IL NUOV0 CIMENTO VoL. 76A, N. 4 21 Agosto 1983

Dynamical Implementation of the Gravitons as Nambu-Goldstone Bosons.

H . SALLEl~

Max-I)la~wk-Institut ]i~r Phys ik und Aslrophysik ~ Werner-l~eisenberg I,nstitut fiir Phys ik ~> - Mi~nchen, B .R .D .

(ricevuto il 18 Novembre 1982)

S u m m a r y . - - General re la t iv i ty can be considered as a theory with a nonlinear realization of the symmetry GJL~,R. In analogy to the Nambu- Jona-Lasinio ehiral model, i t is shown how the massless gravitons arise in the pair approximation. The ul t raviolet divergences of the canonical approaches can be avoided by using fields with anomalous dimensions. Such an approach is suggested by the fact tha t the te t rads contain also the di la ta t ion Goldstone field which affects the dimensions of the fields in- volved. The characterist ic mass for the GL4, R condensation proves to be the Planck mass, which in turn is also re la ted to the most dangerous consequence of the subcanonicM structure, the indefinite metr ic in state space. The dynamical incorporat ion of gravi ty shows a way how to determine, by finite gap equations, the rat ios of the characterist ic masses for the in ternal breakdowns (e.g. ehira l i ty or S U2) to the Planck mass.

PACS. 11.30. - Symmetry and conservation laws. PACS. 12.20. - Electromagnetic and unified gauge fields. PACS. 12.25. - Models for gravi ta t ional interactions.

I n t r o d u c t i o n .

I t has b e e n s u g g e s t e d (1,~) a n d shown (") t h a t g e n e r a l r e l a t i v i t y , as i t s t ands ,

a l lows a n i n t e r p r e t a t i o n as a t h e o r y wi th a n o n l i n e a r r e a l i z a t i o n of t h e

(1) W. H~ISE~'n~RG: Introduction to the Uni]ied )Field Theory o] Elementary Particles (London, 1966). (2) Y..NAMBU: Ann . N. ~ . Acad. Sci., 74, 294 (1969). (3) C. J . IsI[~.'~, A. SALA,'~ and J. S~RA'rHD~V.: Ann . Phys. (N. Y.), 62, 98 (1971); II . P. Di)m~: M P I - P A E / P T h 38/82.

43 - II Nuovo Cimento A. 663

6 6 4 It. SALLX~

symmet l T G154, n. In this in terpreta t ion the tc t rad fields, more exact ly the i r 10 symmetr ic massless components found in the metrical tensor, contain the Goldstone fields related to GL4,R/SO~. ~.

Modelled after the four-fermion theory of superconduct ivi ty (4) the chiral model of h-ambu and Jona-Lasinio (5) shows how the mass-zero I~-ambu- Goldstone modes, re la ted to a nonlinear realization of (~( spontaneously broken ,~) ehiral symmet ry , are implemented and can be computed in a pair approxi- mat ion. Deadsick in the ul traviolet region, those considerations nevertheless show how a universal cut-off procedure reproduces all the expected and desired results (subsect. 1"1).

I n this paper (subsect. 1"3) we implement dynamically, b y using fermion fields d la NArrow and Jo~A-LAs1~1o, the gravitons as mass-zero Goldstone fields re la ted to the GL4, n breakdown. Apar t f rom some additional algebraic complications and subtleties connected with the isolation of a linearized theory, all the merits and problems of the chirM approach will be rcfound. The ultra- violet sickness is one degree more severe.

The ul traviolet problems are then approached in a frontul at tack. They are connected, in all the models of sect. 1, with the fact that---ill canonical ~heories-- one lacks the dimension-reducing tools which, e.g. in the chh'al c,~se, lower the dimension of the fermionie product ~((1 + 75)/2)~(x) to tha t of an effec- t ive canonical Higgs field. A theory of a dynamical cut-off (*) is needed. B y Mlowing anomalous dimensions (~) of the condensating fermion fields, sym- metr ical ly dis tr ibuted around the canonical value, a dynamical cut-off can be in t roduced leading in sect. 2 to ul traviolet better-defined versions of the models of sect. 1.

I f one looks ~t the <( weight ~) of the tields in contrast to their dimensions, a concept in t roduced by W~,YL (7), one gets the impression tha t the te t rads (and the i r inverse) with positive and negative weight are the key for under- s tanding the ul t raviolet problems (**). This s trongly suggests a connection between Planek mass and indefinite metric which we t r y to substant ia te in

subseet. 2"3. A dynamical implementat ion of the symmet ry breakdown has the decisive

advan tage - - in comparison to theories with basic ]=[iggs f ie lds- - tha t the arising gap equations allow, at least in principle, the determinat ion of mass ratios. In contrast to the theory of superconductivi ty, however, all those gap equa- tions in purticle thcory are divergent.

(4) J. B~D~E~, ]L. N. CooP~m and D. R. SClmI~I.'~'~R: Phys. l~ev., 108, 1175 (1957). (~) Y. N),MBU and G. JONA-I~ASrSlO: Phys. lr 122, 345 (1961). (~ Given with the aid of the phonons in the theory of superconductivity (e). (e) E.g., A. A. Am~IHOSOV, L. P. GoR~(ov and I. E. DZYALOSHIXflKI: Methods o]Quantr t~ield Theory in Statistical Physics (New York, N. Y., 1965). (') H. W:BYT.: Raum-Zeit-Materie (Berlin, 1923). (**) The tetrad as the phonon of elementary-particle theory?

DYNA~IICAI, IMPL:EM:EKTATION OF THE GRAVITONS ETC. ~ 5

I f the te t rads ~rc the phonons of p~rticle theory, thei r dynamical implemen- ta t ion should show us also the way to obtain finite, meaningful gap equations which relate the characterist ic mass of, e.g., the ehiral breakdown to the characteristic mass of the dilatat ion breakdown, which proves to be the Planek mass (subsect. 2"3). We show in subsect. 3"2 ra ther qual i tat ively how such finite gap eqnations real ly can occur. The quant i ta t ive ~( details ~) for such an approach to the ~( mass tfierarchy problem ~ will have to be studied in fur ther pubhcations.

l . - Dynamical implementation of Goldstone and gauge bosons in canonical theories.

1"]. Chiral Nambu-Goldstone boson (s,s). _ For the chiral condensation in the model of ~Nambu and 5ona-Lasinio, an effective chiral Higgs field r interact ing with a Dirac field ~p(x)

(1 .]) ~(x) ~-~ exp [--i75~51~(x ) ,

in a globally chiral- invariant dynamics

(1.2) ~f:~'(~f, r --= ~y, ' ff ~, y~ -~- '~1--

is replaced by a fel'mion product

r ~-~exp [--2ias]r

+ ~ l~,r (r - ~ ) "

1 + 7 5 ) (1.3) r =: - - 2g~ (4~)'-' v~ ---2- ~ (x).

In the ehiral- invariant pure ly fermionic Lagrangian (*)

i '~ gA(4~) 2 (1 .4) ~ o ( ~ ) = (pT"~ ~ ~- - - ( f ~ P ~ ) ( f ~ P ~ )

' 2

(s) H. ]). Ds and II. SALL]~R: NUOVO Cimento A, 39, 31 (1977); 41, 677 (1977); 48, 505, 561 (1978). (*) Throughout this paper rationalized coupling constants, e.g. g+(4:~) 2, will be used.

666

with the most general four-fermion coupling given by

i F * = : + V gaF @ g,d § a) + gT~(v a) - -

= - - g ~ ( s + p

s = 1 ~ 1 , p = iyo @ iy : , v --= y~, ~ ;,~',

(1.5) v q-2 a) + g ] ( v - - a ) (Fierz

l:I. 8 ~ . L L E R

antisymmetrie),

a = iyu y~ ~ i7~' 76 ,

a chiral breakdown <~l~(x)> r 0 leads to a fermion mass m and a resulting gap equation for this mass

) ' '

(1.7) m

m . . . . 2g~ ( 4 ~ ) , ( ~ l ~ ( x ) > = Sg -~ - - m'-'"

i f = ~ f(l~pnJ p

The integration symbol

(1 .8 )

is assumed to include also necessary regularizations, which allow all the manip- ulations performed.

Regularizing in (1.7) with (A2/(A~--p'~)) "2, one obtains for m e : 0

mz 10 m: ] (1.9) l = S g ~ A : 1 -4--~ g - ~ H- ...

and for the limiting case

(1.10) A ~-*c<), g~.A2 ~ ~ .

The mass-zero Goldstonc boson O(x) ~ O(x) -~ ~z5 is found in tile pMr approxi- mation (~ for (~[(viys~v(x)[O> given by

(1.11) < ~21(~i~ A tf(x)[O> =~ i f dy <DIr ~i~ -F. . . =

= -- 2ig~(4zt)2fdy Tr [iy~ ~L'7'(x-- Y)iy~ ~.~(y -- x)]<~l~,i'/~W(y)IO> +. . .

(*) Couplings to ~u(~yuys~) have been omitted, l-Q) is the ground state.

DYNAMICAL IIt~PLEM]~NTATION OF THE GRAV[TONS ~,TC.

or in self-evident momentum-space representation

667

(1.12)

O(p) = ~(p) O(p) ,

a

q:~ = q ~ p/2 .

For the same regularization ~sed in the gap and the Goldstone boson equations, (1.7) and (1.12), respectively, one obtains

(1.13) ÷ f l

I (p = O) ---- 8gA q2 - -m ~ = 1 ,

I(p) = 8g~(A ~ + cop ~ ÷ . . .) , A 2

eo'~ log ~ .

Therefore, one can establish the mass-zero pole of the Goldstone field. I t s normalization, however, (i.e. its coupling to the fermions) vanishes for A2--> oo.

1"2. U~ gauge boson (9). _ Disregarding all ultraviolet aspects, one m a y t ry to replace an effective gauge boson A~,(x) interacting with a fermion ~p(x):

(1.14) y(x) ~-~ exp [--iot(x)'Jy~(x), A~(x) ~.A~(x) + ~ ( x )

in a U~,~oo-invariant dynamics

(1.15)

~ ( i ~ ) 1 A ~

A ~ = ~ A , ~ ~ A , ,

by a fcrmion composite

(1.16) A~(x) ~-- aA(4~)2v~y~(x) .

The fermionic product with a small separation ~ transforming as

I aA(4~)~(x-)TJ'~(x+) ~-~ (1- - i~a)aA(4z)2(p(x_)?~p(x+) , (1.17) [ x~ = x + 2/2,

(9) J .D . BJO~X]~N: Ann. Phys. (iV. Y.), 24, 174 (1963); H. SALL]~R: .YUOVO Cimento A, 67, 70 (1982).

6 6 8 ~ . s ~ R

can produce an inhomogeneous (~,a)-dependent c-number contribution (*)

(1.18)

1 p2--2m2

The required inhomogeneous te rm for a composite gauge boson is obtained

with the condition

2a ~" p~ - - 2m~ (1.19) a j ( ~ m ~ ~ = 1 .

A cut-off regularization in (1.19) with (A~/(A2--p~)) 2 leads to the limit condi~

t ion

(1.20) A S -+ oo ~ aa "A ~ -+ ½ .

F r o m the Lagrangian with a composite gauge boson (1.16)

(1.21)

the pair approximat ion for the gauge boson (**)

(1.22) ( f21A~(x) It} : i f d y ( QIA~,(x) ~fint(Y)t~} + . . . .

= - - ifdy [A. J,(x - y)(f2]A,(y)]~} + A~,A, (x - - y)(QIJ,'(y)]~}]

gives the field equation

(1.23) A~(p) ~--- - - A~,J~(p)A,(p) - - At, A,,(p)J~'(p) , ( J r J

(1.24) A ~ J ~ ( p ) = a A f T r [ ~ , t ~ q + - - ~ m ~ , ~ g _ I m ] . q

(*) The inhomogeneous gauge behaviour is the most important characteristic of the gauge bosons. There are many approaches for composite gauge bosons which--by neglecting this point--have no definite statements concerning their masses. (**) A~J~(x--y) etc. are double fermion contractions, e.g. ~ ( x ) ~ r ~ p ( y ) .

I I

:DYNAMICA.L I M P l e M E N T A T I O N OF T H ~ G R A V I T O l q S ~ T C . ~

~or p -- 0 the integral in (]..24) connects the gauge invariance condition (1.19) with the mass-zero p roper ty of the gauge boson, provided the same regulariz~-

t ion is used:

(1.25) f 2 q ~ - - 4m'-

- - A.LJ,(0 ) = - - ~7~,A~Jq(0) = ~.~a~3(q. W-m~)~ . . . . V"'"

The integral A , A ~ ( p - 0) i s - -wi th equal regular izat ion--givcn b y aA(47~)2~7~,. Therefore, the gauge boson propagator relat ing gauge boson and current

(1.26) - - A ~A~(p)

(i,i J'(p)

obtains a residue at the mass-zero pole, i.e. a coupling constant, proport ional to 1/log (A2/m ~) -+0.

I u canonical theories the distinction between current J , (] .21) and com- posite gauge boson A~ (] .16) is artificial. The gauge boson should have intrinsic

dimension 1.

1"3. The tetrad ]ields as GLd, R Nambu-Goldstone bosons. - T h e dynamics of a gravi ta t ing fermion field (,0) ~(x)

(1.27) r / i,-~ \ M'- R]

uses the t e t rad fields re la ted to the metrical tensor by

(1 .2 s )

and the inverse te t rad (') hi(x)

(1.29)

v j v j h (x) == h (x) =

1 ~ ~ ~,~

1 ~ o~,,Q~,; ~ det h - - 4 ! ~k,~ o -~ ,o, ,oQ ,v. �9

(lo) E.g., Y. M. C~O: _Phys. Ir D, 14, 2521, 3335 (1976). (*) We use the redundant notation for tetrad h~ and its inverse ~ ah'eady characterized by their Greek GLd,R and Latin SL2, c indices.

670 H. SALLER

R is essentially the scalar curvature (~0). M~ is related to Newton's constant G by

(1.30) .Mz_ ~z _ (2.1.10 ~ GeV) ~

D V~ is the SL~. c covari,~nt der ivat ive containing six SLy. c gauge bosons (~o). The Lagrangian 5 r (1.27) multiplied by d~x shows the (~global,~ (*)

GZ,. R symmetry (invertible t ransformations only)

x~-+x'~--=a~x ", d~x~-~(deta)d~x, ~ - ~ ( a )~ ,

(1.31.) h~'(x) ~-*a ~'~ h;(x') , itS(x) ~ (a-~ ~)~,h,( ~' x ' ) ,

~(x) ~ ~(x ' ) ,

and an obvious SZ2.C,~o ~ invariance. The fermionic par t of the Lagrangiaa (t .27) can be wri t ten as a deter-

minant including the inverse te t rad o111y

(1.32) ,~t - - m~p

3! . ,mo ,o,,oo v } - u hl].

The flat-space approach (11) for the linearization (" ) involves a breakdown

GI%R| SL,~,c ~ S03, ~

(1.33) h~(x) = 5' --]e~(x), <h~(x)} ~- 5' )

with / as an order parameter . One has to take care tha t those symmetries - - n o t broken, but nonlinearly realized in the fi~ll t h e o r y - - a r c m~ntu ined . Up to order ] the linearization le~ds to

.~ , , , (~) = (5,~ (~V ' i ' -* ) (1.34) ~ 2 ~V --m~p -~

" ~e]'( i'~, \

(~ At least with constant parameters a~ in a ncighbourhood. (11) W. TUII~RIN6: fforischr. Phys., 7, 79 (1959); Ann. Phys. (N. ~.), 16, 96 (1961); S. D>:SER: Gen. Relativ. Gravit., 1, 9 (1970). ('~ Thc decomposition (1.33) of the << IIiggs field ,) ]~(x) characterizes the asymmetric R dccomposition (analogue in the U~ Goldstone model ~(x) ---- r -~- ~'(x)). The symmetric U-decomposition (analogue ~(x)= ~(x) cxp[--iO(x)]) would be given by ~ ( x ) = = H~(x)bi(x) with an antisymmetric 6-component bJ(x). The six b~(x) co-operatc in a Higgs mechanism to make the SLy. c gauge bosons massive. The ten H~(x) arc the origin of the ten masslcss g~,(x).

D~AU~CAL ~wr~.~t~a'~Tro~ o~ 'rH~ O~WTO.~S ~TC. 671

T h e r e m n a n t of t h e (~ g loba l )~ G~5~.a s y m m e t r y in (1.3.1) is t he (( local transla-

t ion invaria~ce ,~ (~o) up to order f. W i t h art in f in i tes imal GL,,n t r a n s f o r m a t i o n

(1 35) % ~ ' ]~':

a n d the x - d e p e n d e n t p ~ r a m c t e r

(1.36) ~ ' (x ) - - e~x ~ , ~ ~t'(x) - - s~',

-1 ,,____ 5 " - - ] % , (a ),, ,,

~ ~qe~'(x) =- 0 ,

one ob ta ins i nv~r i ance of t h e l incar iza t ion (1.34) up to order f unde r (1.31) o r - - w r i t t e n in the local t r a n s l a t i o n f o r m - -

I x" e - , x '~ '=x~ ,+]e l~ (x ) , d ' x F-~ d 'x ( ] +]~Q~e) ,

(1.37) [ h~(x) ~--~h~(x ' ) - - l (~et ' )h~(x) ,

which induces the gauge boson or Go lds tone boson b e h a v i o u r for e~(x) - -depend- ing on the g a u g e d t r a n s l a t i o n or b r o k e n GL~, R i n t e r p r e t a t i o n - -

(1.38) e~(~) ~, e y ) + ~ ( ~ ) - - C ~ ' ) + ~' .

F o r ~ t h e o r y w i th a t c t r a d c o m p o s e d b y f e r m i o n iic]ds F(x) (12)

(1.39) ]e~,(x) = dA(47c) 2 ~f7 ~ c~, y~(x)

t he (( c o n s t a n t ,) d A can be d e t e r m i n e d b y the Go lds tone field t r a n s f o r m a t i o n b e h a v i o u r (1.38) as follows.

T h e b i fe rmion ic p r o d u c t (1.39) with smal l s epa ra t i on ~ t r a n s f o r m s for iniin- i t c s imal (( t r ans l a t ions ~> as follows (~3):

(1.4o) le~(x, ~) = a~(t~) -~ ~(x_)7 ~ ~;(x+) = d~(4~)-'i ~ [~(x_)r;(x+)]~

~,(4~)~i ~ [ ~ ( x + t~(x_))r~V(x~,. + l~(x+))] =

T h e i n h o m o g e n e o u s t e r m in (] �9 6~" ~ -,Q~ ~(~,c ) can ar ise f r o m the c - n u m b e r eontr i -

(12) A. ]). SAKIIXttOV: ])okl. Akad. _h'auk SSS.R, 177, 70 (1967) (English translation: Soy. Phys. J E T P , 12, 1040 (t968)). (la) H. SALLER: 2VUOVO Cimenlo A, 71, 17 (1982).

672 H. SALLER

bution in the bifermionic p~cduct for ~--~0

(1A1) ]( ~;.eO) dA(4~)'i ~ [ ~a ~--"~ ~(X_)

= ]:(3,e~ ~) -I- J(OaeO)dA(4~)~i~ ~ - - ~2

~ , ~ o ~(x_) ~ 0 ( x + ) .

The second term on the t.h.s, yields the c-number contr ibution

(].42) dA(4n)2i~ ~

(1.43)

~xJ'e~L' <~(x-)TJ~~ = dAi~ ~ Tr ?i cxp [ - - ip~] =

= P~'PoPJ __ ~7~Jo~ ~ p2(2p~-3rn2) p' j ,

The inhomogeneous term for the Goldstone field, respectively~ t ranslat ion gauge field e~(x) is, therefore, reproduced with the condition

(1.44) da ~pO-(2p: - - 3m 2) 3 j - 1 ,

which for a cut-off regularization with (A ' / (A2- -p2) ) ~ gives the limit condition

(1.45) A 2 -+ oo , dA .A' -~ 3 .

I t is in the spirit of keeping GL4, n and SLy, c invarianee apar t tha t all the other inhomogeneous terms arising from (1.42) are shifted to higher approxima-

tions ('). The pair approximat ion for a composite tc t rad used in the Lagrangian

(1.34)~ i.e. with the interact ion (*')

( 1 . 4 6 )

(1.47)

= le~ O, + [e~.(O, - - m~o) + . . . ,

i.-*

l akes into account only the first te rm of (l.46) with respect to the GL~, n and

1") A convincing formulation of this procedure is not quite clear to me. 1"*) The full thcory is given by the GL4, R | SL2,C, lo ~ invariant Lagrangian .Lf(~p) _~

DYNAMICA$, IMPLEMENTATION OF TIlE GRAVITO.NS Y~TC. 673

SL2, c not mixing contributions. The homogeneous par t of the resulting t e t rad field equation

(1.48) (Y2[/e.(x)lh,~ '~\ ~- idA(4~p y (~F/'~ ~,'P)(x) wv, 5 ~,~ (y) <Y2IYe;'(x)',h} , I I

(1.4~) i~(p) = ~:,~(p) ~:'(p) ,

involves the integTaI

[ . . . . . . ] (1.50) ~" q+ - - m y q_ - - m I,,~(p) = - - d A ~qu T r 7 ~ 1 i 1

q

Only the contributions proport ional to ~zj are in accordance ~ i th the separa-

t ion of GL~, R and S/,~, c propert ies; one obtains

(~ .51) I,Z~(O) = ~ f ff.qa[2q~q ~ - n~J(q ~ - m 2)

q

_ _ . , ~l~.vZjq~(q.. m ~ ) _ 1 . ~ J i . 2 ~ -3-q,,~u J ~j d.i f q2(2q2 - - 3m -~)

q q

The equal regularization in the t ransformat ion condition (1.44) and the field equat ion (1.51) establish mass-zero gravitons as Nambu-Goldstonc bosons.

2. - D y n a m i c a l i m p l e m e n t a t i o n o f the P l a n c k m a s s as a phys ica l cut-off .

All considerations of the former section suffer J:rom the fact tha t the ul traviolet divergences are cured by ad hoe piocedures. Especially ~hc pre- scription to use the same regulariz,~tion in the (( gap equa t ions , (1.7), (1.19), (]..14) and in the corresponding pair approximations (].12), (].2.1), (].51) is not stringent.

We think tha t those technical cut-off procedures b~ve to be replaced by

formalisms which allow a physical in terpre ta t ion (*). In this sense wc think t ha t the theory of the gauge bosons and the gravitons is at the same t ime theory of the physical cut-off.

We sh~ll review now subsect. 1"1-1"3 with ~ physically incorporated cut-off.

(*) As a model we think always of the theory of superconductivity whcre the tech- nically applied cut-off at tile Dcbyc frequency for the four-fermion interaction can be physically substantiated in a formulation of the thcory with elcctron-phonon interac- tion (6).

674 II. SALLER

2"1. UI gauge boson (~'). - The composile gauge boson of canonical theo-

ries (1.]6) has not really intrinsic dimension I, it is too closely related to the

current with its dimension 3. In theories b~volving subcanonical fermion fields,

i.e. fields with dimension ~--n (n = 1, 2, ...), those difficulties can be avoided.

By means of a subcanonical fermion field ~(x) with dimension �89 a U~ gauge

boson A,(x) with dimension :l can be defined as a finite-part product (: :)(~5)

(2 . ] ) A , (x ) ---- a(.in) 2 :~7~ v~:(x).

The canonical linearized theory of a subcanonical field ~ is given by

+ (;~, + ~r + A~- , ,~ (~,

p, ~ in (2.2) can be considered to be the first- and second-order der iva t ive

of (~ with

(2.3) d i m ( ~ , ~ p , ~ ) - ~ + ( - - 1 , 0 , 1 ) - - - - ( ~ , ~ , :~).

We have included mass t e rms in such a w~ty as to yield the fermion prop,~-

ga tor ( ')

1 (2.4)

~f3(p) - - (p - - m)(A'~ _ p : ) "

For the UI.~o c t ransformat ion

(2.5) (~, ~, ~)(x) ~ exp [ - i ~ ( x ) ] ( ~ , ~,, ~))(x),

the composi te gauge field (2.1) t ransforms as

(2.6) A , ( x ) ~ A~,(x) + ~ , ~ ( x ) ,

if the cons tant a is chosen as

(2.7) a - � 8 9

(14) H. SAILER: Nuovo Cimenta A , 24, 391 (1974); 68, 324 (1982). (is) l[. P. DgraR and N. J. WIXT:-aR: Nuavo Cimento A , 70, 467 (1970). (') A 2 is a physically meaningful cut-off (subsect. 2"3). We assume m2/A 2 << 1.

D Y , N A M I C A L I M P L E M E . - ' q T A T I O N Old" T H E G R A V I T O N S E T C . 675

Zro divergences occur in establishing the condition (13,14) (2.6), (2.7). The pair approximation for the composite gauge boson with the four-fermion interac- t ion (of dimension 4)

(2.8) .Lfin t = - - J a A . ,

(2.9) J~, - - - (~PF~, r '-- ~+~ ~/ ' ~Pr~,F)

leads to the gauge boson equation

(2.Jo) [~1~ + A , , J , ( p ) ]A,'(p ) = - - A~ A~(p )J~(p ) .

The integral replacing (1.2-11) in canonical theories

(2.11) A~,J~(p) = - - a Q=4 Tr Y~, q + - - m a y ' q - - 1 real --] q

incorporates dynamically the fermionic structures at A'~;

3 (2.12) ~_~(p) = ~ iA :1 A = I p - - m , ~

1 A + o 1 A - - o lIA] A " - - m ~- ~ . - -m 2A ~o--A-- 2--A ) - ~

v 2 (2.13) ~ ~.4 == 0 , ~ ~., m~ = 0 , ~ ~am~ -- - - 1 . A A A

The gauge boson current relation involving tile gauge boson propagator

(2.~4)

(2.15)

- - A~,A~(p) ( ) Az(p) -- ~ , , . A z J , ( p ) J - - , ~I,,---P-pP~ (4~)~q~(P~')'~ " q ~ ) j (p)

q,(p~) = _ 1 2 p2 A ~ ((p2),,) 3 A T - m-; Jog ~., + o ~-i ,

1 (mO-p ') ~(p~) . . . . . . . : . 0 2 A 2 , ~.,., ~ ,

exhibits a mass-zero trans~:ersal gauge bosou and its coupling constant

(2.16) [ p~p~ 3(4z)~/4 log (A2/m 2) m 2

A~,(p) = ~ - - - p 2 - ) _ p~ J"(p) , A- ~ << 1 .

:h~o divergences occur in this ~pproximatioa (not even logarithmic ones).

6 7 6 m SALLIs

2"2. Chiral Goldstone boson and Higgs mechanism (~3,~,)._ [['he linear Lagran- gian of a subcanonical field ~p with only quadratic mass terms

~i i~_.~ _ i ,~ (2.17) .s __ _ V;~y 3 -- ~,~gV-- ~ o + (~o + ~r + ~ (~V + ~ )

is globally chiral invariant

I (~' r ~ e x p [-- i~](~, r (2.18) [ ~(x) ~-~ exp [ -t ias]yJ(x) �9

I t can bc made locally c, hiral invariant by coupling the composite ehiral gauge boson

(2.19) A~,(x) = a ( t ~ ) " ~ y . y ~ : ( x ) ~-+4 s + ~.a 5 a = ' /* ~ 2 .

The interaction induced by gauge inrariance

(2.2o) ~e,o, = - J~A~.,

tt

carries in itsel] the possibility o] a chiral condensation <~1 ~(x)} # 0, as shown by

(2.22) s ~t, J ~A = a(4n)'~(~yuy~ ~v)(~y, ys(v) + . . . -

= a ( ~ ) ' ( ~ ) ( ~ ) + . . . . m ( ~ ) + . . . .

The condensation (2.22) adds to the linearizcd theory the linear mass terms

(2.23) ~ . , ( ~ ) - - m(~q, + ~ ) .

The resulting propagator

] v ~ (2.21)

~v_~(p) = -(p " - m ) [ A ' - - p(p -- m)]

leads to the- -ye t logarithmically divergent--gap equation

f [ 1 ,] (2.25) m = a(ln)~<~pl~(x)>=--a Tr ( /~_m)(A 2 _ p ( p _ m ) "

The dynamical Yiiggs phenomenon in the pair approximation gives a massive

DY_N-AMICAL IMPLEMENTATION OF THE GRAVITONS FJI~C. 6 7 7

t ransversal gauge boson with the pole bchaviour

(2.26) As,As,.(p). = ~ - - - - P " + 6 m ~ - , A--~, " << 1 .

] [ere no divergences occur.

2"3. Cut-o i l as the P l a n c k mass (~3) (*). _ The minim~] subcanonical extcn- siort to establish a dimensionless t e t rad hJ,(x) uses a subcanonical theory with fermions ~ of dimension - - ~ :

(2.27) /e~.(x) = a(4~)-'" ~z~ ~ ~,, ~ " (x) .

The linear subcanonical theory

(2.28) ~eo(~) = ~ + ~ + ~ , ~ + ~ ~ g ? , ~ - -

+ +

dim (~, ~f, ..., ~) ---- ~ - i - ( - - 2 , - - ] , ..., 2) ---- (--�89 ..., =~) (2.29)

gives the leading small-distance belmviour

(2.30) ~ (~ ) ~ (x ) -+ i , (8~)~. 7 " ~ l o g ~ 2 , ~ --> 0 .

To obtain the correct gauge Goldstone-boson behaviour of the t e t r ad (2.27), one starts f rom the analogue to (1.42) in canonical theories (*')

(2.31) d(4n)~" 'i~" c~, ~ o (~(x_)7 ~ ~(xq )) ---- - - d~ ~ ~ (~J log ~) ---- - - 5 ~ "

The v~lue for d is given by

(2.32) d - - - - - -3 .

(') Some qualitative and quantitative errors in the paper (13) arc corrected in the following. (~ For the small-distance limit wc use always the space-time-averaged lilait

~-~o ~2 = 4~i1:' lira . . . . . . . 24~i~zm"

6 7 8 H. S~.LL~r~

In Che pair approximation for the te t rad with the interaction

(2.33) ~'al,~t = teuO j q- , I I /s " ' "

(2.34) 0~ -- ~y'2- ~,,W + ... + r 0u~,

one has for the GL,. R and SL:. c not mixing contributions the field equation

(2.35) [~],,,~t, § f~ zO~(p) ] ~ ( p ) = ~h,~Z ",,(p)O~,(p)'

with the integral

h~O,,(p) = ~Aq, q~, Tr 7 t -

q

1 1 l - - Y~ / �9

q+- -ma q_--m~ ]

Here the pole decomposition for the fermion field ~ has been used

5 1 (2.37) ~ ( v ) = ~ & - - -

A=I P -- ~'nA

(2.38)

~: ~(m~)" = o,

:E &(m~)' = 1. A

n = 0 , 1 , 2 , 3 ,

Expanding (2.36) around p -~ 0 and taking into account only the terms pro- portional to ~,j, one obtains (without divergences) the result (~3)

(2.39) " l ) 2 hz~Ou(p) = - - ~"[%, + (~1,.~ p - - p,, p,) ~_, 2 ~ m~ log m~ + . . . ] . A

:For the special case of a dit)ole cut-off

_ 1

(2.40) ~3(p) p ( p 2 _ A2) 2 ,

the massless gravitons obey the field equation

(2.41) 2 - ( 4 7 t ) 2

~'s(~1"uP~--P~P~')-~ "h~ -- A" Ot~(P) '

whith relates ;Newton's constant G (Planck mass) and the mass A 2, charac- teristic for the GL4, R breakdown

(2.42) gIG = 2A 2 .

DYNAMICAL IMPLEMENTATION OF TIFE GRAVITONS ETC. 679

3 . - Di latat ion b r e a k d o w n and finite gap equat ion.

3"1. Dilatation breakdown and indefinite metric. - The fermionic massless par t of the ~ a v i t y Lagrangian

(3.1) 1 v

C~f~r~.(~p) __-- __ 3.' ~ j ~ O ~ h~ h o ha yr/, ~ o~p

~ e f ! / \ shows in addit ion to the dilatation invariance of d*x ~ ) contained in the global GL4. R related to the co-ordinates

(3.2) x~ ~ x ' . = exp [~olX,', /~,(x) ~ c . ~ p [-;,o]~(x'), vJ(x) ~ f ( x ' )

also the Weyl invariance (~), which defines the weight w. Up to the gauge bosons and the t c t rad the weight coincides with the usual propagator- induced dimension. The weight t ransformations do not affect the co-ordinates and are given by

x , ~ x , , ~(x) ~ e x p [.~;.]~(x), h~(x) ~ e x p [--~]h'~(x),

(3.3) -~ w(h) = - - 1 w ( ~ ) = ,

%n the theory of subseet. 2"3 with noncanonieal fields (~, ..., ~) the weights a re given by

(a.4) w(~, . . . , ~) = ( - . L ... , ~i).

Already the (( free )) field J~agrangian of the subcanonical theory (2.28) can be nndcrstood as a linearization of a theory with a pure de te rminant interact ion

(3.5)

~f(~) = det (]~, ~t, h, 0) ,

0 " ~ 7 ~ ~ § ... + ~ ,

where the c-number values (]~} = 1, ( ~ } -- 1 have to be taken into account t o obtain the linearization (2.28). In such an approach the in terpreta t ion of ~o(~) (2.28) as a theory with fifth-order derivatives is only possible af ter the linearization.

We have the recurrent (~6) impression t h a t the fcrmion theory with the

(le) IL SALLEI~,: -~lto~o C i m e n t o A, 34, 99 (1976); 42, 189 (1977).

4,4 - 1l JVuovo Cfmento A.

6 8 0 H. SALLER

noneanon iea l set (~, . . . , ~) is a consequence of, not a precondition/or the dilata- tion breakdown ( ') , i l lus t ra ted in the fol lowing.

I f one s t a r t s f rom a basic L a g r a n g i a n of one dimen,.ionless ]ermion field 7.(x) wi th a ~, g lobal ,~ GL,, R i nva r i ance for the L a g r a n g i a n

(3.6) d ' x .LZb"(;r = d ' x , let J ~ 0u~ ,

a l inea r i za t ion ef fec ted b y a GL,,R b r e a k d o w n induces a d i l a t a t ion Go lds tone

field e(x) ~ o(x) - - 20

(3.7) ZY ~ c~ Z(x)- - exp [(~(x)] ~7 ' i .-. e~ ~(~)

a n d t h e redressed field

(3.8) ~(x) = exp [ - - ~ ] Z(x ) .

T h e weigh t can now be i n t e r p r e t e d as ref lect ing the power of t he d i l a t a t ion dress ing o p e r a t o r exp [0(x)] ( " ) .

I n the r e a r r a n g e d fo rm of t he L a g r a n g i a n (3.6)

C (3.9) ~b"(X) = cxp [4o] de t ~yJ ~ 0". ,

one can shif t the weight , s t a r t i n g f rom 6(x) , ove r 4 un i t s ( ' " ) - - f r o m - - ~ to

(3.10)

- ~ ~ ~ i

(~ With respect to dilatation invariance this seems to be an example for a suspected 9eneral proli]eration principle (1,17): A symmetry breakdown gives rise to a prolifera- tion for the linearized theory--where the proliferated set of fields is connected by con- densations involving the broken symmetry. (17) H. SALL~t~: NUOVO Cimenta A, 30, 541 {1975); II. P. DORR and H. S~mLm~: Phys. Left. l~, 84, 336 (1979); Phys. ]~ev. D, 22, 1176 (1980). ('~ The weight and the dimension would const i tutc--with respect to di la ta t ion-- a similar situation as hypcrehargc and 3rd component of isospin for internal sym- metries W). (**') With this interpretation the field ~(x) with dimension ~ would not play a spccial role-- in contrast to other approaches (xs). (is) H. SALLER: .u Cimento A, 42, 189 (1977); II. P. DORR: Phys. Zett. B, 115, 33 (1982); preprints IMP.PAE/PTh 28/82 and 38/82.

DY_NAMICAL 1MPLEMENTA'I'ION OF 'I'IIE GRAV1TONS ]ETC. ~ 1

with the dressed fermion fields

w exp[~,.,lz, ~ exp 7 = ---- [ ~ e ] Z ,

(3.11) ~f = exp [~o]Z ~ , ~ -- cxp [-~-o]Z ~ , c ---- conjugated .

Wi th the rearrangement of the dilatation properties (3.10) the Lagrangian for the extended set (~, ..., ~) (3.5) can be constructed.

The considerations above are presented only as a qualitative hint for a possible next level of fundamenta l i ty in which only one dimensionless fermion field (*) X is the origin of the proliferated set (~ , . . . , ~) which induces the indefinite metric in the linearization.

3"2. Incorporation o] gravity leading to ]inite gap equations. - I f we con- centrate on the dilatation aspect of gravi ty izi a theory of interacting non- c~nonic~l fermions (~f, . . . , ~) with local chiral invariance~ the relevant Lagran- gian is given by

Here ~ is a scalar field of weight --1 (tp) replacing the te t rad field h~

(3.13) ~(x): : do(4u) 2 ~y~ ~ ~,,~, w(~) = -- 1 ,

with some constant do. ~o(~, ~) is tim "Weyl-invariant kinetic fermion te rm

= v , ~ v + .. .-:- r ~,p-~(~ + ~ + ~7r + ~).

The Wey] invariance breakdown effected by

(3.15) < ~ > -- - i

connects the fields of the noncanonicM set ( ~ ...~ ~) in the lincarized Lagran- gian (2.28) b y derivatives

(3. ,6) ~o(~) = ~ z e o ( ~ , ~)i~-~ �9

(*) A Wcyl field is enough. (19) H. SALLER: -LVI~0~O Cimento A, 74, 159 (1983).

682

The chiral gauge boson A ~ has to be weightless: /s

(3.17) x~(~) = ~ , (x )~( . , )

The factor A t has weight 1 o i

(3.18) w(A~) = O,

.... ~ ( x ) X ~ ( x ) .

w(A~) = 1 .

H . S A L L E R

In a theory with fields (~, ..., ~) a composite gauge boson A~(x) can be con- s t ru t t ed by adding two contributions (*)

2 (x). (3.19) A~(x) = a [

After multiplication by the te t rad (or ~ (3.13)) the gauge boson arises as a 4-]ermion product.

The interaction in .~eo(~, ~) can give rise to a spontaneous dilatation break- dowa (~9), where we take as a characteristic example

(3.20) <~w + ~ > ~ A . .

Adding to .W~ (3.16) the quadratic mass term induced by the breakdown (3.20)

(3.21)

one obtains a fermion propagator with dipole regularization

1 (3.22)

L~(P) = p(p, - - A'-)'-"

The dilatation breakdown (3.20) in addition to the condition <~> = l (3.15) allows in the four-fermion gauge boson product the isolation of a two-fermion expression

(3.23)

= A~(x) -~- b(4~t)2A -0 ~y.y~ ~(x) + ... 0

with a well-determined numerical factor b. Therefore, the embedding of the

(*) This symmetric superposition (3.19) is not essential for the following qualitative considerations.

D Y N A M I C A L IMPLEMI. iNTATION OF T i l e ( iRAVITONS ETC. 6 8 3

i n t e r n a l gauge bosons in to g r a v i t y g e n e r a t e s a d d i t i o n a l (, so]t ,> contribution.~

(3..24) d i m A ~ = I d i m ~ 7 , 7 5 ~ - - - - 1 0 I z ~

T h e chira.1 c o n d e n s a t i o n in t h e i l l t e r ac t i o n w i th t hose <, sof t ,) t e r m s l eads to

m a s s t e r m s o b e y i n g ] iMte g a p eq.uati(ms

(3.'25) f , , , , - - - - 4 ~ d ~ = - - b ( 4 . ~ ) = . l ' ( } 7 , , 7 ~ ) ( ~ y , ' ~ , ~ , ) i - . . . =

(3.26) m = b(4 , - t )~ :12(~1 ~(.r);~ .

I f th(; c o n d ( m s a t e d mass t e r m (3.25) p r o d u c e s a d re s sed p r o p a g a t o r rc i ) lae ing

the, b a r e fo rm (3.22)

( 3 . 2 7 ) ~ , j , ( p ) - -

(p - - m)(p~ - - A=): '

t h e n t h e f in i te g a p e q u a t i o n (3.26) r e a d s

f m A a (a.e8) , . - - 4 b ( P ~ _ i , i g G , ~ - A ' - ' ) ~

r e l a t i n g th(., f e rmion mass m to t h e <, t ' l a n e k mass ~ zD.

The q u a l i t a t i v e c o n s i d e r a t i o n s of r ids subsec t io l l should i l l u s t r~ te how the

i n c o r p o r a t i o n of t he ( IL , , R Sl,l'Ue{,ui'c, e spec i a l l y i ts d i la ta . t ion part. , a lso for t h e

i n t e r n a l symnl ( , t r i es l eads to soft ( t ini te) p a r t s in t h e g a p e q u a l i o n s .

F r o m a quan t . i t a t ive po in t of view the gap e q u a t i o n (3.26) l acks a n u m e r i c a l

d e t e r m i n a t i o n of t he c o n s t a n t b a.nd ,~ st.ruetur'~l d e t e r m i n a t i o n of t h e fo rm

of t h e prol)a.gator ( ' ) . Those qnan t i t a . t i ve aq)ecl~s r e l a t e d to t he <, mass hier-

a r c h y l ) rob lem ,> wilt I)(, t r e a t e d in subscqu(mt p u b l i c a t i o n s .

v ~ ( ') A small constant b and a ~, soft .~ dipole ~,~.(p) L(P--~)(1 F ' - m2)(P~ "*

can produce a hu'ge mass ratio, b log (A~/m -') =

�9 R I A ~ q S 1 7 N T O (')

La relat.ivith generale si pus eousiderare una t.eoria con uua realizzazione non linearc della sin,metria (4LLR. In anah)gia al modello ehirale di Nambu-Joim-Lasinio, si mo, t ra eoIne i g, 'avitoni privi di zm~ssa eompaiano n('ll"~pprossimazione della coppia. Le diver-

(*) Traduzione a eura della l, 'edazio.e.

684 ~t. SaL~Z~

genze u l t rav io lc t te degli approeei canoniei si possono ev i ta re usando campi con dimen- sioni anomale. Qucsto approccio ~ suggeri to dal fa t to che le t e t rad i contengono anche i l campo di d i la taz ione di Goldstone che influenza le d imensioni dei campi impl ica t i . Si p rova c h e l a massa cara t te r i s t ica del la condensazione GLt, R 5 la massa di P lanck , ehe a sun vo l ta ~ anche legata al la conseguenza p i6 pericolosa del la s t ru t t u r a subcanonica, la metric.~ indef ini ta nello spazio degli s tat i . / ? i nco rpo ramen to d inamico della grav i t~ mos t ra una maniera per de terminare , median te le equazioni ad in tc rva l lo f ini te, i rap- por t i t r a le masse cara t te r i s t iche delle ro t tu re in terne (per esempio la chiral i t~ o SU2) c la massa di Planek.

~SHaMnqecKoe nocTpoense rpaaHTOHOa, RaK 60~OHOB HaM6y-FOa~lCTOyHa.

Pe3mMe (*). - - O6~tla~ Teopm~ OTHOCHTeJI'bHOCTH MO~eX pacCMaTpnBaTbCn, I~aK Teopr~t c HemtHefiHO~ peaJm3auHel~ CHM~erprm GL4,n. I Io aHaJ'IOFHH C Kltpa.nbHO~ MO~,eJlbIO HaM6y-lT"[oHa-fla3rn~rto, rIoKa3bmaeTcfl, KaK 6e3MaCCOB},le rpaB/dTOnbI BO3HHKalOT B IlapHoM rlpn6ml~enrrrL Yn~,rpadpHOneTOBble pacxo~aMOCTri Kanorm~ecKHX rto/Ixo~1oB MOryT 6blab ycTpaHeHbI, ec.na I.tCnO.rl.b3OBaTb rio.nil c attoMa.r~HbtM-~ pa3MepaocrsMH. Taro~t no~xxo~ i1pe~nonaraeT, ~t.TO TeTpa~bI co~ep~aT Tax,~e none I"O~CTOyHa, roTopoe H3MertseT pa3MepHocTR noneS. ~oKa3btBaeTc~, qTO xapaKTepaaa Macca ~J-fft GL~, R rorr~er~catmH rfpe,~crasnseT Maccy I-laaHKa, KOTOpa~l, B CBO~O o,~epe~,, cB~r'3aHa c rmH60nee or~ao~btM c.'aeacT~ae,ct cy6gaHoruaaecro~ crpy~"ryp~,l, ma,ae~a8~rHo~t MeTpngo~ B npoc'rpascTBe COCTO~nIH~, ,/~rlttaMHqecKoe BK.rI/Otleritte rpaBHTallltrl yKa3btBaeT nyTb, KaK or~pe~em~Tb OTHomeHHe xapagTeprmlX Mace ~aa BHyTpeHHHX napymerml~ g Macce I-lnaHKa.

(*) IlepeseOeno pec)ar4ue~t.