Dynamical implementation of the gravitons as Nambu-Goldstone bosons

  • Published on
    23-Aug-2016

  • View
    215

  • Download
    2

Embed Size (px)

Transcript

<ul><li><p>IL NUOV0 CIMENTO VoL. 76A, N. 4 21 Agosto 1983 </p><p>Dynamical Implementation of the Gravitons as Nambu-Goldstone Bosons. </p><p>H. SALLEl~ </p><p>Max-I)la~wk-Institut ]i~r Physik und Aslrophysik ~ Werner-l~eisenberg I,nstitut fiir Physik ~&gt; - Mi~nchen, B.R.D. </p><p>(ricevuto il 18 Novembre 1982) </p><p>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. </p><p>PACS. 11.30. - Symmetry and conservation laws. PACS. 12.20. - Electromagnetic and unified gauge fields. PACS. 12.25. - Models for gravitational interactions. </p><p>I n t roduct ion . </p><p>I t has been suggested (1,~) and shown (") that genera l re la t iv i ty , as i t stands, </p><p>al lows an in terpretat ion as a theory with a non l inear rea l i zat ion of the </p><p>(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. </p><p>43 - II Nuovo Cimento A. 663 </p></li><li><p>664 It. SALLX~ </p><p>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~. ~. </p><p>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). </p><p>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. </p><p>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. </p><p>If one looks ~t the </p></li><li><p>DYNA~IICAI, IMPL:EM:EKTATION OF THE GRAVITONS ETC. ~5 </p><p>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. </p><p>l . - Dynamical implementation of Goldstone and gauge bosons in canonical theories. </p><p>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) </p><p>(1 .]) ~(x) ~-~ exp [--i75~51~(x ) , </p><p>in a globally chiral-invariant dynamics </p><p>(1.2) ~f:~'(~f, r --= ~y,' ff ~, y~ -~- '~1-- </p><p>is replaced by a fel'mion product </p><p>r ~-~exp [--2ias]r </p><p>+ ~ l~,r (r -~)" </p><p>1+75 ) (1.3) r =: - - 2g~ (4~)'-' v~ ---2- ~ (x). </p><p>In the ehiral-invariant purely fermionic Lagrangian (*) </p><p>i '~ gA(4~) 2 (1.4) ~o(~) = (pT"~ ~ ~- - - ( f~P~)( f~P~) </p><p>' 2 </p><p>(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. </p></li><li><p>666 </p><p>with the most general four-fermion coupling given by </p><p>i F* =: + V gaF @ g,d a) + gT~(v a) -- </p><p>= - -g~(s +p </p><p>s = 1 ~ 1 , p = iyo @ iy: , v --= y~, ~ ;,~', </p><p>(1.5) v q-2 a) +g] (v - -a ) (Fierz </p><p>l:I. 8~.LLER </p><p>antisymmetrie), </p><p>a = iyu y~ ~ i7~' 76 , </p><p>a chiral breakdown r 0 leads to a fermion mass m and a resulting gap equation for this mass </p><p>) ' ' (1.7) </p><p>m m . . . . 2g~ (4~) , (~ l~(x)&gt; = Sg -~ - - m'-'" </p><p>i f=~ f(l~pnJ p </p><p>The integration symbol </p><p>(1.8) </p><p>is assumed to include also necessary regularizations, which allow all the manip- ulations performed. </p><p>Regularizing in (1.7) with (A2/(A~--p'~)) "2, one obtains for me:0 </p><p>mz 10 m: ] (1.9) l=Sg~A: 1 -4--~ g -~ H- ... </p><p>and for the limiting case </p><p>(1.10) A ~-*c given by </p><p>(1.11) &lt; ~21(~i~ A tf(x)[O&gt; =~ i f dy </p></li><li><p>DYNAMICAL IIt~PLEM]~NTATION OF THE GRAV[TONS ~,TC. </p><p>or in self-evident momentum-space representation </p><p>667 </p><p>(1.12) </p><p>O(p) = ~(p) O(p) , </p><p>a </p><p>q:~ = q ~ p/2 . </p><p>For the same regularization ~sed in the gap and the Goldstone boson equations, (1.7) and (1.12), respectively, one obtains </p><p>(1.13) f l </p><p>I(p = O) ---- 8gA q2--m ~=1, </p><p>I(p) = 8g~(A ~ + cop ~ ...), A 2 </p><p>eo'~ log ~. </p><p>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--&gt; oo. </p><p>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): </p><p>(1.14) y(x) ~-~ exp [--iot(x)'Jy~(x), A~(x) ~.A~(x) + ~(x) </p><p>in a U~,~oo-invariant dynamics </p><p>(1.15) </p><p>~( i~ ) 1A~ </p><p>A~ = ~A, ~ ~A, , </p><p>by a fcrmion composite </p><p>(1.16) A~(x) ~-- aA(4~)2v~y~(x) . </p><p>The fermionic product with a small separation ~ transforming as </p><p>I aA(4~)~(x-)TJ'~(x+) ~-~ (1-- i~a)aA(4z)2(p(x_)?~p(x+), (1.17) [ x~ = x + 2/2, </p><p>(9) J .D. BJO~X]~N: Ann. Phys. (iV. Y.), 24, 174 (1963); H. SALL]~R: .YUOVO Cimento A, 67, 70 (1982). </p></li><li><p>668 ~. s ~ R </p><p>can produce an inhomogeneous (~,a)-dependent c-number contribution (*) </p><p>(1.18) </p><p>1 p2--2m2 </p><p>The required inhomogeneous term for a composite gauge boson is obtained with the condition </p><p>2a ~" p~ - - 2m~ (1.19) a j (~ m~ ~ = 1. </p><p>A cut-off regularization in (1.19) with (A~/(A2--p~)) 2 leads to the limit condi~ tion </p><p>(1.20) A S -+ oo ~ aa "A ~ -+ . </p><p>From the Lagrangian with a composite gauge boson (1.16) </p><p>(1.21) </p><p>the pair approximation for the gauge boson (**) </p><p>(1.22) ( f21A~(x) It} : i f dy( QIA~,(x) ~fint(Y)t~} + . . . . </p><p>= - - ifdy [A. J,(x - y)(f2]A,(y)]~} + A~,A, (x - - y)(QIJ,'(y)]~}] gives the field equation </p><p>(1.23) A~(p) ~--- - - A~,J~(p)A,(p) - - At, A,,(p)J~'(p) , ( J r J </p><p>(1.24) A~J~(p)=aAfTr [~, t~q+- -~m~,~g_ Im] . q </p><p>(*) 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) . </p><p>I I </p></li><li><p>:DYNAMICA.L IMP leMENTATION OF TH~ GRAVITOlqS ~TC. ~ </p><p>~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: </p><p>(1.25) f2q ~ - - 4m'- </p><p>- - A.LJ,(0 ) = - - ~7~,A~Jq(0) = ~.~a~3(q. W-m~)~ . . . . V"'" </p><p>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 </p><p>(1.26) - - A ~A~(p) </p><p>(i,i J'(p) </p><p>obtains a residue at the mass-zero pole, i.e. a coupling constant, proportional to 1/log (A2/m ~) -+0. </p><p>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. </p><p>1"3. The tetrad ]ields as GLd, R Nambu-Goldstone bosons. -The dynamics of a gravitating fermion field (,0) ~(x) </p><p>(1.27) r / i,-~ \ M'- R] </p><p>uses the tetrad fields related to the metrical tensor by </p><p>(1.2s) </p><p>and the inverse tetrad (') hi(x) </p><p>(1.29) </p><p>v j v j h (x) == h (x) = </p><p>1 ~ ~ ~,~ </p><p>1 ~ o~,,Q~,; ~ det h -- 4 ! ~k,~ o -~ ,o, ,oQ ,v. 9 </p><p>(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. </p></li><li><p>670 H. SALLER </p><p>R is essentially the scalar curvature (~0). M~ is related to Newton's constant G by </p><p>(1.30) .Mz_ ~z _ (2.1.10 ~ GeV) ~ </p><p>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,~ (*) </p><p>GZ,. R symmetry (invertible transformations only) </p><p>x~-+x'~--=a~x ", d~x~-~(deta)d~x, ~-~(a )~ , </p><p>(1.31.) h~'(x) ~-*a ~'~ h;(x') , itS(x) ~ (a-~ ~)~,h,( ~' x'), </p><p>~(x) ~ ~(x') , </p><p>and an obvious SZ2.C,~o ~ invariance. The fermionic part of the Lagrangiaa (t.27) can be written as a deter- </p><p>minant including the inverse tetrad o111y </p><p>(1.32) ,~t -- m~p </p><p>3! . ,mo ,o,,oo v}- u hl]. </p><p>The flat-space approach (11) for the linearization (") involves a breakdown </p><p>GI%R| SL,~,c ~ S03, ~ </p><p>(1.33) h~(x) = 5' --]e~(x), </p></li><li><p>D~AU~CAL ~wr~.~t~a'~Tro~ o~ 'rH~ O~WTO.~S ~TC. 671 </p><p>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 </p><p>(1 35) % ~ ' ]~': </p><p>and the x -dependent p~ramcter </p><p>(1.36) ~'(x)-- e~x ~ , ~ ~t'(x) - - s~', </p><p>-1 ,,____ 5" - - ]% , (a ),, ,, </p><p>~ ~qe~'(x) =- 0 , </p><p>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- - </p><p>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 - - </p><p>(1.38) e~(~) ~, ey)+ ~(~) - - C~' )+ ~'. </p><p>For ~ theory with a tc t rad composed by fermion iic]ds F(x) (12) </p><p>(1.39) ]e~,(x) = dA(47c) 2 ~f7 ~ c~, y~(x) </p><p>the (( constant ,) d A can be determined by the Goldstone field t rans format ion behav iour (1.38) as follows. </p><p>The bi fermionic product (1.39) with small separat ion ~ t rans forms for iniin- i tcs imal (( t ranslat ions ~&gt; as follows (~3): </p><p>(1.4o) le~(x, ~) = a~(t~) -~ ~(x_)7 ~ ~;(x+) = d~(4~)-'i ~ [~(x_)r;(x+)]~ </p><p>~,(4~)~i ~ [~(x + t~(x_))r~V(x~,. + l~(x+))] = </p><p>The inhomogeneous te rm in (] 9 6~" ~ -,Q~ ~(~,c ) can arise f rom the c -number eontri- </p><p>(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). </p></li><li><p>672 H. SALLER </p><p>bution in the bifermionic p~cduct for ~--~0 </p><p>(1A1) ]( ~;.eO) dA(4~)'i ~ [ ~a ~--"~ ~(X_) </p><p>= ]:(3,e~ ~) -I- J(OaeO)dA(4~)~i~ ~ - - ~2 </p><p>~, ~o ~(x_) ~0(x+) . </p><p>The second term on the t.h.s, yields the c-number contribution </p><p>(].42) dA(4n)2i~ ~ </p><p>(1.43) </p><p>~xJ'e~L' </p></li><li><p>DYNAMICA$, IMPLEMENTATION OF TIlE GRAVITO.NS Y~TC. 673 </p><p>SL2, c not mixing contributions. The homogeneous part of the resulting tetrad field equation </p><p>(1.48) (Y2[/e.(x)lh,~ '~\ ~- idA(4~p y (~F/'~ ~,'P)(x) wv, 5 ~,~ (y) </p></li><li><p>674 II. SALLER </p><p>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. </p><p>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) </p><p>(2.]) A, (x) ---- a(.in) 2 :~7~ v~:(x). </p><p>The canonical linearized theory of a subcanonical field ~ is given by </p><p>+ (;~, + ~r + A~-,,~ (~, </p><p>p, ~ in (2.2) can be considered to be the first- and second-order derivative of (~ with </p><p>(2.3) d im(~,~p,~)- ~+( - -1 ,0 ,1 ) - - - - (~ ,~, :~). </p><p>We have included mass terms in such a w~ty as to yield the fermion prop,~- </p><p>gator (') </p><p>1 (2.4) </p><p>~f3(p) - - (p - - m)(A'~ _ p:) " </p><p>For the UI.~o c transformation </p><p>(2.5) (~, ~, ~)(x) ~ exp [ - i~(x)] (~, ~,, ~))(x), </p><p>the composite gauge field (2.1) transforms as </p><p>(2.6) A, (x ) ~ A~,(x) + ~,~(x) , </p><p>if the constant a is chosen as </p><p>(2.7) a - 89 </p><p>(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 </p></li><li><p>DY,NAMICAL IMPLEME. - 'qTAT ION Old" THE GRAVITONS ETC. 675 </p><p>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) </p><p>(2.8) .Lfin t =- - JaA . , </p><p>(2.9) J~, - - - (~PF~, r '-- ~+~ ~/ ' ~Pr~,F) </p><p>leads to the gauge boson equation </p><p>(2.Jo) [~1~ + A , , J , (p ) ]A,'(p ) = - - A~ A~(p )J~(p ) . </p><p>The integral replacing (1.2-11) in canonical theories </p><p>(2.11) A~,J~(p) = - - a Q=4 Tr Y~, q+- -ma y 'q - -1 real --] q </p><p>incorporates dynamically the fermionic structures at A'~; </p><p>3 (2.12) ~_~(p) = ~ iA :1 A=I p - - m,~ </p><p>1 A+o 1 A - -o lIA] A"- -m ~- ~.--m 2A ~o--A-- 2--A ) -~ </p><p>v 2 (2.13) ~ ~.4 == 0, ~ ~., m~ = 0, ~ ~am~ -- -- 1. A A A </p><p>The gauge boson current relation involving tile gauge boson propagator </p><p>(2.~4) </p><p>(2.15) </p><p>- - A~,A~(p) ( ) Az(p) -- ~, , . Az J , (p ) J - - , ~I,,---P-pP~ (4~)~q~(P~')'~ " q~) j (p) </p><p>q,(p~) = _12 p2 A ~ ((p2),,) 3 A T - m-; Jog ~., + o ~-i , </p><p>1 (mO-p ') ~(p~) . . . . . . . : . 0 2A 2 , ~.,., ~ , </p><p>exhibits a mass-zero trans~:ersal gauge bosou and its coupling constant </p><p>(2.16) [ p~p~ 3(4z)~/4 log (A2/m 2) m 2 A~,(p) = ~- - -p2- ) _ p~ J"(p) , A- ~ </p></li><li><p>676 m SALLIs </p><p>2"2. Chiral Goldstone boson and Higgs mechanism (~3,~,)._ [['he linear Lagran- gian of a subcanonica...</p></li></ul>

Recommended

View more >