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Unified theories for quarks and leptons based on Clifford algebras
Volume 90B, number 1,2
PHYSICS LETTERS
11 February 1980
UNIFIED THEORIES FOR Q U A R K S A N D LEPTONS B A S E D ON CLIFFORD A L G E B R A S ~
R. CASALBUONI 1 CERN, Geneva, Switzerland and R. GATTO Ddpartement de Physique Thdorique, Universitd de Gendve, 1211 Geneva 4, Switzerland Received 27 November 1979
The general standpoint is presented that unified theories arise from gauging of Clifford algebras describing the internal degrees of freedom (charge, color, generation, spin) of the fundamental fermions. The general formalism is presented and the ensuing theories for color and charge (with extension to N colors), and for generations, are discussed. The possibility of further including the spin is discussed, also in connection with generations.
In this paper we shall extend the unified description of quarks and leptons that we have already proposed [1]. We shall first present the general formalism and discuss in detail its application to the description of the electric charge and the three colors as Clifford degrees of freedom. The case of N colors will also be studied. We shall then discuss the extension to fermion generations and examine the possibility of including the spin degree of freedom in the overall Clifford algebra. This last extension suggests unification with gravitation. General formalism. Our general standpoint is to describe quantum operators having a finite spectrum (charge, color, generation, spin) in terms of Clifford algebras [2]. Quarks and leptons are then thought of [ 1] as different states of a unique physical object with a dependence on a set of Clifford variables {~A } satisfying
[~, ~] + = 28A8.
(1)
If the Clifford algebra of eq. (1)is of even order 2N,
its automorphism group is O(2N), with generators
SA8 = ~-i [~A,
~81- •
(2)
If the Clifford algebra is of order 2N + 1 it can always be realized nonlinearly in terms of the Clifford algebra of order 2N, C2N , by expressing one of the generators ~i in terms of the others; for instance ~2N+l = iN~l ~2 "'" ~2N"
(3)
Therefore, in any case we shall be led to orthogonal groups of even dimensions, O (2N). More specifically we are led to classify the fundamental fermions according to spinor representations of such an O(2N), since these are indeed the representations of the Clifford algebras. Corresponding to O(2N), the maximal unitary subgroup U(N) (which also has rank N ) plays a crucial role, as can be seen by rewriting eq. (1) in the complex basis (i = 1,2 ..... N ) _1 di - ~ [~2i-1 - i~2i1 , (4) d/? _ 1 - 5 [ ~ 2 i - 1 + i~2i] ,
giving the algeb~ra of N Fermi oscillators, Supported in part by the Swiss National Science Foundation. 1 On leave of absence from, and address after January 1980: INFN, Sezione di Firenze, 50125 Florence, Italy.
[di, dil + = [ati, 4 1 + = O,
. [d i, d 7 ] + = ~ii,
(5)
for which U(N) is the automorphism group. These operators generate the Fock space: 81
PHYSICS LETTERS
Volume 90B, number 1,2 10),
d~,t0),
..., d~
dtNI0}.
The algebra (5) admits the discrete automorphisms (charge conjugations) d i -+ dti, which are generated by C i = d i + d}. In particular, one can construct an overall charge conjugation given by
C = C 1 ... CN .
(6)
The Fermi algebra (5) can be represented on analytic functions of a set o f N Grassmann variables {Oi}* a
dixlt(O )
=
Oixlt(O),
d~P(O)
=
(7)
The O(2N) generators of eq. (2) are:
for U(1):
N -1
Ok O0k , a
for SU(N):
0 i g a - N - l Sil
for the c o s e t s * 2 0 ( 2 N ) / U ( N ) :
~
(8)
Ok aOak ,
(8')
OiO1, ~0ai 00ai
.(8")
The function xP(0) can be split into an even and an odd part, * + (0), ~I'_ (0), with respect to the Grassmann algebra, both parts transforming irreducibly under O(2N). xI,_+(0) are "superfields" with respect to the U ( N ) basis, for instance: G _ ( x ; 0) = * i ( x ) O i + ~i/k(X)OiO/Ok + ....
(9)
The expansion (9) terminates with the monomial o f degree N or N - 1 according to whether N is odd or even. The monomial o f degree k (k odd) transforms as the totally antisymmetric irreducible representation of SU(N) o f dimensionality ( N ) , a . The generators (8) act as raising and lowering operators, respectively, within such sequences o f representations. ,1 In a classical theory the Grassmann coordinates 0 i themselves play the role of dynamical variables [2, 3]. Gauge fields can be introduced, belonging to the Lie algebra of the group of the canonical transformations, leading to the full set of equations of motion for the Grassmann variables and the gauge fields. ,2 The symmetric riemannian space O (2N)/U (N) reduces to the two-dimensional sphere S2 forN = 2, and to CP3 for N = 3 [41. 82
A "chiral" superfield, like G (0) in eq. (9), can be quantized consistently, with commutators or anticommutators according to the case, without running into contradictions, because it has a unique parity in 0 (definite grading). Finally, a local gauge theory is obtained by considering the generators of the Clifford algebra as "moving frames" on s p a c e - t i m e [6, p. 381]. The related connection takes values in the Lie algebra o f the automorphism group of the Clifford algebra (SO(2N) for the real basis, eq. (1), and U ( N ) for the complex basis, eq. (5) , 4 ) .
axp(o)/ao i ,
[0 i, a/ao i] + = 8ij .
~k
11 February 1980
The states o f the first generation. T h e simplest model based on such a description was presented in ref. [1 ]. In correspondence to the only exact symmetries, color and charge, one introduces four Fermi oscillators, a i, i = 1, 2, 3 (for red, white and blue) and b for the electric charge, satisfying the algebra (5). We assign charge 1/3 to each al? and - 1 to b t ; and we assign B - L = 2/3 to each a/* and 0 to b t . The complete Fock space is (for a "ground state" with Q = 0 and B- L =-1): I0) -- (re) L ,
b # IO) - ( e - ) L ,
a~ IO) ------(di) L ,
btati IO) -- (fii)L '
1 t t 5ei]kajaklO)=(Ui)L,
1~ b t ~ t a t m \ 5Wk ~] k W ,
a t1a t2a t3 0 ) - (e+)L ,
b t a t1a t2a t310)~(ff)L . (10)
(di)L,
,a The maximal monomial in the Oi's, 01 ... ON, transforms as a singlet under SU(N) and belongs to the eigenvalue 1 of the U (1) generator defined in eq.t(8). ,~orrespondingly one can construct the product c-~= dt l ... dj)h which is a component of the tensor representation 5 (#iv) of O (2N).C~ commutes with the generators of SU (N) defined in eq. (8') but breaks U(1) of eq. (8). ThereforeQ'~ breaks O (2N) to SU(N). This term is responsible for the Majorana mass [5]. ,4 We notice that O(2N) is the group of the inner automorphisms of C2N. This group admits a Z2-graded extension in the following way: C2N possesses exterior automorphisms of the type ~A ~ ~A + ell where [eA, ~A]+ = leA', eB]+ = 0. The inf'uiitesimal generators are aA ~A~2N+ 1 + J3A~A + #=N+I ~2N+I (etA are c-numbers and ~A, ~ are Grassmann variables~. This structure defines a Z2-graded Lie algebra by putting g i[~A, ~B] , ~A~=N+I in the even sector and ~A and ~2N+l in the odd sector. We notice that in this graded Lie algebra the generators of the Clifford algebra itself appear. It is worth noticing that if we do not grade the algebra, then the previous generators give rise, under commutation, to the Lie algebra of O (2N +2 ).
11 February 1980
PHYSICS LETTERS
Volume 90B, number 1,2
The states in eq. (10) transform as 8 • 8 under the automorphism group 0 ( 8 ) of our Clifford algebra C 8. They transform [1 ] as an irreducible 16 under the group O(10) [7] obtained by a nonlinear extension of 0(8). Such an O(10) can be regarded as the automorphism group of a Clifford algebra C10 obtained from C 8 by a linearization procedure which is analogous to the introduction of homogeneous variables in describing projective spaces [8]. By using the C10 basis, the states (10) are described by a single O(10) superfield ~ _ (x; 01 . . . . . 05) (c~ = 1,2 is a Weyl index), each component transforming irreducibly under SU(5) [9] (5 • 1_.9_0"• 1_).
Extension to N colors. Recent work has shown the usefulness of studying QCD theories [10] for a number of colors N 4: 3. Within our framework we only have to replace the set of Fermi oscillators a 1 , a 2, a 3 with the set al, a2, ..., aN , s . When we demand that the SU(N) singlets appearing in the spectrum (see table 1) be identified with the leptons (ve, e ± , ~), we find that the electric charge and B - L must be assigned as follows: to a~ charge 1IN and B - L = 2IN; to bt charge - 1 and B - L = 0. Again, the "ground state" has charge zero and B - L = - 1. It is clear that N has to be odd. In fact for any N we can identify the weak isospin generator T - as
state, uiuJfiij, the total charge of one is obtained as 2(N - 1)/N- ( N - 2)/N, whereas for the second state it is obtained as ( N - 1)IN- I/N+ 2IN. Similarly; for the neutron one has the two states: diu/fiij, didl~dij, Again, one has to take N odd in order that the proton and neutron be doublets of weak isospin. In the discussion of the large-N limit [ 10,13 ], the possibility of representing baryons out of three, rather than N, fundamental fermions is definitely advantageous. For the smallest value of N, N = 1, the spectrum contains only the leptons. Correspondingly, one can set up an 0 ( 6 ) gauge theory describing only leptons (belonging to the representation 4, which decomposes as 3 • 1 under SU(3)). The unrenormalized Weinberg angle is given in this purely leptonic model by sin20w = 1/4. For N = 3, as we know, one has the familiar spectrum of quarks and leptons of the first generation with corresponding O(10). We show in table 1 the spectrum for general odd N ; correspondingly one can set up an O(2N + 4) gauge theory. For such a general case one can work out the value of the Weinberg angle as a function of N. We obtain
sin20w 1
Tr(T2) Tr(Q2)
1 (N-l)/2 4 k=0
N)~kk~=0
k 2 N
N
21+N
N
1
T- =~[l +exp(iTri~=l atiai)lb t It follows that e + in table 1 is a singlet only for N odd. In table 1 we have written explicitly only the states which survive for N = 3. In fact, for N ~ 3 we have di di, dii -~ Uk, u ij ~ dk, u i ~ ui .6. Out of such states for any N one can form mesons, each of them in two possible ways: for instance, for rr- we have in addition to the usual fii di also fii/d i]. Baryons can be formed for any N out o f three fundamental fermions, each of them in two possible ways. For instance, for the proton, we have the two states: utdldij. In the first
uiuJ~lij,
, s In the classical t.beory one introduces, corresponding to the oscillators a set of Grassmann variables and gauge fields (see footnote 1). The quantization of the theory by functional integration leads [ 11 ] to the Wilson loop which theiefore acquires a classical interpretation. ,6 These states coincide with those proposed by Corrigan and Ramond [ 12] to represent fermions for N > 3.
a],
which indeed f o r N = 1 gives 1/4, f o r N = 3 gives 3/8, for N = 5 gives 5/12, etc., increasing monotonically to sin20w = 1/2 for N ~ ~. The Z 0 coupling to the electron correspondingly goes from purely A in the leptonic model (N = 1), to purely V + A for N -+ ,~.
Fermion generators. The most straightforward way to introduce fermion generations within our general frame is to add to the set (ai, b) a set of colorless neutral Fermi oscillators c a, a = 1 ..... rn, leading in total to the Clifford algebra C10+2 m , with corresponding automorphism group O(10 + 2m) [14]. The Fock space o f eq. (10) then gets enlarged in the following way: for each state I~> o f e q . (10) one has now a sequence of 2 m states: I~>,
c~[~),
c~ctil~> . . . . .
c~...Ctml~V>, (11)
so that in all we get 2 m +4 states. In this total of 2 rn +4 states we find 2 m - 1 times the states in eq. (10), and 83
11 February 1980
PHYSICS LETTERS
Volume 90B, number 1,2 Table 1 The spectrum of states for N-color Fermi oscillators.
SU(N)
g - L
Ve
I0~
1
0
-1
(1il
a t 10>
N
1IN
- ( N - 2)IN
- i2 dil
ail t ai210) t
½ N ( N - 1)
2IN
- ( N - 4)IN
uiN-liN
at
½ N ( N - 1)
(N - 2)IN
(N-4)]N
u iN
at a~ O> il"" iN-1
N
(N-
(N-2)/N
e÷
atI . . .aNt 0'~
1
1
e-
btlo>
1
fiil
bta~ll O>
N
- ( N - 1)IN
-(N-
2)IN
-
btat at O"
½ N ( N - 1)
- ( N - 2)IN
-(N-
4)IN
z,t
½N(N - 1)
-2IN
(N - 4)IN
N
-I/N
( N - 2)IN
1
0
ix"" atiN-2 I 0>
uili2 o,.
il
.
.
i21
~
o!N_,lo>
2 m - 1 times their complex conjugates. The latter would, however, have a V + A weak coupling and one must assume that they lie at a high mass [15]. A simple argument showing the appearance of the V + A couplings follows from the fact that now the weak isospin T - must be identified with the O(10 + 2m) generator [8] : T - = b t ½(1 + ~9+2m) -m
t
+_3
t
.
1 [1 +(--1)(Za=lCaCa Z i = l a i a p ] b t
(12)
where m
1)IN
-1
1 -1
.
,,iN_t, v
1
that if the state I ~ ) transforms as a doublet (singlet) under weak isospin, the state c~ ...c~ _ I~> transforms as a singlet (doublet), so that exactl3~ ~ f of the states must have V + A couplings. The maximal monomial c ~ in the Clifford algebra C10+2 m (see footnote 2) breaks O(10 + 2m) to SU(5 + m)* 7. For m = 0 this breaking is particularly appealing because it splits off (ff)L from the other states of eq. (10), thus leaving us with a single two-component neutrino. This appealing property is, however, lost for m 4: 0, since only one of the (~)L contained in the various 16 of the O(10) subgroup is split off.
3
~9+2rn = I-I ( 1 - 2e~ea) .l-I ( 1 - 2 a ~ i ) ( 1 ,~=1 t=l
2bib),
(13) in the notations of eq. (10) for the set (a i, b). We see 84
Q
7 In our formalism the construction of all the breaking terms contained in the product of two spinors only amounts to writing all possible monomials in the Fermi oscillators [8].
Volume 90B, number 1,2
PHYSICS LETTERS
Inclusion o f spin in the Clifford algebr~ A Clifford description of the spin offers the interesting possibility of describing exactly two families, both with the correct V - A coupling. To describe the spin we introduce two Fermi oscillators ~a, a = 1, 2, which is equivalent to introducing a real C4 algebra for the Dirac "7" matrices. In terms of these oscillators we get the undotted Weyl spinor r/ta 10), and the dotted spinor (10), 17~r~?2[0)). On such a basis the generators of the Lorentz group O (3, 1) are M = { r~te 72, N ÷ -~7t~1 - 1 t2, N 3 --
N-
(14)
= 7/2/71 ,
71 +
-
1).
The total Clifford algebra (ai, b, rlc~) gives the followhag Fock space: rl~ IV), IV),
(15) rT] r/t2l~),
(15')
where IV) runs over the set of states in eq. (10). The automorphism group of the above Clifford algebra C12 is O(12) .8 which extends to O(14) in the way already described in ref. [1 ]. The weak isospin generator T - is now T - = b t ~1 (1 _x
-5
[1
--
-
(-1)[
~13) t +xai=1 ata a=lnc~na i i ,' ] b #
x2
,
(16)
where 2 ~13 = H ,~=1
3 (1 - 2r/?ar/~) H
(1 - 2a~ai)(1 - 2 b ' b ) .
11 February 1980
sult is similar to using four-component spinors instead of Weyl spinors, to describe generations [16], except that now one is led to a large group O (14). In this group O(14) (strictly O(13, 1)), besides the transformations belonging to O(10) and to the Lorentz group, we have transformations mixing dotted and undotted indices, and therefore with Lorentz vector character. The gauging of this theory can be done following the general remarks made in the introduction. The gauging for the oscillators r/~ gives rise to a connection taking values in the Lie algebra of 0(3, 1), i.e., the automorphism group for the ~a's. Such a connection is the standard one for the gauging of the Lorentz group (eventually to be expressed in terms of the gravitational field and possibly of the torsion). Analogously the gauging for the remaining oscillators gives rise to the standard connection taking values in the Lie algebra of O(10). Our unified description of the total Clifford algebra suggests the gauging of the entire automorphism group O(I 3, 1), with a connection which includes additional gauge fields with both O(10) and Lorentz indices. This introduces additional couplings of the fermion currents to spinless gauge fields carrying O(10) indices. The picture that emerges [8] is one of unification of the gravitational interaction with the other fundamental interactions; however, the problem of a unified description of the full set of gauge fields still remains open. Finally we note that to break the unifying O(13, 1) to O(10) ® 0(3, 1) one has to introduce a breaking through an antisymmetric tensor of order four whose geometrical meaning is to select in the 14-dimensional space a four-dimensional subspace corresponding to the Lorentz indices ,9. Such an antisymmetric tensor is a pseudoscalar of the form
t=l
(17) Under T - in eq. (16) the states in (15) and (15') have opposite behavior. However, at the same time they also have opposite helicity. Therefore, all the teft-helicity states have the correct chiral coupling to the weak isospin. This is to be contrasted with the situation which we obtained by introducing the c a oscillators for generations, where it was inescapable to have half of the states with the wrong chiral coupling. Formally the re-
u0
where
(,~)v = 8~v
fu, v = 0, 1, 2, 3).
I
Such a breaking can be made arbitrarily large without affecting the fermion degeneracy because it does not appear in the direct product of a spinor representation of O(13, 1) with itself. .9
,8 Strictly speaking we are dealing with O(11, 1) because the Clifford algebra C12 was not real, due to the hermiticity properties of the Dirac matrices.
A U1 A U 2 A U 3 ,
The required antisymmetric tensor corresponds to a Higgs field taking values in the coset space O(13, 1)/O(10) ®0(3, 1), which is the Grassmann manifold G(4, 10) [6, p. 212]. 85
Volume 90B, number 1,2
PHYSICS LETTERS
We w o u l d like to t h a n k Bruno Z u m i n o for reading the manuscript and for his useful c o m m e n t s .
References [1] R. Casalbuoni and R. Gatto, Phys. Lett. 88B (1979) 306. [2] A. Barducci, R. Casalbuoni and L. Lusanna, Lett. Nuovo Cimento 19 (1977) 581; R. Casalbuoni, Proc. Seoul Symp. on Elementary particles (1979) p. 421. [3] F.A. Berezin and M.S. Marinov, Ann. Phys. 104 (1977) 336; R. Casalbuoni, Nuovo Cimento 33A (1976) 115; A.P. Balachandran, P. Salomonson, B.S. Skagerstam and J.O. Winnberg, Phys. Rev. D15 (1977) 2308. [4] J.A. Wolf, Spaces of constant curvature (McGraw-Hill, New York, 1967) p. 283; S. Helgason, Differential geometry and symmetric spaces (Academic Press, New York, 1962) p. 350. [5] M. Gell-Mann, Po Ramond and R. Slansky, unpublished; P. Ramond, preprint CALT-68-709 (February 1979). [6] W. Greub, S. Halperin and R. Vanstone, Connections, curvature, and cohomology, VoL II (Academic Press, New York, 1973).
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11 February 1980
[7] H. Fritzsch and P. Minkowski, Ann. Phys. (NY) 93 (1974) 193; H. Georgi, Paricles and fields, ed. C.E. Carlson (ALP, New York, 1975) p. 575; M. Chanowitz, J. Ellis and M.K. GaiUard, Nucl. Phys. B128 (1977) 506. [8] R. Casalbuoni and R. Gatto, to be published. [9] H. Georgi and S.L Glashow, Phys. Rev. Lett. 32 (1974) 438. [10] G.'t Hooft, Nucl. Phys. B72 (1974) 461; E. Witten, NucL Phys. B149 (1979) 285. [11] S. Samuel, Nucl. Phys. B149 (1979) 157; I. Ishida and A. Hosoya, Prog. Theor. Phys. 62 (1979) 544. [12] E. Corrigan and P. Ramond, Phys. Lett. 87B (1979) 73. [13] E. Witten, Harvard preprint HUTP-79~A007. [14] F. Wilczek and A. Zee, Princeton preprint (1979). [15] F. Wilczek, talk given at the Lepton-photon conf. (Fermilab, August 1979), to be published. [16] R. Gatto, Nuovo Cimento 28 (1963) 567; A. Salam and J.C. Ward, Phys. Lett. 13 (1964) 168; S. Weinberg, Phys. Rev. D5 (1972) 1962.
PHYSICS LETTERS
11 February 1980
UNIFIED THEORIES FOR Q U A R K S A N D LEPTONS B A S E D ON CLIFFORD A L G E B R A S ~
R. CASALBUONI 1 CERN, Geneva, Switzerland and R. GATTO Ddpartement de Physique Thdorique, Universitd de Gendve, 1211 Geneva 4, Switzerland Received 27 November 1979
The general standpoint is presented that unified theories arise from gauging of Clifford algebras describing the internal degrees of freedom (charge, color, generation, spin) of the fundamental fermions. The general formalism is presented and the ensuing theories for color and charge (with extension to N colors), and for generations, are discussed. The possibility of further including the spin is discussed, also in connection with generations.
In this paper we shall extend the unified description of quarks and leptons that we have already proposed [1]. We shall first present the general formalism and discuss in detail its application to the description of the electric charge and the three colors as Clifford degrees of freedom. The case of N colors will also be studied. We shall then discuss the extension to fermion generations and examine the possibility of including the spin degree of freedom in the overall Clifford algebra. This last extension suggests unification with gravitation. General formalism. Our general standpoint is to describe quantum operators having a finite spectrum (charge, color, generation, spin) in terms of Clifford algebras [2]. Quarks and leptons are then thought of [ 1] as different states of a unique physical object with a dependence on a set of Clifford variables {~A } satisfying
[~, ~] + = 28A8.
(1)
If the Clifford algebra of eq. (1)is of even order 2N,
its automorphism group is O(2N), with generators
SA8 = ~-i [~A,
~81- •
(2)
If the Clifford algebra is of order 2N + 1 it can always be realized nonlinearly in terms of the Clifford algebra of order 2N, C2N , by expressing one of the generators ~i in terms of the others; for instance ~2N+l = iN~l ~2 "'" ~2N"
(3)
Therefore, in any case we shall be led to orthogonal groups of even dimensions, O (2N). More specifically we are led to classify the fundamental fermions according to spinor representations of such an O(2N), since these are indeed the representations of the Clifford algebras. Corresponding to O(2N), the maximal unitary subgroup U(N) (which also has rank N ) plays a crucial role, as can be seen by rewriting eq. (1) in the complex basis (i = 1,2 ..... N ) _1 di - ~ [~2i-1 - i~2i1 , (4) d/? _ 1 - 5 [ ~ 2 i - 1 + i~2i] ,
giving the algeb~ra of N Fermi oscillators, Supported in part by the Swiss National Science Foundation. 1 On leave of absence from, and address after January 1980: INFN, Sezione di Firenze, 50125 Florence, Italy.
[di, dil + = [ati, 4 1 + = O,
. [d i, d 7 ] + = ~ii,
(5)
for which U(N) is the automorphism group. These operators generate the Fock space: 81
PHYSICS LETTERS
Volume 90B, number 1,2 10),
d~,t0),
..., d~
dtNI0}.
The algebra (5) admits the discrete automorphisms (charge conjugations) d i -+ dti, which are generated by C i = d i + d}. In particular, one can construct an overall charge conjugation given by
C = C 1 ... CN .
(6)
The Fermi algebra (5) can be represented on analytic functions of a set o f N Grassmann variables {Oi}* a
dixlt(O )
=
Oixlt(O),
d~P(O)
=
(7)
The O(2N) generators of eq. (2) are:
for U(1):
N -1
Ok O0k , a
for SU(N):
0 i g a - N - l Sil
for the c o s e t s * 2 0 ( 2 N ) / U ( N ) :
~
(8)
Ok aOak ,
(8')
OiO1, ~0ai 00ai
.(8")
The function xP(0) can be split into an even and an odd part, * + (0), ~I'_ (0), with respect to the Grassmann algebra, both parts transforming irreducibly under O(2N). xI,_+(0) are "superfields" with respect to the U ( N ) basis, for instance: G _ ( x ; 0) = * i ( x ) O i + ~i/k(X)OiO/Ok + ....
(9)
The expansion (9) terminates with the monomial o f degree N or N - 1 according to whether N is odd or even. The monomial o f degree k (k odd) transforms as the totally antisymmetric irreducible representation of SU(N) o f dimensionality ( N ) , a . The generators (8) act as raising and lowering operators, respectively, within such sequences o f representations. ,1 In a classical theory the Grassmann coordinates 0 i themselves play the role of dynamical variables [2, 3]. Gauge fields can be introduced, belonging to the Lie algebra of the group of the canonical transformations, leading to the full set of equations of motion for the Grassmann variables and the gauge fields. ,2 The symmetric riemannian space O (2N)/U (N) reduces to the two-dimensional sphere S2 forN = 2, and to CP3 for N = 3 [41. 82
A "chiral" superfield, like G (0) in eq. (9), can be quantized consistently, with commutators or anticommutators according to the case, without running into contradictions, because it has a unique parity in 0 (definite grading). Finally, a local gauge theory is obtained by considering the generators of the Clifford algebra as "moving frames" on s p a c e - t i m e [6, p. 381]. The related connection takes values in the Lie algebra o f the automorphism group of the Clifford algebra (SO(2N) for the real basis, eq. (1), and U ( N ) for the complex basis, eq. (5) , 4 ) .
axp(o)/ao i ,
[0 i, a/ao i] + = 8ij .
~k
11 February 1980
The states o f the first generation. T h e simplest model based on such a description was presented in ref. [1 ]. In correspondence to the only exact symmetries, color and charge, one introduces four Fermi oscillators, a i, i = 1, 2, 3 (for red, white and blue) and b for the electric charge, satisfying the algebra (5). We assign charge 1/3 to each al? and - 1 to b t ; and we assign B - L = 2/3 to each a/* and 0 to b t . The complete Fock space is (for a "ground state" with Q = 0 and B- L =-1): I0) -- (re) L ,
b # IO) - ( e - ) L ,
a~ IO) ------(di) L ,
btati IO) -- (fii)L '
1 t t 5ei]kajaklO)=(Ui)L,
1~ b t ~ t a t m \ 5Wk ~] k W ,
a t1a t2a t3 0 ) - (e+)L ,
b t a t1a t2a t310)~(ff)L . (10)
(di)L,
,a The maximal monomial in the Oi's, 01 ... ON, transforms as a singlet under SU(N) and belongs to the eigenvalue 1 of the U (1) generator defined in eq.t(8). ,~orrespondingly one can construct the product c-~= dt l ... dj)h which is a component of the tensor representation 5 (#iv) of O (2N).C~ commutes with the generators of SU (N) defined in eq. (8') but breaks U(1) of eq. (8). ThereforeQ'~ breaks O (2N) to SU(N). This term is responsible for the Majorana mass [5]. ,4 We notice that O(2N) is the group of the inner automorphisms of C2N. This group admits a Z2-graded extension in the following way: C2N possesses exterior automorphisms of the type ~A ~ ~A + ell where [eA, ~A]+ = leA', eB]+ = 0. The inf'uiitesimal generators are aA ~A~2N+ 1 + J3A~A + #=N+I ~2N+I (etA are c-numbers and ~A, ~ are Grassmann variables~. This structure defines a Z2-graded Lie algebra by putting g i[~A, ~B] , ~A~=N+I in the even sector and ~A and ~2N+l in the odd sector. We notice that in this graded Lie algebra the generators of the Clifford algebra itself appear. It is worth noticing that if we do not grade the algebra, then the previous generators give rise, under commutation, to the Lie algebra of O (2N +2 ).
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Volume 90B, number 1,2
The states in eq. (10) transform as 8 • 8 under the automorphism group 0 ( 8 ) of our Clifford algebra C 8. They transform [1 ] as an irreducible 16 under the group O(10) [7] obtained by a nonlinear extension of 0(8). Such an O(10) can be regarded as the automorphism group of a Clifford algebra C10 obtained from C 8 by a linearization procedure which is analogous to the introduction of homogeneous variables in describing projective spaces [8]. By using the C10 basis, the states (10) are described by a single O(10) superfield ~ _ (x; 01 . . . . . 05) (c~ = 1,2 is a Weyl index), each component transforming irreducibly under SU(5) [9] (5 • 1_.9_0"• 1_).
Extension to N colors. Recent work has shown the usefulness of studying QCD theories [10] for a number of colors N 4: 3. Within our framework we only have to replace the set of Fermi oscillators a 1 , a 2, a 3 with the set al, a2, ..., aN , s . When we demand that the SU(N) singlets appearing in the spectrum (see table 1) be identified with the leptons (ve, e ± , ~), we find that the electric charge and B - L must be assigned as follows: to a~ charge 1IN and B - L = 2IN; to bt charge - 1 and B - L = 0. Again, the "ground state" has charge zero and B - L = - 1. It is clear that N has to be odd. In fact for any N we can identify the weak isospin generator T - as
state, uiuJfiij, the total charge of one is obtained as 2(N - 1)/N- ( N - 2)/N, whereas for the second state it is obtained as ( N - 1)IN- I/N+ 2IN. Similarly; for the neutron one has the two states: diu/fiij, didl~dij, Again, one has to take N odd in order that the proton and neutron be doublets of weak isospin. In the discussion of the large-N limit [ 10,13 ], the possibility of representing baryons out of three, rather than N, fundamental fermions is definitely advantageous. For the smallest value of N, N = 1, the spectrum contains only the leptons. Correspondingly, one can set up an 0 ( 6 ) gauge theory describing only leptons (belonging to the representation 4, which decomposes as 3 • 1 under SU(3)). The unrenormalized Weinberg angle is given in this purely leptonic model by sin20w = 1/4. For N = 3, as we know, one has the familiar spectrum of quarks and leptons of the first generation with corresponding O(10). We show in table 1 the spectrum for general odd N ; correspondingly one can set up an O(2N + 4) gauge theory. For such a general case one can work out the value of the Weinberg angle as a function of N. We obtain
sin20w 1
Tr(T2) Tr(Q2)
1 (N-l)/2 4 k=0
N)~kk~=0
k 2 N
N
21+N
N
1
T- =~[l +exp(iTri~=l atiai)lb t It follows that e + in table 1 is a singlet only for N odd. In table 1 we have written explicitly only the states which survive for N = 3. In fact, for N ~ 3 we have di di, dii -~ Uk, u ij ~ dk, u i ~ ui .6. Out of such states for any N one can form mesons, each of them in two possible ways: for instance, for rr- we have in addition to the usual fii di also fii/d i]. Baryons can be formed for any N out o f three fundamental fermions, each of them in two possible ways. For instance, for the proton, we have the two states: utdldij. In the first
uiuJ~lij,
, s In the classical t.beory one introduces, corresponding to the oscillators a set of Grassmann variables and gauge fields (see footnote 1). The quantization of the theory by functional integration leads [ 11 ] to the Wilson loop which theiefore acquires a classical interpretation. ,6 These states coincide with those proposed by Corrigan and Ramond [ 12] to represent fermions for N > 3.
a],
which indeed f o r N = 1 gives 1/4, f o r N = 3 gives 3/8, for N = 5 gives 5/12, etc., increasing monotonically to sin20w = 1/2 for N ~ ~. The Z 0 coupling to the electron correspondingly goes from purely A in the leptonic model (N = 1), to purely V + A for N -+ ,~.
Fermion generators. The most straightforward way to introduce fermion generations within our general frame is to add to the set (ai, b) a set of colorless neutral Fermi oscillators c a, a = 1 ..... rn, leading in total to the Clifford algebra C10+2 m , with corresponding automorphism group O(10 + 2m) [14]. The Fock space o f eq. (10) then gets enlarged in the following way: for each state I~> o f e q . (10) one has now a sequence of 2 m states: I~>,
c~[~),
c~ctil~> . . . . .
c~...Ctml~V>, (11)
so that in all we get 2 m +4 states. In this total of 2 rn +4 states we find 2 m - 1 times the states in eq. (10), and 83
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Volume 90B, number 1,2 Table 1 The spectrum of states for N-color Fermi oscillators.
SU(N)
g - L
Ve
I0~
1
0
-1
(1il
a t 10>
N
1IN
- ( N - 2)IN
- i2 dil
ail t ai210) t
½ N ( N - 1)
2IN
- ( N - 4)IN
uiN-liN
at
½ N ( N - 1)
(N - 2)IN
(N-4)]N
u iN
at a~ O> il"" iN-1
N
(N-
(N-2)/N
e÷
atI . . .aNt 0'~
1
1
e-
btlo>
1
fiil
bta~ll O>
N
- ( N - 1)IN
-(N-
2)IN
-
btat at O"
½ N ( N - 1)
- ( N - 2)IN
-(N-
4)IN
z,t
½N(N - 1)
-2IN
(N - 4)IN
N
-I/N
( N - 2)IN
1
0
ix"" atiN-2 I 0>
uili2 o,.
il
.
.
i21
~
o!N_,lo>
2 m - 1 times their complex conjugates. The latter would, however, have a V + A weak coupling and one must assume that they lie at a high mass [15]. A simple argument showing the appearance of the V + A couplings follows from the fact that now the weak isospin T - must be identified with the O(10 + 2m) generator [8] : T - = b t ½(1 + ~9+2m) -m
t
+_3
t
.
1 [1 +(--1)(Za=lCaCa Z i = l a i a p ] b t
(12)
where m
1)IN
-1
1 -1
.
,,iN_t, v
1
that if the state I ~ ) transforms as a doublet (singlet) under weak isospin, the state c~ ...c~ _ I~> transforms as a singlet (doublet), so that exactl3~ ~ f of the states must have V + A couplings. The maximal monomial c ~ in the Clifford algebra C10+2 m (see footnote 2) breaks O(10 + 2m) to SU(5 + m)* 7. For m = 0 this breaking is particularly appealing because it splits off (ff)L from the other states of eq. (10), thus leaving us with a single two-component neutrino. This appealing property is, however, lost for m 4: 0, since only one of the (~)L contained in the various 16 of the O(10) subgroup is split off.
3
~9+2rn = I-I ( 1 - 2e~ea) .l-I ( 1 - 2 a ~ i ) ( 1 ,~=1 t=l
2bib),
(13) in the notations of eq. (10) for the set (a i, b). We see 84
Q
7 In our formalism the construction of all the breaking terms contained in the product of two spinors only amounts to writing all possible monomials in the Fermi oscillators [8].
Volume 90B, number 1,2
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Inclusion o f spin in the Clifford algebr~ A Clifford description of the spin offers the interesting possibility of describing exactly two families, both with the correct V - A coupling. To describe the spin we introduce two Fermi oscillators ~a, a = 1, 2, which is equivalent to introducing a real C4 algebra for the Dirac "7" matrices. In terms of these oscillators we get the undotted Weyl spinor r/ta 10), and the dotted spinor (10), 17~r~?2[0)). On such a basis the generators of the Lorentz group O (3, 1) are M = { r~te 72, N ÷ -~7t~1 - 1 t2, N 3 --
N-
(14)
= 7/2/71 ,
71 +
-
1).
The total Clifford algebra (ai, b, rlc~) gives the followhag Fock space: rl~ IV), IV),
(15) rT] r/t2l~),
(15')
where IV) runs over the set of states in eq. (10). The automorphism group of the above Clifford algebra C12 is O(12) .8 which extends to O(14) in the way already described in ref. [1 ]. The weak isospin generator T - is now T - = b t ~1 (1 _x
-5
[1
--
-
(-1)[
~13) t +xai=1 ata a=lnc~na i i ,' ] b #
x2
,
(16)
where 2 ~13 = H ,~=1
3 (1 - 2r/?ar/~) H
(1 - 2a~ai)(1 - 2 b ' b ) .
11 February 1980
sult is similar to using four-component spinors instead of Weyl spinors, to describe generations [16], except that now one is led to a large group O (14). In this group O(14) (strictly O(13, 1)), besides the transformations belonging to O(10) and to the Lorentz group, we have transformations mixing dotted and undotted indices, and therefore with Lorentz vector character. The gauging of this theory can be done following the general remarks made in the introduction. The gauging for the oscillators r/~ gives rise to a connection taking values in the Lie algebra of 0(3, 1), i.e., the automorphism group for the ~a's. Such a connection is the standard one for the gauging of the Lorentz group (eventually to be expressed in terms of the gravitational field and possibly of the torsion). Analogously the gauging for the remaining oscillators gives rise to the standard connection taking values in the Lie algebra of O(10). Our unified description of the total Clifford algebra suggests the gauging of the entire automorphism group O(I 3, 1), with a connection which includes additional gauge fields with both O(10) and Lorentz indices. This introduces additional couplings of the fermion currents to spinless gauge fields carrying O(10) indices. The picture that emerges [8] is one of unification of the gravitational interaction with the other fundamental interactions; however, the problem of a unified description of the full set of gauge fields still remains open. Finally we note that to break the unifying O(13, 1) to O(10) ® 0(3, 1) one has to introduce a breaking through an antisymmetric tensor of order four whose geometrical meaning is to select in the 14-dimensional space a four-dimensional subspace corresponding to the Lorentz indices ,9. Such an antisymmetric tensor is a pseudoscalar of the form
t=l
(17) Under T - in eq. (16) the states in (15) and (15') have opposite behavior. However, at the same time they also have opposite helicity. Therefore, all the teft-helicity states have the correct chiral coupling to the weak isospin. This is to be contrasted with the situation which we obtained by introducing the c a oscillators for generations, where it was inescapable to have half of the states with the wrong chiral coupling. Formally the re-
u0
where
(,~)v = 8~v
fu, v = 0, 1, 2, 3).
I
Such a breaking can be made arbitrarily large without affecting the fermion degeneracy because it does not appear in the direct product of a spinor representation of O(13, 1) with itself. .9
,8 Strictly speaking we are dealing with O(11, 1) because the Clifford algebra C12 was not real, due to the hermiticity properties of the Dirac matrices.
A U1 A U 2 A U 3 ,
The required antisymmetric tensor corresponds to a Higgs field taking values in the coset space O(13, 1)/O(10) ®0(3, 1), which is the Grassmann manifold G(4, 10) [6, p. 212]. 85
Volume 90B, number 1,2
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We w o u l d like to t h a n k Bruno Z u m i n o for reading the manuscript and for his useful c o m m e n t s .
References [1] R. Casalbuoni and R. Gatto, Phys. Lett. 88B (1979) 306. [2] A. Barducci, R. Casalbuoni and L. Lusanna, Lett. Nuovo Cimento 19 (1977) 581; R. Casalbuoni, Proc. Seoul Symp. on Elementary particles (1979) p. 421. [3] F.A. Berezin and M.S. Marinov, Ann. Phys. 104 (1977) 336; R. Casalbuoni, Nuovo Cimento 33A (1976) 115; A.P. Balachandran, P. Salomonson, B.S. Skagerstam and J.O. Winnberg, Phys. Rev. D15 (1977) 2308. [4] J.A. Wolf, Spaces of constant curvature (McGraw-Hill, New York, 1967) p. 283; S. Helgason, Differential geometry and symmetric spaces (Academic Press, New York, 1962) p. 350. [5] M. Gell-Mann, Po Ramond and R. Slansky, unpublished; P. Ramond, preprint CALT-68-709 (February 1979). [6] W. Greub, S. Halperin and R. Vanstone, Connections, curvature, and cohomology, VoL II (Academic Press, New York, 1973).
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[7] H. Fritzsch and P. Minkowski, Ann. Phys. (NY) 93 (1974) 193; H. Georgi, Paricles and fields, ed. C.E. Carlson (ALP, New York, 1975) p. 575; M. Chanowitz, J. Ellis and M.K. GaiUard, Nucl. Phys. B128 (1977) 506. [8] R. Casalbuoni and R. Gatto, to be published. [9] H. Georgi and S.L Glashow, Phys. Rev. Lett. 32 (1974) 438. [10] G.'t Hooft, Nucl. Phys. B72 (1974) 461; E. Witten, NucL Phys. B149 (1979) 285. [11] S. Samuel, Nucl. Phys. B149 (1979) 157; I. Ishida and A. Hosoya, Prog. Theor. Phys. 62 (1979) 544. [12] E. Corrigan and P. Ramond, Phys. Lett. 87B (1979) 73. [13] E. Witten, Harvard preprint HUTP-79~A007. [14] F. Wilczek and A. Zee, Princeton preprint (1979). [15] F. Wilczek, talk given at the Lepton-photon conf. (Fermilab, August 1979), to be published. [16] R. Gatto, Nuovo Cimento 28 (1963) 567; A. Salam and J.C. Ward, Phys. Lett. 13 (1964) 168; S. Weinberg, Phys. Rev. D5 (1972) 1962.