Dynamical implementation of the gravitons as Nambu-Goldstone bosons

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  • 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 Physik und Aslrophysik ~ Werner-l~eisenberg I,nstitut fiir Physik ~> - Mi~nchen, B.R.D.

    (ricevuto il 18 Novembre 1982)

    Summary . - - General relativity can be considered as a theory with a nonlinear realization of the symmetry GJL~,R. In analogy to the Nambu- Jona-Lasinio ehiral model, it is shown how the massless gravitons arise in the pair approximation. The ultraviolet divergences of the canonical approaches can be avoided by using fields with anomalous dimensions. Such an approach is suggested by the fact that the tetrads contain also the di latation Goldstone field which affects the dimensions of the fields in- volved. The characteristic mass for the GL4, R condensation proves to be the Planck mass, which in turn is also related to the most dangerous consequence of the subcanonicM structure, the indefinite metric in state space. The dynamical incorporation of gravity shows a way how to determine, by finite gap equations, the ratios of the characteristic masses for the internal breakdowns (e.g. ehiral ity 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 gravitational interactions.

    I n t roduct ion .

    I t has been suggested (1,~) and shown (") that genera l re la t iv i ty , as i t stands,

    al lows an in terpretat ion as a theory with a non l inear rea l i zat ion of the

    (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~: MPI -PAE/PTh 38/82.

    43 - II Nuovo Cimento A. 663

  • 664 It. SALLX~

    symmetl T G154, n. In this interpretation the tctrad fields, more exactly their 10 symmetric massless components found in the metrical tensor, contain the Goldstone fields related to GL4,R/SO~. ~.

    Modelled after the four-fermion theory of superconductivity (4) the chiral model of h-ambu and Jona-Lasinio (5) shows how the mass-zero I~-ambu- Goldstone modes, related to a nonlinear realization of (~( spontaneously broken ,~) ehiral symmetry, are implemented and can be computed in a pair approxi- mation. Deadsick in the ultraviolet region, those considerations nevertheless show how a universal cut-off procedure reproduces all the expected and desired results (subsect. 1"1).

    In this paper (subsect. 1"3) we implement dynamically, by using fermion fields d la NArrow and Jo~A-LAs1~1o, the gravitons as mass-zero Goldstone fields related to the GL4, n breakdown. Apart from 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 ultraviolet problems are then approached in a frontul attack. 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 that of an effec- t ive canonical Higgs field. A theory of a dynamical cut-off (*) is needed. By Mlowing anomalous dimensions (~) of the condensating fermion fields, sym- metrically distributed around the canonical value, a dynamical cut-off can be introduced leading in sect. 2 to ultraviolet better-defined versions of the models of sect. 1.

    If one looks ~t the

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

    I f the tetrads ~rc the phonons of p~rticle theory, their dynamical implemen- tation should show us also the way to obtain finite, meaningful gap equations which relate the characteristic mass of, e.g., the ehiral breakdown to the characteristic mass of the dilatation breakdown, which proves to be the Planek mass (subsect. 2"3). We show in subsect. 3"2 rather qualitatively how such finite gap eqnations really can occur. The quantitative ~( details ~) for such an approach to the ~( mass tfierarchy problem ~ will have to be studied in further 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 interacting 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+75 ) (1.3) r =: - - 2g~ (4~)'-' v~ ---2- ~ (x).

    In the ehiral-invariant purely 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~.LLER

    antisymmetrie),

    a = iyu y~ ~ i7~' 76 ,

    a chiral breakdown 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 me:0

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

    and for the limiting case

    (1.10) A ~-*c given by

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

  • 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. Its normalization, however, (i.e. its coupling to the fermions) vanishes for A2--> oo.

    1"2. U~ gauge boson (9). _ Disregarding all ultraviolet aspects, one may try 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~ ) 1A~

    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).

  • 668 ~. s ~ R

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

    (1.18)

    1 p2--2m2

    The required inhomogeneous term 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~ tion

    (1.20) A S -+ oo ~ aa "A ~ -+ .

    From the Lagrangian with a composite gauge boson (1.16)

    (1.21)

    the pair approximation for the gauge boson (**)

    (1.22) ( f21A~(x) It} : i f dy( 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)=aAfTr [~, t~q+- -~m~,~g_ Im] . 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 IMP leMENTATION OF TH~ GRAVITOlqS ~TC. ~

    ~or p -- 0 the integral in (]..24) connects the gauge invariance condition (1.19) with the mass-zero property of the gauge boson, provided the same regulariz~- tion is used:

    (1.25) f2q ~ - - 4m'-

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

    The integral A ,A~(p- 0) is--with equal regularization--givcn by aA(47~)2~7~,. Therefore, the gauge boson propagator relating 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, proportional to 1/log (A2/m ~) -+0.

    Iu 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. -The dynamics of a gravitating fermion field (,0) ~(x)

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

    uses the tetrad fields related to the metrical tensor by

    (1.2s)

    and the inverse tetrad (') 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 derivative containing six SLy. c gauge bosons (~o). The Lagrangian 5r (1.27) multiplied by d~x shows the (~global,~ (*)

    GZ,. R symmetry (invertible transformations 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 part of the Lagrangiaa (t.27) can be written as a deter-

    minant including the inverse tetrad 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),

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

    The remnant of the (~ global )~ G~5~.a symmetry in (1.3.1) is the (( local transla- tion invaria~ce ,~ (~o) up to order f. Wi th art inf initesimal GL,,n t rans format ion

    (1 35) % ~ ' ]~':

    and the x -dependent p~ramcter

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

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

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

    one obtains inv~riance of the l incarizat ion (1.34) up to order f under (1.31) o r - -wr i t ten in the local t rans lat ion fo rm- -

    I x" e- ,x '~'=x~,+]el~(x) , d'x F-~ d'x(] +]~Q~e) , (1.37) [ h~(x) ~--~h~(x')-- l(~et')h~(x), which induces the gauge boson or Goldstone boson behaviour for e~(x)--depend- ing on the gauged trans lat ion or broken GL~, R in terpretat ion - -

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

    For ~ theory with a tc t rad composed by fermion iic]ds F(x) (12)

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

    the (( constant ,) d A can be determined by the Goldstone field t rans format ion behav iour (1.38) as follows.

    The bi fermionic product (1.39) with small separat ion ~ t rans forms for iniin- i tcs imal (( t ranslat 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+))] =

    The inhomogeneous te rm in (] 9 6~" ~ -,Q~ ~(~,c ) can arise f rom the c -number eontri-

    (12) A. ]). SAKIIXttOV: ])okl. Akad. _h'auk SSS.R, 177, 70 (1967) (English translation: Soy. Phys. JETP , 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 contribution

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

    (1.43)

    ~xJ'e~L'

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

    SL2, c not mixing contributions. The homogeneous part of the resulting tetrad field equation

    (1.48) (Y2[/e.(x)lh,~ '~\ ~- idA(4~p y (~F/'~ ~,'P)(x) wv, 5 ~,~ (y)

  • 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 derivative of (~ with

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

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

    gator (')

    1 (2.4)

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

    For the UI.~o c transformation

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

    the composite gauge field (2.1) transforms as

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

    if the constant a is chosen as

    (2.7) a - 89

    (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

  • DY,NAMICAL IMPLEME. - 'qTAT ION Old" THE GRAVITONS ETC. 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- tion (of dimension 4)

    (2.8) .Lfin t =- - JaA . ,

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

    q,(p~) = _12 p2 A ~ ((p2),,) 3 A T - m-; Jog ~., + o ~-i ,

    1 (mO-p ') ~(p~) . . . . . . . : . 0 2A 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) = ~- - -p2- ) _ p~ J"(p) , A- ~

  • 676 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 ~exp [--i~](~, r (2.18) [ ~(x) ~-~ exp [ -t ias]yJ(x) 9 It 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

  • DY_N-AMICAL IMPLEMENTATION OF THE GRAVITONS FJI~C. 677

    transversal gauge boson with the pole bchaviour

    (2.26) As,As,.(p). = ~- - - -P" +6m~- , A--~, " 0.

    To obtain the correct gauge Goldstone-boson behaviour of the tetrad (2.27), one starts from 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"

  • 678 H. S~.LL~r~

    In Che pair approximation for the tetrad 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 (p2_ A2) 2 ,

    the massless gravitons obey the field equation

    (2.41) 2 - (47t ) 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 latation breakdown and finite gap equation.

    3"1. Dilatation breakdown and indefinite metric. - The fermionic massless part of the ~av i ty Lagrangian

    (3.1) 1 v

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

    ~ef! / \ shows in addition 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 tctrad the weight coincides with the usual propagator-induced dimension. The weight transformations do not affect the co-ordinates and are given by

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

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

    %n the theory of subseet. 2"3 with noncanonieal fields (~, ..., ~) the weights are 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 determinant interaction

    (3.5)

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

    0 "~7~ ~ ... + ~ ,

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

    We have the recurrent (~6) impression that the fcrmion theory with the

    (le) IL SALLEI~,: -~lto~o C imento A, 34, 99 (1976); 42, 189 (1977).

    4,4 - 1l JVuovo Cfmento A.

  • 680 H. SALLER

    noneanonieal set (~, ..., ~) is a consequence of, not a precondition/or the dilata- tion breakdown ('), i l lustrated in the following.

    I f one starts from a basic Lagrangian of one dimen,.ionless ]ermion field 7.(x) with a ~, global ,~ GL,, R invar iance for the Lagrangian

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

    a l inear izat ion effected by a GL,,R breakdown induces a di latat ion Goldstone field e(x) ~ o(x) - - 20

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

    and the redressed field

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

    The weight can now be interpreted as reflecting the power of the di latat ion dressing operator exp [0(x)] (") .

    In the rearranged form of the Lagrangian (3.6)

    C (3.9) ~b"(X) = cxp [4o] det ~yJ ~ 0". , one can shift the weight, s tart ing f rom 6(x), over 4 units ( ' " ) - - f rom - - ~ 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 constitutc--with respect to di latation-- 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.

    With 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 fundamental ity 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 gravity 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 tetrad field h~

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

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

    = 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) by derivatives

    (3. ,6) ~o(~) = ~zeo(~, ~)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. SALLER

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

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

    After multiplication by the tetrad (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) ~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.

  • DYNAMICAL IMPLEMI. iNTATION OF T i le ( iRAVITONS ETC. 683

    in terna l gauge bosons into grav i ty generates add i t iona l (, so]t ,> contribution.~

    (3..24) d imA ~ = I d im ~7,75~ - - - - 1 0 I z ~

    The chira.1 condensat ion in the i l l teract ion with those

  • 684 ~t. SaL~Z~

    genze ultraviolctte degli approeei canoniei si possono evitare usando campi con dimen- sioni anomale. Qucsto approccio ~ suggerito dal fatto che le tetradi contengono anche il campo di di latazione di Goldstone che influenza le dimensioni dei campi implicati . Si prova che la massa caratterist ica della condensazione GLt, R 5 la massa di P lanck, ehe a sun volta ~ anche legata alla conseguenza pi6 pericolosa della struttura subcanonica, la metric.~ indefinita nello spazio degli stati. /? incorporamento dinamico della gravit~ mostra una maniera per determinare, mediante le equazioni ad intcrval lo finite, i rap- port i tra le masse caratterist iche delle rotture interne (per esempio la chiralit~ 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.

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