Leptons may be assigned the six flavour quantum numbers: electron number, muon number, tau number, and corresponding numbers for the neutrinos.

• These are conserved in strong and electromagnetic interactions, but violated by weak interactions.
• Therefore, such flavour quantum numbers are not of great use. A separate quantum number for each generation is more useful: electronic lepton number (+1 for electrons and electron neutrinos), muonic lepton number (+1 for muons and muon neutrinos), and tauonic lepton number (+1 for tau leptons and tau neutrinos).
• However, even these numbers are not absolutely conserved, as neutrinos of different generations can mix; that is, a neutrino of one flavour can transform into another flavour.

The strength of such mixings is specified by a matrix called the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix). (Wikipedia)

Gell-mann matrices are a complete set of Hermitian noncommuting trace-orthogonal matrices. In addition, they also play an important role in physics where they can be thought to model the eight gluons that mediate the strong force quantum chromodynamics, an analogue of the Pauli matrices well-adapted to applications in the realm of quantum mechanics. (Wolfram)

The action of C⊗O on itself can be seen to generate a 64-complex-dimensional algebra, wherein we are able to identify two sets of generators for SU(3)c.

• Furthermore, we show that these three-generation results can be extended, so as to include all 48 fermionic U(1)em charges.
• The 64-dimensional octonionic chain algebra splits into two sets of SU (3) generators of the form iΛν and −iΛ * ν * , six SU (3) singlets j , six triplets q k , and their complex conjugates.
• These objects are sectioned off above into four quadrants according to their forms: νaν, ν * aν, νaν * and ν * aν * for a in the chain algebra.

Transforming particles into anti-particles, and vice versa, requires only the complex conjugate i → −i in our formalism. (Standard Model from an algebra - pdf)

The pseudoscalar meson nonet. Members of the original meson “octet (8)” are shown in green, the singlet in magenta.

• Although these mesons are now grouped into a nonet (9), the Eightfold Way name derives from the patterns of eight for the mesons and baryons in the original classification scheme.
• The Eightfold Way classification is named after the following fact:
  • If we take three flavors of quarks, then the quarks lie in the fundamental representation, 3 (called the triplet) of flavor SU(3).
  • The antiquarks lie in the complex conjugate representation 3.
• The nine states (nonet) made out of a pair can be decomposed into the trivial representation, 1 (called the singlet), and the adjoint representation, 8 (called the octet).
• The notation for this decomposition is 3⊗3=8⊕1.

Figure below shows the application of this decomposition to the mesons. (Wikipedia)

In order to be four-spinors like the electron and other lepton components, there must be one quark component for every combination of flavour and colour, bringing the total to 24 (3 for charged leptons, 3 for neutrinos, and 2·3·3 = 18 for quarks). Each of these is a four (4) component bispinor, for a total of 96 complex-valued components for the fermion field. (Wikipedia)

The origin of multiple generations of fermions, and the particular count of 3, is an unsolved problem of physics.

• Generations of matter: Why are there three generations of quarks and leptons? Is there a theory that can explain the masses of particular quarks and leptons in particular generations from first principles (a theory of Yukawa couplings)?
• String theory provides a cause for multiple generations, but the particular number depends on the details of the compactification of the D-brane intersections.
• Additionally, E8 grand unified theories in 10 dimensions compactified on certain orbifolds down to 4‑D naturally contain 3 generations of matter.
• This includes many heterotic string theory models.

In standard quantum field theory, under certain assumptions, a single fermion field can give rise to multiple fermion poles with mass ratios of around eπ≈23 and e2π≈535 potentially explaining the large ratios of fermion masses between successive generations and their origin. (Wikipedia)

Why do we see certain types of strongly interacting elementary particles and not others? This question was posed over 50 years ago in the context of the quark model.

• M. Gell-Mann and G. Zweig proposed that the known mesons were qq¯ and baryons qqq, with quarks known at the time u (“up”), d (“down”), and s (“strange”) having charges (2/3,–1/3,–1/3).
• Mesons and baryons would then have integral charges. Mesons such as qqq¯q¯ and baryons such as qqqqq¯ would also have integral charges. Why weren’t they seen?
• They have now been seen, but only with additional heavy quarks and under conditions which tell us a lot about the strong interactions and how they manifest themselves.

The present article describes recent progress in our understanding of such “exotic” mesons and baryons. (Multiquark States - pdf)

These are called hypercomplex numbers, such as, quaternions (4D), octonions (8D), sedenions (16D), pathions (32D), chingons (64D), routons (128D), and voudons (256D). These names were coined by Robert P.C. de Marrais and Tony Smith. It is an alternate naming system providing relief from the difficult Latin names, such as: trigintaduonions (32D), sexagintaquattuornions (64D), centumduodetrigintanions (128D), and ducentiquinquagintasexions (256D). (Wordpress.com)

The set of points in Euclidean 4-space having the same distance R from a fixed point P0 forms a hypersurface known as a 3-sphere where R is substituted by function R(t) with t meaning the cosmological age of the universe. Growing or shrinking R with time means expanding or collapsing universe, depending on the mass density inside (Wikipedia).

Throughout his life, Einstein published hundreds of books and articles. He published more than 300 scientific papers and 150 non-scientific ones. On 5 December 2014, universities and archives announced the release of Einstein’s papers, comprising more than 30,000 unique documents (Wikipedia).

On the other hand, one does not yet have a mathematically complete example of a quantum gauge theory in 4D Space vs Time, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so! (Clay Institute’s - Yang Mills Official problem description).

Equivalently, they are the linear transformations fixing that hyperboloid of two sheets. If we discard one of the sheets, we obtain the orthochronous (time-preserving) subgroup.

• From the perspective of the centre of the cone, the hyperboloid looks like an open disc. The orthochronous Lorentz transformations precisely correspond to distance-preserving transformations of the hyperbolic plane. These are themselves determined uniquely by a conformal (or anticonformal) transformation of the ‘circle at infinity’.
• Adding an extra dimension, the orthochronous Lorentz group O^{+}(3,1) is isomorphic to the group of distance-preserving transformations of hyperbolic 3-space, which is again isomorphic to the group of (anti-)conformal transformations of the ‘sphere at infinity’, namely our index-2 supergroup of the Möbius group.
• Moreover, this nicely generalises: the group generated by geometric inversions on the n-sphere is abstractly isomorphic to the orthochronous Lorentz group O^{+}(n+1,1).

And when n = 24, we get a very beautiful discrete subgroup, namely the automorphism group of the II(25,1) lattice intimately related to the Leech lattice. (Complex Projective 4-Space)

There are 8 different types of tiny particles, or ‘states’, that we can find in a special kind of space that has 6 dimensions and involves both real and imaginary numbers. These particles include:

• The Higgs field, which doesn’t spin and is represented by 0.
• Fermions, which are particles like electrons, having a spin of plus or minus a half.
• Bosons, like photons, which have a spin of plus or minus 1.
• Anti-fermions, which are like fermions but have a spin of plus or minus two-thirds.
• The graviton, believed to be responsible for gravity, with a spin of 2.

In a diagram at the top left, this 6-dimensional space is shown to be curved. In another diagram at the bottom right, we see two waves that are perpendicular to each other, representing the motion of a particle in a ‘Dirac harmonic oscillator’ – a concept in quantum mechanics. (Physics In History)

Four of the dimensions are the usual four of spacetime. The six (or perhaps seven) extra dimensions are rolled up to be almost unobservable.

• First, let’s see why they exist at all. If N=8 Supersymmetry is correct the universe must be 10 or 11 dimensional.
• Let D be the actual dimensionality of space time. Let d be the apparent dimensionality. (We know d = 4, but let’s think generally.) Then there is a nice relation between D, d and N.
• It follows from the number of spinor dimensions required by the Dirac equation, which is The s mean round down to the nearest whole number. So plugging in d=4 and N=8 (which is the highest value N can have) we get D = 10 or 11. String theory has D=10, M-theory has D=11.
• One dimension is reserved for time, leaving space with 9 or 10 dimensions.

We don’t see 6 (or 7) of these extra dimensions because - we assume - they are rolled up a la Kaluza–Klein theory into a 6 dimensional Calabi–Yau space

In particle physics, SO(10) refers to a grand unified theory (GUT) based on the spin group Spin(10). The shortened name SO(10) is conventional[1] among physicists, and derives from the Lie algebra or less precisely the Lie group of SO(10), which is a special orthogonal group that is double covered by Spin(10).

SO(10) subsumes the Georgi–Glashow and Pati–Salam models, and unifies all fermions in a generation into a single field. This requires 12 new gauge bosons, in addition to the 12 of SU(5) and 9 of SU(4)×SU(2)×SU(2).

• Left: The pattern of weak isospin, W, weaker isospin, W’, strong g3 and g8, and baryon minus lepton, B, charges for particles in the SO(10) model, rotated to show the embedding of the Georgi–Glashow model and Standard Model, with electric charge roughly along the vertical. In addition to Standard Model particles, the theory includes 30 colored X bosons, responsible for proton decay, and two W’ bosons.
• Right: The pattern of charges for particles in the SO(10) model, rotated to show the embedding in E6.
• The matter representations come in three copies (generations) of the 16 representation. The Yukawa coupling is 10H 16f 16f. This includes a right-handed neutrino.

It has been long known that the SO(10) model is free from all perturbative local anomalies, computable by Feynman diagrams. However, it only became clear in 2018 that the SO(10) model is also free from all nonperturbative global anomalies on non-spin manifolds — an important rule for confirming the consistency of SO(10) grand unified theory, with a Spin(10) gauge group and chiral fermions in the 16-dimensional spinor representations, defined on non-spin manifolds. (Wikipedia)

It was based on representations of the noncompact groups SO(3,1) or SL(2,C), so the spin foam faces (and hence the spin network edges) were labelled by positive real numbers as opposed to the half-integer labels of SU(2) spin networks. (Wikipedia)

In four spacetime dimensions, N = 8 supergravity, speculated by Stephen Hawking, is the most symmetric quantum field theory which involves gravity and a finite number of fields.

• It can be found from a dimensional reduction of 11D supergravity by making the size of seven (7) of the dimensions go to zero.

• It has eight (8) supersymmetries, which is the most any gravitational theory can have, since there are eight half-steps between spin 2 and spin −2. (The spin 2 graviton is the particle with the highest spin in this theory.)

• More supersymmetries would mean the particles would have superpartners with spins higher than 2.
• The only theories with spins higher than 2 which are consistent involve an infinite number of particles (such as String Theory and Higher-Spin Theories).
• Stephen Hawking in his Brief History of Time speculated that this theory could be the Theory of Everything.
• However, in later years this was abandoned in favour of string theory.
• The theory contains 1 graviton (spin 2), 8 gravitinos (spin 3/2), 28 vector bosons (spin 1), 56 fermions (spin 1/2), 70 scalar fields (spin 0) where we don’t distinguish particles with negative spin.
• These numbers are simple combinatorial numbers that come from Pascal’s Triangle and also the number of ways of writing n as a sum of 8 nonnegative cubes A173681.
• One reason why the theory was abandoned was that the 28 vector bosons which form an O(8) gauge group is too small to contain the standard model U(1) x SU(2) x SU(3) gauge group, which can only fit within the orthogonal group O(10).

There has been renewed interest in the 21st century, with the possibility that string theory may be finite. (Wikipedia)

There exist scenarios in which there could actually be more than 4D of spacetime. String theories require extra dimensions of spacetime for their mathematical consistency. These are situations where theories in two or three spacetime dimensions are no more useful.

In string theory, spacetime is 26-dimensional, while in superstring theory it is 10-dimensional, and in M-theory it is 11-dimensional.

This classification theorem identifies several infinite families of groups as well as 26 additional groups which do not fit into any family. (Wikipedia)

M Theory and/or Loop Quantum Gravity hold the promise of resolving the conflict between general relativity and quantum mechanics but lack experimental connections to predictability in physics.

• A connection is made to these and other theories vying for the title of a “Theory of Everything” by questioning the value of the traditional Planck unit reference point for the scales at which they operate.
• It also suggests a cosmological model which has acceleration as being fundamental.
• It provides for an intuitive understanding of the Standard Model and its relationship to particle masses and the structure of the atom.

The prediction of particle mass and lifetimes is a good indicator for its validity. (TOE - pdf)