Contents

# Contents

## Idea

The standard model of particle physics is a model (in particle physics): a quantum field theory that describes the fundamental particles currently experimentally known, containing

as well as three of the four fundamental forces as currently known, which, somewhat roughly, are

It is defined as a local Lagrangian field theory which is an Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory.

The main ingredient missing from the standard model is the quantum version of the field of gravity. For decades, a large part of theoretical physics has been absorbed with attempts to understand how this last of the known fundamental forces might fit into the picture.

As a quantum field theory, the standard model is in particular a Yang–Mills gauge theory with spinors in Yang–Mills theory.

Although there are several approaches to formulate a mathematically precise definition of what a quantum field theory is, there is no rigorous formulation (yet) that comprises the whole standard model.

## Properties

### Gauge group

The exact gauge group of the standard model is not quite the product group

$U(1) \times SU(2) \times SU(3)$

of the circle group with special unitary groups, but is the quotient group

(1)$G_{SM} \;=\; \big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6$

of that by a cyclic group $\mathbb{Z}_6 \subset U(1) \times SU(2) \times SU(3)$ which is the subgroup generated from an element of the form

$(q_6, q_2 \mathbf{1}_2, q_3 \mathbf{1}_3) \;\in\; U(1) \times SU(2) \times SU(3) \,,$

where $q_n \in U(1)$ denotes an $n$th primitive root of unity, i.e.

$\left( e^{2 \pi i \tfrac{1}{6}} \;,\; e^{2 \pi i \tfrac{1}{2}} \mathbf{1}_2\;,\; e^{2 \pi i \tfrac{1}{3}} \mathbf{1}_3 \right) \;\in\; U(1) \times SU(2) \times SU(3) \,.$

(See Baez-Huerta 09, p. 33-36 for a fairly comprehensive discussion. See also e.g. HMY 13, p. 2.)

Strikingly, this exact gauge group (1) of the standard model of particle physics happens to coincide with…

1. …the group

$G_{SM} \;\simeq\; S\big(U(2) \times U(3)\big) \;\subset\; SU(5)$

of determinant-1 elements in the direct product group $U(2) \times U(3)$, which makes manifest that the standard model gauge group is a subgroup of the simple Lie group SU(5).

This is the basis of “grand unified theories” (GUT), speculative extensions of the standard model trying to understand its gauge group as being a spontaneously broken simple Lie group-symmetry.

2. …the subgroup of the Jordan algebra automorphism group of the octonionic Albert algebra that “stabilizes a 4d sub-Minkowski spacetime” (see there for details).

More concretely, it is identified with the subgroup of Spin(9) which respects a splitting $\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3$ (Krasnov 19, see also at SO(10)-GUT)

This is part of ongoing speculation that octonionic exceptional structures might be behind the standard model.

### Fermion content

flavors of fundamental fermions in the
standard model of particle physics
generation of fermions1st generation2nd generation3d generation
quarks
up-typeup quarkcharm quarktop quark
down-typedown quarkstrange quarkbottom quark
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino

## Tension with experiment and Incompleteness

The standard model of particle physics is supported by a tremendous level of confirmation by experiment, lately by the LHC experiment. But there are indictations of discrepancies, pointing to “new physics”. See at

Not quite in tension with the standard model, but still possibly pointing to unaccounted effects is the

Then there are experimental phenomena which are known to be implied by the standard model by way of computer simulatrion (lattice QCD), but which are not conceptually understood:

Finally there is the evident and notorious fact that the standard model simply does not include any quantum field theory of the fourth known fundamental force, gravity. See at

for more on this.

## Variations and generalizations

There is a plethora of attempts and suggestions for variations and generalizations of the standard model into models that may resolve the tension with experiment mentioned above and which may be conceptually more satisfying from the point of view of models in theoretical physics.

### Kaluza–Klein theory

Shortly after the conception of general relativity, it was observed by Kaluza and Klein that the force of gravity alone may effectively appear – if considered on a spacetime that is a bundle whose fiber has a tiny volume (as meaured by the Riemannian metric) – as the field of gravity coupled to gauge fields on the base of the bundle. For details on this see Kaluza-Klein mechanism.

The huge conceptual simplification that this observation suggested had excited theoreticians early on, but a problem of Kaluza–Klein models is that not only does the “compactified” theory of gravity as if by magic emulate gauge fields, but it also always contains further scalar fields that are not experimentally observed.

For that reason interest in Kaluza–Klein theories had decreased in the middle of the last century. Physics departments saw a major revival of the idea when string theory (see below) gained interest, since that theory necessarily exhibits a Kaluza–Klein mechanism. Incidentally, the problem of the spurious fields – the moduli – was still present in this approach. For more on this see the entry landscape of string theory vacua.

### GUTs

One of the oldest studies of variations of the standard model is the investigation of grand unified theories (GUTs), which are Yang–Mills theories that instead of the standard model gauge group have a bigger gauge group which is however a simple group.

### Noncommutative geometry

A widespread perception is that some of the conceptual problems with the standard model point to the fact that some basic assumption of 20th century physics on the nature of reality is oversimplified. According to the approach of noncommutative geometry, modeling spacetime as a smooth manifold is an oversimplification that makes itself felt when the quantization of the force of gravity becomes relevant.

In a class of “noncommutative” generalizations of the standard model, spacetime is therefore replaced more generally by a spectral triple that models a possibly “noncommutative space”. One of the more successful approaches in this direction is the Connes-Lott-Chamseddine model. This effectively is a Kaluza-Klein theory (see above), but with the crucial difference that the fiber in the KK-picture is a highly non-classical non-commutative space, whose classical dimension is that of a point, but whose intrinsic dimension is 6. (This is incidentally the same value of the internal dimension as suggest by string theory.)

For more on this see

### String theory

A more drastic theoretical modifications to the standard model is proposed in the context of string theory, where the entire concept of quantum field theory is proposed to be refined by something else. As opposed to GUTs, this approach at least suggests a way in which also the fourth remaining force field of gravity could be incorporated into the picture.

See

standard model of particle physics and cosmology

theory:Einstein-Yang-Mills-Dirac-Higgs
gravityelectroweak and strong nuclear forcefermionic matterscalar field
field content:vielbein field $e$principal connection $\nabla$spinor $\psi$scalar field $H$
Lagrangian:scalar curvature densityfield strength squaredDirac operator component densityfield strength squared + potential density
$L =$$R(e) vol(e) +$$\langle F_\nabla \wedge \star_e F_\nabla\rangle +$$(\psi , D_{(e,\nabla)} \psi) vol(e) +$$\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$

### General

Textbook accounts:

Further review and outlook:

Review specifically with an eye towards grand unified theory is in

and specificlly with an eye towards intersecting D-brane models in

### Anomaly cancellation

Discussion of anomaly cancellation on the standard model of particle physics:

• Takaaki Hashimoto, Mamoru Matsunaga, Kenta Yamamoto, Quantization of hypercharge in gauge groups locally isomorphic but globally nonisomorphic to $SU(3)_c \times SU(2)_L \times U(1)_Y$ (arXiv:1302.0669)

• Nakarin Lohitsiri, David Tong, Hypercharge Quantisation and Fermat’s Last Theorem (arXiv:1907.00514)

(relating to Fermat's last theorem)

### History

The big international conference of $[$1974$]$ in London was a turning point $[$$]$ Ellis’ catalog well reflected the state of theoretical confusion and general disarray in trying to interpret the $e^+ e^-$ data. But in the midst of all of this was a talk by John Iliopoulos (I think I was there too). With passionate zealotry, he laid out with great accuracy what we call the standard model. Everything was there: proton decay, charm, the GIM mechanism of course, QCD, the $SU(2)\times U(1)$ electroweak theory, $SU(5)$ grand unification, Higgs, etc. It was all presented with absolute conviction and sounded at the time just a little mad, at least to me (I am a conservative).