a geometric foundation for a unified field theory

Foundations o f Physics, Vol. 14, No. 2, 1984

A Geometric Foundation for a Unified Field Theory

Nathan Rosen I and Gerald E. Tauber 2

Received December 28, 1982

Generalizing the work of Einstein and Mayer, it is assumed that at each point of space-time there exists an N-dimensional linear vector space with N >/5. This space is decomposed into a four-dimensional tangent space and an ( N - 4 ) - dimensional internal space. On the basis o f geometric considerations, one arrives at a number of fields, the field equations being derived from a variational principle. Among the fields obtained there are the electromagnetic field, Yang-Mills gauge fields, and fields that can be interpreted as describing matter. As a simple example, the ease N = 6 is considered.

1. I N T R O D U C T I O N

Since Einstein published his general theory of relativity, which provided a geometric description of gravitation, physicists have been looking for a unified field theory which would describe other fields, such as electromagnetism, in addition to gravitation, in terms of geometric concepts. An early attempt to set up such a unified field theory was that of Kaluza. (2) In this theory he enlarged space-time to a five-dimensional Riemannian space, assuming all geometric quantities to be independent of the fifth coordinate (cylindricity), and he took as the field equations essentially the Einstein equations in five dimensions. In this way he was able to get the Einstein-Maxwell equations if he interpreted certain components of the five- dimensional metric tensor as the electromagnetic potentials.

Einstein was interested in Kaluza's theory, but was critical of certain of its aspects. In 1931 he published a paper with Mayer, (1) in which they

i Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel. 2 Department of Physics and Astronomy, Tel-Aviv University, Ramat Aviv, Israel.

171

0015-9018/84/0200-0171503.50/0 © 1984 Plenum Publishing Corporation

172 Rosen and Tauber

modified Kaluza's approach. They took space-time to be a four-dimensional Riemannian space, but they assumed that at each point there was a five- dimensional linear vector space. By assigning suitable geometric properties to this space and choosing suitable field equations, they were able to arrive at the Einstein-Maxwell equations in a way which they considered to be superior to that of Kaluza.

However, the equations which they obtained were those in the absence of matter. This meant that with these equations matter could be described only by means of singularities in the field. Einstein considered this unsatisfactory; he believed that it should be possible to describe matter in terms of geometric quantities involving smooth functions, free from singularities. Hence he and Mayer published a second paper ~3) in which they modified their formalism so as to obtain the Maxwell equations with sources on the right-hand side. It seems, however, that these equations were not satisfactory, since they made no use of them thereafter.

The present work refers to the first paper of Einstein and Mayer (1) and has as its purpose to generalize their approach by considering the linear vector space at each point of space-time to have N dimensions instead of 5, with N/> 5. It is hoped thus to lay the foundation for a possible unified field theory of gravitation and the other fields that exist in nature. It might be mentioned that generalizations of the Kaluza theory are found in the literature, ~4-6) but the formalisms are different from ours.

In Section 2 of this paper we develop the geometry of the linear vector space and arrive at several different kinds of fields, including gauge fields (for N > 5). Our method of getting the field equations differs from that of Einstein and Mayer in that we make use of a variational principle. This is done in Section 3. To get some idea of what one can expect to find in the theory, some simple examples are treated in Section 4. What is interesting is that the geometry considered leads to fields that can be interpreted as describing matter, in addition to gauge fields. Although, following Einstein and Mayer, we take the linear vector space to be real, it is possible to describe it with the help of a complex formalism if N is even.

2. G E O M E T R Y

Following the general procedure of Einstein and Mayer, (1) although wi th some change of notation, let us assume that there exists a four- dimensional Riemannian space (corresponding to our space-time and to be referred to as the base space), the points of which are labeled by coordinates x j ( j = 1, 2, 3, 4), and that at each point there is a linear vector space of N dimensions, with N ) 5 . For a given basis in this vector space there are

A Geometric Foundation for a Unified Field Theory 173

contravariant and covariant vectors with components such as a " , b , ~ , v = 1, 2,..., N). Under a transformation of the basis, the vector components are transformed,

a" ~ a ' " = M " ~ a ~ (1)

b ~ b ' u = ( M - 1 ) ~ b~ (2)

where the elements of the (nonsingular) transformation matrix can be functions of the coordinates x k. In the vector space one can define a metric tensor with components 7 . . , 7 "", that can be used to lower and raise indices.

In the base space we have vectors with components such as a J, bk, which transform under the coordinate transformation x i ~ x ' ; ,

a J ~ a , i = (C~X'J/C~xk)a k

bj- bj = (exVex'i)b

(3)

(4)

and there is a metric tensor with components gjk, gjk. In order to relate quantities in the two spaces, we have a mixed tensor,

or projector, h k , . To a given vector a" one associates a vector a k according to the relation

a k = hk , a" (5)

If a k is given, a" is not uniquely determined. For example, if we have a vector A k = 0, then (5) gives

h k . A " = 0 (6)

and this will have nonvanishing solutions for A". If we assume that the matrix (hk , ) has the rank 4, Eq. (6) will have n = N - 4 linearly independent solutions. Linear combinations of these will also be solutions. Labeling the solutions with an index P = 1, 2,..., n, we can take the vectors Ap" satisfying

hkuA1" ~' = 0 (7)

so as to form an orthonormal set, i.e.,

AP uA o_u = 3t'o (8)

Let us assume that the two metric tensors are related in a way corresponding to (5), i.e.,

g J k = h f h ~ 7 , , (9)


Raising and lowering indices, we can write

= h " h ~" gjk j k Yuv

and

hJa hk ~ = 6~

Let us now consider the tensor

A ~ = h k . h k ~

From (11) we get

or

hJ~A ~ = hJ~,

P hJ.( G - A ~) = 0

Comparing with (7), we can write (summing over P)

v v p 5~, -- A , = C e . A v

Multiplying by AQ~ and using (8), (12), and (7) we get

Cou = AQu

so that (15) can be written

hkuhk ~ + Am, A e ~ = 6~

o r

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

ae = Ae, a" (20)

We see that, if a k is known, a a is given by (19), where the coefficient ae is arbitrary.

where

~,v = ©khS~,hkv + A v u A v , (18)

which one can put into various forms by raising and lowering indices. From Eq. (5), with the help of (17), one finds

a :~ = hkaa k + a e A j t (19)


The above formalism can be interpreted as serving to separate the linear vector space into two subspaces. For fixed coordinates x ~ one can consider hk. ( k = 1 ..... 4) as the components of four vectors spanning a four- dimensional subspace, the tangent space, and then there is the n-dimensional subspace spanned by the vector Ae. . Let us refer to the latter as the "internal vector space."

A vector ~. lying in the internal space can be written

O. = OeA.. (21)

and can therefore be represented by the coefficients Op as its components in the orthonormal basis formed by the vectors Ap. .

We have seen that one can have basis transformations in the vector space as in (1) and (2) and coordinate transformations in the base space as in (3) and (4). In the case N > 5, another kind of transformation is possible in the internal space, one in which the vector Ae. are replaced by linear combinations of them. Let us consider the transformation

Ap. Aeu = SpQAo. (22)

where A~. , which satisfies (7), is taken to satisfy (8) and therefore provides a new orthonormal basis in the internal space. It follows that Seo is an element of an orthogonal matrix,

SeQ = (S-~)oe (23)

If we now write the vector 0 . ,

(24)

then, by (23)

= Oo (25)

or, in matrix notation

, ' -- (26)

From (22) and (25) we see that the index P, which was previously regarded simply as a numbering label, has now become an index associated with a type of covariance.

Let us now consider covariant differentiation. For this purpose we have the Christoffel symbols in the Riemannian base space, but in the vector


space we have to introduce a connection, or three-index symbol. Let us write, for example, for the covariant derivative of T " k ,

T A i.la _ t t m (27) ZUkllJ = TUk,j + k ;~j T re{M}

where a c o m m a denotes an ordinary partial derivative. As a special case,

akllj = ak;j (28)

where a semicolon denotes a Riemannian covariant derivative. One also has

guIIk --= gu;k -= 0 (29)

Let us assume that

Y,~ Ilk = 0 (30)

Let us now go back to Eq. (1) giving the t ransformat ion of the vector a" under a change of basis. I f we also require that its covariant derivatives t ransform in the same way,

a'"ll~ = M"va~l l k (31)

then it is readily found that

F ' " a k = M"or°~k (M-1) t r . , t - - M " o . k ( M - ' )~x (32)

Wha t about vectors in the internal space? Let us write

O, Ilk = O,.k - c?oBaek (33)

and let us require that under the t ransformation (25)

O'vllk = SvaOo Ilk (34)

One finds that one must then have

Blpok = S v u B u v k ( S - 1 ) v o - SpR.k (S-1)R o (35)

a relation similar to (32). Corresponding to the matrix notat ion of (26), we can write (33) and (35)

#Ilk = •,k - - *Bk ( 3 6 )

B' k = SBk S-1 -- S ,kS-J (37)


From (37) we see that the matrices B k can be regarded as describing a gauge field.

If one requires that, for two arbitrary vectors 0P and ~te,

(Oe ~Ue)l~, = (0e ~%),k (38)

one gets Bpok = - -Boek (39)

so that (36) can also be written

O[Ik ---- q.k + Bkq (40)

Now let us go back to the general linear vector space, and let us try to determine the F a . k . For this purpose it is convenient to consider first a simpler quantity, to be denoted by Fauk, giving the covariant derivative

F . , k = F . , k - - r ~ I ~ . k (41)

The definition of this connection is to be such that

h~.lk = O, A~ulk = 0 (42)

One finds that these give the relations

hJ,~I~'~ k = hJ. ,k + hl.{~k} (43)

A e,~f'~uk = A pu,k -- A o_uB op k (44)

from which, with the help of (17), one gets

f a h ahJ hat , z jj t . k = J . , k + A / ~ A e . . k + i " .~Zk~ + A p a A q . B e o k (45)

From (42) and (18) it follows that

7 .~ l k = 0 (46)

Now let us write

F a . k = / ~ a k + Va k (47)

Then

7. , , I l k = 7 . v l k - - 7o, v l / ° ~ . k - - 7uo, V°~vk ( 4 8 )

so that, by (30) and (46),

V.~ k + V~. k = 0 (49)

Rosen and Tauber

We note that

7.~ IlJk -- 7.~ I[~/: Pulik + P..jk = 0 (56)

by (30). This gives us a symmetry property of the curvature tensor. One can form tensors of lower order

P . j = hkaPa . j k , P = h J " P . j (57)

Corresponding to (55), we have

h~. IlJk - hk . Ik~J = PuJ - - h m . R m j (58)

Multiplying by h j" gives

P - R = hJ"(hk . ]lJk - - hk . [IgJ) (59)

The right side can be calculated by making use of (42), (47), and (50). One finds

P = R + wiJkwu~ + Fj.JkF~jk (60)

178

The general form of V.v k is given by

V.~k = h i . hY o Wij k + (ApuhJ~ - A ~ v h J u ) F e j k + Ae, ,Ao~Det?k (50)

with

Wij k ~ - - - Wfik , Dpe k = --DQp k (51)

However, the last term in (50) is redundant since a term of the same nature appears as the last term in (45). Let us therefore take

Dpo k = 0 (52)

Now let us turn to the curvature. In the base space we have the Riemann-Christoffel tensor R'jkt and, by contraction,

Rjk : R mjkm, R = RJj (53)

In the linear vector space we can define the curvature tensor

Paujt, = Fa,a,k --}- Fa .k , j jr_ FAod.Fauk -- ]-'AakFauj (54)

One finds, for example

Q. i l l J k - QuiIIM : Q,~iP'~.Jk + Q.mRmijk (55)


It should be noted that on the right side there are no terms involving Bpo k appearing in (45) [or Dpo k present in (50) if one takes it different from zero ].

In the internal space we can also define a curvature tensor in analogy to (54),

Bp& k = --Bpej, k + BpQk,j + BpRjBRQk -- BpRkBnQ j (61)

Using the matrix notation as in (36) and (37), one can write this

Bjk = --Bj,~ + Bk, J + [B i, Bk] (62)

with the transformation law

Bj k = SBjk S -1 (63)

If we go back to (61), we note that we cannot form contractions analogous to those of (57) since we do not have a quantity analogous to hka that would relate the base space to the internal space. If we want to form a scalar corresponding to P, it is necessary to go to a quadratic expression,

Y = BeQ'ikBeo.jk (64)

We have developed a geometric framework. The next step is to set up field equations.

3. FIELD EQUATIONS AND VARIATIONAL PRINCIPLE

As mentioned earlier, in order to get the field equations, we will follow a procedure different from that used by Einstein and Mayer for N = 5; we will work with a variational principle. For this purpose it is natural to make use of the scalar curvature P given by (60). However, as we see from (60), P does not contain any contribution from the curvature of the internal space, so that something more is needed. Since we have the scalar Y given by (64), which is related to the internal-space curvature, it is reasonable to add it to P. We take therefore as our variational principle

I = y (P + Y)(--g)l/2 d4x, 3 1 = 0 (65)

with the usual conditions on the variations of the quantities appearing in the integrand.


From the expressions in (60) and (64) one readily finds that varying h ik gives the field equations

Rjk -- ½ gjk R = --8~rTjk

where

(66)

Tjk = (3/8ZO(Wjmn Wk m" -- ~gjk glZpqr wPqr)

+ (1/4=)(FpjmFvkm _ 1 gakFm~Fpmn)

+ (1/47c)(BpojmB~,ok m -- ¼gjkB,omnBpo mn) (67)

We see that we have here the Einstein field equations with Tjk, the energy-momentum density tensor, as the sum of contributions from the fields

W j m n , Fpjm, and Beo/m. If we vary Wij k and Fej k, we obviously get Wig k = Fej k = 0. To avoid

this let us express the fields in terms of derivatives of potentials. Consider WUk. For the sake of simplicity, let us write it in a covariant

form that does not involve Christoffel symbols, which, if present, would vitiate Eq. (67). We can do this by assuming that

Wij k = - W , k j (68)

so that, in view of (51), Wuk is now completely antisymmetric. One can then introduce the vector W m dual to Wo. k by writing

Wijk=eijm Win, 3 [ W m = e i J l ' m W u k (69)

where eijkm is the completely antisymmetric tensor

ei jkm = ( - -g ) l /2~ , i j km , e ijkm ~- - - ( - - g ) - l / Z ~ i j k m (70)

eijt, m being the Levi-Civita symbol. In (60) one now has

m i j k m i j k = 3 [ W m m m (71)

To introduce potentials let us write

= jk (72) W J c P j + U ;k

where an underlined index is to be raised, and

U Jk = - U kJ (73)

If we vary • in (65), making use of (60), (71), and (72), we get

Wk;k = 0 (74)


On the other hand varying U jk gives

wj,k - w ,j = 0 (75)

We see that Wj is a gradient. Hence we can discard the last term in (72) and write simply

W] = ~ , j , WUk = eukm q~,_rn (76)

so that (74) gives the field equation

gjk¢IJ;j k = 0 (77)

Thus q~ satisfies the wave equation. Let us set @ = 0. Now let us consider Fej k. The simplest way to express it in terms of

derivatives is to assume

and to write

/~)ik = --Fpkj (78)

F p j k = Fp j j l k - - Fpk j l E

= F~j,k -- Fek,j + BeokFoj -- Be~jFok (79)

I f we vary Fej in (65), we get

Ffkipk = Fjk;k + BpQkFo jk = 0 (80)

Finally, if we vary BvQj in (65), taking into account (60), (64), (61), and (79), we obtain

B jk - - F JkF " (81) PO Prk = l(Ft 'JkFok O Pk)

To summarize, we now have as field equations: the Einstein equations, (66) and (67), the equations for Wij k, (76) and (77), the equations for Fpjk, (79) and (80), and the equations for Bpojk, (61) and (81).

4. SIMPLE EXAMPLES

Let us consider some simple cases. If we take N = 5, n = 1, the case treated by Einstein and Mayer , (1) we have only one vector A1. = A . , so that the internal space is one-dimensional, and there is one tensor F~j k = Fjk in addition to the scalar ~. No t ransformations of the form (22) are possible,


and therefore no gauge field Bpey exists, as is confirmed by (39). Then (79) becomes

Fjk = Fy,k -- Fk, j (82)

and (80) reads

FJk:t, = 0 (83)

SO that we have the Maxwell equations and can interpret Fj~ as the electromagnetic field tensor. We have the usual gauge transformation

r j -+ F j --- r j + 2 j (84)

which does not change Fjk. This gauge transformation has no effect on the internal space.

Now let us take N---6, n = 2, so that we have a two-dimensional internal space spanned by the vectors Ap,, ( P = 1, 2). We then have two tensors Fej k = (Ftnjk, F(z) jk ) and also the gauge field B I 2 k -~ - - B 2 1 k "= B k.

The orthogonal transformation in (25) and (26) can be expressed in terms of the matrix

{ cos 0 - s in 0 ) S = \ sin 0 cos 0

(85)

which is a representation of the group 0(2). From (25) one has

~'1 = 01 cos 0 - ¢2 sin 0 (86)

0~ - 01 sin 0 + ~2 cos 0

From (85) it follows that

S,kS -1

and one finds that (37) can be written

so that

Bk

B ' k = B k + O j, (89)

This is again the gauge transformation of electromagnetism, but now it is associated with the transformation (86) of quantities in the internal space.


Writing Bjk=B2u k in (61), we get

Bj. k = B j, k - Bk, j (90)

as in electromagnetism if Bjk is the field tensor. Now let us look at the field equations. Equation (79) gives

F ( I ) j k = F(a) j , k - - F(1)k , j ~- BkF(2 u - - B j F ( z ) k (91)

F(2) j k : F ( 2 ) j , k - - F(2)k , j - - B~F(~ u + B j F ( 1 ) k

while (80) can be written

F(liJk;k + F(2/kBk : 0 (92)

F(z)Jk;k - - F( l )Jkok = 0

Finally, from (81) one gets

BJk;k = 4 7 J j (93)

where j l jk 4rd = ~[(F(2 ~ F(llk-- F~l)JkF(21k] (94)

From (90) and (93) we see that Bj. k satisfies the Maxwell equations with an electric charge-current density JJ given by (94). Hence Fpj k describes a field carrying charge, i.e., a field which, from the quantum-theoretical point of view, consists of charged particles. Making use of (91) and (92) one gets

Jk;k = 0 (95)

as was to be expected. The preceding discussion takes on a simpler form if we go over to a

complex formalism. Suppose we have a vector in the internal space given by (21). This can be characterized by the components ~p= (~l,0Z). Let us combine these into a complex quantity

0 = 2-1/2(01 + i02) (96)

At the same time we can combine the two vectors Apu = (Alu,A2~,) into a complex vector,

A, = 2-U2(A~u + iA2, ) (97)

which, by (8), satisfies the relations

A~,A = 1 , AuA u=O (98)

825/14/2-6


We can write OeAm, = O*A. + 0 A * (99)

From (86) we now have

0 ' = ei°0 (100)

so that e i°, which determines the transformation, is a representation of the group U(1). It is readily verified that (33) can be written

0Ilk = O,k -- iBkO (101)

Equations (89), (100) and (101 are of course very familiar from quantum mechanics.

To discuss F~j k, let us write

qJj = F(1)j + iF(2)j (102)

qJjk = Fmjk + iF(2)jk

It is convenient to introduce the operator D k such that when it acts on a geometric object .(2 (scalar or tensor) it gives

DkX? = ~ k -- iBkY2

From (91) one then gets

and from (92)

qJjk = D k q/j -- Dj ~k

(lO3)

(104)

(105) Dk ~flk = 0

as the field equations. Equation (94) can be rewritten

JJ = (i/167Q(q/*JkqJk --- q/jk~*) (106)

The term in (60) which, through the variational principle, leads to (105) is

r s k rpjk = ~U~ ~ k (107)

with ~t i and ~u* being varied independently. From the standpoint of quantum theory, (105) is the equation for a field

consisting of charged vector msons having zero rest mass. If one wants to have mesons with nonzero rest mass, one can add to the term (107) appearing in the integrand of (65) a term

--lz2FejFe j = - - / d 2 ~ j lff j (~/ = const) (108)

although it is hard to justify this from the geometric standpoint. In that case (105) is replaced by

Dk q/jk + pzqjj = 0 (109)


In the case of an inner space of higher dimensions the complex formalism leads to a number of interesting results. These will be discussed, together with some applications of the theory to physical problems, in a subsequent paper.

5. D I S C U S S I O N

Generalizing the work of Einstein and Mayer, we have assumed that at each point of space-time there exists an N-dimensional linear vector space. Following these authors, we have decomposed this space into two subspaces: the four-dimensional tangent space at the point, spanned by hk u, and an (N-4)-dimensional vector space, the internal space, spanned by an orthonormal set of vectors Ap r. One can then carry out two kinds of transformations: coordinate transformations, which induce vector transformations in the tangent space, such as (3), and (for N~> 6) orthogonat transformations of the internal-space basis vectors. The latter induce inhomogeneous transformations of the internal-space connections, the gauge transformations.

On the basis of geometric considerations, one is led to introduce a number of fields. The field equations are determined in the present work by means of a variational principle. There are several kinds of fields.

(a) There is the space-time metric tensor gjk describing gravitation and determined through the Einstein equations by the energy and momentum density of all the other fields.

(b) There is a scalar field q~ satisfying the wave equation. It interacts only with the gravitational field. One can get a Klein-Gordon equation for if, in (65), one adds to P a term proportional to q~2. However, from the geometric standpoint this is an arbitrary step.

(c) There is the gauge field (or internal-space connection) Bp~ k which undergoes the gauge transformation (35) or (37) when the basis vectors of the internal space are subjected to an orthogonal transformation. The present formalism is based on a real internal space with real basis vectors. However, if the internal space has an even number of dimensions n, one can combine the basis vectors into pairs, each pair forming a complex basis vector, and in this way obtain a complex internal space having n/2 dimensions. In that case the transformations of the new basis vectors will be unitary, and the gauge fields will be similar to those of the Yang-Mills theory. (7) A very simple example of this was given in the previous section for n = 2. It was found that the gauge field could be identified with the electromagnetic potential, and the gauge transformation was associated with the unitary transformation group U(1).


In the general case, one can imagine an internal space consisting of a number of subspaces, each transforming independently and having its own gauge field. In this way one could have a number of gauge fields, the group of transformations of the internal space being a product of the groups associated with the various subspaces.

(d) The gauge fields under discussion are not new. The new element in the present approach is the existence of the fields Fpj k. These fields satisfy gauge-covariant field equations, or one can say that they interact with the gauge fields BpQ k. We see from (81) that they provided sources for the gauge fields. They can therefore be interpreted as fields describing matter. In the previous section it was found that for N = 6 one had a charged meson field. As in that example, so in the general case, one can have a mass term appearing in the field equation (80) for Fpj k by adding a suitable term to the integrand of the variational principle (65), subject to the reservation mentioned in the example.

It was remarked previously that Einstein and Mayer, in their second paper, ~3) tried to modify their earlier formalism so as to have in their field equations a matter source described by geometric quantities, and that they were apparently unsuccessful in this. We see now from the examples considered in Section 4 that, to get a matter field, it is desirable to take a linear vector space of at least 6 dimensions.

It is possible to generalize the geometry considered in the present work by assuming that, instead of a linear vector space at each point of space- time, we have a more general fiber-bundle space with a Riemannian geometry. This will be investigated in a future paper.

N O T E A D D E D IN P R O O F

Through an oversight Eq. (60) was written in the form corresponding to the assumptions made later, Eqs. (68) and (78).

R E F E R E N C E S

1. A. Einstein and W. Mayer, Sitzungsber. Preuss. Akad. Wiss., p. 541 (1931). 2. Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss., p. 966 (1921). 3. A. Einstein and W. Mayer, Sitzungsber. Preuss. Akad. Wiss., p. 130 (1932). 4. R. Kerner, Ann. Inst. H. Poincard 9, 143 (1968). 5. A. Trautman, Rep. Math. Phys. 1, 29 (1970). 6. Y. M. Cho, J. Math. Phys. 16, 2079 (1975). 7. C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954).

a geometric foundation for a unified field theory

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