Fortschritte der Physik / Progress of Physics: Band 29, Heft 9 [Reprint 2022 ed.] 9783112656006

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 9 • 1981 . B A N D 29

A K A D E M I E - V E R L A G

31728

EVP 1 0 , - M

.

B E R L I N

ISSN 0 0 1 5 - 8 2 0 8

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an eine Buchhandlung oder an den AKADEMIE-VERLAG, D D R - 1080 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Berlin (West) an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber OHG, D - 7000 Stuttgart 1, Wilhelmstraße 4—6 — in den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber GmbH, CH - 8008 Zürich/Schweiz, Dufourstraße 51 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, D D R - 7010 Leipzig, Postfach 160; oder an den AKADEMIE-VERLAG, D D R - 1080 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der P h y s i k " Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Arthur Lüsche, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Oesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1080 Berlin, Leipziger Straße 3 - 4 ; Fernruf: 2236221 und 2236229; Telex-Nr.: 114120; B a n k : Staatsbank der D D R , Berlin, K.onto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 1040 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: VEB Druckhaus „Maxim Gorki", D D R - 7400 Altenburg, Carl-von-Ossietzky-StraOe 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der P h y s i k " erscheint monatlich. Die 12 Hefte eines Jahres bilden einen B a n d . Bezugspreis je Band 180,— M zuzuglich Versandspesen (Preis f ü r die D D R : 120,— M). Preis je Heft 15,— M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/29/9. (c; 1981 by Akademie-Verlag Berlin. Printed in the German Democratic Republic. AN (EDV) 57618

ISSN 0 0 1 5 - 8 2 0 8 Fortschritte der Physik 29, 3 8 1 - 4 1 1 (1981)

Furry Picture for Quantum Electrodynamics with Pair-Creating External Field E . S. FRADKIN

P. N. Lebedev Physical Institute, Moscow, USSR and D . M . GITMAN

Pedagogical Institute, Tomsk,

USSR

Abstract In the paper the perturbation theory is constructed for QED, for which the interaction with the external pair-creating field is kept exactly. An explicit expression for the perturbation theory causal electron propagator is found. Special features of usage of the unitarity conditions for calculating the total probabilities of radiative processes in the case are discussed. Exact Green functions are introduced and the functional formulation is discussed. Perturbation theory for calculating the mean values of the Heisenberg operators, in particular, of the mean electromagnetic field is built in the case under consideration. Effective Lagrangian which generates the exact equation for the mean electromagnetic field is introduced. Functional representations for the generating functionals introduced in the paper are discussed. I. Introduction

During the recent years the growing interest is attached to the problems of quantum electrodynamics (QED) with intense electromagnetic field. To some extent this interest is due to the achieving of strong fields in experimental conditions, further growth of the laser intensities and recognition of some situations in astrophysics where the values of the effektive fields are tremendous, indeed. This interest is also provoked by the existence of analogies with problems in gravitation and in gauge theories with spontaneous symmetry breaking. In this connection solving similar problems in QED may be thought of as, in a way, the first step in this sphere in the mentioned theories. Finally, results for specific problems in QED with intense electromagnetic field are important for checking its validity in the extreme domains of parameters and undoubtly are of general scientific value. In the present paper we will consider special features of constructing QED formalism, which are connected with the possibility of particle creation in an intense electromagnetic field. The paper to a great extent is a generalization and review of a part of general results which were obtained in ( G I T M A N [ 1 ] ( 1 9 7 6 , 1 9 7 7 ) ; G I T M A N , G A V R I L O V [2] ( 1 9 7 7 ) ; F R A D K I N , GITMAN [ 5 ] ( 1 9 7 8 , 1 9 7 9 ) ; GAVRILOV, GITMAN, SCHWARTSMAN [ 4 ] ( 1 9 8 0 ) ) .

Thus, if one discusses problems of QED with an intense electromagnetic field in the frame of QED with an external field, then one of the most important is here the problem of how to keep exactly the interaction with the external field to all the orders of per1

Zeitschrift „Fortachritte der Physik", Bd. 29, Heft 9

382

B . S . F K A D K J N a n d D . M . GITMAN

turbation expansion. 1 ) This problem has been investigated well, e.g., for the spinor or scalar charged fields interacting with the external electromagnetic field (FEYNMAN [6] (1949); SCHWINGER [7] (1951, 1954 a, b). During the recent years the growing interest is attached to it due to the examination of processes of particle creation from the vacuum b y the external field both in electrodynamics and in gravitation. (NIKISHOV [,

— z(out| = /(out| V.

(6)

The perturbation analysis of (5) creates a number of differences from the relations which are usually obtained in this way. The matter ist that the propagators for perturbation expansions of the matrix elements (out| |in) and (in| |in) are different; in the first case it is the generalized chronological coupling (2.43) out(0|

T y(x) f{y)

|0) i n • O r 1 =

-iS = in(0| S-^xJ X A{Zl) 4)

-

x

2/ z )

••• tp(xn) f (i/i) ••• f (ym) A(z,) T

-

ip(x'n.) f(yi')

-

f(y'm.) A&)

-

A{z'v.) S |0) in .

The construction of the perturbation expansion and diagrammatic technique for the Green functions of the type of ^ooo.MMV i n statistical physics was considered in (KELDYSH [25], (1964)) by ordering along a contour.

404

E . S . E R A D K I N a n d D . M . GITMAN

The diagrammatic technique in terms of the matrix quantities (9) has the JTevnman form. Thus e.g., the expansions for the mean field (1) (A(x)) in the cases when the initial state is the vacuum state or a single-electron state have, respectively, the form a)

lin) = |0) in ,

f 2V c t (z - y) J (y) dy + - ,

(A(x)) =

D0Tet(x - y) = d(x° - y») D0(x - y ) = D0°(x - y) + D0(+)(x -

y).

|in) = a„+(in) |0) in ,

b)

(A(x)) =

+

= f D0*\x

¿

L

^

j

^

- y) [J (y) + My)] dy + •

jn{x) = e+yn(x) y+>pn(x), fly

n o = +2 =

e

(16)

, €

After making the change of variables (14—16) in the integral (10) and acting on the result obtained according to the rules (8) we will get the final expression for the f unctionalZ. Z = J-1 j exp iL • ZMj • DA2 • DVl • Dy>2 • Df{ • D%p2, L = j

A •' A +

y+qQ-.y.

•' = ?'(Aext) =

$(AeKt)

Qwip . AQaA + IAQaA + tpQ^

•,

$ $(Aext),

• D0 •

J ^ ( 4 e x t ) —^(Aext)

+

(17)

406

E . S . F R A D K I N a n d D . M . GITMAN

The result (17) might have been obtained in a simpler way if, instead of the standard representation (4.9), we should have made use of the representation: exp|— in

exp^i

= J-1J

Vi> Wz i n the integral (17) are interdependent. They are defined by the formulae (14—16). For the generating functional Z one can get the following set of the functional equations.

_ ÖZ

d2 Z try-z—

I Z - e

or] drj

(18)

Ö \ dZ ( A — iA —) —r = —iriZ. ext

Ol j 07}

The equations (18) under X — 1 coincide formally with the equations (4.13). The equations (18) generate a set of equations for the Green functions. Let us introduce, as usual, the functional W = i\nZ. (19) which is the generating functional for the connected Green functions and the following definitions

dW ~öfij=JJ=0

d2w dldl >7=n =0 D,

a,

d2W

0 c> 0

250, (7.14)

ifinE

c/ln En,

252, (8.4) and (8.5)

Aa(E)

[Aa{E)\

252, r.h.s. of (8.5)

lim £-» oo

— lim

252, (9.1)

E

Z

Table 1, (6.5) Table 1, (8.4)

E-* oo

lim

Lim

E—> oo

E—> oo

Aa(E)

ln \Ao(E)\

ISSN

0015-8208

Fortschritte der Physik 29, 4 1 3 - 4 4 0 (1981)

Multidimensional Unified Theories CLAUDIO A . ORZALESI

Istituto di Fisica dell'Università di Parma, 43100 Parma, Italy, and Istituto Nazionale di Fisica Nucleare, Sezione di Milano

Contents Abstract

413

1. Introduction

414

2. Projective formalism for extended spacetime with isometries 2.0. Notation 2.1. The projection 2.2. The vertical Lie algebra 2.3. The vertical isometry group 2.4. if-frames 2.5. Gauge fields and extended metric

416 416 417 419 420 422 424

3. Multidimensional unified theories 3.1. Coordinate and gauge transformations 3.2. Riemann connection and curvature 3.3. Einstein-Hilbert action principle 3.4. The hierarchy of constraints

426 427 429 430 433

4. Summary and concluding remarks

437

Acknowledgements

439

References

439

Abstract An extended spacetime, Ml+N, is a Riemannian (4 + iV)-dimensional manifold which admits an jV-parameter group G of (spacelike) isometries and is such that ordinary spacetime M* is the space Mi+NIO of the equivalence classes under (^-transformations of Mi+Jf. A multidimensional unified theory (MZJT) is a dynamical theory of the metric tensor on Mi+N, the metric being determined from the Einstein-Hilbert action principle: in absence of matter, the Lagrangian is (essentially) the total curvature scalar of M i + N . A MUT is an extension of the Cho-Freund generalization of Jordan's five-dimensional theory. A MUT can be faithfully translated in four-dimensional language : as a theory on M*, a MUT is a gauge field theory with gauge group G. A unifying aspect of MUT's is that all fields occur as elements of the metric tensor on Mi+N. When the isometry generators are subjected to strongest constraints, a MUT becomes the De Witt-Trautman generalization of Kaluza's five-dimensional theory; in four-dimensional language, this is the theory of Yang-Mills 3

Zeitschrift „Fortschritte der Physik", Bd. 29, Heft 9

414

CLAUDIO A . ORZALESI

gauge fields coupled to gravity. With weaker constraints, a MUT appears to be more natural than a Yang-Mills theory as a physical realization of the gauge principle for an exact symmetry of gauged confined color. Such weakly-constrained MUT leads to bag-type models without the need for ad hoc surgery on the basic. Lagrangian. The present paper provides a detailed introduction to the formalism of multidimensional unified gauge field theory.

1. Introduction The Yang-Mills (Y-M) construction [1] of a gauge field theory (GFT) originally started from a matter Lagrangian invariant under a global symmetry group; gauge fields were then introduced from the requirement of invariance under spacetime-dependent group transformations. As Yang and Mills indicated1), this requirement of local gauge invariance can be motivated on general grounds. A Y-M GFT corresponds to a well defined geometric structure [2— 5] which is, however, defined independently of the matter fields. Therefore, it becomes more natural and conceptually clearer to consider the GFT geometry as structurally given ab initio, and the invariant interaction with matter fields as a successive construct 2 ). Indeed, this attitude is adopted in the standard geometric formulation of GFT's, where the geometry of the theory is identified with a principal fibration with connection determined by the Y-M potentials [3—5]. However, this identification is somewhat imperfect, because it sets aside the physically important geometric property that spacetime has a Riemannian metric structure (be it preassigned, as in special-relativistic theories, or dynamical, as in theories including gravity). As was first emphasized by D E W I T T [6] and by TRAUTMAN [3], the Riemann metric of the base space can be extended to the total bundle space of the Y-M G F T : the extended metric can be obtained by a suitable combination of the spacetime metric with an invariant metric on the structural group manifold. The resulting theory [6—8] is then equivalent to a multidimensional unified theory (MUT) which generalizes the KaluzaKlein (K-K) theory of electromagnetism and gravitation [9, 10] to a general compact gauge internal Lie group. In essence, a MUT is a theory of the metric tensor in an extended (i.e., multidimensional) spacetime having certain preassigned symmetries. The unifying aspect of a MUT lies not so much in the reduction of coupling parameters as in the unified role of gauge and gravitation potentials, which all occur as elements of the metric tensor on the extended spacetime. The geometry of a (generalized) K - K MUT is that of an extended spacetime with a (gauge) group of constrained isometries3). The total curvature scalar of such extended spacetime yelds the Y-M Lagrangian with gavitations included, so that the Y-M GFT is equivalent to the Einstein-Hilbert "gravitation" on an extended spacetime with constrained isometries4). See Ref. [2]: "The orientation of the isotopic spin is of no physical significance . . . As usually conceived, however, this arbitrariness is subject to the following limitation: once one chooses what to call a proton, what a neutron, at one space-time point, one is then not free to make any choices at other space-time points. I t seems that this is not consistent with the localized field concept that underlies the usual physical theories". 2 ) Of course, only the geometric structure is independent of the matter fields: e.g., the value of the bundle curvature form (the gauge field strength) depends on whether or not matter is present. In a way, the attitude of considering the gauge geometiy per se is a return to the roots, since the gauge invariance in Maxwell's theory was originally found and discussed without referring to the sources. 3 ) Essentially, an isometry is constrained when the isometry generators are subject to constraints; the K - K constraints fix the length of the generators, see sect. 3.4 below. 4 ) As such, a gauge theory can be founded on a generalized equivalence principle, see ref. [22] for a heuristic discussion.

Multidimensional Unified Theories

415

The K - K constraints leading to the Y-M GFT appear rather artificial for an exact gauge symmetry 5 ) and it seems more natural to consider a MUT where, saving the bundle structure of GFT's, these constraints are abandoned. The resulting theory is the CHOFREUND [12, 13] nonabelian generalization of t h e JORDAN-THIRY [14, 15] five-dimen-

sional theory." Again, this is a theory of Einstein-Hilbert "gravitation" on an extended spacetime with isometries [1S\; as in Kaluza's theory, the gauge fields appear as the Killing covector fields generating the isometries. Local gauge invariance is simply the reflection of the covariance of the theory under isometric coordinate transformations [1I\. The theory can be rewritten as a theory on ordinary spacetime, and in this language it appears as a generalization of a Y-M GFT, where additional scalar fields occur with covariant and gauge-covariant interactions 6 ). When one translates a GFT into a MUT, the usual attitude, of considering ordinary spacetime as "primordial" and the GFT as a construct on the four-dimensional base, gradually loses its focus. As soon as the translation of the GFT into MUT language is completed, it becomes rather more natural to consider the extended spacetime as "primordial" and ordinary spacetime as a construct. Such a shift in attitude is not without consequences, for it naturally leads to questions which are not as clearly formulated within the conventional setup. For example, one wonders what the origin of Y-M GFT's might be: in MUT language, this amounts to asking for a mechanism to generate the K - K constraints, so t h a t a partial answer could be obtained from a study of the Cho-Freund theory, where these constraints are not imposed ab initio. More generally, one might ask how the isometries are generated from a theory where no preassigned symmetry is imposed on the extended spacetime. Since the isometry is responsible for the equivalence of a MUT with a four-dimensional gauge theory, answering questions such as these might deepen our understanding of the origin of the gauge interactions occurring in Nature and of the four-dimensional appearance of the physical world. 7 ) I n any case, and leaving aside matters of mathematical elegance and general questions, the fact remains t h a t a MUT can be regarded as "yet another formulation" of a GFT (and, geometrically at least, a very natural formulation). Since it usually pays to look at the same physical theory from various viewpoints and with varying techniques, MUT's are candidate helpers for a better understanding of GFT's. This is sufficient motivation for the present paper, which aims at providing a systematic and detailed introduction to the Riemannian multidimensional formulation of GFT's. I n sect. 2 we discuss the projective formalism for an extended spacetime Mi+N with an _/V-dimensional isometry group G. This formalism generalizes to (4 + N) dimensions the five dimensional projective theory developed in ref. [16]. The projection leading from Mi+N to ordinary spacetime Ml is discussed first, then we consider the isometry G: as always in gauge theories, Ml is the quotient Mi+N/G. To prepare the code for translating a MUT into four-dimensional language, we introduce a class of preferred frames on Mi+N: we call-them K-frames and use them to study the theory by means of associated ^-connections on Mi+N. I n sect. 3, we first discuss gauge transformations, seen here as originating from the covariance of l£-frames under certain coordinate transformations on M i + N . The Riemann connection and Ricci tensor are then calculated in a /C-frame. A MUT is defined (at last!) as a dynamical theory of the metric tensor on Mi+N, where the metric is determined from an Einstein-Hilbert action principle, with the Lagrangian identified with the total curva5 ) Indeed, in this case the constraints appear to contradict the general motivation outlined in footnote 1 ); see also sects. 3.3, 3.4 and 4 below for further discussion. 6 ) See sect. 3.3 below. 7 ) See also the general discussion in ref. [12], where gauge theories are looked upon as the result of spontaneous breaking of general co variance in 4 + N dimensions.

3*

416

C l a u d i o A. O r z a l e s i

ture scalar of Mi+N (modulo a possible cosmological term). This theory is an extension of the-Cho-Freund generalization of Jordan's theory. We explicitly construct the associated GFT on M*. The fact that, because of the ¿^-dimensional isometry group, a MUT on M i + N can be translated into four-dimensional language, is ultimately responsible for the "tactile" unobservability of the additional N dimensionals, which play a role of "internal" coordinates (see sect. 4 for further discussion of this point). Various models, which originate from a general MUT by imposing certain constraints on the isometry generators, are discussed in sect. 3.4. The strongest constraints lead to a Y-M GFT with gravity included. The possibly remarkable physical interest of theories with weaker constraints is enphasized: on the one hand, such theories appear as more natural candidates than Y-M GFT's when dealing with an exact gauged symmetry of confined color: on the other hand, they lead [17] to bag-type models without the need of the drastic ad hoc modifications made in the derivation of such models from a QCD-type Lagrangian. The concluding sect. 4 contains a reasoned summary, some remarks and indications on possible developments and uses of MUT's. Conceptually, sects. 2.2, 2.3, 3.2 and parts of 2.5 and 3.3 are an elaboration and reorganization of ideas discussed in refs. [6, 8, 12, 13]. Our presentation is, however, perhaps simpler and more detailed. On the other hand, our K-frames and associated iT-coordinates(sect. 2.4) lead to more explicit constructions than those in the literature quoted: in particular, we use them to spell out in sect. 2.5 the dependence of the metric on the vertical coordinates, insect. 3.1 the correspondence between coordinate and gauge transformations and in sect. 3.3 the faithful translation of a MUT into a corresponding gauge theory in four dimensions. By the same token, our hierarchy of constraints (sect. 3.4) can be discussed directly in four-dimensional language. Thus, our methods can be of practical use, and in particular we found them helpful to understand how matter fields can be introduced [17], covariantly and gauge-covariantly, in MUT's.

2. Projective Formalism for Extended Spacetime with IsometriesjJ 2.0. Notation Extended spacetime, Mz, is a ^-dimensional differentiable manifold 8 ) with symmetric Riemannian metric y having signature (Z — 2). For £ 6 Mz, we denote by rl\(Mz) and Tf*(Mz) the tangent and cotangent space to Mz at f. Greek indices run from 1 to Z = 4 -f- N, N > 0. Local coordinates £ -> (£") 6 Rz on Mz near £ define coordinatebased generators and d£", forming the natural frame (SJ on T((MZ) and coframe (d^) on T(*(MZ) associated to the (£"). For a vector u (and covector u*) the natural frame components are denoted by u" (and «„*), so that u = u"da (respectively-, u* = ua* d$"). For a generic frame {ex) on 1\{MZ), its dual coframe (e*a) is defined from e*"(e^) = bf:

b/af

= 6f.

(2.0.1)

In particular, for the triangular matrix < 8

= Mi is surjective, then [n{Uj) cz Mi, j 6 J) is a covering for an .atlas on M4. When global properties are stated in terms of coordinates, it is understood that coordinate patches forming an atlas are being used and that the usual compatibility properties hold in intersecting patch domains. Capital Latin indices run from 4 + 1 to Z — ^ N and will label coordinates in the "vertical" internal space, see below. Ordinary

2.1. The Projection The first property assumed of the extended spacetime Mz is that the ordinary spacetime Ml is somehow "contained" in Mz and can be obtained from Mz by a projection 9 ): I: There exists a surjective differentiable map ti : Mz called the projection of Mz onto ilf 4 , such that

Assumption

—> MA by x = jr(f),

(i): the four covector fields h* a : h* a = h * a d i \

h*a =

=

,

(2.1.1)

z

which are smooth on M , are such that the matrix (h*a • h*b) = h*aya^*b

= gab

(2.1.2)

is nondegenerate at all £ 6 Mz and such that gab(£) = gab{£') for TZ{£) = jr(f'). (ii): n{Mz) with local coordinates a;0 = and metric tensor g, with components (2.1.2) at x = 7t(S) in the natural basis (dxa), is isomorphic to ordinary spacetime M*; (iii): for each x € ilf 4 , the fibre n~1(x) cz Mz of the projection over x is an iV-dimensional differentiable metric submanifold of Mz, with metric G, induced by y as in eq. (2.1.6) below, having signature N. a) Let denote the linear span of the h*a. The nondegeneracy of gab guarantees that the h*a are linearly independent; hence, II * is 4-dimensional and isomorphic to Tx*(Mi): a natural isomorphism is defined from

Remarks:

dx" = dna = k*ad¥

=

.

(2.1.3)

9 ) A mapping n : Mz —> Ml with the properties (i) — (ii) in Assumption I is sometimes called a Riemannian submersion, see ref. [20].

418

Claudio A. Oezalesi

From bilinearity, (2.1.2) defines a nondegenerate symmetric (co-)tensor g on Ht* and, by the natural isomorphism, on Tx*(Mi). By (ii), n(Mz) equipped with this metric g is identified with ordinary spacetime M*. b) The functions na^ form a p-vector [16], i.e. they transform as the components of a 4-vector under the reparametrizations n" —> n' a (n b ) of n. These reparametrizations correspond to coordinate changes xa —> x'a(xb) on Mi. Similarly, gab is a second-rank y-tensor, while the four covectors h*a define Z ^-vectors hx*a, ..., hz*a. c) With gab the inverse matrix of gab, we have gab = ( k - h b ) = ?ab,

(2-1-4)

where ha is the dual vector of h*a (see below) and yab are the components of y along kkTo form a coframe (k*a) on T*f(Mz), together with the four covectors h*a we must consider N additional covectors V*A, A = 5, ..., Z. Without loss of generality, we choose them to be orthogonal to the h*a: . p*A)

=

¿OA =

frtayipp^A

=

0.

(2.1.5)

Choosing the V*A to be linearly independent, the matrix QAB

(f*A .

=

=

f *AY«fSy

=

pAB

( 2 .1.6)

is nondegenerate (because y is nondegenerate) and by (iii) defines (from bilinearity) a metric G with signature N on the ^-dimensional "vertical" subspace n~i(x) a Mz. For the spaces Mz to be considered by us, the main nontrivial input from (iii) is t h a t G has signature N: indeed, the definition (2.1.6) shows that the metric G on 7T1{x) is just the metric y of Mz restricted to the subspace n'^ix) of Mz. To summarize, Assumption I amounts locally to the existence of a coframe (k*') on Te*(Mz), (it*") = (h*a, V*A), (2.1.7) such that (2.1.1—6) are satisfied. The associated dual frame, (k) = (K,Va),

(2-1-8)

&*•&„) = i f .

(2.1.9)

is defined from the conditions

In terms of natural components, it is given by ha" = V^gabh*b,

V/ = y^GABV^,

(2.1.10)

AB

where GAB is the inverse matrix of G . The components of y in this frame (fcj are given by 9ab = (k-k)

= gab,

YaA = (k-vA)

=

o,

Eqns. ( 2 . 1 . 9 — 1 1 ) imply the completeness relation 8J = h*ah/ + t *

A

yAB = (vA • vB) = gab. V/,

(2.1.11) (2.1.12)

and we can express the natural basis components yap, y*P of y in terms of the components in the (k), {&*") bases: JV = gatk*ah*h

+

y* = gabhaakf

+ GABVA*tB?.

(2.1.13)

Multidimensional Unified Theories

419

To see how the projection works, note that any vector u = (w^SJ can be decomposed as follows into "horizontal" and "vertical" components (uh°, uvA): u* = ha"uha +tA'uvA:uha

= h*aW,

uvA = V*Au"\

(2.1.14)

this decomposition is such that [u • w) = ley^v)» = ukagabwhb + uvAGABwvB.

(2.1.15)

Covectors are similarly decomposable. To go from (fe0) to (it*"), one inverts (2.1.10): h*a =

V* A =

(2.1.16)

In general, the frames (fc„), (7c*") are not coordinates-based, and the generally nonvanishing commutators [ha, hb], [VA, VB] and [ha, VA] have generally nonvanishing components along hc as well as along Vc. 2.2. The Vertical Lie Algebra We anticipate that we shall be interested in the case when the fibres n~1(x) are, for all x € M*, isomorphic to a Lie group 0. To proceed by steps, we first investigate some consequences of assuming a corresponding weaker property on the vertical vectors V A : Property if: The linear span F f of the F^'s is closed under commutation and is, for all I £ Mz, isomorphic to a Lie algebra % with structure constants C°AB. This property is summarized by the commutation relations [VA, F*] = C°ABVD

(2.2.1)

or, in components, by V a ^ L - Vb'K. =

(2.2.2)

Locally, Property "8 states that the commutators [V A ,V B ] are vertical; globally, it states that each component CDAB of these commutators along VD is ^-independent. To see what (2.2.1) implies for the remaining components of the fea and fe*°, first expand the N2 covectors K j f B ' W = (Pii -

^B'd&>

(2.2.3)

a

in the basis (k* ); by keeping in mind that =

=

(2.2.4)

it is not difficult to show that (2.2.1) implies H ^ b ' = C*Ktf.

(2.2.5)

From (2.2.4—5), it is easily found that (2.2.1) implies that the commutators [ha, VA] must vanish and that [n0> /¿¡,] must be purely vertical 10 ). Calling the component of 10

) E.g., to prove that [ha, VA] vanishes, note that (2.1.9) implies icp (2.1.12) as follows with the components of [ha, VA]:

-

=(

+

VB

-

j = —

0 and use

tsU*)

the first term vanishes because of (2.2.4), the second by (2.2.5) and (2.1.9). The verticality of [ha, hb] is similarly proved analytically; for a geometric proof, see ref. [5] or eq. (2.4.23) below.

420

Claudio A. Orzalesi

[tia, hb] along VD, the consequences of Property % can therefore be summarized as follows: [t,a, hb] = -%DabVry, The (so far arbitrary) quantities

lk,VA]

= 0;

[VA, VB] = C%BVD.

(2.2.6)

will later be related to the gauge field strengths.

2.3. The Vertical Isometry Group An important feature of a Riemannian manifold is its isometry group, i.e. the group of differentiable motions on the manifold which leave the metric invariant. Our next assumption on Mz leads to a notion of gauge potentials as covectors of fields generating isometries of Mz. It will later be seen that our notion is indeed equivalent to the standard one. Our discussion is limited to compact connected Lie isometry groups because we later introduce an Einstein-Hilbert action principle on Mz\ the group manifold shall be not only metrizable, but also compact in order to have a finite action. If the latter requirement is abandoned (or if another action is used), the discussion can be generalized to connected semisimple Lie groups, and indeed all our results up to sect. 3.3 retain their validity in this case and even more generally. We recall that a Killing vector field V on Mz generates a local [18, 19] one-parameter group of isometries of Mz. This happens if and only if the Lie derivative ly of y vanishes. V is complete if it generates a global isometry group; equivalently, V is complete when it is generated by a global isometry one-parameter group. An integral curve of F through £ is a local curve £ on Mz such that ffy,f(0) = £ and dav,i(t)/dt = F({(i)) in a neighborhood of t = 0. When V is complete, it has a unique global integral curve (Gr.i(t), t € H) through each f ; conversely, a global (smooth) curve describing an isometric motion on Mz generates a (smooth) Killing vector field, on the points of the curve, as the tangent vector the to curve. If a smooth covering of Mz by isometric curves (one through each £) is given, this defines a smooth Killing vector field on Mz. We give two versions of the second basic property assumed of M z ; a short version is as follows: Assumption II: (i) The bundle (Mz, Mi, n) is a principal fibre bundle with connected compact structural group 0, and (ii) the action of G on Mz is an isometry of Mz. A longer version of Assumption I I may help in understanding what this assumption involves: Assumption IT: (i) The N vector fields VA are smooth and complete Killing vector fields on Mz; (ii) the N vector fields VA satisfy Property , with '$ the Lie algebra of a connected compact Lie group G, and (iii) for any f 6 Mz, the span of the integral curves of (all vectors in the algebra of) the i V s through f covers the subspace 7t_1(a;), where x = n(£). We now outline how I I follows from I I ' : consider first the restriction VA\X of VA to 7i'l(x). This is a smooth Killing vector field on n~1(x) and, by Il'(ii), the algebra of the F^z's is (for each fixed x) isomorphic to '§. Since G and n~\x) are both iV-dimensional, the above properties are sufficient to conclude [ JS] that the group of transformations generated by the VA]X is a faithful realization of G and that G acts effectively on n'1(x). Now, by II'(iii) it is clear that G acts on 7i_1(a;) without fixed points and that, in conclusion, G and n~1(x) are isomorphic. That G acts smoothly and without fixed points on Mz now follows from the smoothness of the F A 's. Definition: A Riemannian space Mz with metric y having signature Z — 2 and satisfying Assumptions I and I I is called an extended spacetime with (vertical) isometry G and projection n and is denoted by (Mz, G) for short.

Multidimensional Unified Theories

421

Clearly, i f 4 is the quotient space MZ!G: thus, points of Mi correspond to an entire fibre, isomorphic to G, in Mz. That Assumption II' follows from II is clear from Assumption I and from the basic defining properties of a principal fibre bundle. Indeed, the vertical vector fields VA are the fundamental vector fields [i9] of the bundle (Mz, M4, n). Looking back at Assumption I, now one realizes that it is, in essence, the statement that the global metric y is compatible with the fibre bundle structure of (Mz, G): now n is seen as the projection Mz -> MZIG, and g and G are the metrics induced through n by y from Mz onto Mz/G and onto n^ix) respectively. Concerning the isometry aspect, II (ii), we recall that the condition of vanishing Lie derivatives, 'f ^ (y) = 0, can be written as follows in natural components [18, i.9]: K*y».. +

+ v J u = 0;

(2-3-1)

indeed, the left hand side gives the components of X$ (y) because y has vanishing covariant derivative along any vector. Before resuming our study of the properties of {Mz, G), we recall some notions and results on Lie groups [6, 18~\. Let X, Y, X • Y be elements of G and XA, YA, (X • Y)A be their coordinates in a local chart over G. We recall that the leftand right auxiliary functions of G, LBA(X) and RBA(X), are defined as follows (I denotes the identity of (?): LBA(X)

= d(Y-

X)A/DYB\y=r,

Rba{X)

= d(X • Y)A/8Y*\Y=I.

(2.3.2)

The differential equations characterizing the Bba are RLARDA A

-

RLARCA

= CA1)CRAB

,

(2.3.3)

A

and LB is related to RB by

LBA(X)=RBA(X~i).

(2.3.4)

Knowledge of the left (or right) auxiliary functions of G is equivalent to the full knowledge of G. The adjoint representation of G is defined by DBA{X)

= L-^(X)RBC{X).

(2.3.5)

We will also need the inverse adjoint matrices, D-^B = B ~ W ,

(2.3.6)

from which the coadjoint representation is obtained by transposition. The differential equations characterizing DBA and £) _1B are as follows: D i e = L ^ d c C \ d D b a = D d a C d e b R-I$, D-^cRjf = -C

a

dcD^b,

(2.3.7) (2.3.8)

showing that (CB)CA = CABC are the generators of the adjoint representation. Finally, we recall the following useful results,: C b aeD b ° = C ' W A A

(2.3.9)

= C a c e D-^D-^ d ,

(2.3.10)

L-'Ib.c] = - C ' a b e L - ^ b L - ^ .

(2.3.11)

422

CLAUDIO A . ORZALESI

2.4. Jf-Frames Physical measurements and observables are usually referred to ordinary spacetime; contact between extended and ordinary spacetimes can be made natural by adopting new coordinates on M z. The coordinate transformation used, f

=

£'"(£'),

(2.4.1)

is such that the first four new coordinates are precisely the functions n a expressing the projection, while the remaining coordinates are coordinates on the fibre n^(x), x — 7i(|). The new coordinates are compatible with the bundle structure (M z, M 4, n) assumed for (M z, G) and n on f is the canonical projection n b{£' a, £'A) = £'b. Dropping primes, we shall hereafter adopt local bundle coordinates, which are characterized by the conditions

(t) = so that

S A) = (x°, X A): ji a{x b, X») = n a,

(2.4.2)

dji a/d£* = da a = h* a.

(2.4.3)

The previously defined generators ka, it.*" can now be referred to the natural basis associated with (2.4.2). The orthogonality relations (2.1.9) and (2.4.3) now imply the conditions h a b = dab> Va" = 0, I> B *rf c B = d c * , (2.4.4) so that, in particular, the N x N matrix PB* A is the inverse of VB J. From (2.4.4) and (2.2.2) with p = B one obtains

H J D

A

- H J C

A

= C%CVA B ;

(2.4.5)

now, observe, that this equation has exactly the same form as (2.3.3). Therefore, we can solve (2.4.5) by making the identification

= Ra b> RA* = RA b(X c) .

(2.4.6)

With this identification, by (2.4.4) the vector fields V A are independent of the coordinates x a. Clearly, the identification (2.4.6) amounts to a particular constructive definition of the bundle coordinates X A (still unspecified as fibre coordinates by (2.4.2)): indeed, the natural generators associated with the (x a, X A) can now be expressed in terms of the generators (h a , V A ) satisfying (2.4.4) and (2.4.6). This correspondence is one-to-one; we call a frame (fea) and a set of coordinates ( x a , X A), such that (2.4.2), (2.4.4) and (2.4.6) are satisfied, a it-frame and a set of K-coordinates. The natural basis associated with such coordinates, (na) = (da) = (da,dA), (2.4.7) will be called the natural K-basis associated to the J^-frame (ka). On account of (2.4.4) and (2.4.6), eq. (2.2.5) gives (0 = D)

R-^cmRB0

= V

E

,

(2.4.8)

which is of course equivalent to (2.3.3). For /3 = b, eq. (2.2.5) gives V*ARB» = - C A B c t b * c ,

(2.4.9)

Multidimensional Unified Theories

423

while the orthogonality relations (2.1.9) also imply that Pb*A = - h ^ R - ^ i .

(2.4.10)

To summarize, the ¿-frame and coframe components are characterized by the following properties: VAb = 0, VA* = Rab;

(ka) = (K, VA): ha" = dab, k B = -P*ARAB; (k*°) = (h* a , V * A ) : h$*a = V ;

Vb* a = R ^ i ' ,

RBA,a =

(2.4.11) (2.4.12)

Note that the existence of -it-coordinates follows from the properties of the V A and the isomorphism of JC' (X) and 0 (see also sect. 2.5 below). To actually determine a ¿-frame, functions V a * A satisfying (2.4.9) have to be given. Eq. (2.4.11) gives the useful relation 1

V

= V

- JV®ff/ Mi we have a local decomposition of Mz as a direct product M* x G, i.e. an isomorphism between (the algebra of vectors in) Te(Mz) and the direct product Tx(Ml) x V( g^ TX(MA) X Correspondingly, a canonical projection n': Tt(Mz) —f TJM4') is defined by n'(u) = uaha, where u = (uaha, uAVA). Now, note that (2.4.16) defines a linear isomorphism between the linear span H( of the ha and the linear span of the ha. Of course, this is not an algebraic isomorphism, because H( is not closed under commutation. The linear isomorphism (2.4.16) is the canonical projection ri applied to Hf. The inverse isomorphism [S, 19] a : He -» He by ha = ha — fra*BVB, eq. (2.4.13), is called the horizontal lift of the vector ha, here considered as a vector in TJM^). Parenthetically, these remarks explain the geometric origin of the first set of eqs. (2.2.6): indeed, since [5] n'{[K> /»»]) = [«'(*«), n{hb)} = [ha, fcs] = 0, (2.4.23) the commutator [h a , h b ] has vanishing horizontal components, i.e. it is purely vertical. A further remark is that all our definitions, of it- and associated K-frames, etc., were given in local terms; unless the bundle structur of ( M z , G) and the topology of 0 are trivial, more than one coordinate patch is needed to form an atlas on Mz; clearly, this is true a fortiori of a Ji-atlas. 2.5. Gauge Fields and Extended Metric By using ^-coordinates and associated it-frames, we now resume our analysis of the consequences of Assumptions I and II. Recall that, for each fixed x, n~1{x) is isomorphic to G. Let cpx denote this isomorphism, cpx: n~1(x) -> G by = g°»t VTK) = 9AH> Yab =

gab>

YM = yaA

=

-GABAABD-II, _g*bLFAAbF

yAB = gEF D^iD^;

= 0,

t

Y^ yAB =

(2.5.11')

= YAB + A

B

LE LF (g

EF

(2.5.12') + g

ab

E

F

Aa Ab ),

(2.5.13')

where gEF is the inverse of the matrix gEF. W e see that, when a iC-frame or a natural if-basis are used, the gauge potentials explicitly appear in the matrixelements of y, and therefore play essentially the same role as the spacetime metric components gab. I n A-frames, this role is less apparent, in t h a t the AaB do not appear explicitly in ya? but rather in the basis (fcj where yals are calculated. Of course, we found that the gauge potentials also have the role of determining the Killing covector fields generating the Cr-isometry of (MZ, G), i.e. the V*A: this role of the gauge potentials as components of the bundle connection is clear in all frames, and indeed it provides a frame-independent definition of the meaning of gauge potentials.

3. Multidimensional Unified Theories With our detailed picture of the geometric structure (M z , G) at hand, we now proceed to define some possible physical theories on (M Z , G). First, we show that gauge potentials and fields as we defined them indeed have the usual gauge transformation laws; in (MZ, G), the latter acquire a richer meaning than in the standard formulation on M4: in fact, we show below that the gauge transformation properties of the AaB are simply the reflection of the covariant transformation of the bundle connection under certain coordinate transformations in the extended spacetime. We also make some remarks on "admissible" frames on MZ. We then compute the Riemann connection and curvature of MZ. Finally, we arrive at a definition of a MUT as a theory on (MZ, G) with the metric y dynamically determined from an Einstein-Hilbert action principle. MUT's are generalizations of Y-M GFT's; the latter are recovered by imposing further constraints on the Killing vector fields of (M Z , G). The possible, geometrically natural, constraints

427

Multidimensional Unified Theories

which can be imposed on a MUT are discussed and classified in a hierarchy. Some interesting properties of theories based on weaker constraints than those leading to the Y-M *tFT are emphasized. 3.1. Coordinate and Gauge Transformations We introduce the following definition: a local coordinate system (£") near | € Mz is K-admissible if a ¿-frame (k„) can be defined using the f" as coordinates and eqs. (2.4.11 — 12) for the frame generators (with kA = VA the Killing vectors generating the isometry G of Mz). In other words, (£") = (£«, £A) is inadmissible if (i) the | a xa are local coordinates on Ml near x = and the are fibre coordinates on n~1(x), and (ii) the £A = XA are such that (ha, VA), {h*a, V*A) are expressed as in (2.4.11—12) in terms of their components in the natural basis associated with the (£"). If one starts with ^-admissible coordinates (f°) and performs a coordinate transformation, = !'•(£•), (3.1.1) in general the (£'"') will not be ^-admissible. We wish to find the conditions for (3.1.1) to be a K-admissible transformation, i.e. for the (£'") to be also iT-admissible. Differently stated, our problem is to find the natural covariance group of the set of all A-frames at f or, equivalently, of all natural if-bases at f. Let (ka), (k*a) be the ii-frame and dual coframe associated to the if-coordinates (f®) anr" let (ka'), (k'*a) be the transformed frames under (3.1.1). In order that (fc/) be a ¿-.frame, the first condition to be fulfilled ist that the be local coordinates on M*; thus, they must be obtained from the z a = by a coordinate transformation: £'a = x'°(xb),

i.e.

= 0.

(3.1.2)

It follows that (3.1.1) must be of the form r a = (x'a, eA):

x'a = z ' V ) ,

=

XB) •

(3.1.3)

By inverting the first four equations, we see that a if-admissible transformation can generally be written as = xa{x'b), = £'A{xb, XB). (3.1.4) Therefore, it is clear that a if-admissible transformation is always the product of a general ^-transformation xa -> x'a on M* times an ^"-dependent transformation of the XA. The former is unrestricted by if-admissibility. Since general y-covariance on Ml is well understood, we can limit our attention to the special transformations x'a = x f ,

£'A = £'A{xb, XB).

(3.1.5)

Recall the transformation laws of vectors and covectors under coordinate changes: in natural components, these are u*

u'"

= £*tfv?\

u*

ua'* =

£fa =

(3.1.6)

Consider the infinitesimal form of (3.1.5): £'" = £«+