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What is special about the group of the standard model?
Volume 223, number 3,4
PHYSICS LETTERS B
15 June 1989
WHAT IS SPECIAL ABOUT T H E GROUP OF T H E STANDARD MODEL? H.B. NIELSEN and N. BRENE Nields Bohr Institute, DK-2100 Copenhagen 0, Denmark Received 17 March 1989; revised manuscript received 11 April 1989
The standard model is based on the algebra of U~ × SU2 X SU3. The systematics of charges of the fundamental fermions seems to suggest the importance of a particular group having this algebra, viz. S (U2 X U3). This group is distinguished from all other connected compact non semisimple groups with dimensionality up to 12 by a characteristic property: it is very "skew". By this we mean that the group has relatively few "generalised outer automorphisms": One may speculate about physical reasons for this fact.
1. Introduction
The standard model [ 1 ] assumed to provide the basis of the physics of elementary particles has two aspects. ( 1 ) It describes the forces acting between the gauge panicles by means of a field theory based on the algebra OfUl X SU2X SU3. (2) It describes the set of fundamental fermions (quarks and leptons) between which the forces act. This set appears to consists of at least three generations, three repetitions of a particular collection of representations for the above Lie algebra and thereby of the universal covering group ~ X S U 2 × S U 3. We shall see that this collection points to a particular group, S (U2 × U3). Among all possible representations of the product group × SU2 X S U 3
( 1)
Nature has chosen a set for which the representations of the three factors in the group product satisfy the relation Y+ ½D+ I T = 0 (mod 1 ).
(2)
Here Y is the weak hypercharge (the scale is chosen such that the lefthanded electron doublet has Y= - ½), D is the duality (0 for a weak isosinglet, 1 and 0 for isodoublet and isotriplet respectively) and 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
finally T is triality (0 for leptons, + 1 for quark and antiquark triplets). The charge quantization rule (2) is characteristic for one definite group with the algebra U~ X SU2 X SU3 [ 2 ]. This group is obtained from the covering group R × SU2 × SU3 by division with a discrete invariant subgroup (determined from the charge quantization rule [2,3] ). It may be termed S (U2 X U3) because it can be represented by 5 X 5 block diagonal matrices with determinant 1 and 2X2 and 3 × 3 diagonal blocks representing the groups U2 and U3. We shall denote it SMG, the standard model group. The purpose of this paper is to investigate what is special about SMG, hoping that such investigation may shed light on the mechanism by which Nature selected ,just this group. It is worth noting that the algebra U~ X SU2 X SU3 which has dimensionality 12 corresponds to a total of thirteen connected Lie groups of which nine are compact [ 4 ]. The total number of connected compact groups with dimensionality 12 is 71 and adding all compact groups of smaller dimensionality we have 230. Does the SMG show unique properties when compared to this background? Two properties [ 5 ] of the SMG are evident: ( 1 ) Its Lie algebra is composed of all the three lowest possible Lie algebras U~, SU2, SU3 (low with respect to dimensionality), and (2) each algebra occurs only once. There are in total eight Lie algebras (useful for compact groups) of dimensionality 12 and the total number of algebras of dimensionality 12 or less is 44. 399
Volume 223, number 3,4
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A lower algebra has a construction analogous to that of the standard model algebra: U~ ×SU2 of dimensionality 4. There are only two algebras of dimensionality 4 and in total six of dimensionality 4 or less. Clearly such properties alone put the standard model algebra in a distinct series of algebras U~, UI ×SU2, UI × S U 2 × S U 3 , U 1 X S U 2 × S U 3 × S O 5 , ... but they do not point to any particular group. The second of the two above properties signifies a lack of symmetry of the algebra compared to algebras allowing permutation of subalgebras as does e.g. SU2 × SU2. This leads us to consider the degree of symmetry of a group. In the following we shall therefore propose some measure for the degree of symmetry of a group and demonstrate that the SMG is distinguished with respect to this measure. In fact the SMG is distinguished among the 192 connected non semisimple groups of dimensionality 12 or less by having the absolutely lowest degree of symmetry; we may say that it is the most skew of these groups. One may then speculate about possible physical reasons for the skewness of the SMG.
15 June 1989
Axiom 2. (a, b) ~ R ~ (a -1, b -~ )~R. It is evident that i f ~ A a n d / ~ B , then the relation def Rab-~-----{ ( ~ a ~ - ' , b~b~-l)l(a, b)~R} is also group compatible. In the particular case that the groups A and B are equal there exists a trivial (diagonal) relation RtriviaI = { ( a , a ) l a i R } ,
(4)
with associated relations of the form Rtrivial,~/7= ( ( a a a -1, ~ a ~ - 1) l a~m} = { ( a , / 7 ~ - ~a(/7~ -~ ) -1)la~A}
(5)
(for ~, ~eA). They correspond to inner automorphisms for A. We need some notation before we present theorems for group compatible relations.
Definition 1. 2. Classification of relations between groups
PA(R) To obtain a measure for the symmetry of a group we define relations between groups compatible with the group composition laws. The number of such relations between two groups A and B is a measure for the similarity of these groups. In particular, the number of relations between a group and itself (selfrelations) is a measure for the selfsimilarity or symmetry of the group. It is therefore important to classify and enumerate all the (self)relations for a given group; we shall see that these relations represent isomorphisms between factor groups. Consider two (Lie) groups A and B. A relation R between A and B is a subset of their direct product R _ ~ A × B = { ( a , b ) l a ~ A , bEB}.
(3)
We define a relation R to be compatible with the group structure i f - for all values of a, a~, a2~A and b, b~, b2EB - it satisfies the following two axioms:
Axiom 1. ( ( a l , bl ) e R A (a2, bz)~R)~(ala2, blb2)~R. 400
PB(R)
def
- - { a ~ A ] ~b~B [(a, b)~R]}, def
{b~BI 3a~A [ (a, b)~R]}, def EA(R) - - {aeAI (a, 1 )eR}, def EB(R) - - {beB l(1, b)~R}. The sets EA(R ) and Ea(R) are obviously invariant subgroups Of PA(R) and Pn(R).
Theorem 1. Let R__ A × B be a group compatible relation. Then P A ( R ) / E A ( R ) ---Pa(R)/EB(R). If this isomorphism is expressed
f:PA (R ) /EA (R ) ~ P B ( R ) /EB(R ), t h e n f c a n be chosen so that R = { (a, b ) ~ P A ( R ) × P B ( R ) I f ( a E A ( R ) ) = b E a ( R ) } . For the set of relations between two groups (com-
Volume 223, number 3,4
PHYSICS LETTERSB
patible with group structure) there exists partial ordering "inclusion" R~ ~ R 2
meaning VaeA, b~B
[(a,b)~Rt~(a,b)~R2].
The inclusion means that R~ is the stronger relation in the sense that it contains more information to know that (a, b)~R~ than (a, b)~RR. In this paper we are mainly interested in relations for which P A ( R ) = A and P a ( R ) = B . A relation is called minimal if and only if no other relation with this property can be included in it.
Theorem 2. Let Rj and R2 be two group compatible relations between the groups A and B. Assume furthermore that PA(RI) =PA(R2) = A
and
PB(R,) =PB(R2) =B.
Then RI ~R2 implies the existence of an invariant subgroup H of the factor group A/EA(R~ ) such that A/EA(R2) -- (A/EA(RI) ) / H --- (B/EB (R~) )/f~ ( H ) ----B/EB(R2), where f~ is the isomorphism of theorem 1 applied to Rp 3. A measure for the symmetry of a group
From a given group A we obtain new groups by division with invariant subgroups H of order n (H). As we are interested in the possibility that the factor group is isomorphic with the original group, we consider discrete invariant subgroups only; it is then easy to show that H must be a subgroup of the center of the group A. A relation R between two groups A and B is according to theorem 1 given by an isomorphism A/HA ~ B / H B .
(6)
The group B may well be identical to the group A; then the relation is called a selfrelation. Thus a selfrelation for a group corresponds to an isomorphism between two of its factor groups
A/Hi~-A/Hj,
15 June 1989 (7)
where one or both of the invariant subgroups may consist of one element only (the unit element). Now we can characterise a relation by a level pair (n~, nj), the orders of the two invariant subgroups H~ and Hi. All groups have the trivial selfrelation (4). Nonabelian groups also have the associated inner automorphisms (5). We shall ignore these relations which occur unavoidably on the ( 1, 1 ) level. Likewise we shall ignore relations which are not minimal. We consider all such relations trivial. A group with simple algebra has a finite discrete center; hence a group with simple algebra has a finite number of possible level pairs. A group with continuous center (e.g. Ul ) has an infinite number of level pairs. Some groups have nontrivial relations occurring already at the lowest possible level pair ( 1, 1 ), i.e. outer automorphisms. There are many examples: SU2 (Peter) × SU2 (Paul) is obviously symmetric under the exchange of Peter and Paul, an outer automorphism (we denote it pp-symmetry). Many simple groups also have outer automorphisms, e.g. SU3 and higher SUN groups for which the symmetry corresponds to charge conjugation (we denote it cc-symmetry). Also the group U~ has cc-symmetry. We also know groups which have no selfrelations at all, e.g. SU2: It has just one factor group SU2/ Z2~SO3. These two groups have no outer automorphisms and they are not isomorphic. It would have been very attractive to use the lowest level at which selfrelations occur as a measure for the skewness, lack of symmetry. If we do that groups like SU2 or SU2 × SU2/Z2 would win over all groups with continuous center (including the SMG) in the competition of being most skew. We would, however, like to consider such groups as disqualified as they are not enlightened by at least a single UI in their algebra. This means that we shall close our eyes to the occurrence of a single cc-symmerry in combination with all inner automorphisms when we look for the lowest level pair exhibiting symmetry. A second cc-symmetry (which occurs when the algebra has several U l'S) shall, however, not be overlooked. With this qualification we propose that a useful measure for the skewness (lack of symmetry) of physical groups is the lowest level pair (n~, nj) 401
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at which a relation occurs; actually we consider pairs (1, nj).
4. Results In our investigation of symmetry properties we have considered all connected compact groups of dimensionality up to 12. Among the 230 groups very few (semisimple) groups have no nontrivial selfrelations; seven non semisimple groups have first relations at the level pair ( 1, 2): UI, UI × SU2 × SO3, U3, U 1×Spins, U 1×SO5, SU2×U3, SO3×U3, five have first relations at the level pair (1,3): U2, SU2XU2, U2× SO3, U2× SU2xSO3, (R× Spins)/{ (2n,-)P} ~1, and one group has first relations at the level pair ( 1, 5 ): SMG. All others (179) have nontrivial relations already at the level ( 1, 1 ). We may state the result in a slightly different way: count the number of connected compact non semisimple groups (of dimensionality less than 13) that have their first relation at a level (1, n) or higher. The result is n=1~192, n=4~l,
n=2~13,
n=3~6,
n=5~l.
The group occurring for n = 4 and 5 is of course the SMG. Groups of structure similar to that of the SMG are distinguished by a relatively high degree of skewness, i.e. the first relation occurs at a high level pair. Thus for the two analogs of SMG corresponding to the algebra U I × S U 2 × S U 3 X S U 5 the lowest symmetry bearing level pair is ( 1, 11 ). The group SMG is the most skew of connected compact groups of dimensionality up to at least 12 when we disregard semisimple groups. The non semisimple groups contain at least one Ut as invariant subalgebra. A single UI has generalized symmetries, i.e. relations for any level pair (n, m) because the group Ut/Zn is isomorphic to U~ itself and thus to U~/Zm. This makes any group with a UI as a factor very symmetric. Nevertheless some groups that do contain U~'s in the algebra are rather skew. The rea~ The generating element of the subgroup divided out is characterised by a step in the R-groupand the non-unit center element of the Spins-group. 402
15 June 1989
son is that such groups are obtained from their covering group by division by a discrete invariant subgroup with nontrivial projections in both the UI factor and other group factors. Therefore the original generalized symmetry of U I is destroyed; an example is U2 =~(Ut × SUE ) / ( ( 2 rt, - )P} which has the first relation on the level (1, 3). This mechanism plays a decisive role in the selection of SMG.
5. Some physical remarks Mathematical considerations have shown us that SMG has a special status among compact connected non semisimple groups when we apply the criterion of skewness. We may therefore assume that skewness has physical significance when we go to models beyond the standard model. In grand unification models like the SU5 model of Georgi and Glashow [ 6 ] skewness seems to play no role(?). The group itself and the special Higgs field that break it are taken out of the air to give the right break down products, but without any further justification; it appears to be a bare accident that the surviving SMG is skew. We feel tempted to discard such models and instead look for models for which skewness is essential. Such models should include some mechanism that looks for the symmetry of groups and capable of breaking down a group if it does show some symmetry. One suggestion is to generalise models for confusion. In confusion models [3] there are hypersurfaces in space (confusion surfaces) along which some degrees of freedom are continued in a modified way, e.g. the degrees of freedom on one side of the hypersurface are continued on the other side in their images under some automorphism. The effect of confusion is to prevent a local gauge symmetry from being also a global symmetry. In a generalized confusion model one imagines the continuity conditions modified to respect some group compatible relations. This might be formulated in terms of two paths approaching each other from either side of a confusion surface
Volume 223, number 3,4
PHYSICS LETTERS B
Z--E
gAu(x)dxU),
(limexp(i ~ \E~O
\
gAu(x)dxU))eR.
[imexp(i I
(8)
y+E
Here E is a four-vector along the oriented normal of the confusion surface. These conditions may prevent a global symmetry. Another suggestion is related to the observation that a more skew object has more possibility for alignment than a less skew object. The skew object should, of course, be translated to a skew group whereas the possibilities are translated to the different ways it might act on some system, e.g. a field theory. We shall here introduce a little formalism: Assume a system L with elements leL and relations Q e L × L , and a group G with elements g. The group may act on the system in many possible ways each described by a function F:G×L--,L for which
F(g,/)eL is the image of l under the operation representing g. These functions are required to satisfy the conditions
F(g~g2,l)=F(gl, F(g2, l)), if (1~, 12)eQ
then(F(g,l~),F(g, lz))eQ.
(9)
One can show that the number of such functions F(g, l) must (under plausible conditions) be the same for groups of equal volume. We then seek, for all groups G having a given volume, corresponding sets of functions F(g, l) such that G is a symmetry of the system L, i.e. (9) is satisfied. The group G may have an outer automorphism g ~ a (g). If this is the case one can from one F define another by def
Fa(g, l) ~F(a(g), l), but this new F a is already a member of the set of F s . Thus two of the functions are used to implement the symmetry G in the same way up to an inessential automorphism.
15 June 1989
In general the a priori number of potential ways of implementing a group is reduced by a factor equal to the number of outer automorphisms of the group. This shows that a group with more automorphisms has smaller chance to be implemented as a symmetry group for a system than a group with fewer automorphisms. We need some mechanism for amplifying this effect, e.g. by letting it take place in any little spacetime region. One may note that the break down of the standard model associated with the break down of the SalamWeinberg theory cannot be related to any of these mechanisms as the surviving group U3 is less skew than U2.
6. Concluding remarks We have found a criterion that singles out the SMG (S (U2 × U3 ) ) among a large collection of groups potentially useful for construction of gauge theories with associated fermion generations. The criterion is skewness expressed by means of the first level pair at which a group has nontrivial selfrelations or generalized outer automorphisms. Actually the SMG is the most skew among all connected compact groups with continuous center and dimensionality up to and including 12. There are 192 such groups. Unfortunately we have had to add the word "nontrivial" to avoid the ever present identical relation as well as symmetry under charge conjugation both combined with inner automorphisms. These symmetries are shared by all groups having a continuous center and hence a U~ subalgebra. Thus we have considered the symmetry under complex conjugation as trivial on the same footing as symmetry under the identity relation. That we have looked only among groups with continuous center is certainly a weak point. On the other hand this point may probably inspire new development. In spite of this weak point the fact that the criterion of skewness chooses the SMG uniquely among 192 possible groups strongly suggests that skewness has some physical significance. Arguments concerning grand unification models with break down to the standard model appear not to apply skewness; this is one reason to consider them with scepticism. 403
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We propose two possible ways to explain that Nature apparently prefers a skew gauge group. One is a generalisation of confusion models [ 3 ]. The other is an abstraction from the fact that it is easier to close a suitcase with five liter of wine in ordinary bottles than one with the same q u a n t i t y in one spherical ( a n d highly symmetrical) container. Both ways may appear natural in models involving randomness at some energy level.
Acknowledgement One of the authors (H.B.N.) wants to acknowledge inspiration following discussions with B. D u r h u u s years ago.
References [ 1] S.L. Glashow, Nucl. Phys. 22 ( 1961 ) 579; A. Salam, Proc. 8th Nobel Symp. (Stockholm, 1968), ed. N. Svartholm (Almquist & Wiksell,Stockholm, 1968);
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S. Weinberg,Phys. Rev. Lett. 19 (1967) 1264; H. Fritsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. 30 (1973) 1346. [2]L.O'Raifeartaigh, Group structure of gauge theories (Cambridge U.P., Cambridge, 1986); L. Michel, Group theoretical concepts and methods in elementaryparticle physics,ed. F. G~irsey(Gordon & Breach, New York, 1962). [3] H.B. Nielsen and N. Brene, in: Proc. XVII1 Intern. Symp. (Ahrenshoop) (Institut f'tir Hochenergiephysik, Akademie der Wissenschaftender DDR, Berlin-Zeuthen, 1985). [4] R. Gilmore, Lie groups, Lie algebras, and some of their applications (Wiley, New York, 1974). [ 5 ] H.B. Nielsen, Random dynamics and relations between the number of fermion generations and the fine structure constants, in: Proc. Zakopane Summer School (1988 ), Acta Physica Polon., to appear; D. Bennett, H.B. Nielsen, N. Brene and L. Mizrachi, Confusion and the heterotic string, Proc. Paris-Meudon Colloq. (September 1986) (Wolrd Scientific, Singapore, 1987). [6] H. Georgiand S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438.
PHYSICS LETTERS B
15 June 1989
WHAT IS SPECIAL ABOUT T H E GROUP OF T H E STANDARD MODEL? H.B. NIELSEN and N. BRENE Nields Bohr Institute, DK-2100 Copenhagen 0, Denmark Received 17 March 1989; revised manuscript received 11 April 1989
The standard model is based on the algebra of U~ × SU2 X SU3. The systematics of charges of the fundamental fermions seems to suggest the importance of a particular group having this algebra, viz. S (U2 X U3). This group is distinguished from all other connected compact non semisimple groups with dimensionality up to 12 by a characteristic property: it is very "skew". By this we mean that the group has relatively few "generalised outer automorphisms": One may speculate about physical reasons for this fact.
1. Introduction
The standard model [ 1 ] assumed to provide the basis of the physics of elementary particles has two aspects. ( 1 ) It describes the forces acting between the gauge panicles by means of a field theory based on the algebra OfUl X SU2X SU3. (2) It describes the set of fundamental fermions (quarks and leptons) between which the forces act. This set appears to consists of at least three generations, three repetitions of a particular collection of representations for the above Lie algebra and thereby of the universal covering group ~ X S U 2 × S U 3. We shall see that this collection points to a particular group, S (U2 × U3). Among all possible representations of the product group × SU2 X S U 3
( 1)
Nature has chosen a set for which the representations of the three factors in the group product satisfy the relation Y+ ½D+ I T = 0 (mod 1 ).
(2)
Here Y is the weak hypercharge (the scale is chosen such that the lefthanded electron doublet has Y= - ½), D is the duality (0 for a weak isosinglet, 1 and 0 for isodoublet and isotriplet respectively) and 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
finally T is triality (0 for leptons, + 1 for quark and antiquark triplets). The charge quantization rule (2) is characteristic for one definite group with the algebra U~ X SU2 X SU3 [ 2 ]. This group is obtained from the covering group R × SU2 × SU3 by division with a discrete invariant subgroup (determined from the charge quantization rule [2,3] ). It may be termed S (U2 X U3) because it can be represented by 5 X 5 block diagonal matrices with determinant 1 and 2X2 and 3 × 3 diagonal blocks representing the groups U2 and U3. We shall denote it SMG, the standard model group. The purpose of this paper is to investigate what is special about SMG, hoping that such investigation may shed light on the mechanism by which Nature selected ,just this group. It is worth noting that the algebra U~ X SU2 X SU3 which has dimensionality 12 corresponds to a total of thirteen connected Lie groups of which nine are compact [ 4 ]. The total number of connected compact groups with dimensionality 12 is 71 and adding all compact groups of smaller dimensionality we have 230. Does the SMG show unique properties when compared to this background? Two properties [ 5 ] of the SMG are evident: ( 1 ) Its Lie algebra is composed of all the three lowest possible Lie algebras U~, SU2, SU3 (low with respect to dimensionality), and (2) each algebra occurs only once. There are in total eight Lie algebras (useful for compact groups) of dimensionality 12 and the total number of algebras of dimensionality 12 or less is 44. 399
Volume 223, number 3,4
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A lower algebra has a construction analogous to that of the standard model algebra: U~ ×SU2 of dimensionality 4. There are only two algebras of dimensionality 4 and in total six of dimensionality 4 or less. Clearly such properties alone put the standard model algebra in a distinct series of algebras U~, UI ×SU2, UI × S U 2 × S U 3 , U 1 X S U 2 × S U 3 × S O 5 , ... but they do not point to any particular group. The second of the two above properties signifies a lack of symmetry of the algebra compared to algebras allowing permutation of subalgebras as does e.g. SU2 × SU2. This leads us to consider the degree of symmetry of a group. In the following we shall therefore propose some measure for the degree of symmetry of a group and demonstrate that the SMG is distinguished with respect to this measure. In fact the SMG is distinguished among the 192 connected non semisimple groups of dimensionality 12 or less by having the absolutely lowest degree of symmetry; we may say that it is the most skew of these groups. One may then speculate about possible physical reasons for the skewness of the SMG.
15 June 1989
Axiom 2. (a, b) ~ R ~ (a -1, b -~ )~R. It is evident that i f ~ A a n d / ~ B , then the relation def Rab-~-----{ ( ~ a ~ - ' , b~b~-l)l(a, b)~R} is also group compatible. In the particular case that the groups A and B are equal there exists a trivial (diagonal) relation RtriviaI = { ( a , a ) l a i R } ,
(4)
with associated relations of the form Rtrivial,~/7= ( ( a a a -1, ~ a ~ - 1) l a~m} = { ( a , / 7 ~ - ~a(/7~ -~ ) -1)la~A}
(5)
(for ~, ~eA). They correspond to inner automorphisms for A. We need some notation before we present theorems for group compatible relations.
Definition 1. 2. Classification of relations between groups
PA(R) To obtain a measure for the symmetry of a group we define relations between groups compatible with the group composition laws. The number of such relations between two groups A and B is a measure for the similarity of these groups. In particular, the number of relations between a group and itself (selfrelations) is a measure for the selfsimilarity or symmetry of the group. It is therefore important to classify and enumerate all the (self)relations for a given group; we shall see that these relations represent isomorphisms between factor groups. Consider two (Lie) groups A and B. A relation R between A and B is a subset of their direct product R _ ~ A × B = { ( a , b ) l a ~ A , bEB}.
(3)
We define a relation R to be compatible with the group structure i f - for all values of a, a~, a2~A and b, b~, b2EB - it satisfies the following two axioms:
Axiom 1. ( ( a l , bl ) e R A (a2, bz)~R)~(ala2, blb2)~R. 400
PB(R)
def
- - { a ~ A ] ~b~B [(a, b)~R]}, def
{b~BI 3a~A [ (a, b)~R]}, def EA(R) - - {aeAI (a, 1 )eR}, def EB(R) - - {beB l(1, b)~R}. The sets EA(R ) and Ea(R) are obviously invariant subgroups Of PA(R) and Pn(R).
Theorem 1. Let R__ A × B be a group compatible relation. Then P A ( R ) / E A ( R ) ---Pa(R)/EB(R). If this isomorphism is expressed
f:PA (R ) /EA (R ) ~ P B ( R ) /EB(R ), t h e n f c a n be chosen so that R = { (a, b ) ~ P A ( R ) × P B ( R ) I f ( a E A ( R ) ) = b E a ( R ) } . For the set of relations between two groups (com-
Volume 223, number 3,4
PHYSICS LETTERSB
patible with group structure) there exists partial ordering "inclusion" R~ ~ R 2
meaning VaeA, b~B
[(a,b)~Rt~(a,b)~R2].
The inclusion means that R~ is the stronger relation in the sense that it contains more information to know that (a, b)~R~ than (a, b)~RR. In this paper we are mainly interested in relations for which P A ( R ) = A and P a ( R ) = B . A relation is called minimal if and only if no other relation with this property can be included in it.
Theorem 2. Let Rj and R2 be two group compatible relations between the groups A and B. Assume furthermore that PA(RI) =PA(R2) = A
and
PB(R,) =PB(R2) =B.
Then RI ~R2 implies the existence of an invariant subgroup H of the factor group A/EA(R~ ) such that A/EA(R2) -- (A/EA(RI) ) / H --- (B/EB (R~) )/f~ ( H ) ----B/EB(R2), where f~ is the isomorphism of theorem 1 applied to Rp 3. A measure for the symmetry of a group
From a given group A we obtain new groups by division with invariant subgroups H of order n (H). As we are interested in the possibility that the factor group is isomorphic with the original group, we consider discrete invariant subgroups only; it is then easy to show that H must be a subgroup of the center of the group A. A relation R between two groups A and B is according to theorem 1 given by an isomorphism A/HA ~ B / H B .
(6)
The group B may well be identical to the group A; then the relation is called a selfrelation. Thus a selfrelation for a group corresponds to an isomorphism between two of its factor groups
A/Hi~-A/Hj,
15 June 1989 (7)
where one or both of the invariant subgroups may consist of one element only (the unit element). Now we can characterise a relation by a level pair (n~, nj), the orders of the two invariant subgroups H~ and Hi. All groups have the trivial selfrelation (4). Nonabelian groups also have the associated inner automorphisms (5). We shall ignore these relations which occur unavoidably on the ( 1, 1 ) level. Likewise we shall ignore relations which are not minimal. We consider all such relations trivial. A group with simple algebra has a finite discrete center; hence a group with simple algebra has a finite number of possible level pairs. A group with continuous center (e.g. Ul ) has an infinite number of level pairs. Some groups have nontrivial relations occurring already at the lowest possible level pair ( 1, 1 ), i.e. outer automorphisms. There are many examples: SU2 (Peter) × SU2 (Paul) is obviously symmetric under the exchange of Peter and Paul, an outer automorphism (we denote it pp-symmetry). Many simple groups also have outer automorphisms, e.g. SU3 and higher SUN groups for which the symmetry corresponds to charge conjugation (we denote it cc-symmetry). Also the group U~ has cc-symmetry. We also know groups which have no selfrelations at all, e.g. SU2: It has just one factor group SU2/ Z2~SO3. These two groups have no outer automorphisms and they are not isomorphic. It would have been very attractive to use the lowest level at which selfrelations occur as a measure for the skewness, lack of symmetry. If we do that groups like SU2 or SU2 × SU2/Z2 would win over all groups with continuous center (including the SMG) in the competition of being most skew. We would, however, like to consider such groups as disqualified as they are not enlightened by at least a single UI in their algebra. This means that we shall close our eyes to the occurrence of a single cc-symmerry in combination with all inner automorphisms when we look for the lowest level pair exhibiting symmetry. A second cc-symmetry (which occurs when the algebra has several U l'S) shall, however, not be overlooked. With this qualification we propose that a useful measure for the skewness (lack of symmetry) of physical groups is the lowest level pair (n~, nj) 401
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at which a relation occurs; actually we consider pairs (1, nj).
4. Results In our investigation of symmetry properties we have considered all connected compact groups of dimensionality up to 12. Among the 230 groups very few (semisimple) groups have no nontrivial selfrelations; seven non semisimple groups have first relations at the level pair ( 1, 2): UI, UI × SU2 × SO3, U3, U 1×Spins, U 1×SO5, SU2×U3, SO3×U3, five have first relations at the level pair (1,3): U2, SU2XU2, U2× SO3, U2× SU2xSO3, (R× Spins)/{ (2n,-)P} ~1, and one group has first relations at the level pair ( 1, 5 ): SMG. All others (179) have nontrivial relations already at the level ( 1, 1 ). We may state the result in a slightly different way: count the number of connected compact non semisimple groups (of dimensionality less than 13) that have their first relation at a level (1, n) or higher. The result is n=1~192, n=4~l,
n=2~13,
n=3~6,
n=5~l.
The group occurring for n = 4 and 5 is of course the SMG. Groups of structure similar to that of the SMG are distinguished by a relatively high degree of skewness, i.e. the first relation occurs at a high level pair. Thus for the two analogs of SMG corresponding to the algebra U I × S U 2 × S U 3 X S U 5 the lowest symmetry bearing level pair is ( 1, 11 ). The group SMG is the most skew of connected compact groups of dimensionality up to at least 12 when we disregard semisimple groups. The non semisimple groups contain at least one Ut as invariant subalgebra. A single UI has generalized symmetries, i.e. relations for any level pair (n, m) because the group Ut/Zn is isomorphic to U~ itself and thus to U~/Zm. This makes any group with a UI as a factor very symmetric. Nevertheless some groups that do contain U~'s in the algebra are rather skew. The rea~ The generating element of the subgroup divided out is characterised by a step in the R-groupand the non-unit center element of the Spins-group. 402
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son is that such groups are obtained from their covering group by division by a discrete invariant subgroup with nontrivial projections in both the UI factor and other group factors. Therefore the original generalized symmetry of U I is destroyed; an example is U2 =~(Ut × SUE ) / ( ( 2 rt, - )P} which has the first relation on the level (1, 3). This mechanism plays a decisive role in the selection of SMG.
5. Some physical remarks Mathematical considerations have shown us that SMG has a special status among compact connected non semisimple groups when we apply the criterion of skewness. We may therefore assume that skewness has physical significance when we go to models beyond the standard model. In grand unification models like the SU5 model of Georgi and Glashow [ 6 ] skewness seems to play no role(?). The group itself and the special Higgs field that break it are taken out of the air to give the right break down products, but without any further justification; it appears to be a bare accident that the surviving SMG is skew. We feel tempted to discard such models and instead look for models for which skewness is essential. Such models should include some mechanism that looks for the symmetry of groups and capable of breaking down a group if it does show some symmetry. One suggestion is to generalise models for confusion. In confusion models [3] there are hypersurfaces in space (confusion surfaces) along which some degrees of freedom are continued in a modified way, e.g. the degrees of freedom on one side of the hypersurface are continued on the other side in their images under some automorphism. The effect of confusion is to prevent a local gauge symmetry from being also a global symmetry. In a generalized confusion model one imagines the continuity conditions modified to respect some group compatible relations. This might be formulated in terms of two paths approaching each other from either side of a confusion surface
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Z--E
gAu(x)dxU),
(limexp(i ~ \E~O
\
gAu(x)dxU))eR.
[imexp(i I
(8)
y+E
Here E is a four-vector along the oriented normal of the confusion surface. These conditions may prevent a global symmetry. Another suggestion is related to the observation that a more skew object has more possibility for alignment than a less skew object. The skew object should, of course, be translated to a skew group whereas the possibilities are translated to the different ways it might act on some system, e.g. a field theory. We shall here introduce a little formalism: Assume a system L with elements leL and relations Q e L × L , and a group G with elements g. The group may act on the system in many possible ways each described by a function F:G×L--,L for which
F(g,/)eL is the image of l under the operation representing g. These functions are required to satisfy the conditions
F(g~g2,l)=F(gl, F(g2, l)), if (1~, 12)eQ
then(F(g,l~),F(g, lz))eQ.
(9)
One can show that the number of such functions F(g, l) must (under plausible conditions) be the same for groups of equal volume. We then seek, for all groups G having a given volume, corresponding sets of functions F(g, l) such that G is a symmetry of the system L, i.e. (9) is satisfied. The group G may have an outer automorphism g ~ a (g). If this is the case one can from one F define another by def
Fa(g, l) ~F(a(g), l), but this new F a is already a member of the set of F s . Thus two of the functions are used to implement the symmetry G in the same way up to an inessential automorphism.
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In general the a priori number of potential ways of implementing a group is reduced by a factor equal to the number of outer automorphisms of the group. This shows that a group with more automorphisms has smaller chance to be implemented as a symmetry group for a system than a group with fewer automorphisms. We need some mechanism for amplifying this effect, e.g. by letting it take place in any little spacetime region. One may note that the break down of the standard model associated with the break down of the SalamWeinberg theory cannot be related to any of these mechanisms as the surviving group U3 is less skew than U2.
6. Concluding remarks We have found a criterion that singles out the SMG (S (U2 × U3 ) ) among a large collection of groups potentially useful for construction of gauge theories with associated fermion generations. The criterion is skewness expressed by means of the first level pair at which a group has nontrivial selfrelations or generalized outer automorphisms. Actually the SMG is the most skew among all connected compact groups with continuous center and dimensionality up to and including 12. There are 192 such groups. Unfortunately we have had to add the word "nontrivial" to avoid the ever present identical relation as well as symmetry under charge conjugation both combined with inner automorphisms. These symmetries are shared by all groups having a continuous center and hence a U~ subalgebra. Thus we have considered the symmetry under complex conjugation as trivial on the same footing as symmetry under the identity relation. That we have looked only among groups with continuous center is certainly a weak point. On the other hand this point may probably inspire new development. In spite of this weak point the fact that the criterion of skewness chooses the SMG uniquely among 192 possible groups strongly suggests that skewness has some physical significance. Arguments concerning grand unification models with break down to the standard model appear not to apply skewness; this is one reason to consider them with scepticism. 403
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We propose two possible ways to explain that Nature apparently prefers a skew gauge group. One is a generalisation of confusion models [ 3 ]. The other is an abstraction from the fact that it is easier to close a suitcase with five liter of wine in ordinary bottles than one with the same q u a n t i t y in one spherical ( a n d highly symmetrical) container. Both ways may appear natural in models involving randomness at some energy level.
Acknowledgement One of the authors (H.B.N.) wants to acknowledge inspiration following discussions with B. D u r h u u s years ago.
References [ 1] S.L. Glashow, Nucl. Phys. 22 ( 1961 ) 579; A. Salam, Proc. 8th Nobel Symp. (Stockholm, 1968), ed. N. Svartholm (Almquist & Wiksell,Stockholm, 1968);
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S. Weinberg,Phys. Rev. Lett. 19 (1967) 1264; H. Fritsch, M. Gell-Mann and H. Leutwyler, Phys. Lett. 30 (1973) 1346. [2]L.O'Raifeartaigh, Group structure of gauge theories (Cambridge U.P., Cambridge, 1986); L. Michel, Group theoretical concepts and methods in elementaryparticle physics,ed. F. G~irsey(Gordon & Breach, New York, 1962). [3] H.B. Nielsen and N. Brene, in: Proc. XVII1 Intern. Symp. (Ahrenshoop) (Institut f'tir Hochenergiephysik, Akademie der Wissenschaftender DDR, Berlin-Zeuthen, 1985). [4] R. Gilmore, Lie groups, Lie algebras, and some of their applications (Wiley, New York, 1974). [ 5 ] H.B. Nielsen, Random dynamics and relations between the number of fermion generations and the fine structure constants, in: Proc. Zakopane Summer School (1988 ), Acta Physica Polon., to appear; D. Bennett, H.B. Nielsen, N. Brene and L. Mizrachi, Confusion and the heterotic string, Proc. Paris-Meudon Colloq. (September 1986) (Wolrd Scientific, Singapore, 1987). [6] H. Georgiand S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438.