NONCOMMUTATIVE GEOMETRY OF THE STANDARD MODEL

WALTER D. VAN SUIJLEKOM

Noncommutative geometry proposes an intrigu-ing broadening of the concept of geometry. As a keyapplication to physics, it allows for a geometric de-scription of the Standard Model of particle physics,thus putting the Standard Model on the same foot-ing as Einstein’s general theory of Relativity.

This short paper gives an introduction to theuse of noncommutative geometry in particle physicsand describes how it unifies the four fundamentalforces in nature.

1. Geometry in mathematics and physics

Geometry first started to play a serious role inphysics through the work of Albert Einstein. Hisgeneral theory of relativity is based on a field inmathematics developed in the 19th century, essen-tially by Bernhard Riemann. This Riemannian ge-ometry describes in addition to flat spaces also non-Euclidean spaces, such as spherical and hyperbolicsurfaces (see Figure 1).

Figure 1. Riemannian geometryin two dimensions: spherical, hy-perbolic and flat surfaces

In higher dimensions the idea of a Riemannianmanifold is intuitively described by it being locallyflat (=Euclidean) at first order. This mathematicalframework is tailor-made for an accurate descrip-tion of the physical principle of general covariancein relativity.

In this way Riemannan geometry describes grav-ity and the question that immediately comes tomind is whether there is a geometrical theory forthe other three fundamental forces. In this articlewe will show that noncommutative geometry pro-vides such a generalization of Riemannian geom-etry. Indeed, the full Standard Model of particlephysics can be described geometrically, albeit by anoncommutative space. Before we try to explainwhat this means, we consider the physical inputneeded to describe a particle propagating in curvedspacetime.

First of all, curved spacetime is described bya four-dimensional (pseudo) Riemannian manifoldM . This means that there are local coordinatesx0, x1, x2 and x3, comparable with the four vectorsin special relativity. Then, if we want to describethe propagation in curved spacetime of a fermionof mass m, we should solve the Dirac equation.In compact form this equation can be written as(/∂M −m)ψ = 0, where /∂M is the Dirac operator.It is the general relativistic analogue to the oper-ator that Dirac found in his search for a specialrelativistic version of Schrodinger’s equation. Thewave function ψ thus describes the fermion propa-gating through curved spacetime.

A question which Einstein already played withwas whether it is possible to geometrically describealso the other fundamental forces (at his time: theelectromagnetic force). In this way one would be letto a unified theory of gravity and electromagnetism,and possibly even the nuclear forces. An elegantsolution could have been Kaluza–Klein theory inwhich spacetime is replaced by spacetime times acircle, but this was rejected for physical reasons.

Our claim here —following Alain Connes and co-authors [4]— is that the full Standard Model of par-ticle physics can be unified with general relativityby replacing spacetime by the product of spacetimewith a noncommutative space. We will thus look atthe noncommutative space

M × F

where we consider F as an internal, noncommu-tative space. This is comparable to Kaluza–Kleintheory in philosophy, but without its physical ob-jections.

The noncommutative space F is described bygiving coordinates, just as we did for M with xµ.

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The difference with spacetime is that the coordi-nates on F do not commute anymore. In fact,they are typically given by matrices. An example isgiven by the complex 3× 3 matrices, for which in-deed ab 6= ba given two of such matrix ‘coordinates’a and b.

The ‘propagation’ of a particle through the inter-nal space F is described by a Dirac-type operator/∂F , which in this case is nothing but a symmetricmatrix.

Example 1 (Electroweak theory). A ‘coordinate’of the space F for the Glashow–Weinberg–Salamelectroweak theory consists of

• a complex number z;• a quaternion q = q0 +

∑i qiσi, where σ1, σ2

and σ3 are the Pauli matrices.

Finally, the Dirac operator on F is given by thematrix

/∂+F =

(ϕ1 ϕ2

−ϕ2 ϕ1

).S

and the negative chirality part is /∂−F = (/∂

+F )†. No-

tice the Higgs-type form of the matrix /∂F , which isno coincidence. We will get back to this point later.

But how then does this lead to physics? In gen-eral, given a noncommutative spacetime M×F and

Figure 2. French mathematicianAlain Connes, the founder of non-commutative geometry [7]

Dirac operator /∂M and /∂F , let us search for a La-grangian describing the dynamics and interactionsof the physical model.

It turns out that there is an extremely simpleprescription for such Lagrangians if we take a spec-tral point of view. Namely, if we count the numberof eigenvalues of /∂M + /∂F up to a certain cutoffscale Λ:

(1) SΛ = Trace

(f

(/∂M + /∂F

Λ

))The function f is a (smooth version of) the functionthat is 1 between −1 and 1 and vanishes elsewhere.

This so-called spectral action [1] gives us the La-grangian of the theory. Only the spectrum of therelevant operator is needed to get to the physicalaction (or Lagrangian). The coupling constants ofthe theory are related to the form of the chosencutoff function. Let us illustrate how this works forEinstein’s general theory of relativity.

1.1. Commutative NCG. If there is no noncom-mutative space F then the above recipe applied to/∂M gives the Einstein–Hilbert Lagrangian of Gen-eral Relativity, whose equations of motion are

Rµν − 12gµνR = 0,

i.e. the Einstein equations in vacuum. Indeed, along calculation [2] (see also [10]) based on so-called‘heat-kernel’ expansions shows that in this case

SΛ =Λ2f2

48π2

∫M

√gR dx+ · · ·

where f2 is the area under the graph of f . If we

now identify Λ2f224π2 with 1

8πG where G is Newton’sconstant, this is precisely the Einstein–Hilbert La-grangian of General Relativity! As said, the equa-tions of motion of this Lagrangian are preciselythe Einstein equations. There are some additionalterms in the spectral action, which are of interestin themselves; we refer to [9] for more information.

As a conclusion to this paragraph let us add aremark on the transition from metric to spectrum[8]. That this is not such a crazy idea can be seenfrom the historical transition of the definition ofthe meter in 1791 by a platinum bar, to the def-inition in terms of the frequency —a property ofthe spectrum— of a certain transition radiation inCaesium.

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2. Noncommutative geometry of theStandard Model

The success of the above noncommutative ap-proach gets even more appreciated for the Stan-dard Model of particle physics. In fact, it turnsout that there exists a noncommutative spacetimeF that allows for a derivation of the full StandardModel, minimally coupled to gravity and includingthe Higgs mechanism. For all computation detailswe refer to [4] and the book [9] (see also the review[11] that also includes a noncommutative descrip-tion of electrodynamics and the electroweak the-ory).

The internal space F is described by the follow-ing non-commuting ‘coordinates’

• a complex number z;• a quaternion q = q0 +

∑i qiσi;

• a complex 3× 3 matrix a.

Compare this to Example 1 for the electroweakmodel.

Next, /∂F is a 96 × 96 dimensional matrix. Thenumber 96 is the number of fermionic degrees offreedom in the Standard Model: 3 families × (2leptonen + 2 quarks, each in 3 colors), of whichleft and right-handed components and to the to-tal we also add the anti-particles. The mathemati-cal conditions that noncommutative geometry putson the matrix /∂F assures that it, fortunately, con-tains mostly zeroes, and for the remaining part isparametrized by all bosons in the Standard Model,including Higgs boson.

Now, if we apply the spectral action principleto the operator /∂M + /∂F a long calculation yieldsthe full Lagrangian of the Standard Model, includ-ing the Higgs mechanism and minimally coupled togravity.

Figure 3. The elementary parti-cles in the Standard Model

As said, the (single) cutoff function f enters inthe different coupling constants of the theory, inthis case the Standard Model. It follows that thecouplings g1, g2 and g3 (electromagnetic, weak andstrong interaction) are related via the GUT-relation

g23 = g2

2 =5

3g2

1 .

We interpret this result as follows. Even thoughthe particle content of the noncommutative modelis equal to that of the Standard Model, and in-cludes their mutual couplings, the spectral actionimplements the GUT-relation for the coupling con-stants. Just as in eg. SU(5)-grand unification, thenoncommutative model describes a unified theorywith particle content equal to that of the StandardModel. In comparison to the usual SU(5)-grandunification this has the advantage that the noncom-mutative model avoids the notorious leptoquarks.

But, when does such a relation hold? In Fig-ure 4 we show the ‘running’ of the coupling con-stants αi = g2

i /4π in the Standard Model, depend-ing on the energy scale µ. This dependence is dic-tated by the renormalization group equations of theStandard Model. We conclude that the noncommu-tative model sits at GUT-scale, near the so-calledGUT-triangle.

Figure 4. The ‘running’ of thethree coupling constants of theStandard Model.

Another, important relation that our Lagrangianimplies is between the Higgs self-coupling and g3:

λ ∼ 4

3g2

3

As our field theory sits at GUT-scale, we interpretthis relation as being valid at the same GUT-scale.If we then let the Higgs self-coupling run accord-ing to the renormalization group equations of theStandard Model, with boundary condition given bythe above relation, we obtain a value for λ at lowerenergy. Here, we assume the big desert: no new par-ticles arise up to GUT-scale besides those presentin the Standard Model. At lower energy (at the

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mass of the Z-boson) λ is related to the mass ofthe Higgs boson, for which we then determine thevalue:

m2H = 8λ

M2W

g2∼ 170 GeV ( at MZ)

If we take into account neutrino mixing (as also de-scribed by the noncommutative model) we find asomewhat lower expected value: 168 GeV. A simi-lar computation allows for a postdiction of (an up-per limit of) the mass of the top quark, to wit mt <180 GeV, compatibly with its measured value.

A few remarks on the predicted Higgs mass arein order, as it is incompatible with the measuredvalue at CERN, which seems to falsify the noncom-mutative approach. However, it is the big deserthypothesis that is is unlikely to hold, and it is nat-ural to look for noncommutative models that de-scribes theories beyond the Standard Model. Theconverse analysis is also possible: find a noncom-mutative model that predicts a Higgs mass of eg.125 GeV. This could then lead to a prediction ofnew particle content, described by the noncommu-tative model. This resulted in [3] in an extension ofthe Standard Model with a real scalar particle, al-lowing for a lower Higgs mass and at the same timeguaranteeing the Higgs vacuum stability. The sym-metry of this model is as in Pati–Salam unification,and has been described in detail in [6, 5].

References

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for noncommutative geometry actions: Unifications ofgravity and the standard model. Phys. Rev. Lett. 77

(1996) 4868–4871.

[2] A. H. Chamseddine and A. Connes. The spectral actionprinciple. Commun. Math. Phys. 186 (1997) 731–750.

[3] A. H. Chamseddine and A. Connes. Resilience of theSpectral Standard Model. JHEP 1209 (2012) 104.

[4] A. H. Chamseddine, A. Connes, and M. Marcolli. Grav-

ity and the standard model with neutrino mixing. Adv.

Theor. Math. Phys. 11 (2007) 991–1089.[5] A. H. Chamseddine, A. Connes, and W. D. van Sui-

jlekom. Beyond the Spectral Standard Model: Emer-

gence of Pati-Salam Unification. (2013).[6] A. H. Chamseddine, A. Connes, and W. D. Van Sui-

jlekom. Inner fluctuations in noncommutative geome-try without the first order condition. J.Geom.Phys. 73

(2013) 222–234.

[7] A. Connes. Noncommutative Geometry. AcademicPress, San Diego, 1994.

[8] A. Connes. On the foundations of noncommutative

geometry. In The unity of mathematics, volume 244of Progr. Math., pages 173–204. Birkhauser Boston,

Boston, MA, 2006.

[9] A. Connes and M. Marcolli. Noncommutative Geome-try, Quantum Fields and Motives. AMS, Providence,

2008.

[10] G. Landi. An Introduction to Noncommutative Spacesand their Geometry. Springer-Verlag, 1997.

[11] K. van den Dungen and W. D. van Suijlekom. Par-ticle Physics from Almost Commutative Spacetimes.

Rev.Math.Phys. 24 (2012) 1230004.

Institute for Mathematics, Astrophysics and Par-

ticle Physics, Faculty of Science, Radboud UniversityNijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The

Netherlands

E-mail address: waltervs@math.ru.nl

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