MATTER AND FORCES

MATILDE MARCOLLI

Contents

1. The Standard Model of Particle Physics 22. The Geometrization of Physics 82.1. Evolution of the Kaluza-Klein idea 103. What is a good mathematical model? 20

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1. The Standard Model of Particle Physics

As we mentioned in a previous chapter, the Standard Modeldescribes all the known elementary constituents of matter and allthe forces (except gravity) through which they interact.

The model consists of a set of elementary particles (fermions)that are the basic constituents of matter, subdivided into leptonsand quarks, and a set of particles (bosons) that include the forcecarriers of the electoweak and strong forces (gauge bosons) andthe Higgs field. Quarks interact through the strong force (gluons)and the electromagnetic and weak forces (photons and Z and Wbosons), while the charged leptons (the electron and its partnersin the other two generations, muon and tau) interact throughelectromagnetism and weak force, while the neutrinos interactvia the weak force only. The Higgs field gives mass to particlesit interacts with though the symmetry breaking phenomenon we

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discussed briefly in a previous chapter. All currently known mat-ter is built out of these elementary building blocks.

Some of the questions that are central to contemporary high-energy physics involve what lies beyond the Standard Model:massive neutrinos, supersymmetric partners, dark matter, darkenergy? An even more crucial question is the unification of thisvery successful theory of matter with gravity: what is quantumgravity? Can it be successfully modeled by strings, branes, loops,noncommutative spaces, or something else still?

There are other important questions that involve the StandardModel itself. The model depends on a certain number of param-eters, which include the masses of the particles and the “mixingangles” (which allow particles from the three different generationsto mix). There are 19 such parameters in the minimal version ofthe Standard Model and many more in various possible exten-sions that incorporate candidates for new physics. The values ofthese parameters are not predicted by the theory and are knownonly through experimental observations in particle accelerators.

There are puzzling properties of the observed values of theseparameters. For example, if one looks at the masses, one seesthat there is a very wide range of masses among the particlesof the Standard Model. Particles with the same characteristicsin different generations can have very different masses, and over-all the masses range from extremely small values for neutrinos(which were previously believed to be massless), to the heavi-est top quark, a single elementary particle that weights aboutas much as a whole atom of Tungsten (atomic number 74). Itwould be highly desirable to have a priori theoretical reasons forthe values of these parameters to be what they are.

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Particle colliders are like giant microscopes, by going to in-creasingly high energies they can probe increasingly small scalesin the structure of matter. Collision events in particle accelera-tors create large composite particles that decay into their lighterconstituents. By analyzing the “debris” produced in these colli-sions, the momenta of the resulting particles that hit detectorsin the accelerator chamber, one can reconstruct the original par-ticles formed in the collisions of the accelerator beams. That ishow new particles are discovered and their properties analyzed.

The role of theory is to produce a sophisticated computationalsetting where the result of events can be predicted and comparedwith what is observed in the accelerators. Quantum Field Theoryis the theoretical model behind all these computations.

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Figure 1. Gregory Allen, Cern Atomic CollisionPhysics And Colliding Particles, 2007

Artists have incorporated the visual structure of collision eventsin particle accelerators, like the CERN Large Hadron Collider(LHC), in the form of abstract paintings. We have already men-tioned the work “Collisions II” by Dawn Meson in a previouschapter, as a painting that shows the close connection betweenparticle physics theory and the smooth structure of the 4-dimensionalspacetime manifold. An example that more closely follows thepatterns of lines drawn by particle trajectories in accelerators isGregory Allen’s painting reproduced here.

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The laws that govern the interactions of particles and all thepossible observable events that can occur in experiments are de-rived from the Lagrangian of the Standard Model.

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This has a long and complicated expression where all the possi-ble forms of interaction between the fundamental particles listedin the table above are specified. We have a formula: does itmean that we understand everything? Here is where the task ofthe physicist and the mathematician differ. While the main taskof the physicist is to extract computational predictions from theStandard Model Lagrangian, the mathematician should pursue adifferent type of question: is there a simple principle behind thecomplicated expression of the Standard Model Lagrangian? Doesthe expression follow from a small set of simple principles by com-putation? Not what can it predict, but what does it mean? Wewill see that, in the course of modern physics, geometry has beena crucially important guiding principle for tackling complexity.

2. The Geometrization of Physics

Shortly after the geometrization of gravity achieved by Ein-stein’s General Relativity, the first instance of geometrization ofthe physics of matter and forces occurred with Kaluza–Klein the-ory. In General Relativity gravity is realized by the metric on a4-dimensional spacetime manifold. In Kaluza–Klein theory, elec-tromagnetism is incorporated in the picture by adding a fifthdimension, curled up into circles: a circle bundle over spacetime.

The fundamental idea of the Kaluza–Klein theory is that whatwe exprience as forces acting in out ambient 4-dimensional space-time (in this case the electromagnetic force) are in fact geometryin additional dimensions, which are not extended like the space-time directions, but compactified (in circles in the electromag-netism case). These additional dimensions are responsible for

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Figure 2. Dawn N. Meson, Kaluza-Klein (Invis-ible Architecture III), ca. 2007

the “internal degrees of freedom” of the physical theory beingmodeled, the type of forces that can act and the type of particlesthat interact through those forces.

These invisible curled up directions, these compactified addi-tional dimensions, that are responsible for the particles and forcesin the original Kaluza-Klein model and in its modern extensions

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are potrayed in the painting of Dawn Meson, “Kaluza-Klein (In-visible Architecture III)”. The complex shape of the extra di-mensions plays a crucial role in more modern theories like StringTheory, as well as physical models based on NoncommutativeGeometry.

2.1. Evolution of the Kaluza-Klein idea. Other forces actingon elementary particles, the weak and the strong force, have adifferent geometry. The fact that electromagnetism gives riseto a circle bundle in the Kaluza-Klein model is related to thefact that the carrier of this force, the photon, can be describedmathematically as a U(1) gauge potential, where U(1) is thecircle, viewed as the complex numbers of modulus one. The gaugebosons of the weak and strong force obey similar rules, but thecorresponding gauge groups are more complicated groups, SU(2)and SU(3), respectively. The general form of the geometry issimilar, however. It can be described in terms of vector bundlesover the 4-dimensional spacetime manifold, where a vector spacelies on top of each point of the spacetime manifold, describingthe inner degrees of freedom.

The geometry of these vector bundles describes the physics ofthe particles of matter and the forces: connections and curvaturedetermine the gauge potentials and the force fields, sections ofthe vector bundle determine the matter particles (fields), and

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Figure 3. Regina Valluzzi, Vector Field, 2012

the symmetries of the bundle are the gauge symmetries of thephysical theory, with the correct symmetry groups replacing thecircle of electromagnetism.

A section of a vector bundle is a consistent choice of a vectorover each point of the underlying spacetime. Such a section de-scribes a (classical) field, a type of matter particle, depending onwhat model of matter the geometry of the bundle describes. Theconcept of a vector field is captured in the painting by ReginaValluzzi.

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Figure 4. Marcia Lyons, RED Force Fields, 2011

A connection on a vector bundle is a consistent choice of “hor-izontal directions” and the curvature of the vector bundle mea-sures how much the choice of horizontal directions fails to beglobally flat. The presence of curvature results in a force fieldthat is felt at all points in spacetime where the curvature doesnot vanish. The concept of a force field inspired the depictiongiven in Marcia Lyons’ work.

In the models of particle physics based on the geometry ofgauge theories described above in terms of vector bundles, sec-tions, connections, and gauge groups, composite particles (for

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instance baryons) arise in terms of elementary particles (quarks)through the mathematical theory of representations of Lie groups(in this case the group SU(3), gauge group of the strong force).

This way of obtaining composite particles from elementary par-ticles corresponds, at the level of the geometry of vector bundles,

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Figure 5. Aram Mekjian, Physical Processes, 1997

to ways of constructing vector bundles from group representa-tions and correspondingly obtaining the composite particles assections.

The physical processes that take place in the interactions be-tween particles of matter and forces can then be seen geomet-rically (at the classical level) as sections of vector bundles thatare solutions to certain equations involving the connection andcurvature. A pictorial view of the richness of physical processesarising from the interactions of matter and forces and their geo-metric nature is depicted in the work of the physicist and painterAram Mekjian.

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Another more recent evolution of the original Kaluza-Kleinidea gave rise to the geometry underlying String Theory. Inthis setting, instead of a bundle of circles over the 4-dimensionalspacetime, the extra dimensions are given by a 6-dimensionalgeometry, chosen among a class of manifolds called Calabi-Yau3-fold (the 3 here refers of complex dimension, which means 6 realdimensions). Several popularization books about String Theoryhave rendered people familiar with the corresponding image ofthe resulting geometry, which we reproduce here below.

The different modes of vibration of strings in such geometries de-termine the types of matter particles and forces. As we mentionedin a previous chapter, the very large number of possible choicesof the geometry gives rise to a multiverse of possible universeswith different forms of elementary particles.

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Figure 6. Craig Clarke, Chaos in Wood, ca. 2010

Inspired by the concept of string vibrating in different back-ground geometries, the sculptor Craig Clarke created large in-stallations of tangled strings in various background landscapes,such as “Chaos in Wood” reproduced here.

Another development of the Kaluza-Klein idea gave rise re-cently to an approach to particle physics models based on Non-commutative Geometry. In this setting the extra dimensions areneither circles, nor other manifolds. They are a different kind

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of geometry entirely, a noncommutative geometry, which as wediscussed in a previous chapter is a kind of geometry that incorpo-rates quantum mechanical rules like the Heisenberg uncertaintyprinciple.

Non-commuting variables arise naturally in Quantum Mechan-ics because observables are represented by operators, which meansby matrices in a finite dimensional case, and matrix multiplica-tion does not commute:(

a bc d

)(u vx y

)=

(au + bx av + bycu + dx cv + dy

)6=

(au + cv bu + dvax + cy bx + dy

)=

(u vx y

)(a bc d

)The fact that observables in quantum mechanics usually do notcommute, and in particular the variables of posiiton and velocity(momentum) do not commute, is the basic mathematical rea-son behind the Heisenberg “uncertainty principle”: these twoquantities cannot be simultaneously computed because the corre-sponding operators cannot be simultaneously diagonalized (theirspectra being the possible observed values).

What is then a noncommutative space? An example arisesnaturally when we consider the composition law for spectral lines.It has been one of the fundamental observations leading to thedevelopment of quantum physics, that absorption and emission oflight by matter (say, the hydrogen atom) happens only at certainfrequencies, which come structured in certain series of spectrallines that can be observed in experiments. These correspond tothe amounts of energies that are needed for an electron to jumpbetween energy levels of its bound state as part of the atom.That is, the energy levels are quantized.

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Two successive jumps between energy levels can take place whenthe level reached after the first jump is the starting level of thesecond jump. There is a mathematical structure that accountsfor this type of composition of successive transitions. It is calleda groupoid. Two transitions (arrows) in a groupoid can be com-posed whenever the endpoint of the first is the starting point ofthe second. The spectral lines of atoms are structured so that agroupoid law describes all the possible transitions.

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An advantage of the noncommutative geometry approach toparticle physics is that the complicated Standard Model Lagrangianarises from a very simple geometric setting, consisting of thechoice of a zero-dimensional (in a sense, but 6-dimensional in an-other sense: noncommutative spaces have more than one notionof dimension) noncommutative space as the “extra-dimensions”and a simple action functional called the spectral action. Thenon-commmutative space that gives rise to the Standard ModelLagrangian has algebra of functions given by

A = C⊕H⊕M3(C)

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where the complex numbers C, the quaternions H, and the ma-trices M3(C) are responsible, respectively, for electromagnetismand weak force and for the strong force.

3. What is a good mathematical model?

All the approaches we discussed above are geometric in natureand all have a common fundamental idea: the existence of moredimensions than the 4-dimensions of space and time, which areresponsible for the type of matter and forces that exist in theuniverse and their interaction.

When is a mathematical model a good model of the physi-cal world? There are some fundamental requirements that aregenerally viewed as a favorable indication.

• Simplicity: difficult computations should follow from sim-ple principles;• Predictive power: the model should provide new insight

on physics and give rise to new testable calculations;• Minimalism: Ockham’s razor “entia non sunt multipli-

canda praeter necessitatem” means that a model shouldnot introduce unnecessary assumptions or an excessivenumber of new entities and objects that are not strictlyrequired to achieve consistency and satisfy the previoustwo requirements;• Elegance: there is a general aesthetic guiding principle in

mathematics, which expects mathematical theories andexplanations of physical reality to be not only consistentand testable, but intrinsically beautiful.

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More than one mathematical model may be needed to explain dif-ferent aspects of the same physical phenomenon, just as physicsat different energy scales is described by different mathematics.

In addition to the very general list of properties mentionedabove, the construction of a good model should be driven by spe-cific questions. For example, one of the pressing questions is theunderstanding of dark matter. The existence of a new type ofelementary particles is one of the proposed explanation for theexcess of matter in the cosmos that is not visible by direct as-tronomical observations (does not emit light) but is known toaffect the gravitational motion of galaxies and larger structures.If supersymmetric extensions of the Standard Model would beconfirmed experimentally, then the lightest supersymmetric par-ticle would be stable and a possible constituent of dark mat-ter. Other proposals for possible dark matter constituents haveranged from small astronomical objects like brown dwarf startsto modifications to the laws of gravity of general relativity, toexotic smoothness. The dark matter question may become oneof the most significant guidelines in the development of particlephysics models.

A view of forms of dark matter that do not emit light butgravitate is realized by Cornelia Parker’s installation, “Cold DarkMatter, An Exploded View”.

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Figure 7. Cornelia Parker, Cold Dark Matter, AnExploded View, 1991