Fortschritte der Physik / Progress of Physics: Volume 34, Number 3 [Reprint 2022 ed.] 9783112613702, 9783112613696

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FORTSCHRITTE DER Volume 34-1986 Number 3 PHYS K PROGRESS OF PHYSICS Board of Editors

F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J. Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J. JLopuszanski, Wroclaw A. Salam, Trieste D. Y. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J. Zinn-Justin, Saclay

CONTENTS: F . KASCHLUHN

The Quadratic Divergencies and Confinement in Non-Abelian Gauge Theories

119—143

D . W . EBNER

Energy-Momentum Tensor and Equations of Motion of Glashow-Salam-Weinberg-Theory in Curved Space-Time

H P

AKADEMIE-VERLAG • BERLIN

ISSN 0015 - 8208

Fortschr. Phys., Berlin 84 (1986) 3, 119-166

145—166

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from " 1 " onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the author's name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred too in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written so small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vectors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p, ...), elementary mathematical functions like Re, Im, sin, cos, exp,...): black underlined Greek letters: red underlined Boldface Greek letters: red underlined twice Upright Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as c C, k K, o 0, p P, a S, u U, v V, w W, x X,yY,z Z). It will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, b\, M-j, Wn£ Please differentiate between following symbols: a,«; a,

(19)

A

where we also have the pole compensation which requires strictly real e. We notice that the principal value integrals ¿1 J

a x"

2 J

dx

(;x — ie)n

J _ ie)V}'

(x -

n

n=

1,2,...

(20)

are well-defined for all even n but diverge for odd n (so for n = 1 logarithmically). The superregularization yields well-defined limits for all n whereas the dimensional regularization (e = 0) leads to the pole 1/(1 — /?) for j3 = 1 (but is good for n = 2, 3, ...). Within the general frame work of distribution theory [5] we may proceed as follows. We define the supervalue by (0(x) is the step function, 0(x) = 0 for x < 0, 0(x) = 1

124

F . KASCHLUHN, The Quadratic Divergencies

for x ^ 0) X

=

&(*)

A

\x\

(21)

8 1 / 1 \x\ ~ 2

+

Re a < 1, « — 1 + ie

*r);

fulfilling =

| dx

= J dx -

dx ( i

U(x) -

/(0)]

=

1

dx — [fix)

f(x) « — +

dx — f{x) x

0 1 1 -

+•

/(0)

/(0)]

dx — f(x) x

(22)

where f(x) is a test function. The supervalue represents a well-defined distribution for which also other operations like differentiations are defined. We add the remark that the supervalue (as any other distribution) can be explained without an explicit regularization, i.e. can be defined by a certain functional relation which in our case is given by The above definition of the supervalue of a first-order pole fixes the subtraction constant occurring in a general renormalization scheme. A free constant referring to the logarithmic divergency of the originally ill-defined integral (16) for n = 1 can be incorporated simply by replacing in (21) S(c) \x\

Re,

Jx\~2

1 , « - > 1 + ie, c > 0. (23)

In (18) one has then to replace In A by In A/c. Of course, it suffices to multiply the «-part of S/\x\ by [1 + ( d% j

[(//" 1 I t y p - * + (/¿-i

;

Re

< 1, «,- -> 2 + is

(27)

and symmetrize after momentum integration the occurring singular factors in a,-, 2 + is all poles come out in the form

i.e. there is a complete compensation. I n addition we use some further prescription for fixing the subtraction scheme as it will be discussed below. Needless to say our method applies also to Green functions and scattering amplitudes. We add the remark that throughout this paper we are only concerned with the ultraviolet divergencies, but the infrared divergencies can be treated by the superregularization, too. First let us consider the 2-loop integral 2 ). As far as the leading logarithmic divergencies are concerned we have to superregularize the ill-defined integral (k = \k\)3) 00

d

(29)

/ *nr

A

The integration over one of the loops is already performed leading to the logarithmic factor in (29). According to our rule we have to replace (29) by (kj — \kj\) oo

/*2 +

is.

Similarly one may treat supervalue integrals for poles of higher order. Of course, for linearly and higher divergent integrals the standard dimensional regularization would be sufficient as mentioned previously. ' Now we are in the position to discuss that the renormalized loop functional Wnn(G) with a cusp of angle

g'M = 2 (in A ) 0 V ) •

(44)

The relation (43) replaces (8) for the finite theory. For smooth C, i.e. q> = 0, we have Z9 = 1 in (43). However, renormalization invariance of Wren(C; a, fi) can only hold in this case (provided 1/a 4= 0) if the subtraction parameter a is considered as ^-dependent, i.e. a = a(fi), and this in such a way that the /¿-dependence of y(ji) in the exponential factor of (43) is compensated. Concluding this section we discuss the derivatives of the loop functional. If we perform the functional derivative on the regularized loop functional Wtt%(G) we get the wellknown result [1] 0. E.g. if the limit 6C —> 0 performed in the sense of a scale reduction SC —> XdC, X —> 0 we find for smooth 0. Actually also the first term on the right-hand side of (50), the Stokes term, will not vanish in general. In the case that C and 8C are not smoothly connected with each other there are subtraction terms in the renormalized expressions referring to the logarithmic divergencies of the originally ill-defined integrals. These terms will be independent of X, too. Thus the local derivatives of the renormalized loop functional PFren(0) considered in (49) will diverge in this case. In order to circumvent such difficulties we restrict ourselves to smooth global variation of the renormalized loop functional Wren(C') XdL/a).

x^s)

- > x^s)

+

epipis)

Fig. 2. .Smooth global variations

130

F . KASCHLUHN, T h e Q u a d r a t i c Divergencies

where 0 the variation x or z —> y, respectively, is concerned. Of course, the solution (56) fulfils the Makeenko-Migdal equation ¡(59) c Now we show that the first term on the right-hand side of (56) has only linear divergencies as the highest ones whereas the last two terms are even quadratically divergent. These last two terms are actually non-abelian since they are absent in the abelian case where we have G(Cxy, Cyx) = y^WiC). We shall see below t h a t only these terms will contribute to the area law as we should expect it. In order to determine the singular structure of (56) we have to know the behaviour of G(Czy, Cyz) for z —> y. From (58) we see that actually the behaviour of W(Czy, Cyz) for z y is required. For calculating the linearly and quadratically divergent terms in (56) the functional G{Czy, Cyz) or W(Czy, Gyz), respectively, can be deregularized first since the superregularization ignores such divergencies at all.4) But it includes the referring subtraction constants. Since the theory is asymptotically free we may employ renormalization group relations. Thus putting Cyz = C — Czy we have to study the renormalized functional I Tr Tr \ W{Czy, Cyz) = W{Czy> C-G, ,)= {^U{Czy)-V{G (60) -Gzy)^ 4

) In order to get rid of the logarithmic singularities one has to apply the symmetrization rule discussed in the preceding section.

132

F . Kaschltjhn, The Quadratic Divergencies

for 2 —> y. We may approximate (60) in the form W(Czy,

C -

Gzy)

=

A[g(n),

\z -

y\ p)

(61)

W(C)

where W{C) requires no further treatment here. To get (61) from (60) we neglected CGzycontractions. They have no endpoint singularities and remain finite for z —> y. The function A[g(jji), \z — y\ juj involves all leading logarithmic singularities of W(Gzy, Cyz). We may sum up them by the renormalization group solution in lowest approximation of the /J-function. Since A(g(/u), |z — y\ pj renormalizes multiplicatively according to 5 ) ¿W)>

I* - V\ f*') = Z ^(/i), In

A(g(p),

\z - y\ p)

(62)

the renormalization group equation reads in our case P-

dp

r(g(p))

8g(p)

A{g(p),

\z -

y\p)

=

0

(63)

where dzr{gW)

/t •

=

(64)

dp'

is the anomalous dimension function. In lowest order we evaluate for SU(N) ,man gauge = 1

z

-

w

-

^

in Feyn-

(65

J

>

and get for (64) n

r(g{/i))

~ n

(66)

2 7t2

Then by employing standard methods we find from the solution of (63) A(g(v),

Iz -

> C(st(m)) ( - in |z -

y\ ¡j)

y\

/*Y

(67)

with N

~ N

00 was not assumed. Concluding let us compare our results with those of the lattice approach to nonabelian gauge field theories [6]. Here one arrives at the following expression for the string tension. "(«> 9o2) =

(100)

where cL is a numerical factor, a the lattice constant, g0 the bare coupling constant and f}0 the lowest approximation of the /^-function.9) Expression (100) is certainly a non-perturbative result (g0 enters in an essentially singular manner) and represents a quadratic divergency of the theory if one performs the limit a —> 0 for g02 > 0. Actually the theory says nothing about the connection between g0 and a when the continuum limit is taken. Thus one may try also a smooth limit to the continuum as it is generally done in this context. A smooth limit (where in particular divergencies are avoided) is indeed possible by requiring e.g. that the coupling constant including all radiative corrections equals its physical value already in the regularized theory z(ln OA0 &>»(«) =9L2(t*)-

(101)

This value is held fixed when the limit a —.> 0 is taken. Since z(ln ap) diverges for a —.> 0 the bare coupling constant 0. The details are given by the renormalization group relation

(102)

9

) Since we are interested only in the limit a -s* 0 higher expansion terms in (100) involving p v /?2 etc. are unimportant for us. This is discussed in more detail in appendix D.

139

Fortschr. Phys. 34 (1986) 3

valid in the ease of asymptotic freedom. The parameter AL can be expressed in the form Al = ^-tMg^w

(103)

where dL is a certain integration constant which cannot be determined by means of the asymptotic solution (102). The z-factor in (101) has then the asymptotic form z(ln a/i) = d

L

- 2/30gL2(fi) In ap =

~gL2{fi)

2 f t In aAL.

(104)

If gL2{p.) is given AL is in principle determined by the theory. If (102) is inserted into (100) one gets for the string tension in the continuum limit the physical value a = lim a[a, g02(a)] = cLAL2. a—>0

(105)

From (105) on might conclude that a is in principle also fixed by the theory since Ah does it. However, we have to ask whether the continuum limit (required only to be smooth) can indeed be unique. The limit discussed above was based on (101). We stress once more that the theory has no receipt how to connect g0 and a when the continuum limit is performed. There is obviously no reason that the coupling constant involving all radiative corrections must have its physical value already in the (unphysical) regularized theory. I t would be sufficient that this holds only in the limit a —> 0. Thus let us perform a smooth continuum limit as follows. We replace in (102) AL by a certain other parameter AL', i.e. g02(a) by

=

"

, ( a )

+

WJ^

•

( 1 0 6 )

Then we get instead of (101) for small a z'(ln o/«) g0'2(a) =

+ 0{ 1/ln aAL)

(107)

since now we have (cf. (104)) z'(ln an) = -gL2{n)

2 f t In aAL' + 0(1) = z(ln a/x) + 0 ( 1 ) .

(108)

In (107) there are included again all radiative corrections. However, (107) is as good as (101) for a smooth continuum limit since the last term in (107) vanishes for a —> 0 and in the continuum theory the observable coupling constant gL{fx) occurs in the desired manner. For such a limit the string tension comes out in the form

2,

3, . . .

(C.2)

0

We added for generality an arbitrary subtraction constant c referring to the divergency at the lower integration limit. The analytic continuation of the integral on the right-hand side of (C.2) to ft = 2, 3, . . . can be performed via the transformation (Re ft < 1) a

In15 ar 1

dx

1 ft-

InM-1 1

A*-

1

1 1

P ~ 1 P

-

-

In"*

i

^

d In*'1

x

A"-1

+

J

x-1

In 5 " 1 1

1 1

dx

A'1

1

1 'ft

f

W (ft -

- 1 ) In*"2 A- 1 l)'

• In1» A- 1

L ft - l d l n

dfo-ik

A'1

A'1

lnd"1 .

1

a(k;d) —ift-l

(C.3)

142

F. KASCHLUHN, The Quadratic Divergencies

with

1 a(k;d)=

71

s M " 5 - 1 r((5 + l ) s i n - ^ .

(C.4)

2

If 8 = —1, —3, ... we take the superregularization with respect to 8, i.e. we replace in (C.3) In 3 a r 1 ->

A

(ln i+ie a r 1 + In 6 "" a r 1 ) .

(C.5)

I t is then obvious from (C.3) and (C.4) that the analytic continuation to has a well-defined limit for all (5.

= 2, 3, ...

Appendix D Let us assume instead of (100) the more general expression for the string tension * =

(D.l)

where the dimensionless function like

f{g02)

behaves for g02 —> 0 (within exponential accuracy)

/(¡¡to2)

fi-1"-'.

(D.2)

We discuss now t h a t it is always possible to choose a smooth limit to the continuum by requiring

so t h a t (D.l) takes the form (cf. (109)) 0 using (D.2) * • = *'•

= - 2 J ^ a ! "

Repeating the considerations (106)ff. we see t h a t for a H> 0 the renormalized coupling constant g^p) comes out in the desired manner independent of the value of AL'. This argumentation shows quite generally that the physical string tension a is not fixed by

Fortschr. Phys. 34 (1986) 3

143

References [ 1 ] J . GERVAIS, A . N E V E U , P h y s . L e t t . 8 0 B , 2 5 5 ( 1 9 7 9 ) ; Y . NAMBIT, P h y s . L e t t . 8 0 B , 3 7 2 ( 1 9 7 9 ) ;

A. M. POLYAKOV, Nucl. Phys. B 164,171 (1979); Yu. M. MAKEENKO, A. A. MIGDAL, Phys. Lett. 88 B , 135 (1979). For a review including a rather complete list of references see; H. DORN, Fortschr. Phys. 88 (1985) 11. [2] Ytr. M. MAKEENKO, A. A. MIGDAL, Phys. Lett. 97 B , 253 (1980). [3] F . KASCHLUHN, Proceedings Symposium Ahrenshoop 81, P H E 81-7, 39; 83, P H E 83-13, 53.

[4] V. S. DOTSENKO, S. N. VERGELES, Nucl. Phys. B 169, 527 (1980); I. YA. AREF'EVA, Phys. Lett. B 93, 34 (1980); R . A. BRANDT, F . NERI, M. SATO, Phys. Rev. D 24, 879 (1981). [5] I. M. G ELF AND, G. E . SHILOV, Generalized Functions, New York, Academic Press, 1964. [6] C. P. KORTHALS-ALTES, Lectures on Lattice Gauge Field Theory in "Non-Perturbative Aspects of Quantum Field Theory", World Scientific, Singapore 1982.

Fortschr. Phys. 34 (1986) 3, 145-166

Energy-Momentum Tensor and Equations of Motion of Glashow-Salam-Weinberg-Theory in Curved Space-Time DIETER W . EBNER

Physics Department. University of Konstanz, Konstanz, Federal Republic of Germany1)

Abstract The equations of motion of the unified gauge theory of weak and electromagnetic interactions, when minimally coupled to the gravitational field, are given.

Contents 1. Introduction 2. Conventions for Spinors 3. Conventions for Gauge-Theories 4. Particle Contents of the GSW-Theory 5. Spinors and Gauge-Theories in Curved Space-Time 6. Equations of Motion for the GSW-Theory Appendix A: Auxiliary Calculations Appendix B: Comparison with other conventions References

1.

145 146 147 . 149 151 153 156 164 165

Introduction

The Glashow-Salam-Weinberg theory (GSW-theory) [1]—[20] is a highly successful unified gauge theory of weak and electromagnetic interactions. Since general relativity has formal similarity with a gauge theory of the Poincare-group, the search for a unified gauge theory of electroweak and gravitational interactions is an outstanding problem in theoretical physics. However, because of fundamental difficulties (non uniqueness of quantization, renormalization, geometrical interpretation of gauged translations, etc.), such a unification will probably be possible only after a radical change in our ideas. There have been a lot of attempts in this directions [21]—[35]. In this paper we will follow the most conservative (but also the most secure) approach: Minimal coupling of the GSW-theory to the gravitational field. This minimal coupling prescription in the case of the electromagnetic interactions alone has been exploited in several investigations [36]—[40]. This paper is particularly prepared for readers with the intention to find classical solutions of the GSW-theory in curved space-time. J

) D-7750 Konstanz, Box 5560

146

D. EBNER, Energy-Momentum Tensor

2.

Conventions for Spinors [41]—[47]

We use 2-component (van der Waerden) spinor calculus in matrix notation : yR = {yÀ),

yL = {yÀ)\

A = 1,2.

(2.1)

The "right-handed" spinor yR and the "left-handed" spinor yL are considered columns. They are related by YR=-eyL*,

U Ó)'

yL = eyR* e

2=-l,

yL'

=.ÜyL

(2.2) e

r

=

_

(2.3)

£ -

They transform as yR=UyR,

U = (U+r\

(2.4)

U £ SL{2, C).

The connection between SL(2, C) and the Lorentz-group $0(3,1)° is assumed as U — |cosh with

— ma sinh

|cos

+ ma sin -^-j

tanh^ = w

(2.5 a) (2.5 b)

where the Lorentz-transformation is given as a spatial rotation with axis n, angle p' = Dtp

(3.9)

D „ y = 0{D a W )

(3.10)

A: = 0{Aa + ig-1 a.]

(3.11)

tp (and Dayi) are particle fields, Aa is a gauge-field: particle-fields: local (homogeneous linear transformation law) gauge-fields: non-local (non-homogeneous linear transformation law). Curvature tensor: -igF.e

(3.12)

with F^ = F^aTa = 8aA? - d?Aa - ig\Aa, Af]. 2)

The Pauli-matrices are denoted by ra when they are used with respect to the iso-spin When used in conjunction with SL(2, C) they will be denoted by o a .

(3.13) STJ(2).

Fortschr. Phys. 34 (1986) 3

149

In components: (3.13a)

Fx?a = d^Afr - 8^Aaa + gCbcaAabApc f' a „ = U P . f U - i = 4.

(3.14) - iglA^,

.

(3.15)

Particle Contents of the GSW-Theory In the GSW-theory the underlying group is3) G = SL(2, C) X SUW(2) X Uy( 1).

(4.1)

We have labeled these groups with subscripts (W = weak isospin, Y = weak hypercharge) to distinguish them from the same abstract groups occuring in other applications. 4 ) A matrix V £ G can best be labeled with double indices (4.2)

UAp.Bq = UABUwpqUv where U = (UAB) £ SL(2, C), Uw = (Uwpq) £ 8UJ2),

Uy £ Uy(l).

{A, B = l , 2 ; p , g = I, II). (Uy is a complex number, U and Uw are complex 2 x 2 matrices). (We call A, B Lorentzspinor indices, JJ, q iso-spinor-indices) In the GSW-theory the following 3 particles are assumed:5)

- M i : ) '

- ( : ) •

Written with full indices they are eR = eR(x) = {eRA(x)); L = L(x) = {LAp(x)) = ( j ^ j ^ j ) 5 9»(®) = (?*(*))

(4-4)

with the following transformation rules6) e'RA(x') = UABeRB(x) • Uy~*-, eR'=UeR.Uy~\

A = 1, 2

(4.5a') (4.5a")

In words: eR is a right-handed Lorentz spinor field, an isosinglet (trivial representation of SU(2)) with hypercharge —2. 3)

In quantum ohromodynamics an additional factor x SUC(3) is assumed. 8UW(2) is the weak isospin group. SU(2) occurs also as the strong isospin group in the theory of strong interactions transforming a proton into a neutron, or a «-quark into a (¿-quark. An U(t) occurs also as the group of electromagnetic gauge transformations. 6) e is the XR and e is the

«(0)(n)Gn — ~r iu>«(n)(m)£nmlGl = ~7 ««(AOMffL^W 0 ^ = 2 ^

(5.10)

w«(|1)(,| = ecM;«^/ = e x M ; a e M i

(5.11)

CUa(/i)(>) = —0)a(v)(/t)

(5-12)

D«yL

(5.13)

= 8*yL ~ r+yL.

W e have the following rules: 1) product rule for covariant derivative. 2) rjWW, e,

ffi^',

e^/"' are covariant constant:

= 0

n^ww Dae = 0

¿ W

0

= 0

(5.14)

DaoLW = 0 D.e/0 = 0. 3) The covariant derivative of a space-time vector is the ordinary covariant derivative of Riemannian geometry. Gauge

Fields

The ra is the gauge-field of the Lorentz-group (or 8L(2, C). Because of the direct product structure (comp. (4.1)), the gauge-fields of the internal symmetries coexist. All particle fields (spinor-fields, vector-fields, etc.), are referred to the tetrads and thus transform under the gauged 8L(2, C) X G{, where Gt is a certain internal group (in GSW-theory: Gi = SUW(2) X Uy(l)). All formulae of § 3 are valid9). The covariant derivatives, field strengths, etc. may be referred to the coordinates or to the tetrads, e.g. -tW =

9)

=

eUDSP>

re)

As an example consider [D„, D„1 = - i g f See Appendix A 1.

a f

[Dw,

D(f)]

=

—igFitl)M.

=

(5-15)

Portschr. Phys. 34 (1986) 3

6.

153

E q u a t i o n s of Motion f o r the G S W - T h e o r y

I n t h e G S W - t h e o r y t h e following L a g r a n g i a n d e n s i t y is a s s u m e d 1 0 ) .

-^gsw = i — g Ä + =

D

=

( d , -

i g

D / p =

( a ,-

i g

L

u

v

— ¿g-yF^ + W ,

w

w

a

W ,

a

-

) -

1

y ' Y p , +

-

r,t) e

i L -

+

o

L

M

D

( a ]

=

m

L + i e

g t ( L +< p e B +

R

+ G

eR+cp+L)

R

^ B

( a )

e

R

(6.1) (6.2a)

R

|

_

»

7 „ -

|

+

*

7 „ )?

/ y )

£

(6.2b) (6.2c)

+ S^Y, -

Y ^ w

W / ' W ^

(D„9»)+ ( D "

D ^ r m

~

(6.2d)

d^Wva — d

r

W

g w s ^ W ^ W , ,

+

M a

(6.2e)

g Y , g w , g e , X H , ¡ i H are coupling constants. e R , L ,

A+1,

iyA+1> 8aW*

=

d„y>A — i

80ipA+1).

Thus 8L

Here

8L

o

8a

=

Sif dL B

„.„

8L ,„,

,

" 8{8aif)

8L

— — -

'

8a

=

Sf*

" 8(8„if*)

'

/dL)

± = (8±)

8W

F / =const.

if and if* are treated as independent variables (i.e; we forget, that if* is the complex conjugate of if). A3: With

Z(y>A,

d„ipA)

=

DoWc =

L(yjA,

DayiA)

doVc — igAoiPc

=

dofc

—

igAacAVA

//

18L\ =

/

const.

8LdL

\

^ccVa — const. (6 6)

-

=

„

R,oAc

+

^ V c =const.

/

dL \ ——\8\8oVc)!vc=const.

igAnA]

In the notation of footnote 15) we have with

(A 3.1)

e~iga**T'>:

U =

a A T **acA — -1