23 The Standard Model

Ear­lier we defined “ordi­nary objects” as
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  • having spa­tial extent (they “occupy space”),
  • being com­posed of a (large but) finite number of objects without spa­tial extent (par­ti­cles that do not “occupy space”),
  • and being stable (they nei­ther explode nor col­lapse as soon as they are created).

The sta­bility of such objects rests on the sta­bility of atoms and mol­e­cules, and this requires (among other things) that the rel­a­tive posi­tions and momenta of their con­stituents be fuzzy (with a lower limit for the product of their respec­tive “uncer­tain­ties”), that the Pauli exclu­sion prin­ciple should hold, and that (as a con­se­quence) the con­stituents of atoms and mol­e­cules be fermions.

More is required. The sta­bility of atoms and mol­e­cules rests on a stable equi­lib­rium between two ten­den­cies — the ten­dency of fuzzy posi­tions to grow fuzzier, which is due to the fuzzi­ness of the cor­re­sponding momenta, and the ten­dency of fuzzy posi­tions to get less fuzzy, which is due to the elec­tro­static attrac­tion between the atomic nucleus and its sur­rounding elec­trons. Without the elec­tro­mag­netic inter­ac­tion, nei­ther atoms nor mol­e­cules would exist.

As you will remember, the proper — math­e­mat­i­cally rig­orous and philo­soph­i­cally sound — way to define and quan­tify a fuzzy phys­ical prop­erty is to assign non­trivial prob­a­bil­i­ties to the pos­sible out­comes of a mea­sure­ment. This is the reason (or at least one reason) why quantum mechanics is a prob­a­bility cal­culus, and why the events to which it serves to assign prob­a­bil­i­ties are mea­sure­ment outcomes.

Quantum mechanics thus pre­sup­poses the exis­tences of mea­sure­ment out­comes — actual property-​​indicating (or value-​​indicating) events. But for an event to be an actual property-​​indicating event, it must leave a per­sis­tent trace. There must be a per­sis­tent record of the event, and the exis­tence of such a record does not seem pos­sible in the absence of a richly struc­tured envi­ron­ment, which only a planet can pro­vide. But without gravity, planets do not exist.

So the exis­tence of “ordi­nary” objects — stable objects that “occupy space” yet are com­posed of finite num­bers of objects that don’t — requires both elec­tro­mag­netism and gravity. These are also the only forces we know how to incor­po­rate into the quantum for­malism without using multi-​​component wave func­tions. In other words, they are the only forces that can be for­mu­lated in terms of a dif­fer­en­tial geom­etry (the former in terms of a particle-​​specific Finsler geom­etry, the latter in terms of a uni­versal pseudo-​​Riemannian geom­etry), and that there­fore have clas­sical analogues.

Yet elec­tro­mag­netism and gravity are not suf­fi­cient. The exis­tence of “ordi­nary” objects requires at least one other force.

The sim­plest and most straight­for­ward way to intro­duce another type of inter­ac­tion is to gen­er­alize from QED. To see how this is done, we first observe that the QED Lagrangian is invariant under the fol­lowing sub­sti­tu­tions (known as a gauge transformation):

ψ → [1:qα]ψ,   V → V−∂tα,   Ax → Ax+∂xα,   Ay → Ay+∂yα,   Az → Az+∂zα.

The World According to Quantum MechanicsThe invari­ance under the sub­sti­tu­tion ψ → [1:qα]ψ gives us the freedom to asso­ciate a dif­ferent com­plex plane with each space­time point and to inde­pen­dently rotate each of these planes. (If we make use of this freedom, then α is a func­tion of the space­time coor­di­nates t,x,y,z, and the gauge trans­for­ma­tion is referred to as a local trans­for­ma­tion.) This is quite anal­o­gous to the freedom we have to asso­ciate a dif­ferent system of iner­tial coor­di­nates with each space­time point and to inde­pen­dently sub­ject each to a Lorentz trans­for­ma­tion. It is this kind of freedom that makes it pos­sible to for­mu­late an inter­ac­tion. While the latter freedom makes room for gravity, the former makes it not merely pos­sible but, in fact, nec­es­sary to intro­duce the elec­tro­mag­netic “force.”

The trans­for­ma­tions of the form ψ → [1:β]ψ con­sti­tute what math­e­mati­cians call a “group.” We may think of the ele­ments of this group, which goes by the name U(1), as rota­tions in (or of) the com­plex plane, β spec­i­fying the angle of rota­tion. The com­plex plane is a 1-​​dimensional com­plex vector space. The most direct gen­er­al­iza­tion of U(1) is the group of “rota­tions” in a 2-​​dimensional com­plex vec­tors space, which is known as SU(2). These “rota­tions” are them­selves the most direct gen­er­al­iza­tions of rota­tions in a real vector space. (You will recall that the com­po­nents of a vector in a real vector space are real num­bers.) The most direct gen­er­al­iza­tion of SU(2) is — no sur­prise there — SU(3), the group of “rota­tions” in a 3-​​dimensional com­plex vec­tors space.

As it turns out, U(1), SU(2), and SU(3) are all the groups that are needed to for­mu­late the so-​​called Stan­dard Model, which com­prises all well-​​tested the­o­ries of par­ticle physics:

  • the theory of the strong nuclear inter­ac­tions called quantum chro­mo­dy­namics (QCD), which is con­structed just like QED except that U(1) is replaced by SU(3),
  • the so-​​called elec­troweak theory, which is a par­tially uni­fied theory of the weak and elec­tro­mag­netic inter­ac­tions (“par­tially” because it involves two sym­metry groups — U(1) and SU(2) — rather than a single one).

The find­ings of every high energy physics exper­i­ment car­ried out to date are con­sis­tent with the Stan­dard Model. While this has real and/​or per­ceived short­com­ings, its pro­posed exten­sions have so far remained in the realm of pure, untestable speculation.

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QCD

Because the sym­metry group of QCD is SU(3), the strongly inter­acting fun­da­mental par­ti­cles — the quarks — come in three “colors” (hence quantum chromodynamics, from the Greek word for color, chroma). As the elec­tro­mag­netic inter­ac­tion is said to be “medi­ated by the exchange of pho­tons,” so the strong force is said to be “medi­ated by the exchange of gluons,” of which there are 3²−1=8 vari­eties. (This ter­mi­nology involves an ille­git­i­mate reifi­ca­tion of cor­re­la­tions between mea­sure­ment out­comes.) Unlike pho­tons, which are elec­tri­cally neu­tral, the gluons “carry” color charges, owing to the fact that the result of two suc­ces­sive SU(3) trans­for­ma­tions depends on the order in which they are performed.

The two most impor­tant char­ac­ter­is­tics of QCD are asymp­totic freedom and quark con­fine­ment. The first means that the shorter the dis­tance over which quarks interact, the more weakly they interact. The second means that only color-​​neutral com­bi­na­tions of quarks appear in iso­la­tion. Baryons and mesons are such com­bi­na­tions. While baryons — among them the proton and the neu­tron — are bound states of three quarks, mesons are bound states of a quark and an antiquark.

Here is a simple (if not sim­plistic) illus­tra­tion of quark con­fine­ment. Imagine trying to sep­a­rate the quark q from the anti­quark q of a meson qq. The energy required to do this grows with the dis­tance between q and q. Since at some point this energy exceeds the energy required to create another meson, you end up with two sep­a­rate mesons instead of two sep­a­rate quarks.

Studied at suf­fi­ciently high res­o­lu­tion, the atomic nucleus thus presents itself as con­taining quarks that interact via gluons, while at lower res­o­lu­tion it presents itself as con­taining pro­tons and neu­trons that interact via mesons. The force that binds the pro­tons and neu­trons in a nucleus is thus a residue of the strong force that binds quarks.

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The elec­troweak force

According to the Stan­dard Model, the fun­da­mental con­stituents of matter are of two kinds: the quarks, which interact via the strong force, and the lep­tons, which don’t. Besides coming in three “colors,” the quarks come in six “fla­vors”: down, up, strange, charm, bottom, and top. There are six lep­tons (the elec­tron e, the muon μ, the tauon τ, and three cor­re­sponding neu­trinos (νe, νμ, ντ). These par­ti­cles group them­selves into three “gen­er­a­tions.” The first con­tains the down and up quarks, the elec­tron, and the electron-​​neutrino; the second con­tains the quarks with strange­ness and charm, the muon, and the muon-​​neutrino; the third con­tains the bottom and top quarks, the tauon, and the tauon-​​neutrino. Except for their masses, the par­ti­cles of the second and third gen­er­a­tions have the same prop­er­ties, and thus interact in the same way, as the cor­re­sponding par­ti­cles of the first gen­er­a­tion. It will there­fore suf­fice to con­sider only the first generation.

One salient fea­ture of the weak force is that while the “left-​​handed” lep­tons (and there­fore also the right-​​handed antilep­tons) interact via the weak force, the right-​​handed lep­tons are immune to it. To get the hang of this hand­ed­ness busi­ness, recall that all lep­tons have a spin equal to 1/​2, then con­sider a mea­sure­ment of the spin of such a par­ticle with respect to the direc­tion in which it is moving. If this mea­sure­ment yields “up” with prob­a­bility 1, the par­ticle is said to be right-​​handed, and if it yields “down” with prob­a­bility 1, the par­ticle is said to be left-​​handed. The lepton part of the elec­troweak Lagrangian there­fore con­tains two terms, a “right-​​handed” one con­structed so as to make it invariant only under local U(1) trans­for­ma­tions, and a “left-​​handed” one con­structed so as to make it invariant under both local U(1) trans­for­ma­tions and local SU(2) transformations.

Another salient fea­ture of the elec­troweak Lagrangian is the absence of a term con­taining par­ticle masses. The pres­ence of such a term would destroy the invari­ance of the Lagrangian under gauge trans­for­ma­tions and thereby render the theory non-​​renormalizable. (In order to yield num­bers that can be checked against exper­i­mental results, a rel­a­tivistic quantum theory must be renor­mal­iz­able. The charges and masses that appear in the Lagrangian of such a theory — they are often referred to as the theory’s “bare” charges and masses — are not the charges and masses that can be mea­sured; they are phys­i­cally mean­ing­less. Renor­mal­iza­tion makes it pos­sible to the­o­ret­i­cally link the charges and masses that are mea­sured at one momentum scale to the charges and masses that are mea­sured at a another momentum scale. It thus allows us to map the depen­dence of phys­ical masses and charges on the momentum scale at which they are measured.)

Since most of the par­ti­cles the elec­troweak theory deals with have masses, a the­o­ret­ical pro­ce­dure was needed for the pur­pose of giving rise to effec­tive mass terms without destroying the invari­ance of the Lagrangian under gauge trans­for­ma­tions. Unsur­pris­ingly, this pro­ce­dure — usu­ally referred to as “Higgs mech­a­nism” — has been hailed as explaining why par­ti­cles have mass. How­ever, a the­o­ret­ical scheme that gives rise to mass terms without jeop­ar­dizing the theory’s renor­mal­iz­ability is one thing, while a phys­ical mech­a­nism or process leading to the cre­ation of mass would be quite another. It bears rep­e­ti­tion: phys­ical the­o­ries are cal­cu­la­tional tools, not descrip­tions of phys­ical enti­ties or processes. It also pays to bear in mind that mass and matter are entirely dif­ferent things, the former a para­meter in some cal­cu­la­tional schemes, the latter either a folk notion or a philo­soph­ical con­cept nei­ther of which has any part in the the­o­ret­ical for­malism of con­tem­po­rary physics.

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