleasure to express my gratitude to K. YERKNAPETIANand A. VARDANIANfor their support.
R E F E R E N C E S
[1] WEILH., Sitzungsber. d. Berl. Akad. (1918) 465, taken from Albert Einstein and the Theory
of Gravitation (Mir, Moscow) 1979.
[2] YANGC. N. and MILLSR. L., Phys. Rev., 96 (1954) 191. [3] UTIYAMAR., Phys. Rev., 101 (1956) 1597.
[4] TER-KAZARIANG. T., Selected Questions of Theoretical and Mathematical Physics (VINITI, N5322-B86, Moscow) 1986.
[5] TER-KAZARIANG. T., Comm. Byurakan Obs., 62 (1989) 1. [6] TER-KAZARIANG. T., Astrophys. Space Sci., 194 (1992) 1.
[7] TER-KAZARIANG. T., ICTP-Preprint, ICO94O290, Trieste, Italy (1994). [8] TER-KAZARIANG. T., hep-thO9510110 (1995).
[9] DUBROVIN B. A., NOVIKOV S. P. and FOMENKO A. T., Contemporary Geometry (Nauka, Moscow) 1986.
[10] PONTRYAGINL. S., Continuous Groups (Nauka, Moscow) 1984.
[11] WEINBERGS., Gravitation and Cosmology (J. Wiley and Sons, New York) 1972. [12] FOKV. A., Z. Phys., 57 (1929) 261.
Gravitation gauge group
G. T. TER-KAZARIAN(*)
Byurakan Astrophysical Observatory - Armenia 378433
(ricevuto l’11 Luglio 1996; approvato il 20 Novembre 1996)
Summary. — The suggested theory involves a drastic revision of the role of local internal symmetries in the physical concept of curved geometry. Under the reflection of fields and their dynamics from Minkowski to Riemannian space a standard gauge principle of local internal symmetries has been generalized. A gravitation gauge group is proposed, which is generated by hidden local internal symmetries. In all circumstances, it seemed to be of the greatest importance for the understanding of the physical nature of gravity. The most promising aspect in our approach so far is the fact that the energy-momentum conservation laws of gravitational interacting fields are formulated quite naturally by exploiting all the advantages of auxiliary
shadow fields on flat shadow space. The mechanism developed here enables one to
infer Einstein’s equation of gravitation, but only with a strong difference from Einstein’s theory at the vital point of well-defined energy-momentum tensor of gravitational field and conservation laws. The gravitational interaction as well as the general distortion of the manifold G( 2 .2 .3 ) with hidden group Uloc( 1 ) has been considered.
PACS 04.20 – Classical general relativity.
PACS 04.20.Cv – Fundamental problems and general formalism.
1. – Introduction
In spite of its unrivalled simplicity and beautiful features, Einstein’s classical theory of gravitation clashes with some basic principles of field theory. This state of affairs has not changed much up to the present and the proposed abundant models of gravitation are not able to provide an unartificial and unique recipe for resolving the controversial problems of the energy-momentum conservation laws of gravitational interacting fields, the localization of the energy of gravitation waves and also severe problems involved in quantum gravity. It may seem foolhardy to think of the rules governing such issues in the scope of the theory accounting for gravitation entirely in terms of intricated Riemannian geometry. The difficulties associated with this step are notorious; however, these difficulties are technical. Mainly, they stem from the fact
(*) E-mail address: gagoHhelios.sci.am
that Riemannian geometry, in general, does not admit a group of isometries. So, the Poincaré transformations no longer act as isometries. For example, it is not possible to define energy-momentum as Noether currents related to exact symmetries and so on. On the other hand, the concepts of local internal symmetries and gauge fields [1-3] have become a powerful tool and have been successfully utilized for the study of electroweak and strong interactions. Intensive attempts have been made for constructing a gauge theory of gravitation. But they are complicated enough and not generally acceptable. Since none of the solutions of the problems of determination of the gauge group of gravitation and of the Lagrangian of the gravitational field proposed up to now seems to be wholly convincing, one might gain insight into some of the unknown features of the phenomena of local internal symmetries and gravity by investigating new approaches in the hope of resolving such issues. In order to effect a reconciliation, it is the purpose of the present article to explore a number of fascinating features of generating the gauge group of gravitation by hidden local internal
symmetry. This is the guiding principle framing our discussion. In line with this we
think it is of the geatest importance to generalize the standard gauge principle of local internal symmetries. In the meantime, a second trend emerged as a formalism of reflection of physical fields and their dynamics from Minkowski to Riemannian space. There is an attempt to supply some of the answers to the crucial problems of gravitation in a concise form and to trace some of the major currents of thought under a novel viewpoint. In exploiting all the advantages of field theory in terms of flat space, a particular emphasis will be placed just on the formalism of reflection. This is not a final report on a closed subject, but we hope that the suggested theory will serve as a useful introduction and that it will thereby increase our knowledge on the role of local internal symmetries in the physical concept of curved geometry. Of course, much remains to be done before one can determine whether this approach can ever contribute to the larger goal of gaining new insight into the theory of gravity.
2. – Formulation of the principle
In the standard picture, suppose that a massless gauge field Bl(xf) 4TaBla(xf) with
the values in the Lie algebra of group G is a local form of expression of the connection in principle bundle with a structure group G. The set of matter fields are defined as the sections of vector bundles associated with G by the reflection Ff: M4K E, so that
pFf(xf) 4xf, where xfM4is a space-time point of Minkowski flat space specified by
the index (f). The Ff is a column vector denoting a particular component of field taking
values in the standard fiber Fxfupon xf: p 21(U( f )
) 4U( f )
3 Fxf, where U
( f )is a region of
base of principle bundle upon which an expansion into the direct product p21(U( f )
) 4
U( f )
3 G is defined. The various suffixes of Ff are left implicit. The fiber is a Hilbert
vector space on which a linear representation Uf(xf) 4exp[2iTauaf(xf) ] of group G
with structure constants Cabcis given. This space is regarded as a Lie algebra of group
G upon which the Lie algebra acts according to the law of adjoint representation: BDad B FfK [B , Ff]. To facilitate writing, we shall consider, mainly, the most
important fields of spin 0, 1
2 , 1. But the developed method may be readily extended to
the field of arbitrary spin s, since the latter will be treated as a system of 2s fermions of half-integral spin. In order to generalize the standard gauge principle below we proceed with a preliminary discussion.
a) As a starting point we shall assume that under gauge field Bfthe basis vectors
el
f transform into four vector fields
e×m(kB f) 4D× m l (kBf) efl, (2.1) where D×l m
are real-valued matrix functions of Bf, and k is a universal coupling constant
by which the gravitational constant will be expressed
(
see eq. (4.7))
. The double occurrence of dummy indices, as usual, will be taken to denote a summation extended over all their values. The explicit form of D×lm
, which must be defined under concrete physical considerations [4-8] (sect. 5, 6), does not concern us here. However, two of the most common restrictions will be placed upon these functions. As far as all known interactions are studied on the base of commutative geometry, then, first of all, the functions D×l
m
will be diagonalized at given (m , l). As a real-valued function of the Hermitian matrix Bf, each of them is Hermitian too and may be diagonalized by a
proper unitary matrix. Hence Dl m (kBf) 4diag(l m l1, l m l2, l m l3, l m
l4), provided with
eigen-values lmlias the roots of the polynomial characteristic equation
C(lml) 4det (l m lI 2D× m l ) 40 . (2.2) Thus em(kB f) 4D m l (kBf) efl (2.3)
leads to commutative geometry. Consequently, if en(kBf) denotes the inverse matrix
vector V D V c 0 , then aem, e nb 4Dl m Dl n4 dmn, Dl m Dk
m4 dlk. A bilinear form on the vector
fields of sections t of a tangent bundle of R4
: g× : t 7 t KCQ(R4), namely the metric
(gmn), is a section of conjugate vector bundle S2t * (symmetric part of tensor degree)
with corresponding components in basis em(kBf):
gmn4 aem, enb 4 aen, emb 4gnm, em4 gmnen.
(2.4)
In holonomic basis g× 4gmndxm7 dxn. One now has to impose a second restriction on the
functions Dl m
by placing a stringent condition upon the tensor vml (xf) implying
¯flD m k(kBf) 4¯fk
(v
ml (xf) Dmm(kBf))
, (2.5) where ¯lf4 ¯ ¯xl f , so that the v 4vllshould be a Lorentz scalar function of the trace of
curvature form V of connection Bl with the values in the Lie algebra of group G. That
is v 4v( tr V) 4v( d tr B) , (2.6) provided that
.
/
´
V 4!
l Ek Ff lkdxflR dxfk, B 4Bfldxfl, Ff lk4 ¯flBk2 ¯fkBl2 ig[Bl, Bk] . (2.7)Thereupon a functional v has a null variational derivative dv( tr V) dBl
4 0 at local variations of connection BlK Bl1 dBl, namely, v is invariant under Lorentz (L) and
b) Constructing a diffeomorphism xm(xfl): M4K R4, the holonomic functions
xm(xl
f) satisfy the defining relation
emc m l4 elf1 xfl(Bf) , (2.8) where xfl(Bf) 4emx m l4 2 1 2em
o xf (¯fkD m l 2 ¯flD m k) dxfk. (2.9)The following notational conventions will be used throughout: cm
l4 ¯flx
m
, where the indices m, n, l, t, s, k stand for variables in R4, whereas l, k, m, n, i, j refer to M4. A
closer examination of eq. (2.9) shows that the covector xf
l(Bf) realizes the coordinates
xmby providing the criteria of integration
¯fkc m l4 ¯flc m k (2.10)
and undegeneration V c V c 0 [9, 10]. Due to eq. (2.6) and eq. (2.9), one has the Lorentz scalar gauge-invariant functions
x 4 ael f, xflb 4 v 2 2 2 , S 4 1 4 cm l Dm l 4 1 2
g
1 1 v 4h
. (2.11)So, out of a set of arbitrary curvilinear coordinates in R4 the real-curvilinear
coordinates may be distinguished, which satisfy eq. (2.8) under all possible Lorentz and gauge transformations. There is a single-valued conformity between corresponding tensors with various suffixes on R4 and M4, while each co- or contra-variant index
transformed incorporating, respectively, with functions cml or clm. Each transformation
of real-curvilinear coordinates xm
K x8m8 was generated by some Lorentz and gauge transformations ¯x 8m 8 ¯xm 4 c m 8 l 8(B 8f) clm(Bf) Ll 8l . (2.12)
There would then exist preferred systems and groups of transformations of real-curvilinear coordinates in R4. The wider group of transformations of arbitrary
curvilinear coordinates in R4would then be of no importance for the field dynamics. If an inverse function clmmeets the condition
¯cml dxn cG l mncll, (2.13) where Gl
mn is the usual Christoffel symbol in accordance with the metric gmn, then the
curvature of R4does not vanish [11].
In pursuing the original problem further we are led to the principle point of a drastic change of the standard gauge scheme and to assume hereafter that a
single-valued, smooth, double-sided reflection of fields FfK F(Ff): FxfK Fx, namely
Ffl R m(xf), takes place under the local group G, where Ff%Fxf, F % Fx, Fx is the fiber
upon x : p21(U) 4U3F
put forth in illustration of a point at issue. Then
FmRd(x) 4cmlRcdmR(Bf) Fl R mf (xf) f (Rc) m R d
l R mFl R mf (xf) ,
(2.14)
where R(Bf) is a reflection matrix. The idea of general gauge principle may be framed
into the requirement of invariance of the physical system of fields F(x) under the
finite local gauge transformations UR4 R 8cUfRc1 of the Lie group of gravitation GR
generated by G , where R 8c4 Rc(B8f) if the gauge field Bf(xf) is transformed under G in
standard form, while the corresponding transformations of fields F(x) and their
covariant derivatives are written as . / ´ F8(x) 4UR(x) F(x) ,
(g
m(x) ˜ mF(x))
8 4 UR(x)(g
m(x) ˜mF(x))
. (2.15)The solution of eq. (2.15) may be readily obtained as the reflection of covariant derivatives
gn(x) ˜nFm R d(x) 4S(Bf) c
m
lRcdmR(Bf) gkDkFfl R m(xf) ,
(2.16)
where Dl4 ¯lf2 igBl(xf), S(Bf) is the gauge-invariant Lorentz scalar, ˜mis the covariant
derivative in R4: ˜m4 ¯m1 Gm, provided with the connection [11] Gm(x) 4 1
2S
abVn
a(x) ¯mVbn(x), the Sab are generators of Lorentz group, Vam(x) are the
components of the affine tetrad vectors ea in the used coordinate net xm: Vam(x) 4
aem, e
ab; one has gm(x) ¨ em(x) and gl¨eflfor the fields (s 40, 1); but gm(x) 4Vam(x) ga
for the spinor field
(
s 412)
, where gaare Dirac’s matrices. Since the fields Ff(xf) no
longer hold, the reflected ones F(x) will be regarded as the real physical fields. But a conformity eq. (2.14) and eq. (2.16) enables Ff(xf) to serve as an auxiliary shadow field
on the shadow flat space M4. These notions arise basically from the most important
fact that a Lagrangian L(x) of fields F(x) may be obtained under the reflection from the Lagrangian Lf(xf) of the corresponding shadow fields and vice versa. Certainly,
L(x) is also an invariant under the wider group of arbitrary curvilinear transformations x Kx 8 in R4:
JcL(x) Ninv (GR; x Kx 8)4 Lf(xf) Ninv (L ; G), (2.17)
where Jc4 V c V k 4
(
1 12 Vaelf, xfkb V 1Vaxfl, xfkb V)
1 O2(
see eq. (4.1))
, while the internalgauge symmetry G remains hidden symmetry, since it is screened by the gauge group of gravitation GR. We note that the tetrad ea and basis efl vectors meet the
condition . / ´ ra l(x , xf) 4 aea, elfb 4Vma(x) D m l (kBf) , aefa, ebfb 4 aea, ebb 4dab. (2.18)
So, the Minkowskian metric is written hab
4 diag ( 1 , 21 , 21 , 21 ) 4 aefa, ef
b
b 4 aea, ebb.
3. – Reflection matrix
One interesting offshoot of the general gauge principle is a formalism of reflection. A straightforward calculation for the fields s 40, 1 gives the explicit unitary reflection matrix R(x , xf) 4Rf(xf) Rg(x) 4exp [2iUf(xf) 2Ug(x) ] , (3.1) provided that Uf(xf) 4g
0 xf Bl(xf) dxfl, Ug(x) 4 0 x [Rf†GmRf1 c21¯mc] dxm, c f (c m l) , (3.2)where Ug4 0 for the scalar field and Ug1 Ug†4 0 for the vector field, because of Gm1
Gm†4 0 and ¯m(c21c) 40. The function S(Bf) has the form of eq. (2.11), since S(Bf) 4
1 4R †(c m lD l m ) R 414clmDl m
. The infinitesimal gauge transformation UfB 1 2 iTaufa(xf)
yields
UR4 exp
y
igCabcTa0
xf
ubfBlcdxfl1 U8g2 Ug
z
,(3.3)
where the infinitesimal transformation B 8la4 Bla1 CabcubfBlc2 1
g¯l
f
uaf was used. For
example, the Lagrangian of the isospinor-scalar shadow field Wfis in the form
Lf(xf) 4JfL(x) 4Jc[ (em˜mW)†(en˜nW) 2m2W
†
W] 4
(3.4)
4 Jc[S(Bf)2(DlWf)†(DlWf) 2m2Wf†Wf] .
The Lagrangian of the isospinor-vector Maxwell’s shadow field arises in a straight-forward manner: Lf(xf) 4JcL(x) 42 1 4JcF † mnFmn4 2 1 4 JcS(Bf) 2(F( f ) mn)†F mn ( f ) , (3.5) provided that Fmn4 ˜mAn2 ˜nAm, Fmn( f )4 (cknDmf2 cmkDnf) Akf, Dmf4 DmlDl, (3.6)
with an additional gauge-violating term
(3.7) Lf G(xf) 4JcLG(x) 42 1 2z 21 0 Jc(˜mAm)†˜nAn4 4 21 2 z 21 0 JcS(Bf)2(llmDlAfm)†lknDkAfn, llm4 1 2 (d l m1 vlm) ,
where z210 is a gauge-fixation parameter. Finally, the Lagrangian of the
isospinor-ghost fields is written as Lf
gh(xf) 4JcLgh(x) 4Jca(em¯mC)†, en¯nCb 4JcS(Bf)2(DlCf)†DlCf.
Continuing along this line we come to a discussion of the reflection of the spinor field Cf(xf)
(
s 4 1 2)
as a solution of eq. (2.15):.
/
´
C(x) 4R(Bf) Cf(xf) , C – (x) 4C–f(xf) RA † (Bf) , gm(x) ˜mC(x) 4S(Bf) R(Bf) glDlCf(xf) , (˜mC – (x))
gm (x) 4S(Bf)(D
lC – f(xf))
glRA†(Bf) , (3.9) where RA4 g0Rg0, Sab 414[ga, gb], G m(x) 4 1 4Dm , ab(x) g agb, the D m , ab(x) are Ricci’srotation coefficients. The reflection matrix R is in the form of eq. (3.1), provided that we make the change
Ug(x) 4 1 2
0 x R† f ]gmGmRf, gndxn( , (3.10)the ] , ( is an anticommutator. A calculation now gives
S(Bf) 4 1 8 Kcm l ]RA†gmR , gl( 4 inv , (3.11) where (3.12) K 4RA† R 4RAg†Rg4 4 exp
y
21 2 0 x(
]Rf†GAm†gm, gndxn( Rf1 Rf†]gmGmRf, gndxn()
z
, AGm†4 g0Gm†g0.Taking into account that [Rf, gn] 40 and substituting [12]
G A m †gn 1 gnGm4 2˜mgn4 0 (3.13)
into eq. (3.12), we get
K 41 . (3.14) Since UAR†UR4 RAUf†RA8†R 8 UfR†4 RAUf†RA†RUfR†, (3.15) where RA8† R84RA† R 41, then UAR†UR4 g0UR†g0UR4 1 . (3.16)
The Lagrangian of the isospinor-spinor shadow field may be written as (3.17) Lf(xf) 4JcL(x) 4 4 Jc
m
i 2[C – (x) gm(x) ˜mC(x) 2(
˜mC – (x))
gm(x) C(x) ] 2mC–(x) C(x)n
4 4 Jcm
S(Bf) i 2[C – fglDlCf2 (DlC – f) glCf] 2mC – fCfn
.In the special case of the curvature tensor Rmntl 4 0 , eq. (2.8) may be satisfied globally in M4by putting cml4 D m l 4 V m l 4 ¯xm ¯jl , VDV c 0 , x f l4 0 , (3.18)
where jl are inertial coordinates. So, S 4Jc4 1 , which means that one simply has
constructed local G-gauge theory in M4 both in curvilinear as well as inertial
coordinates.
4. – Action principle
Field equations may be inferred from an invariant action
S 4SBf1 SF4
LBf(xf) d 4xf1
kLF(x) d4x .(4.1)
The Lagrangian LBf(xf) of the gauge field Bl(xf) defined on M
4 is invariant under
Lorentz as well as G-gauge groups. But the Lagrangian of the rest of fields F(x) defined on R4is invariant under the gauge group of gravitation G
R. Consequently, the
whole action eq. (4.1) is G-gauge invariant, since GR is generated by G. The field
equations follow at once in terms of Euler-Lagrange variations, respectively, in M4
and R4,
.
`
`
/
`
`
´
dfL Bf dfBla 4 Jal4 2 dfL Ff dfBla 4 2¯g mn ¯Ba l d(kLF) dgmn 4 4 21 2 ¯gmn ¯Ba l k VDV Dkmd (kLF) dDn k 4 2k 2 ¯gmn ¯Ba l Tmn, dLF dF 4 0 , dLF dF– 4 0 , (4.2)where Tmn is the energy-momentum tensor of the fields F(x). Making use of the
Lagrangian of the corresponding shadow fields Ff(xf), in generalized sense, one may
readily define the energy-momentum conservation laws and also exploit all the advantages of field theory in terms of flat space in order to settle or mitigate the difficulties whenever they arise, including the quantization of gravitation, which does not concern us here. Meanwhile, of course, one is free to carry out an inverse reflection to R4whenever it is needed. To render our discussion here more transparent, below we
clarify the relation between gravitational and coupling constants. So, we consider the theory at the limit of Newton’s non-relativistic law of gravitation in the case of static weak gravitational field given by the Poisson equation [11]
˜2g004 8 pGT00. (4.3) In linear approximation g00(kBf) B112kBka(xf) uka, uka4
g
¯D0 0(kBf) ¯kBa kh
0 4 const , (4.4)then
uk
a˜2kBka4 4 pGT00.
(4.5)
Since eq. (4.2) must match with eq. (4.5) at the considered limit, then the right-hand sides of both equations should be in the same form:
kuk a dfL Bf dfBk a 4 kukaJkaB 2 k2 2 uk a¯g00 ¯(kBk a) T004 k2(ukauak) T00. (4.6)
With this final detail carried out, one gets
G 4 k 2 4 pu2 c , u2 cf(ukauak) . (4.7)
At this point we emphasize that Weinberg’s argumentation [13], namely, a prediction of attraction between particle and antiparticle, and repulsion between the same kind of particles, which is valid for pure vector theory, no longer holds in the suggested theory of gravitation. Although Bl(xf) is the vector gauge field, eq. (4.7) just furnishes the
proof that only gravitational attraction exists. One final observation is worth recording. A fascinating opportunity has turned out in case one utilizes the reflected Lagrangian of Einstein’s gravitational field
LBf(xf) 4JcLE(x) 4
JcR
16 pG , (4.8)
where R is a scalar curvature. Hence, one readily gets the field equation
g
Rlm2 1 2d m lRh
Dml ¯Dl l ¯Ba k 4 28 pGUak(kBf) 428pGTlnDnl ¯Dl l ¯Ba k , (4.9)which obviously leads to Einstein’s equation. Of course, in these circumstances it is straightforward to choose gmn as the characteristic of the gravitational field without
referring to the gauge field Bla(xf). However, in this case the energy-momentum tensor
of the gravitational field is well defined. At this vital point the suggested theory strongly differs from Einstein’s classical theory. In line with this, one should use the real-curvilinear coordinates, in which the gravitational field is well defined in the sense that it cannot be destroyed globally by coordinate transformations. Meanwhile, taking into account the general rules of eq. (2.12), the energy-momentum tensor of the gravitational field may be readily obtained by expressing the energy-momentum tensor of the vector gauge field Bla(xf) in terms of the metric tensor and its derivatives:
(4.10) Tmn4 gmlTnl4 gmlclkcinT( f ) ik 4 4 gmlclkcin
g
¯Ba l ¯gm 8 n8 c s i¯sgm 8 n8 ¯(¯tgl 8 t8) ¯(¯fkBla) ¯(JcLE) ¯(¯tgl 8 t8) 2 dkiJcLEh
.At last, we should note that eq. (4.8) is not the simplest one among gauge-invariant Lagrangians. Moreover, it must be the same in all cases including eq. (3.18) too. But in the last case it contravenes the standard gauge theory. There is no need to contemplate such a drastic revision of physics. So, for our part we prefer the G-gauge invariant
