PoS(CORFU2014)045 × ) 2 ( U ( S http://pos.sissa.it/ lly can be characterized as that rd Model Gauge Group l the way to the Planck scale except orkshops on Elementary Particle Physics ion(roughly). At the end I argue for the n e Commons Attribution-NonCommercial-ShareAlike Licence. ∗ as developed through hard phenomenological work historica )) [email protected], [email protected] 3 ( Speaker. The main point is the finding, that properly defined the Standa Lie group, which has the minimal size faithfull representat for some “see saw neutrinoes”. scenario that the Standard Model will remain valid almost al U ∗ Copyright owned by the author(s) under the terms of the Creativ c ° Holger Bech Nielsen Niels Bohr Institutet, Blegdamsvej 15 -21 DK 2100Copenhage Small Representation Explaing, Why Standard Model Group E-mail: and Gravity", 3-21 September 2014 Corfu, Greece Proceedings of the Corfu Summer Institute 2014 "School and W PoS(CORFU2014)045 his fine st other n explain ks at least nt, that the l sized balls sed even by the see-saw as comming d that alone, nly Standard ious field for d it says that it is the attitude oblem. ics fundamental Standard Model by LHC just con- ot give in principle f such bound states ituation would call Holger Bech Nielsen rs -, and so we may saw neutrino physics ral - in pracsis below ll choose rather than the Standard Model he information as esti- ding so far any new fun- , i.e. 2 kg (=100000 ton), which can be understood in terms 8 1. Is it satisfactory to have a theory that is only renormalizable, but does n 5. Do we not need a special inflaton field, r can we use the Higgs field. With o 2. Including also gravity can we even get a renormalizable theory? One thin 6. We must understand also the problem with the Standard Model that if we ha 3. Neutrino-oscillations at least seems to require some new physics, at say 4. Can we understand the dark matter (as essentially “seen” astronomically) from the Standard Model.physicists would like Here to answer: it Yes we iswith can imagine the that dark enormous matter I being mass some and of pear about my 10 collaborators contrary to mo fields. Only bound states - ofare 6 top needed, + but 6 no anti new top fundamental quarks particles.[5, - 36] and condensates o of a new vacuum which results from just Standard Model with no new phys 7. The fine tuning problems would also have to be understood. Here it is that finite results? Model we should use in principlea only bound the state Higgs comming field, from or the perhaps Standard some Model. myster something new has to happen when approaching the Planck scale. the anomaly for conservationphenomenologically of observed excess lepton of and matter baryonour over own antimatter, numbers existence, which would would is have been witnes have washedneeded mea away. for Presumably neutrino the oscillations see- anyway would be able to take care of this pr neutrino scale. of the present article andto really of “solve” several the earlier finetuning worksthem,we problems of by shal finding ours truly some that postulate clever we just symmetrytuning an sha that rule, as ca which simple we as proposethe possible is coupling finetuning what constants we law are call or finetuned Multiple so rule. as Point to T Principle[37] arrange an that there are seve To day we are in the situation that the highest accelator enrgies reprsented Continuation of work with Don Bennett on “What is special about the • • • • • • • firms the at the presentdamental scale particles best - theory, especially the not Standardhave the Model, to much by face hoped not the for possibility fin supersymmetry thatfor partne understanding there the is problems very with long such to pure the Standard new Model physics! scenario Such a s Group”[2, 1] presented by Holgermated Bech by Nielsen. Svend Erik Seeking Rugh, to SyrlykkeGroup. and extract HBN[11] t yet unexplained for 1. Introduction Smallest Representation PoS(CORFU2014)045 8 um (2.1) imagine although in a very like white niverse as equivalent vacua beig so that the el for dark next.). One we have luck for the image e the Standard so we could in et there) : that concept: scape neutrino- 2, 1, 4]! (Earlier l and a weight 10 nsity is fixed to be algebra Holger Bech Nielsen lmost to the Planck ave representations, ase with the bound state hat of the “Adjoint” , Gauge groups ) 2 | )) dg + g inside the Standard Model ( U (with respect to the Lie − ) )) g 3 ( ( U representations. U ( 3 | ( × ) Tr 2 ( ∗ faithful U ( 1 S dim = 2 ds ) wins. ) 3 ( SU × ) 2 ( SU × ) 1 ( U How we at least - Colin Froggat and I - have a chance of proposing a mod All Lie groups (potential gauge groups for the right model for Nature) h How we might get almost all the structure of the Standard Model by requiring to matter[36, 5] inside thethat Standard the Model Standard in Model is ascale the complicated except final for way, answer some so for see-saw very that neutrinoes). long we up could in energy(a How we might escape most of the arguments for there being “new physics”, It is very natural, although slightly convention dependent, to define a volume But at least for dark matter [36, 5]we have a proposal Standard Model Gauge group [3] observed astronomically, energy densities. Each timesmall a / vacuum energy essentially de zero awith this coupling as constant I (combination) shall gets review,imposed then finetuned. small we energy may densities[38, If solve 37, three 12]. finetuning problems by three three - vacua with very small, meaning of the order of the dark energy in the u and are properly represented by the sepecial way that representationsI (of worked the with particles) Niels be BreneModel the on group smallest an [10].) alternative ones[ idea also seeking to characteriz fact take the message fromthere seems LHC to so be far no new seriouslyoscillations physics (but form (even I signaling this must new message though physics). admit is that not we y cannot e of a Lie group being mapped into a representation using a “natural” distance I and Don Bennett invented a “game”[2, 1] between possible in a slightly complicated modelwith with a bound Boson state condensate of ofdwarf-star-stuff, 6 such but top bound surrounded + states. by 6 a anti Ballscondensate skin tops from made seperating a and from phase a a material without vacuum new much this ph vacu condesate and of the size like a pear kg make up the dark mattersuch ball (about fell one in astronmic Tunguska unit in between 1908. one ball and the Crudely the main point is this: • • • • • • • Representation 2. Smallest Volume of a Faithful Represntation Compared to t There are many things I would like to tell about such as: Smallest Representation PoS(CORFU2014)045 (2.2) ough we along the e same all e of unitary aithfull ones esentation can nces tations symbolizes an oups), only/mainly is taken for the entation for the ic. | esetation volume dg ... Holger Bech Nielsen | + ferent. (Truly a faith- g and replacement for an adjoint , ) g 2 | )) dg + is the dimension of the representation g ( U =0 and so the distances become 0/0= “ill dim − ) dim g ( U ( 4 | ( Tr ∗ . The number † 1 dim AA = = 2 2 | ds A | is in correspondance with the group itself and has the (inner) geometry dim . , let us say “Adjoint”(now in quotation marks). ) is a (unitary) representation of the group element g ) ( ) g U ( U Explanation of the Symbolic Figure with the Groups and their Represen Even for Lie groups with Abelian components we may construct a Then the Standard Model Gauge group “wins” by having the smallest repr matrices matrices to be defined as representaton Using the metric Typically any group has aritrarily large representations measured in the metr for a faithful representation comparedgroup. to the volume of the “Adjoint” repres hereto infinitesimally close element, while the numerical value symbol But there is a lower limit to the size of the faithfull representations (while non-f where manifolds metric which only deviates from that on the group by a numerical factor bing th the representation copies of the group look verythe similar size (the of same for these simple images(= gr thefull different faithfull representation representations) considered are as dif Riemannianmatries manifold of order imbedded into the spac along the manifold; but this is true only provided one keeps to only count dista count also the trivialmight representation, argue which it in to have some indeterminedbe sense size; said is the to point have infinitely dimension is small, of that the the alth representation purely zero repr defined”) • • • • • Smallest Representation PoS(CORFU2014)045 (if (2.3) of the pendent esentation representa- roup that it group taken dimension”)th e goal quantity d w.r.t. dimension Holger Bech Nielsen inverse square quantity” The by Nature chosen . G d th root of the volume, so A / ne took a definition being G 2 ave the smallest - relative d ¶ one of course):Thus it selects cted the gauge group of the ) ) tation whatsoever! is the dimensionality of the Lie A F ( ( G (which has the same Lie algebra d )) Vol Vol 3 µ ( ) has the smallest “goal quantity” ) U = can compare representations even for 3 ( G × d su ) / 2 2 ( − ⊕ ) U ¶ ( dimensional space-time leads to the “winning” 2 ) ) 5 ( and for any cross product of this group with itself A d F su ( ( G ⊕ for Vol Vol R ) µ d ( = SO 4 and thus our goal quantity puts itself into the series of explanations = d Selecting Goal Quantity G . The crux of our principle: “goal quantity” 3 or = × ··· × d G .) By this being balanced w.r.t. dimension we have in mind that we should avoid that × G G × ” compared to the same linear size of the “adjoint representation” By comparing any representation size - say the volume taken to the (1/“group We want to use the volume of the representation relative to that of the “adjoint” We want to choose the “goal quantity” ( = “the score”) so as to be balance power - to the “Adjoint” (slightly generalized) we different groups tion. of the (gauge) group being tested; therefore we want to take the group “goal quantity” would have sowould much make dependence this dimension on selcted the rather dimensionstucture. than of some As hoped the a for sign Lie more of g dimensionis such inde a arranged balance to having be been the at same least for attempted a is group that th that it becomes rather say the linear scale size ratio. (Here For quite conventional and accidental reasons we started to consider the G linear - understood in theF metric - size of the “the image of smallest faithful repr Let us remind the reader about why this “goal quantity” is so important: Let us even mention that requiring the minimal “goal quantity” for the Lorentz- (analogue style to Johannes 3.14) • • • • dimension being 2.1 Reminder of the main motivation for the whole game with this “goal gauge group- namely the Standad Model group S for simplicity as the compact one Smallest Representation as the Lie algebra of the Standard Model Nature so loved small (faithfull) representations,true that model it (= even the sele Standard Model)to so the that “adjoint”(slightly the generalized) gauge - group possible could faithfull h represen one uses the definition just above;like but the one one in should earlier ask paper for anor the inverse explains or biggest the inverse if square Standard o of Model the group. present This means that we look at PoS(CORFU2014)045 ) ) 1 ( [14] (2.4) rather U (mean- ) ∈ e 1 ) ( compared . (It might for the Lie δ n of charge e system of r U i ] defines the ( ns[1, 7, 8, 6]. e not quite as st the extension quantum). ups. ns - all the irre- roup the 4=3+1 to win. und the monopole t we call a “quan- ount and managed Holger Bech Nielsen nt charges entations of the group. . In the article [1] I did ) the quantization rule reads 1 )) the weak hypercharge, so that the 3 mod ( Y ( 0 U × = 3 is 1/3 for quarks -1/3 for anti quarks, ) / 3 2 ′′ / ( ′′ where the group element is exp U ( ) δ triality ie 6 in question. rather than just Lie algebras - which is the only ( triality “ rather than only from the Lie algebra is given by a group S + 3 W restrictions on the matter field representations I groups Further “ group G . group + 3 2 W 1 matrix exp I / Treatment of Abelian Part Y + × from its 2 / Y = group Q is the third component of the weak isospin,and 3 W . I R Have in mind that working with A priori there are no adjoint representation to compare with for Abelian gro The charges get only quantized, when one considers a compactified Lie g A priori an Abelian Lie group has a continuum of different representatio A short way is then to define the factor in the “volume of the representation significance of the thing that matters for the Yang-Mills gauge field couplings - O’Raifeartaigh [3 than where Indeed the restriction from the be good to think ofobtained this from type the of existence restrictions of as (Dirac)that generalizations can monopoles. of just the In deliver quantitzatio fact the David restriction Olive for fo the Standard Model [13].) to the inclusion of the quarksrule with of their charge - quantization(charge being somewhat mysterious an - integer multiplum of of the the simple charge Millikan 0 for gluons. (Note that for the Standard Model this “quantization rule” is ju ing a representation by a 1 electric charge is ducible ones are one dimensional representations - namely having differe to that of the replacementallowed representation for under the the adjoint representation” formally by means th (charge) on which representations are allowed may be described by wha tization rule”. Such aalgebra quantization which would rule select is out those the representationsE.g. restriction allowed as in on repres the the case representations of the Standard Model • • • • • • The easiest way to treat the Abelian part: Smallest Representation attempted to be given to theGetting question: only Why do to we an haveimpressive just equal as 3+1 race space if time between 4 dimensio (the 3 phenomenologicalinvent and some value) 4 “improvements” had taking been space crudely sigled time theby out dimensions such whole alone compliation is Poincare - group even of if into cours a acc bit motivated, somewhat less beuatyfull -2.2 to make Special problems with the Abelian invariant subalgebras: PoS(CORFU2014)045 ed here, lian part nsion”th 4 [1] (di- n invariant sily be just numbers”) = rge number of d tandard Model rts hydrogen to sibilities among t one may take a hile the dimension resentations of the Holger Bech Nielsen dard Model Group ombinations, which on-abelian invariant xygen burning with ms from the simple re these numbers of y pregnant regularity about being as lucky re the really true new at one that Nature has ps are required trivial. gestion, does not have t of information not yet ). It will turn out that m of giving the number like even LHC with very ) 3 m Structure of the ing about the Planck scale ( h a selecting a gauge group. SU × ) 2 ( SU × ) 1 ( U 7 towards the true model beyond the Standard Model by seeking . The main point of the present talk is: Higgs ( Lie-algebra-wise equivalent to )) 3 ( U Nature give us a hint × ) The number of allowed representations of both kinds are typically infnite, bu Take the ratio of the “number” of allowed representations of the maximal Abelia subalgebra, when no restrictions is putsubgroups to we which representations use. for Then the compare n thatare to allowed, if the the “number” representations of of those the(Really charge non-abelian there c invariant are subgrou infintely manybut of it both is two not types so difficult of to charge define combinations a mention meaningful and finte ratio of the two infinite “ limit, and then this ratio isas the extra the factor contribution to from put the onroot the abelian and volume part.At the ratio minus the for second the end power non-abe one to then get to takes our the “goal “total quantity”. dime 2 But really the amount of information in the not explained gauge group of the S ( • • Gauge Group U ( 3. Noting Philosophy of Extracting the Teaching of Nature fro - rather unbiased -S a relatively simple property characterizing just the Stan The reader should have in mind that we physicists having only accelerators small energies per elementary particle comparedphysics to may say show the up. Planck scale, Thus whe webeing discovered would at love, our if “low” we (compared could toor guess Planck just from scale) somewhat energy some someth higher ver upas in Dalton energy [19] scale was, physics.rational when This volume he means ratios could we between argue dream gasses,hydrogen for to that the make made only existence water chemical of inone reactions molecules/ato a part (e.g. very oxygen). simple But rational o such ratioonly a of to simple the be volumes: rule, concerning so two numbers, simple pa no, thatThe it we important could can also thing trust concern is its structure, just sug suc which that nature the has number of selected reasonablypossibilities the comparably that so simple were remarkable presented pos one, in ourof is paper digital sufficiently by cifers high. Rugh needed to et specify al. Itchosen. among [11] we We[11] similarly in Rugh reasonable the et choices for al. just th explained called measured this in number of bits. cifers If neededbits, we the it make amoun almost a model, must that beaccidentally in “a true, remarkable this but choice language if of explains nature”. it a la only explains a small number of bits it could ea I can use almost the same idea to characterize the phenomenological dimension To make Smallest Representation is only -according to our, i.e. Rugh et al’s, estimate 8 bits (hardly two letters) w mension were also explained muchFermions[18, 4] earlier and [6, the 7], and even for telling the rep PoS(CORFU2014)045 (3.2) (3.1) odel is like ing up ca 150 . we refered to as [11] )) ge charges in the 3 Holger Bech Nielsen ( : If you assume that al number, character- G U ich I shall not tell you d the game of getting the / e with the same gauge × 1 ticle, then there can only m of fermions a lot, and ) ) j e only have energies very 92 bits have been counted 2 j d ( ) ≈ 2 A U 2 F ther Lie algebras, ( e e ( j Group S ∏ Amount in bits 143 to 155 21 8 3 6 4 Abelian factors 8 ∗ i Nevertheless we shall now seek to get some usefull d i ) A F C C ( i More Precise Strategy: ∏ Unexplained information Parameters Structural Information divided out as: Gauge group 3+1 dimensions Spin-distribution Higgs-representation , and simple groups “goal quantity” = ( simple 3 bits, i.e. about a half letter. ≈ Rughs et al.’s Information Counting; Not Explained Information. 2. takes its biggest value for just the Standard Model 1. is relatively The point then should be that this on groups defined “goal quantity”, a re The information (not explained) in the parameters of the Standard Model mak 4 gives • • = bits is like the information4 in letters. 30 letters, In and all the a 21 half bits line. in (Information the in structure the of Weyl-representations Standard M (the notation of which I shall first explain to you later), which as explained, but that maywe be can exagerated? only The hopeStandard idea to Model we observe because refer “you” fermions, to (=small which here the compared is are to physicists the this ) mass presumed are fundamental protectedbe scale so by at fermions “poor” our the that of disposal w per gau one par charge handedness, quantum when numbers. there This existthus requirement non gives/explains restricts of a of lot the course of opposite thethe information. on 92 syste It bits) is this information/explanation which Don Bennett and I found -about by now complicated - calculations a and quantity speculations, depending wh on/defined for groups, or at first ra Smallest Representation d izes the Standard Model gauge grouphighest by pointing value it for out this as “goal the quantity”. one that wins information out of the “book of nature” this way. PoS(CORFU2014)045 G = en Lie (exclud- 1 95782451 F . 2 C the Standard √ p on the figure aper, while the to be the gauge is the dimension e inverse square is: , just without the G are in fact Dynkin Holger Bech Nielsen d y reached when the en the distances (or A n corresponds to the ou see the relation of C r size ( determined as llest representation in / t ways of describing a square of this meaning t subgroup of which the CF with the smallest F . The natural goal quantity would be ) 2 ( say, which is faithful to the volumeof the U F 9 th root of it to get what we culd call the linear scale ratio 6093199492. . G 0 d are respectively quadratic Casimir operator values on the adjoint = F C get the largest goal quantity, 3.91782... But it is very nearly followed and on the irreducible representation )) A 1 and 3 ( 69345184 group; that means it is in fact A . enumerates the simple groups in the essential cross product making up 2 ) C U i √ . 3 × ( G ) 2 SU ( denotes the dimensions of these simple Lie groups. Of course i U d ( S cinsidered is essentially the cross product. Therefore the ratios with our goal quantity as the goal which is to be the largest to win. You see that root of the volume of the small representation to the adjoint is therefore put u G G Let us immediately on the following figure symbolize the result of the “game” betwe To appresiate the just described correspondance of the two equivalen Here the quantities d 581451905 and e.g. . Model Group groups as the times it tookgoal the photo runners. was Apart taken from is thegroup identified squaring to with then be our the chosen old distance by paper the linear Nature goal size. to quantity. But be To that realized win correponds -the - to quantities requires i.e. getting discussed that the in you our largest get older value the work of [2]. sma the inverted adjoint representation and take the corresponding to the volume ratio.root of This the most goal natural quantitydifferent used goal goal in quantity quantities Bennett’s would expressing mine be the firstsquare th same paper. of it) illustrated On being as the identified the figure withmore relation the y natural goal linear betwe quantity size of of the minetime small and representation the Don compared Bennetts runners to old needed the p for adjointthe the running a specified distance. This ratio of linea to take the ratio volume of the small representation of the full Lie group by the next in the run, which is essentially the Standard Model Gauge group strong interaction indices[15]. The index 0 and the symbol goal quantity one should note that the numbers on the figure are related like th (irreducible) representation ing the non-faithful trivial representation)group for the various simple invarian Smallest Representation PoS(CORFU2014)045 ebra do in el so far up in rd Model with ndamental scale gauge degrees of and also how we Holger Bech Nielsen g group), but only a haracterisation of the small representations menologically needed a: model very far up mental significane, but e of our story of how to modifying the Standard ard Model gauge group, w physics by rather little y extra gauge degrees of .r.t. to the idea that all the relevant then the Standard mainly above looked at the should be essentially no new we just have the following equivalent charge quantization rule groups ↔ relevant model in some very significant region of 10 the were selected by Nautre. If such a selection of indeed the )) ” 3 ( O’Raifeartaigh[3], Group Structure of Gauge theories,University U Telling about certain restrictions on the representations occuring in group × ) that is part of a bigger or more extended model, which just happens to 2 ( U ( S Correspondance group-structure sub-model We can assign more information to telling what the gauge Lie algebra is by telling charge quantization rule “group” also what is the “gauge the model (restrictions comming from that different groups with the same Lie alg general not allow all the representationscertain of subset the of Lie them. algebra(or its coverin (This connection is due to: We have above put forward an attempt to find out, why precisely the Standa But such a speculation meets severeIf problems: see saw neutrinoes needed to explain the observed neutrino oscillations It is therefore at least suggested, that if our above characterization is is equivalent to • • in energy Have in mind, that for our purpose of using gauge Press Cambridge (1986)) ways of assigning a certain further information to a gauge theory Lie algebr 4. Outlook from the philosophy that Standard Model is the true Smallest Representation could obtain an excess of matterfreedom over charges, antimatter then cosmologically, they are may just not without gauge change group, the gauge we group. could Since sayModel we Gauge that Group an would extension not of beselect the a the Standard challenge gauge to Model group. the not So interestfreedom being such and speculated relevanc a could see be o.k. sawfermions neutrino ( But with extension it masses without is very an really not small a comparedof priori to energy o.k. the - w Planck should scale besee-saw - mass neutrinoes identified protected as should w.r.t. a namely to fu be the chargeless gauge singlets group. under the The Stand pheno Model must be a good theory overphysics a at very say large the energy scale, LHC. and there the part already discovered in 2014. increase in the energy of theStandard LHC say, Model. then it The would Standard make Model norather sense would just to namely be find have a a no c speial funda conditions. If namely, as many physicists hope for, there would appear ne Standard Model gauge Group by- some is principle the - truth as behind, aboveenergy, then the that principle it this of means Standard that Model the is Standard indeed Model must be the right mod its special Lie group PoS(CORFU2014)045 so as to in in fact ecomes a hysicists, I e the gauge adjoint repre- m of, why the s contributions, niverse sits just finetuned e are several vacua ams etc., has been Holger Bech Nielsen int principle” really the Standard Model xplain the principle st D. Bennetts talks nd perhaps a scalar ould think of as the planantion for, why ndamental” scale of ne takes the point of tral and there would gnitude for the weak e tuning problems of saw neutrino physics. le and thus should be and my own) multiple ns together in this way en choose to have only ations. ow to obtain dark matter symmetry allows it. of view) energy densities the repsentations realized ve small energy densities) , then the usual problem of, ll (in older formulation of MPP 11 Similarly a pure or minimal supersymmetry extension - which also would not chang If we, however, not only care for the gauge group, but also for that In any case I have, however, personally attempted to seek models, in which First of all one would ask one-self: Is there any mechanism, that coupld e But if we now make this assumption of multiple point principle, then we succeeded Can we have dark matter in the pure Standard Model? Contrary to most other p group - could be accepted without spoiling the value of the above consider as fields in the modelsentations should could be be small, claimed then to theview, not gauginos, that have which one the must has smallest belong to representations, to only unless the take o the smallest representations as far as super Smallest Representation and thus if the Standardthe Model only group gagueg were group, selected theconsequently by be see-saw no some neutrinoes way great would to princip get have them to mass-protected.) be totally neu calculate crudely the energyenergy scale - of taken weak to interactions beinteraction the relative energy Planck to scale[38]. scale some - “fu and obtain in fact a good order5. of ma Can we get any hint of the Theory Beyond the Standard Model ? and my collaborators C. D. Froggatt,...in [36] pure have Standard developped an Model, idea only forpoint extended h with principle[37]. our This (D. mutiple Bennetts andabout point Froggatt principle commodities - being which fixed developped ratherat from a than fir phase intensive quantities transition multiple - point, states wherein that the the coupling u constants have been organize, that there are several different vacuum statesthe with very vacua sma should just have theenergy same densities. energy By introducing densities, such but a not finetuningmeans a principle that as priori this one ha “multiple has po atthe least physics the and hope especially a the“modern priori Standard version” of of Model. explaining the (some “multiple If of) pointand one principle” the they formulates (MPP)[37] all fin what have namely very we that small ther c - (almost a zero version from for high which energy I physics thankwhy point Leonard the Susskind cosmolgical for private constant information is - that so must terribly exist small to comparedabsorbed to the into most cosmological of the constant the multiple from variou cosmological point vaious is types assumption. so of small This loopthe is diagr cosmological several of vacua constant course all proble have not smallmeans really energy that solved densities, we unless but anyway putting have one to the has have assumptio some an assumption ex like the mutiple point one. is everything all the way to theIf Planck one scale or has close to this itassociated ambition except with for of it, some see but no at new leastminimal physics no matter more except field gauge for representation particles like see atsevere in problem. that saw the stage, neutrinoes and Standard a ev Model, then e.g. dark matter b PoS(CORFU2014)045 (5.1) o very ed project e long ago of the matter e we have in epresentation? amely that we mmetry just by way effectively are varied a bit, y accident have al quantity as for e for that say the of the fields than eration is so small ns of the “Adjoint” Holger Bech Nielsen namely be admitted e fields a lot. If you somehow comes out esult of the Standard ast smallest represen- djoint” representation there is of course the ge about the smallest given by our selection of gauge symmetry as e on the dimensionality n roughly. auge symmetry, but that mics in as far as we got d Model group the most tity” means that it varies p itself the various fields lly is a randomly selected general rule that taking it . fields transform exceptionally little ) G lxactification mechanism” working when ´ × ··· × G × 12 G ( gq )= G ( gq Having in mind that the various fields e.g. the fermion fields that belong to these s There may be one hint for what could go on beyond the Standard Model, n This story of this in some sense minimal variation under the gauge transformation could assume that even beyond theprinciple Standard of the Model right the gauge gauge group to groupthat make is our our still goal principle quantity maximal. of It maximizingconstructing should the the goal goal quantity quantity balanced is it notof so completlely well the against unique, Lie simple group, becaus dependenc that wefor have a arranged cross that product our of goalthe a quantity cross group obeys product with the factors.That itself is a to number say of we times had arranged leads to the same go fields could be explained by a modelsome of approximate the gauge type that symmetry Nature could at bebecome first found had exact no by [21]. g accident, and If thenbest one in chance looks some to for find one, anas when accidental possible. the approximate variaton If symmetry, of one the onlyaction fields is varies due approximately the to the fields the same a symmetry aftervary little op the the bit fields shift there a as is lot before, the a thanthen action if much also is one higher the expected varied chanc to action th also will varyan from a approximate contuity lot, while vary gauge if a symmetry the bit.under fields under only one So a that there gauge leads is group the toModel giving best bigger group chance small winning variations the to of variations game b the to fields. havethe the fields biggest This exceptionally value means of little our that and “goal ouracident. has quan r This the could best favour a value philosophy to thatone be the and approximately fundamental that theory a even actua good gauge sy symmetry appeared by some “à it is there only approximately at first. Actually we with D. Første and Ninomiya hav proposed such an exactification mechanism[21].something The that very were idea there of at thinking firstat approximately the in end a random almost theoryof by and Random itself then Dynamics one [17, could 20,possible say faithful 21, is representations 22, very as 23]. a muchthe hint in Thus suggestive pointing explanation the we towards for spirit Random could our Dyna finding of read that my the it long messa came from belov a5.1 random Hint actio about Gauge Group Beyond Smallest Representation of Nature selecting strongly the principletation seeking in the the smallest sense faithful or of at having le the smallest volume compared to thatsmall of representations, the are “Adjoint” shifted r exceptionally little forsomehow a given measured shift by the of “A the grouprepresentation element. defined It distance means that concept counted onfor in the the the gauge by Lie group mea group selected by the “lazy” Nature. Lie group You could in almost the call senseas the that little Standar it as varies possible! for a given variation in the grou PoS(CORFU2014)045 G × )) were 3 where ( ) )) U n 3 ×···× g ( with itself inciple” to × . Long tme G U ) n m MPP and G are orders of ,... 2 g × 2 ourse to usual ( × s the extension has as its only g G ) = , U to be defined by sentation among 2 ( call the diagonal amely one factor 1 ssibility to have a o that there could ( t time not known- representing every S g ··· pt - the representa- Holger Bech Nielsen een any group like ( U × ( = that could be claimed f times. The standard S been the cross product )) 2 uld namely propose one 3 g )) sibilty of using this kind ( 3 unifying) group were such ( = t of standard model groups U 1 U × g relative to that of the adjoint ) × r 2 duct of say a group ) ( 2 ( U ( . U ) S ( S G ( × gq )) 3 ( U × 13 ) 2 ( U , so that we even can compare “sizes” of representations ( are equal to each other, i.e. S G i g . In this AntiGUT model each family of quarks and leptons )) × ··· the elements in it are of course of the form 3 ( )) G 3 U ( × U ) 2 × ( × ··· × ) 2 U G ( ( S U × ( × S G groups. )) × 3 ( )) 3 U ( × . The diagonal subgroup is the group consisting of just those elements in U ) different G 2 × ( ) We have defined a measure for the size of representation The concept of this “adjoint” representation is the straightforward conce Including in the counting only faithful representations (which are the ones element of the group by a seperate element) we seek the very smallest repre representation for the same group of tion on the Lie algebraus - for for the the Abelian non-abelian part. part of the Lie group, but has ∈ 2 U i Thus it were not totally correct that just the Standard Model group 2) We once used it combined with what we would now consider “multiple point pr Actually antiGUT has several advantages: 1) Contrary to simple GUT theorie Here we just stress that strictly speaking our small representation principle ( ( • • • g S U ( fit the fine structure constantsas using a the parameter number and of families inAntiGUT - this background which gauge way theory.[] were we at PREdicted tha the number of families to6. be three Resume for Conclusion Smallest Representation Here we denoted the goal quantity for a group G say as uniquely pointed out by our goalof quantity. Rather any it number could equally of wellS have standard model groups. I.e. it could equally well have b can contribute to the smallmagnitude differences hierarchy, between meaning various it quarks may and leptons. help in explaining that there the possible extension of the Standard Modelof antiGUT to type a theories. larger gauge group the pos for which all the component elements to be “the” selceted one.group This with ambiguity a in larger our dimensionality selctionbe than after a all chance the to gives 12 us take of it theof the as po these Standard an cross extension Model products of group, of themodel s the Standard group Model Standard itself group. Model can we group be co withwith found itself itself as a a in number subgroup o a ofsubgroup few of such the different a cross ways. cross product produc group. In It a can group that for is example a cross be pro found as what we ago we worked with agrand model unified which theory[24] were - called anda AntiGUT it that model cross the - product high in of energy contrastin the or of the original Standard c cross ((anti) Model product group for with each itself family of a quarks number and of leptons times - - n a number of times would so to speak obtain its own set of gauge particles. ··· PoS(CORFU2014)045 y A for n (7.2) (7.8) (7.9) (7.1) (7.5) (7.4) (7.6) (7.7) (7.3) (7.10) D and n B Lie groups. eference[2] we ) ul representation N Holger Bech Nielsen n rule) restricts the ( e called even the Lie ubgroupos. The lat- sidering how strongly groups - SO lian subgroup the ratio 2 t faithful representation ) addition to evaluate fac- ) 1 , for faithful repr. N 1 + ( F n 2 ( these C / SO − A when we talk about simple Lie for the adjoint representation 1 C F A = C 1 2 ) − ) ) ) 1 2 1 1 1 1 ) ) ) + 1 1 1 + + 1 8 − − n n n − − + ( n n + − ( 2 ( 2 2 n n n n 1 4 n n ( ( ( 2 2 24 19 minmial among faithful representations )really it n 16 4 − ( = 16 . 14 = n F 2 = = = 4 2 C ) be a spinor representation or a vector representation. / ) ) ) = = 2 2 1 1 1 1 1 18 13 72 57 n / N 1 1 2 3 n 2 + − 1 + + + − − 8 2 with + = = − − 8 n n n 2 n 2 n n n = = − ( 2 n ( / ( ( n n F 3 4 2 n n 26 57 12 18 2 2 9 6 n 2 2 2 2 4 for the quadratic Casimir F ======C n 6 7 n 4 n n n n 2 / F E E A B B C D D G | | | | | | | | | | A A F A A A A F F F F A F C C C C C C C C C C C C C A A A A or the dimension C C C C F vector F spinor F vector F spinor n C C C C respectively - the faithful representation with the smallest quadratic Casimir ma N the Standard Model gauge group for the faithful repsentation F C Our Ratio of Adjoint to “smallest” Quadratic Casimirs all the different groups; then the Lie group having this very smallest faithf is just For calculating our “goal quantityÂt’Ât’ for various groups one has in depending on the rank is only the trivial representation thatalgebras). is Here we excluded used from the being terminology the groups of which simple are a not bit simple, wrongly if in theyodd as have and simple far Lie even as algebra. w For the tor connected with the abelian invariantthe subgroups associated - restriction something relations done (which byrepresentations essentially con - is the the factor charge associatedter quantizatio with factor the is simple obtained nonabelian byof invariant first the s quadratic evaluating Casimir for for each respectivelyof simple the the invariant adjoint non-abe simple representation and non-abelian tha groupconstructed a that table has of the ratios smallest quadratic Casimir. In the r to that Smallest Representation 7. appendix Therefore we give in the table below from reference [2] two cases for PoS(CORFU2014)045 giv- only 1 (7.17) (7.11) (7.12) (7.13) (7.18) (7.15) (7.16) (7.14) t is the A th root of odel gauge )= G 2 d ( representation ) SU F those dimensions e the Holger Bech Nielsen ked for in the article . the center, we would 4. These ratios mean 12 / / 2 9 one (the small one) to the . ) ratio are 6 figure here let me translate = F 12 F ∗ / ) for the ratio of the volume of 1 C 3 2 ) ( / / 2 8 6 A / ) SU 8 ∗ C ) 4 ) 2 andard Model Group,” / F 4 representations corresponding. The / ) 9 8 / C ( ) 9 F / 4 ( ∗ A / 1 2 ∗ C 9 / ( 2 on to the workshop “Beyond the Standard 3 ( / ) 3 ∗ 3 ) 2 / 3 / ) 8 3 / ) 1 ) 8 1 3 ( + ) 15 / + n n = (( 8 n 2 2 ( (( ( ( which gives 1 symplecticLiegroup 2 = SU SO ( SO = A = = = 30 30 quantity n n n n )= A B C − D = 3 ( 8 quantity E | SU goal − for any faithfull representation of the simple group. Were it not for are the special Lie groups. A F 4 − C C F F C , goal n 3 and E / , 2 8 earlier G = ) 2 ( SU ) F C / A Models”, Bled 2011. arXiv:1304.6051 [hep-ph]. C To avoid too much confusion due to that we used in this article a goal-quantity tha The two simple non-abelian groups for which we need the The notation is this table is the mathematical classification of the Lie algebras: As an example you may use the table to obtain the goal quantity for the Standard M ( [2] Don Bennett and H. B. Nielsen, “Seeking...”, Contributi [1] H. B. Nielsen, “Dimension Four Wins the Same Game as the St the adjoint representation manifold to therule to one multiply for this the volume to ratio the allowing by crudely the speakung number one 6 out comming ofa from 6 the too combinations. quantization strong Now restriction dependence to on avoid as the we dimesnion as of the group, we decided to take th the volume ratio. So if we decide to look for a (linear) size ratio of the the factor involving the abelian part and the division out of a subgroup of obtain the factor - from the simple nonabelian - adjoint our goal quatity - now to be minmized to win - characterized as smallest the squares of thecorresponding scaling to factor the in simple the grouo in group question or of representation the manifolds adjoint of compared to the ing Smallest Representation group: 7.1 An example References and the Lie groups inverse square root of thethis one expression used to in the notation previous of papers the and earlier even papers inthe PoS(CORFU2014)045 AGEN UNIV. - Holger Bech Nielsen he Standard Model?,” Phys. Froggatt, L. V. Laperashvili, R. kewness Of The Standard Model: rXiv:hep-ph/0310127]. SUSY11, ˘ ) 2503-2521 arXiv:0812.0828 A¸Sp. 2/16 ound States of Top and Beauty .5716 [hep-ph] Froggatt, C.D. et al. “ proximately massless bound state” as, C.R. et al. “ New Bound States of E. (1978). “Introduction to Lie 811.2089 [hep-ph]; Richard, Jean-Marc 2009) 65-70 arXiv:0811.2711 [hep-ph] sen, Holger Bech, “ PREdicted the uge Symmetry,” IN *KRAKOW 1987, hicago Press. hematics 9 (Second printing, revised bson, Nathan (1979). “Lie algebras.” ation?” arXiv:hep-ph/9906466v1, 23 Jun and previuos Bled Proceedings. s. Quantum Grav. 14, 1.69 - 1.75 (1997). r of Eden, 12 (1990) 157; NBI-HE-89-38; 3.2146 [hep-ph]; C. D. Froggatt, R. ation from the Standard Model in Order to hematical beauty in the physical world.” p Yukawa coupling and an approximately lation and Measurements and the chal, Conformal Field Theory, 1997 ibution to the workshop “Beyond the Standard versity Press Cambridge (1986)) ich.edu/kane/modern.html 16 200 20 ˇ T September 2, 2011 â ˘ A ˘ ÂS4.ISBNA¸S249. 0-486-63832-4. ´ c (1989) 399. 223 Springer-Verlag New York, ISBN 0-387-94785-X 1999. Springer. p. 81. ISBN 978-0-387-40307-6. Humphreys,Algebras James and Representation Theory.” Graduate Texts ined.). Mat New York: Springer-Verlag. ISBN 0-387-90053-5. Jaco Dover Publications. pp. 243à Determination of Some of its Properties. The University of C NBI-HE-87-28 (87,REC.JUN.) 6p H. B. NielsenPossible and Implications,” Physicalia N. Magazine, Brene, The “S Gardene H. B. Nielsen and N. Brene,Lett. “What B Is Special About The Group Of T PROCEEDINGS, SKYRMIONS AND ANOMALIES*, 493-498 AND COPENH Go Beyond It, Proc. of Conference held on Korfu (1992) Nevzorov and H. B. Nielsen, Nucl.Nevzorov Phys. and B H. 743 B. (2006) Nielsen, 133; Phys.Fermilab, C. Atom. Batavia D. Nucl. IL, 67 USA, (2004) August 582 28 [a â Nielsen, Holger Bech “ PREdicted theRemarkable Higgs coincidence Mass” for arXiv:1212 the top YukawaPhys.Rev. coupling D80 and (2009) an 034033 ap arXiv:0811.2089 [hep-ph]Higgs Niel Mass” arXiv:1212.5716 [hep-ph] Das, C.R.Quarks et at al. the “New Tevatron B and LHC”,Heavy arXiv:0908.4514 Quarks [hep-ph] at D LHC and Tevatron”[hep-ph] Int.J.Mod.Phys. A26 (2011 massless bound state” - Phys.Rev. D80“ (2009) About 034033 the arXiv:0 stability of the dodecatoplet” Few Body Syst. 45 ( Models”, Bled 2013. [3] O’Raifeartaigh, Group Structure of Gauge theories,Uni [4] H. B. Nielsen, “Small Representation Principle”, Contr [7] Ehrenfest,P., 1917 Proc. Amsterdam Acad. [6] Max Tegmark, “On the dimensionality of space time”, Clas [9] Norma Mankoc et al. see many contributions to the present [8] Gordon Kane : http://particle-theory.physics.lsa.um [5] Froggatt, C.D. et al. “Remarkable coincidence for the to [16] C.D. Froggatt and H.B. Nielsen“Why do we have parity viol [13] Oliver, David (2004). “The shaggy steed of physics: mat [15] Philippe Di Francesco, Pierre Mathieu, David SÃl’nÃl’ [14] Millikan, Robert Andrews (1917). The Electron: Its Iso Smallest Representation [11] H.B. Nielsen, S.E. Rugh and C. Surlykke, Seeking Inspir [10] H. B. Nielsen and N. Brene, “Spontaneous Emergence Of Ga [12] C. D. Froggatt, R. Nevzorov and H. B. Nielsen, arXiv:110 PoS(CORFU2014)045 (2. part) Holger Bech Nielsen J. Durhuus and J.L. d Expanded Edition. (2nd ed.). s project from fundamental to human ordita preprint 85/23. amentals of quark models”, In: Proc. . Raitio and J. Lindfors(eds.). Springer, N 978-0-8053-6968-7. r". Physical Review D 10: 275 289. cs and Cosmology, San Juan , Puerto cientiae 1: 91. Harvard Preprint article Forces". Physical Review Letters he Scott.Univ. Summer School in . Andrews, august 1976. I.M. Barbour review”, Elaborated version of talk at the hysRevLett.32.438. .dk/ kleppe/random/qa/qa.html ark and lepton masses”, Nucl. Phys. .275. 8)90214-6. Retrieved 2011-03-21. tability of local gauge symmetry. Creation ): 957. Bibcode:2002Natur.415..957E. hysics B 135 (1): 66G 92. Edward Witten, “Vacuum configurations for 978). "Aspects of the grand unification of . ISBN 978-0-8053-6968-7. cillations in Extended Anti-GUT Model” mic Theory. Garden City, New York: Anchor. n Danish) (meaning:“Do we need laws of enormalizability.html, November 2006 Gamma 37 page 35-46, 1978 17 , “Gauge Theories of the Eighties” In: Proc. of the Arctic 181 (1. part) and (1980) 135 -140 B94 Recent developments in quantum field theory. J. Ambjoern, B. Gamma 36 page 3-16, 1978 In: (1979) 114 - 140. of the Seventeenth Scott. Univ. Summer School in Physics, St and A.T. Davies(eds.), Univ. of Glasgow ,Physics, 465-547 1977) (publ. (CITATION = by NBI-HE-74-15;) t Talk given at the Second TropicalRico Workshop on May Particle 2000. Physi arXiv: hep-ph/0011168v1 Nature?”) B164 H.B. Nielsen and C. D. Froggatt, “Statistical Analysis of qu physics”. of light from chaos.” Phys. Lett. Berlin, 1983,p. 288-354. H.B. Nielsen, D.L. Bennett and N. Brene: “The random dynamic School of Physics 1982, Akaeslompolo, Finland, Aug. 1982. R 32: 438 441. Bibcode:1974PhRvL..32..438G. doi:10.1103/P Pati, J.; Salam, A. (1974). "LeptonBibcode:1974PhRvD..10..275P. doi:10.1103/PhysRevD.10 Number as the Fourth Colo Buras, A.J.; Ellis, J.; Gaillard, M.K.;strong, Nanopoulos, weak D.V. and (1 electromagnetic interactions". NuclearBibcode:1978NuPhB.135...66B. P doi:10.1016/0550-3213(7 Hawking, S.W. (1996). A Brief History of Time: The Updated an Petersen(eds.), Elsvier Sci.Publ. B.V., 1985, pp. 263-351 Georgi, H.; Glashow, S.L. (1974). "Unity of All Elementary P Nanopoulos, D.V. (1979). "Protons Are NotHUTP-78/A062. Forever". Orbis S Ellis, J. (2002). "Physics gets physical".doi:10.1038/415957b. Nature 415 (6875 Ross, G. (1984). Grand Unified Theories. Westview Press. ISB Bantam Books. p. XXX. ISBN 0-553-38016-8. Conf. on Disordered Systems, Copenhagen, September 1984. N superstrings”,NFS-ITP-84-170. [22] H.B. Nielsen, Lecture notes in Physics Smallest Representation [17] RANDOM DYNAMICS: H. B. Nielsen, “Dual Strings,”, “Fund [18] C. D. Froggatt, H.B. Nielsen, Y. Takanishi “Neutrino Os [19] Patterson, Elizabeth C. (1970). John Dalton[20] and the Ato H.B. Nielsen, Har vi brug for fundamentale naturlove(i [21] D. Førster, H.B. Nielsen, and M. Ninomiya, “Dynamical s [24] Ross, G. (1984). Grand Unified Theories. Westview Press [23] See the “home page of Random Dynamics”: http://www.nbi [25] H.B. Nielsen and D. L. Bennett, “The Gauge Glass: A short [26] see e.g. John Baez: http://math.ucr.edu/home/baez/r [27] P. Candelas, Gary T. Horowitch, Andrew Strominger, and PoS(CORFU2014)045 m D. L. Bennett, Holger Bech Nielsen ´ r£¡ 200 2232317 ernational Symposium on the Theory es 179-186 al symmetry. Physics Letters B, 219, clear Physics B, 272, 228-252. Lehto, 20, ... bcode:1985NuPhB.258...46C, tt.53, 2211 (1984) lsen H. B., and Ninomiya, M. (1986). ll arXiv:1403.7177v2[hep-ph] and ealier ˙ I arXiv:hep-ph/9607341. ( on decay theorem at high temperature” n. 36 (2003) 2305 à ional and other Lie algebras with integral r/oldversions/. ../packages/.../color.ps chenergiphysik, Akad. der Wissenschaften s 275-280, ˘ ometric quantum lattice: A general A g in fundamental physics: Predictions for ierarchy-problem and a bound state of 6 t UM-TH-98-01 NIKHEF-98-004 Group s” 18 ˘ AIJPredictions for nonAbe lian fine structure constants fro ˙ I Int. J. Mod. P hys. A 9 (1994) 5155 [arXiv:hep-ph/9311321]. ˘ A ˘ AIJMultiple point criticality, nonlocality an d fine tun- in theory factors for Feynman ... www.nikhef.nl/ form/maindi http://arxiv.org/abs/hep-ph/9311321 http://arxiv.org/abs/hep-ph/9607341 Physics Letters B Volume 178, Issues 2-3, 2 October 1986, Pag H.B. Nielsen and N. Brene, Gauge Glass, Proc. of the XVIII Int eigenvalues of the Casimir operator of Elementary Particles, Ahrenshoop, 1985 (Institutder fur DDR, Ho Berlin-Zeuthen, 1985); Comm. Math. Phys. Volume 93, Number 4 (1984), 483-493. PII: S0305-4470(03)56335-1 Representations of the except discussion. Nuclear Physics B, 272, 213-227. Lehto, M., Nie 87-91. works Diffeomorphism symmetry in simplicial quantum gravity.M., Nu Nielsen H. B., and Ninomiya, M. (1989). Time translation gauge coupling constants gives alpha**(-1) = 136.8 +- 9,â multicriticality,â and 6 anti- t’ International Journal of Modern Physics A, vol â http://www.sukidog.com/jpierre/strings/extradim.htm doi:10.1016/0550-3213(85)90602-9 [33] Physics Letters B Volume 208, Issue 2, 14 July 1988, Page [32] T. van Ritbergen, A. N. Shellekens, J. A. M. Vermaseren, [34] Lehto, M., Nielsen H. B., and Ninomiya, M. (1986). Prege [36] C. D. Froggatt and H. B. Nielsen, Tunguska Dark Matter Ba [35] M. Lehto, H. B. Nielsen, and Masao Ninomiya “A correlati [37] D. L. Bennett and H. B. Nielsen, â [38] Froggatt, CD, Nielsen, HB and Laperashvili, LV 2005, ’H [30] P.H. Damgaard, N. Kawamoto, K. Shigemoto:[31] Phys. Rev. Le A J Macfarlane and Hendryk Pfeiffer, J. Phys. A: Math. Ge [29] See e.g. John M. Pierre, “Superstrings Extra dimension Smallest Representation [28] Candelas...Witten, Nuclear Physics B 258: 46 % G % 74, Bi