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Fermionic dimensions and Kaluza-Klein theory
Volume 202, number 3
PHYSICS LETTERS B
l0 March 1988
FERMIONIC D I M E N S I O N S A N D KALUZA-KLEIN THEORY
R. DELBOURGO and R.B. ZHANG Physics Department, Universityof Tasmania, Hobart, Tasmania, Australia 7005 Received 7 December 1987
Instead of appending extra bosonic dimensions to spacetime and needing to exorcise the higher modes, it is possible to construct Kaluza-Klein models in which the additional coordinates are fermionic and the higher modes do not arise. We erect a unified gravity/Yang-Mills theory on such a grassmannian framework and then discuss possible generalisations to other internal groups.
One of the major nuisances in ordinary Kaluza-Klein (KK) models is the persistence of higher modes associated with the extra bosonic dimensions [ 1 ]. The standard apology is to assume that the extra coordinates, corresponding to the compact(?) internal group, "curl up on themselves" into a very small size, thereby raising the excitations to Planck mass energies - although it must be admitted that the evidence and the mechanism for this contraction is not compelling [2]; in any event the higher modes always enter the quantised version and they do not ameliorate the renormalisability problem of current models [ 3 ]. For this and other reasons [ 4 ], KK models have, for the second time in their history, fallen out of favour and people have turned to string models for unification of forces. In this paper we wish to suggest that if the extra KK coordinates are instead taken as anticommuting, the difficulty with the higher modes essentially disappears and the KK construction may again become viable. (We will shortly present a unified gravity/Yang-Mills theory to substantiate this claim.) The inclusion of additional fermionic dimensions is anyhow a good way of visualising spin [ 5 ] and the trick can also be used to great effect in order to quantise gauge models in a BRST invariant manner [ 6 ]. In fact we are inclined to assert that the "existence" of extra Grassmann coordinates is more than just a computational device and should be taken quite seriously - it may well be the reason why we "cannot directly see" the extra dimensions. Given that superspace leads one to the concept of extra Fermi coordinates [ 7], it is rather surprising that, apart from implementing BRST-invariant quantisation, and of course pursuing the enlargement of multiplets to supermultiplets, nobody has (as far as we know) thought of appending these dimensions in the true spirit ot KK. To show that there are no obstacles to constructing a KK model with extra Grassmann dimensions, we shall append two anticommuting coordinates ~1 and ~2 to the commuting spacetime coordinates x ~'and with the same mass dimension. We need a minimum of two real ( to carry any sort of internal structure, even a U(1 ) group. Further ~ are necessary if the internal symmetry is more elaborate. With conjugation consistently defined as the operation [ 5] ( ~ " ) * = - - ~ m = ~ q .....
~/12=1 ,
the Sp(2) "rotation" operators J,,,,,=i( ~,,,010¢" +~,,0/0¢") , can be put into correspondence with SU(2) generators as follows, and the ~ then comprise a two-spinor representation: Jl2---~io'3,
296
Jll--~O'l-/-io'2 ,
J22---~0"1--io'2 .
Volume 202, number 3
PHYSICS LETTERS B
10 March 1988
A particular subgroup, which we would like to associate with electromagnetism, is generated by U (1) rotations ~exp[ieA~3]~, leaving ({)2= 2 ~ 2 invariant. Given these rules and the basic "raising and lowering" index rules, ~'"=~/'""~n,
q/'n~/,nn=S/n,
etc.,
we are in a position to propose a correct ansatz for combined gravity-electromagnetism. First of all one has to recognise that the supermetric will be bosonic in the Bose-Bose and Fermi-Fermi sectors, but grassmannian in the Bose-Fermi sector. One must also pay heed to the fact that one is dealing with a twofold integration measure which will force us to extract two powers of ~ in any lagrangian by the Berezin integration rules. (With more complicated internal groups higher powers of ~ will be needed.) And last but not least one does not wish to introduce at the classical level, ghost fields with the wrong spin statistics. These restrictions lead us to the metric ansatz "~
(g~,,(x) ( 1 -
X)
e 2~2/2x2) - e 2~2A 2
-e(A~)m
-e(/A~)n)
(1)
having the inverse
GMN(x)
= (g""(x)(1 + e2~2/2K 2) \
e(~A") m
e(~A"ln
~
(1 ')
qm"(l+½~2e2AZ)J"
In (1), A stands for the matrix-valued field iAa3, e is the unit of charge and ~c2 is proportional to the newtonian gravitational constant. Note the similarity with ordinary five-dimensional K K theory. In our case the differential dx 5 is, roughly speaking, replaced by ~ d~ which is a nilpotent to third order: this is the key to the removal of unwanted higher modes. It is helpful to write the line element in the form
ds 2=dx"g~,,dx"(1 -e2~2/2~c2) + [d~'" + ( e~A,)mdx "]r/,,,~[ d~" + ( e~A,)"dx"] . Then one can easily see that under a "phase transformation" of the kind ~" ~ ~n [ exp (ieAa3) ] ',; the field A must transform in the correct manner,
A. ~A. - O.A E~tA and its inverse (E.i4EAN= 5,~N), e(~A~)" 5,~ ) EeA~ '"=( l(+ e z ~ z / 4 x 20 ) ' e(IA~) m )
This also leads one to the sechsbein
E~A=( e"~(1-e2~2/4~2)" 0
,
(2)
as well as the connection
.~, = \ ¢ , /
¢,,/'
~ Our notation is to reserve upper case Latin letters for supercoordinates and indices: lower case Greek letters are for spacetime indices, while lower case Latin letters are for Grassmann indices. Early letters correspond to tangent space, later letters to world coordinates. The flat metric is taken to be
l,,~=(~ ~j~ 0 ).
297
Volume 202, number 3
PHYSICS LETTERS B
10 March 1988
which takes its values in the OSp (4/2) algebra and has the symmetry property (see footnote 1 ) ~ A B -= -- [ A B ] rI)BA
with
cl)AB-- IACq) cB .
It is possible to solve for the connection by ensuring that
DE A = d E m ..~EB~)BA=(_~a d~'"~m e2 0 ) , 2K2
¢c~----e,u°~dx ~ ,
transforms covariantly. After some work we discover that ( q ~ ( g ~ - (~d+d~)C(eK~F~P)c qSAe= \ -- e(F~f~)a dx"
-e(¢F~)bdx"~ edff'
,]'
where ¢(g) is the normal gravitational connection and we have used the differential geometric abbreviation ~4=A~dxJ', etc. All this is with the express aim of finding the curvature tensor RAB= d~.4e+
CI)AC(I)cB= l d x N
d x M ( R M N ) AB ,
with the components expressed in terms of the connection as follows:
(RMN)AB= OM(ON)AB- [MN] C)N(CI)M)A~+[NA] [NC] ( OM)AC( ¢I)g)CB- [MN] [MA ] [MC]( ON)AC( q)M) C~ • After some computation, we arrive at the scalar curvature ,X2e2"~
R= [B]EBNEAM[AN]IaC(RMN)C~=(1 + ~-y~z) R (g) -e2(~F~,a)a(Fa'~)~. Consequently, w h e n w e proceed to construct an action for the gauge fields, in the normal w a y ,
S= f d4xd2~ x/--GR/8e2 , w e inevitably finish up with the unified gravity/electromagnetic lagrangian
--/R
(g)
Furthermore, it is relatively easy to see how to incorporate matter felds within this formalism. Take a charged scalar field first. It must be described by a complexfermionic superfield,
q~(x) =~"~0~(x), o*(x) =¢a~*"(X) in order (a) to carry a U(1 ) representation and (b) to embody the correct ~ factor when performing the subsequent Grassmann integration. One straightforwardly establishes that the K K action yields the "right" answer 2e2- ~0t~o) • S~ = f d6 X,,/G GMNOMq)'oNq)= f d4x x/----g(gU~( O~+ ieA~)~o(O~--ieA,)~o* + --~7 Notice the presence of a mass term for ~0arising from the grassmannian part of the metric in combination with det(G). (Actually, it is possible to modify the value of the mass in the non-abelian generalisation by including additional factors of the type 1 - ~ 2 . ) A similar artifice works for the Dirac field. We define the complex Bose superfield ~ ( x ) =~"~,o(x) ,
298
Volume 202, number 3
PHYSICS LETTERS B
and identify the G r a s s m a n n g a m m a - m a t r i x F a with ( M ~ + O/M)ai75, introducing a mass factor M alised Dirac action then properly reduces to
10 March 1988 ~2.
The gener-
S v, = f d 6 X E t p E , 4 ~ t F A DM ~ = f d n x (det e){ ~7[ e~7~(D~, + ieA u) - iMp,5 ] ~u+ et ea"BF,~iy5 ~u/M}. Notice the occurrence o f a Pauli interaction, just as it appears in the conventional theory. This spells trouble for renormalisation, but that was a lost cause anyway for R-gravity. A m o r e interesting d e v e l o p m e n t is the generalisation to U ( n ) symmetries say. Here we extend the coordinate ~ to an n-fold spinor o f the internal group, n a m e l y (~,~3,~5,...) and similarly deal with the conjugate ~2. Take the S U ( 2 ) case for definiteness where we m u s t cope with a 2@2 spinor ~ " where the metric is
(° ° 1
L,,,,=
o 0
0
o
-1
0
4
The b o t t o m two c o m p o n e n t s transform conjugately to the u p p e r two c o m p o n e n t s and for the electromagnetic matrix A ' a 3 must be substituted the full S U ( 2 ) ® U ( 1 ) m a t r i x Auiri®a3: similarly for its curl, F. Here the coo r d i n a t e invariance o f the line element ~3 ds 2 = d x U g u . ( x ) dx"[ 1 -- 2 ( e 2 ~ 2 / K 2 ) 2 ] q- [d~ m + e(~A.) mdx"] r/.,. [d~ ~ + e(~Au)"dx~](1 - 2 e 2 ~ 2 / x 2) , under SU ( 2 ) rotations o f the fermionic coordinates ~ o e x p ( i ' e - A a 3 ) ~ = U~ i m m e d i a t e l y gives us the n o r m a l t r a n s f o r m a t i o n rule o f the gauge field,
A .U(A-iO/e)
U -1
Also the field F, entering the superconnection, d r o p s out quite simply:
F=dA+eA^A
.
The classical action too is acceptable and, after integrating over the ~, the lagrangian including fermion sources reads,
5 p / x / - g = R ~g)/16x2 - ~Fu,,. F~,,, + q~(e~"iT~V,- iM~ s + ea'~ Fo,a iy s/ m ) gt . U n d o u b t e d l y one can a p p e n d an extra two ( B R S T ) G r a s s m a n n variables in order to quantise the m o d e l a n d bring out the BRST s y m m e t r y most transparently, as has already been done for conventional K K models [ 8 ]. Thus it appears that we have a K K scheme which can a c c o m m o d a t e the gauge and m a t t e r fields without bringing in higher m o d e s associated with extra bosonic dimensions. U n i f i c a t i o n o f c h r o m o d y n a m i c s and gravity is i m m e d i a t e if the fermionic coordinates ( a n d their conjugates) form a six-dimensional entity and no heed is p a i d to the spacetime nature o f the fields, b e y o n d s u m m i n g a p p r o p r i a t e l y over the spinor indices. However if one is envisaging a unification o f gravity a n d electroweak theory then m o r e careful attention must be p a i d to the spin properties o f the ~. Success seems within reach if one uses two left-handed ~ plus one right-handed ~ ( b o t h d o u b l e d for spin); and if one assigns the m a t t e r fields to a supermultiplet
~J(X)=~Rr~I/rR(X)'~-~LI~I/IL(X) ,
r = 1, 2 ;
l = 1, 2, 3, 4 ,
~2The need for a single power of~" in F ~ is clear if the term F"3~ is to reproduce a mass term. The i}'5is needed to ensure {F~,F "} =0, and for the rest [F",F ;'] =2qaI'(2+2~O¢+~2MZ+O~/M2) reduces to the scaling operator when acting on wavefunctions that are linear in ~. However there is more flexibility in identifying F" than we have suggested and the last word on this has surely not been spoken. ~3 Note the additional factors of~ which enter the metric so as to guarantee that the Berezin integral will reduce the action to the right four-dimensional form. 299
Volume 202, number 3
PHYSICS LETTERS B
10 March 1988
a n d t h e Higgs b o s o n t o t h e s u p e r m u l t i p l e t
W e b e l i e v e t h a t t h i s n e w a p p r o a c h t o K K s h o w s c o n s i d e r a b l e p r o m i s e a n d we i n t e n d to t a c k l e s p e c i f i c m o d e l s in a future publication, where details of the scheme, which have inevitably been omitted in this compressed letter, will b e s p e l l e d out. This work was supported by an ARGS grant.
References [ 1 ] A. Salam and J. Strathdee, Ann. Phys. 141 (1982) 316. [2] T. Applequist and A. Chodos, Phys. Rev. D 28 (1983) 772; A. Chodos and E. Myers, Ann. Phys. 156 (1984) 412; M.A. Rubin and B.D. Roth, Nucl. Phys. 226 (1983) 444. [ 3 ] R. Delbourgo and R.O. Weber, Nuovo Cimento A 92 (1986) 347. [4] E. Witten, Shelter Island Conf. Proc. (MIT Press, Cambridge, 1985). [ 5 ] R. Delbourgo, Grassmann Wavefunctions and Intrinsic Spin, Int. J. Mod. Phys., to be published. [6] L. Bonora, P. Pasti and M. Tonin, Nuovo Cimento 64 A (1981) 307; R. Delbourgo and P.D. Jarvis, J. Phys. A 15 (1982) 611; L. Baulieu and J. Thierry-Mieg, Nucl. Phys. B 197 (1982) 477; F. Ore and P. van Nieuwenhuizen, Nucl. Phys. B 204 ( 1982 ) 317; R. Delbourgo, P. Jarvis and G. Thompson, Phys. Rev. D 26 (1982) 775. [ 7 ] A. Salam and J. Strathdee, Nucl. Phys. B 76 (1974) 477; S. Ferrara, J. Wess and B. Zumino, Phys. Lett. B 51 (1974) 239. [8] R. Delbourgo, P. Jarvis and G. Thompson, J. Phys. A 15 (1982) 2813; Y. Ohkuwa, Phys. Lett. B 114 (1982) 315.
300
PHYSICS LETTERS B
l0 March 1988
FERMIONIC D I M E N S I O N S A N D KALUZA-KLEIN THEORY
R. DELBOURGO and R.B. ZHANG Physics Department, Universityof Tasmania, Hobart, Tasmania, Australia 7005 Received 7 December 1987
Instead of appending extra bosonic dimensions to spacetime and needing to exorcise the higher modes, it is possible to construct Kaluza-Klein models in which the additional coordinates are fermionic and the higher modes do not arise. We erect a unified gravity/Yang-Mills theory on such a grassmannian framework and then discuss possible generalisations to other internal groups.
One of the major nuisances in ordinary Kaluza-Klein (KK) models is the persistence of higher modes associated with the extra bosonic dimensions [ 1 ]. The standard apology is to assume that the extra coordinates, corresponding to the compact(?) internal group, "curl up on themselves" into a very small size, thereby raising the excitations to Planck mass energies - although it must be admitted that the evidence and the mechanism for this contraction is not compelling [2]; in any event the higher modes always enter the quantised version and they do not ameliorate the renormalisability problem of current models [ 3 ]. For this and other reasons [ 4 ], KK models have, for the second time in their history, fallen out of favour and people have turned to string models for unification of forces. In this paper we wish to suggest that if the extra KK coordinates are instead taken as anticommuting, the difficulty with the higher modes essentially disappears and the KK construction may again become viable. (We will shortly present a unified gravity/Yang-Mills theory to substantiate this claim.) The inclusion of additional fermionic dimensions is anyhow a good way of visualising spin [ 5 ] and the trick can also be used to great effect in order to quantise gauge models in a BRST invariant manner [ 6 ]. In fact we are inclined to assert that the "existence" of extra Grassmann coordinates is more than just a computational device and should be taken quite seriously - it may well be the reason why we "cannot directly see" the extra dimensions. Given that superspace leads one to the concept of extra Fermi coordinates [ 7], it is rather surprising that, apart from implementing BRST-invariant quantisation, and of course pursuing the enlargement of multiplets to supermultiplets, nobody has (as far as we know) thought of appending these dimensions in the true spirit ot KK. To show that there are no obstacles to constructing a KK model with extra Grassmann dimensions, we shall append two anticommuting coordinates ~1 and ~2 to the commuting spacetime coordinates x ~'and with the same mass dimension. We need a minimum of two real ( to carry any sort of internal structure, even a U(1 ) group. Further ~ are necessary if the internal symmetry is more elaborate. With conjugation consistently defined as the operation [ 5] ( ~ " ) * = - - ~ m = ~ q .....
~/12=1 ,
the Sp(2) "rotation" operators J,,,,,=i( ~,,,010¢" +~,,0/0¢") , can be put into correspondence with SU(2) generators as follows, and the ~ then comprise a two-spinor representation: Jl2---~io'3,
296
Jll--~O'l-/-io'2 ,
J22---~0"1--io'2 .
Volume 202, number 3
PHYSICS LETTERS B
10 March 1988
A particular subgroup, which we would like to associate with electromagnetism, is generated by U (1) rotations ~exp[ieA~3]~, leaving ({)2= 2 ~ 2 invariant. Given these rules and the basic "raising and lowering" index rules, ~'"=~/'""~n,
q/'n~/,nn=S/n,
etc.,
we are in a position to propose a correct ansatz for combined gravity-electromagnetism. First of all one has to recognise that the supermetric will be bosonic in the Bose-Bose and Fermi-Fermi sectors, but grassmannian in the Bose-Fermi sector. One must also pay heed to the fact that one is dealing with a twofold integration measure which will force us to extract two powers of ~ in any lagrangian by the Berezin integration rules. (With more complicated internal groups higher powers of ~ will be needed.) And last but not least one does not wish to introduce at the classical level, ghost fields with the wrong spin statistics. These restrictions lead us to the metric ansatz "~
(g~,,(x) ( 1 -
X)
e 2~2/2x2) - e 2~2A 2
-e(A~)m
-e(/A~)n)
(1)
having the inverse
GMN(x)
= (g""(x)(1 + e2~2/2K 2) \
e(~A") m
e(~A"ln
~
(1 ')
qm"(l+½~2e2AZ)J"
In (1), A stands for the matrix-valued field iAa3, e is the unit of charge and ~c2 is proportional to the newtonian gravitational constant. Note the similarity with ordinary five-dimensional K K theory. In our case the differential dx 5 is, roughly speaking, replaced by ~ d~ which is a nilpotent to third order: this is the key to the removal of unwanted higher modes. It is helpful to write the line element in the form
ds 2=dx"g~,,dx"(1 -e2~2/2~c2) + [d~'" + ( e~A,)mdx "]r/,,,~[ d~" + ( e~A,)"dx"] . Then one can easily see that under a "phase transformation" of the kind ~" ~ ~n [ exp (ieAa3) ] ',; the field A must transform in the correct manner,
A. ~A. - O.A E~tA and its inverse (E.i4EAN= 5,~N), e(~A~)" 5,~ ) EeA~ '"=( l(+ e z ~ z / 4 x 20 ) ' e(IA~) m )
This also leads one to the sechsbein
E~A=( e"~(1-e2~2/4~2)" 0
,
(2)
as well as the connection
.~, = \ ¢ , /
¢,,/'
~ Our notation is to reserve upper case Latin letters for supercoordinates and indices: lower case Greek letters are for spacetime indices, while lower case Latin letters are for Grassmann indices. Early letters correspond to tangent space, later letters to world coordinates. The flat metric is taken to be
l,,~=(~ ~j~ 0 ).
297
Volume 202, number 3
PHYSICS LETTERS B
10 March 1988
which takes its values in the OSp (4/2) algebra and has the symmetry property (see footnote 1 ) ~ A B -= -- [ A B ] rI)BA
with
cl)AB-- IACq) cB .
It is possible to solve for the connection by ensuring that
DE A = d E m ..~EB~)BA=(_~a d~'"~m e2 0 ) , 2K2
¢c~----e,u°~dx ~ ,
transforms covariantly. After some work we discover that ( q ~ ( g ~ - (~d+d~)C(eK~F~P)c qSAe= \ -- e(F~f~)a dx"
-e(¢F~)bdx"~ edff'
,]'
where ¢(g) is the normal gravitational connection and we have used the differential geometric abbreviation ~4=A~dxJ', etc. All this is with the express aim of finding the curvature tensor RAB= d~.4e+
CI)AC(I)cB= l d x N
d x M ( R M N ) AB ,
with the components expressed in terms of the connection as follows:
(RMN)AB= OM(ON)AB- [MN] C)N(CI)M)A~+[NA] [NC] ( OM)AC( ¢I)g)CB- [MN] [MA ] [MC]( ON)AC( q)M) C~ • After some computation, we arrive at the scalar curvature ,X2e2"~
R= [B]EBNEAM[AN]IaC(RMN)C~=(1 + ~-y~z) R (g) -e2(~F~,a)a(Fa'~)~. Consequently, w h e n w e proceed to construct an action for the gauge fields, in the normal w a y ,
S= f d4xd2~ x/--GR/8e2 , w e inevitably finish up with the unified gravity/electromagnetic lagrangian
--/R
(g)
Furthermore, it is relatively easy to see how to incorporate matter felds within this formalism. Take a charged scalar field first. It must be described by a complexfermionic superfield,
q~(x) =~"~0~(x), o*(x) =¢a~*"(X) in order (a) to carry a U(1 ) representation and (b) to embody the correct ~ factor when performing the subsequent Grassmann integration. One straightforwardly establishes that the K K action yields the "right" answer 2e2- ~0t~o) • S~ = f d6 X,,/G GMNOMq)'oNq)= f d4x x/----g(gU~( O~+ ieA~)~o(O~--ieA,)~o* + --~7 Notice the presence of a mass term for ~0arising from the grassmannian part of the metric in combination with det(G). (Actually, it is possible to modify the value of the mass in the non-abelian generalisation by including additional factors of the type 1 - ~ 2 . ) A similar artifice works for the Dirac field. We define the complex Bose superfield ~ ( x ) =~"~,o(x) ,
298
Volume 202, number 3
PHYSICS LETTERS B
and identify the G r a s s m a n n g a m m a - m a t r i x F a with ( M ~ + O/M)ai75, introducing a mass factor M alised Dirac action then properly reduces to
10 March 1988 ~2.
The gener-
S v, = f d 6 X E t p E , 4 ~ t F A DM ~ = f d n x (det e){ ~7[ e~7~(D~, + ieA u) - iMp,5 ] ~u+ et ea"BF,~iy5 ~u/M}. Notice the occurrence o f a Pauli interaction, just as it appears in the conventional theory. This spells trouble for renormalisation, but that was a lost cause anyway for R-gravity. A m o r e interesting d e v e l o p m e n t is the generalisation to U ( n ) symmetries say. Here we extend the coordinate ~ to an n-fold spinor o f the internal group, n a m e l y (~,~3,~5,...) and similarly deal with the conjugate ~2. Take the S U ( 2 ) case for definiteness where we m u s t cope with a 2@2 spinor ~ " where the metric is
(° ° 1
L,,,,=
o 0
0
o
-1
0
4
The b o t t o m two c o m p o n e n t s transform conjugately to the u p p e r two c o m p o n e n t s and for the electromagnetic matrix A ' a 3 must be substituted the full S U ( 2 ) ® U ( 1 ) m a t r i x Auiri®a3: similarly for its curl, F. Here the coo r d i n a t e invariance o f the line element ~3 ds 2 = d x U g u . ( x ) dx"[ 1 -- 2 ( e 2 ~ 2 / K 2 ) 2 ] q- [d~ m + e(~A.) mdx"] r/.,. [d~ ~ + e(~Au)"dx~](1 - 2 e 2 ~ 2 / x 2) , under SU ( 2 ) rotations o f the fermionic coordinates ~ o e x p ( i ' e - A a 3 ) ~ = U~ i m m e d i a t e l y gives us the n o r m a l t r a n s f o r m a t i o n rule o f the gauge field,
A .U(A-iO/e)
U -1
Also the field F, entering the superconnection, d r o p s out quite simply:
F=dA+eA^A
.
The classical action too is acceptable and, after integrating over the ~, the lagrangian including fermion sources reads,
5 p / x / - g = R ~g)/16x2 - ~Fu,,. F~,,, + q~(e~"iT~V,- iM~ s + ea'~ Fo,a iy s/ m ) gt . U n d o u b t e d l y one can a p p e n d an extra two ( B R S T ) G r a s s m a n n variables in order to quantise the m o d e l a n d bring out the BRST s y m m e t r y most transparently, as has already been done for conventional K K models [ 8 ]. Thus it appears that we have a K K scheme which can a c c o m m o d a t e the gauge and m a t t e r fields without bringing in higher m o d e s associated with extra bosonic dimensions. U n i f i c a t i o n o f c h r o m o d y n a m i c s and gravity is i m m e d i a t e if the fermionic coordinates ( a n d their conjugates) form a six-dimensional entity and no heed is p a i d to the spacetime nature o f the fields, b e y o n d s u m m i n g a p p r o p r i a t e l y over the spinor indices. However if one is envisaging a unification o f gravity a n d electroweak theory then m o r e careful attention must be p a i d to the spin properties o f the ~. Success seems within reach if one uses two left-handed ~ plus one right-handed ~ ( b o t h d o u b l e d for spin); and if one assigns the m a t t e r fields to a supermultiplet
~J(X)=~Rr~I/rR(X)'~-~LI~I/IL(X) ,
r = 1, 2 ;
l = 1, 2, 3, 4 ,
~2The need for a single power of~" in F ~ is clear if the term F"3~ is to reproduce a mass term. The i}'5is needed to ensure {F~,F "} =0, and for the rest [F",F ;'] =2qaI'(2+2~O¢+~2MZ+O~/M2) reduces to the scaling operator when acting on wavefunctions that are linear in ~. However there is more flexibility in identifying F" than we have suggested and the last word on this has surely not been spoken. ~3 Note the additional factors of~ which enter the metric so as to guarantee that the Berezin integral will reduce the action to the right four-dimensional form. 299
Volume 202, number 3
PHYSICS LETTERS B
10 March 1988
a n d t h e Higgs b o s o n t o t h e s u p e r m u l t i p l e t
W e b e l i e v e t h a t t h i s n e w a p p r o a c h t o K K s h o w s c o n s i d e r a b l e p r o m i s e a n d we i n t e n d to t a c k l e s p e c i f i c m o d e l s in a future publication, where details of the scheme, which have inevitably been omitted in this compressed letter, will b e s p e l l e d out. This work was supported by an ARGS grant.
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