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Supersymmetry at 100 There are two types of symmetries in momentum of a particle (remaining even at difference is clarified in the so-called spin- nature: external (or space-time) symmetries rest) is called spin. (Particle spin is an ex- statistics theorem that states that bosons and internal symmetries. Examples of inter- ternal symmetry, whereas isotopic spin, must satisfy commutation relations (the nal symmetries are the symmetry of isotopic which is not based on Lorentz invariance, is quantum mechanical wave function is sym- spin that identifies related energy levels of not.) metric under the interchange of identical the nucleons (protons and neutrons) and the In quantum mechanics spin comes in inte- bosons) and that fermions must satisfy a.nti- more encompassing SU(3) X SU(2) X U(1) gral or half-integral multiples of a fundamen- commutation relations (antisymmetnc wave symmetry of the standard model (see “Par- tal unit h (h = h/2z where h is Planck’s functions). The ramification of this simple ticle Physics and the Standard Model”). constant). (Orbital angular momentum only statement is that an indefinite number of Operations with these symmetries do not comes in integral multiples of h.) Particles bosons can exist in th~ same place at the change the space-time properties of a par- with integral values of spin (O, h, 273, . .) are same time, whereas only one fermion can be ticle. called bosons, and those with half-integral in any given place at a given time (Fig,, 1). External symmetries include translation spins (r5/2, 3fz/2, 5h/2, . .) are called fer- Hence “matter” (for example, atoms) is invariance and invariance under the Lorentz miens. Photons (spin 1), gravitons (spin 2), made of fermions. Clearly, if you can’t put transformations. Lorentz transformations, and pions (spin O) are examples of bosons. more than one in any given place at a time, in turn, include rotations as well as the Electrons, neutrinos, quarks, protons, and then they must take up space. If they are also special Lorentz transformations, that is, a neutrons—the particles that make up or- observable in some way, then this is exactly “boost” or a change in the velocity of the dinary matter—are all spin-% fermions. our concept of matter. Bosons, on the other frame of reference. The conservation laws, such as those of hand, are associated with “forces.” For ex- Each symmetry defines a particular opera- energy, momentum, or angular momentum, ample, a large number of photons in the tion that does not affect the result of any are very useful concepts in physics. The fol- same place form a microscopically ob- experiment. An example of a spatial transla- lowing example dealing with spin and the servable electromagnetic field that affects tion is to, say, move our laboratory (ac- conservation of angular momentum charged particles. celerators and all) from Chicago to New provides one small bit of insight into their Mexico. We are, of course, not surprised that utility. Supersymmetry. The fundamental prop- the result of any experiment is unaffected by In the process of beta decay, a neutron erty of supersymmetry is that it is a space- the move, and we say that our system is decays into a proton, an electron, and an time symmetry. A supersymmetry operation translationally invariant. Rotational in- antineutrino. The antineutnno is massless alters particle spin in half-integral jumps, variance is similarly defined with respect to (or very close to being massless), has no changing bosons into fermions and vice rotating our apparatus about any axis. In- charge, and interacts only very weakly with versa. Thus supersymmetry is the first sym- variance under a special Lorentz transforma- other particles. In short, it is practically in- metry that can unify matter and force, the tion corresponds to finding our results un- visible, and for many years beta decay was basic attributes of nature. changed when our laboratory, at rest in our thought to be simply If supersymmetry is an exact symmetry in reference frame, is replaced by one moving at nature, then for every boson of a given mass a constant velocity. rz+p+e- there exists a fermion of the same mass and Corresponding to each symmetry opera- vice versa; for example, for the electron there tion is a quantity that is conserved. Energy However, angular momentum is not con- should be a scalar electron (selection), for the and momentum are conserved because of served in this process since it is not possible neutrino, a scalar neutrino (sneutrino), for time and space-translational invariance, re- for the initial angular momentum (spin 1/2 quarks, scalar quarks (squarks), and so forth. spectively. The energy of a particle at rest is for the neutron) to equal the final total Since no such degeneracies have been ob- its mass (E= mc2). Mass is thus an intrinsic angular momentum (spin 1/2 for the proton served, supersymmetry cannot be an exact property of a particle that is conserved be- ~ spin 1/2 for the electron* an integral value symmetry of nature. However, it might be a cause of invariance of our system under for the orbital angular momentum). As a symmetry that is inexact or broken. If so, it space-time translations. result, W. Pauli predicted that the neutrino can be broken in either of two inequivalent must exist because its half-integral spin ways: explicit supersymmetry breaking in Spin. Angular momentum conservation is a restores conservation of angular momentum which the Lagrangian contains explicit terms result of Lorentz invariance (both rotational to beta decay. that are not supersymmetric, or spontaneous and special). Orbital angular momentum re- There is a dramatic difference between the supersymmetry breaking in which the La- fers to the angular momentum ofa particle in behavior of the two groups of spin-classified grangian is supersymmetric but the vacuum motion, whereas the intrinsic angular particles, the bosons and the fermions. This is not (spontaneous symmetry breaking is 100 Summer/Fall 1984 LOS ALAMOS SCIENCE .Supersymmetryat 100 GeV bine Internal symmclrics and space-~lmc symmetries In a nontrivial way, This com- bination IS nw a necessary feature of supcr- symmclry (in fact. it is accomplished b! ek- tendinglhc algebra of Eqs. 2 and 3 In ‘“Super- symmetry and Quantum Mechanics”” 10 ln - cludc more supcrsymmcvr> generators and internal s>mmclry generators), However, an \ Important conscqucrrcc ofsuch an c~[ension A might bc Ihat hosons and fcrmions In dtf- r fcrcmt rcprcscnlalions of an lnlc.rnril s>m- mclry group arc rclawd. For ckample. quarks $:(;=01=0 (fcrmions) arc in [riplcis in the strong-l ntLv- ac[ion group S[1(3), whereas the gluons (bos- ens) arc in OCICIS.Perhaps !hcy are all rclaled in an cxtcndcd supcrsymmcir!. Ihus prol ld- (a) (b)1’““!ng a unifwd dcscrip[ion ofquarks and [heir forces. Second. supcrsymrnctry can provide a the- Fig. 1. (a) An example of a symmetric wavefunction for a pair of bosons and (b) an ory of gravit}, [f supcrs}mmclr> IS glohisl, antisymmetric wave function for a pair of fermions, where the vector r represents [hen a given supers> mmctry rotation must the distance bet ween each pair of identical particles. Because the boson wave be the same over all space-time. However, if function is symmetric with respect to exchange (t+, (r) = yr,(-r)), there can be a supcrsymmct~ is local. the system IS in- nonzero probability (y;) for two bosons to occupy the same position in space (r = varian[ under a supersymme[ry rotation tha~ O), whereas for the asymmetric fermion wave function (VF (r) = –VF (–r)) the may be arbitrarily different a{ even poln(, probability (y;) oft wo fermions occupying the same position in space must be Because the various generators (supersym - zero. mcln charges. four-m omcn[um ~rans]a- lmnal gcncralors. and Lorcntz generators for both ro[atlons and boosIs) satlsf} a common explalncd In Notes 3 and 6 of ‘“Leclurc lhcrc arc many prcdictmrss tha( can be vcri - olgcbra 01’ commutation and anllcwmmula- Notes—From Simple Field Theories to the fmd in [hc next generation of high-energy IIon rclalwns. consistcnc! M>qu(rrs tha[ all Standard Model””). Either way will Iif! the accelerators. The second possibility would the symmc(rlcs arc local. (In facl. ~hc anti - boson-fcrmlon dcgcncracy, bul the latter way nol ncccssanly lead m any new Iow-m-wrgy commutator of Iwo supers} mmctr> gen- will introduce (In a somewhat analogous way conscqucnccs. erators IS a Iranslallrm gcncralor. ) Thus dif- 10 lhc Higgs boson of weak-interaction sym- We will also discuss [be role gravity has ferent points In space-llrnc can transform In metry breaking) a new particle. the Gold- played in the description of low-energy differcnl ways: puI simply. this can amount SIOIIC fermion. (Wc develop ma[hcma[lcally supersymmetry. This connection bctwcecn 10 accclcratlon hclwccm points. whtch. in some of Ihc ideas of this paragraph in physics al the largest mass scale in nature turn. is cqutvalcnt to gra~lt>. In fac~. the ““Supcrsymmt’[ry and Quantum Mechan- (the Planck scale: ‘Mpl= (h~’/(;N)”2 = 1.2 X thcov of local translations and Lorentz ics’-, ) 1019 CJeV/C2. where (;N is Newlon”s gravila- Transformations isjust general relatlvtt}. ~ha[ .+ qucs(ion of extreme imporrancc is lhe [ional constant) and physics at tbe low is. Einstcin”s theory of gravity. and a supcr- scale of supcrsymmctry breaking. This scale encv-gies of the weak scale (.kf~ = 83 GeV/c2 symmc[ric [heo~ ofgravlty IS called supcr- can be characterized in terms of lhe so-called where ,Jf,+ is the mass of the U’ boson re- gravily. 11 IS just the theory invariant under fI(prr,w/J.
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