The papers on the Foundations of Physics are the basis of everything I have ever done, and they represent the earliest layer in my thinking. Overall, they aim to show that physics is based on a universal totality zero and is entirely explained in terms of just four abstract parameters (mass, time, charge and space) whose properties are entirely determined by the four algebras that represent them (real numbers, complex numbers, quaternions and multivariate vectors). The most striking result is that, together, they form a perfect group of order 4 (Klein 4), whose exact symmetry gives the immediate explanation of many fundamental physical laws. This key discovery has never been refuted.
Many of the papers here also continue into other areas of physics I have investigated, and many of the papers in other sections also contain foundational elements. In addition, the series of books listed on the home page are strongly foundational in structure and contain some of the fullest investigations of foundational questions.
There is a single algebra at the heart of physics. Technically, it is a Clifford algebra (Cl(6,0)), a group of order 64. It can be represented as a complexified double quaternion algebra (R ⊗ C ⊗ H ⊗ H) or as a double vector algebra (R ⊗ (C ⊗ H) ⊗ (C ⊗ H)) or as quaternion vector algebra (R ⊗ H ⊗ (C ⊗ H)). Essentially, it is a double Clifford (Cl(3,0)) algebra, or most significantly, the algebra of a DOUBLE SPACE. It is identical to the gamma algebra of the Dirac equation, a double Pauli algebra that defines the quantum mechanics of all fundamental particles or fermions. It is also the algebra that emerges when a complexified octonion is reduced to left- or right-multiplication to preserve group properties.
It would be strange if Nature should arbitrarily choose one particular complicated algebra out of an infinite number of possibilities. BUT IT DOESN’T. This algebra emerges from combining the basic algebras that define the four fundamental parameters of the ‘Periodic Table’: real numbers (R) for mass (unit 1), complex numbers (C) for time (unit i), quaternions (H) for charge (units i, j, k) and vectors or complexified quaternions (C ⊗ H) for space (units i, j, k). These algebras themselves are not arbitrary as they emerge from zero when we apply the Universal Rewrite System. Taken together, the first five units (1, i, i, j, k) define a vector space (i, j, k), fully equivalent to but different from the real space that we are familiar with and that defines our system of measurement (i, j, k). The 8 units have the same structure as a ‘broken octonion’, a left- or right-multiplied complexified octonion, which, by removing the octonion’s antiassociativity, necessarily breaks down to a Cl(6,0) structure.
Taken together the 8 units (1, i, i, j, k, i, j, k) produce 64 products:
1 i -1 –i
ii ij ik ik j –ii –ij –ik –ik –j
ji jj jk ii k –ji –jj –jk –ii –k
ki kj kk ij i –ki –kj –kk –ij –i
iii iij iik ik j –iii –iij –iik –ik –j
iji ijj ijk ii k –iji –ijj –ijk –ii –k
iki ikj ikk ij i –iki –ikj –ikk –ij –i
This table has extraordinary relevance to many aspects of physics at the fundamental level. There are 12 groups of 5 units or pentads. Each pentad can be regarded as providing the generators of the entire algebra. Each is also equivalent to the respective matrix coefficients of the Dirac algebra, γ1, γ2, γ3, γ0, γ5; to the respective coefficients of momentum, energy and mass in the fermion amplitude in nilpotent quantum mechanics; and to the broken symmetry of the strong, weak and electric charges.
In one representation, the 48 units containing quaternion coefficients have the symmetry structure of the 48 fermions / antifermions of the Standard Model. In another the 60 remaining units after the exclusion of 1, i, –1, –i have the same symmetry structure of the 48 fermions and antifermions and the 12 gauge bosons. Each individual pentad can also be seen as a model for one of the 12 types of quark/antiquark (with the additional 3 colour structure provided by the unit vectors i, j, k) and one of the 12 types of lepton / antilepton. These results, which are long established in my work, can be compared with more recent claims at deriving Standard Model physics from a complexified left-multiplied octonion reconstructing itself as Cl(6,0).
Remarkably the table also provides an extremely convenient way of representing the 64 triplet codons provided by the 4 bases of DNA, with each term in the table representing a triplet made from the combinations of (1, i, –1, –i) ⊗ (1, i, j, k) ⊗ (1, i, j, k). Many other developments from the algebra can be seen in the papers incorporated on the site.
The ‘Periodic Table’ or the Klein-4 Group
The Cl(6,0) algebra or group of order 64 occurs at the LOCAL level where point-like particles are created and annihilated, but it depends on a prior symmetry between the four component parameters at the GLOBAL level. Here the four parameters form a kind of ‘Periodic Table’, like that of chemistry but much simpler.
Globally, charges have an unbroken symmetry, but, locally, their symmetry is broken, because the local condition reduces the 8 basic units to 5 composite ones, which then become generators of the Cl(6,0) group of order 64.
Globally, the parameters are divided between 3 primary properties and 3 antiproperties: real / imaginary; conserved / nonconserved; commutative / anticommutative. The division between these is arbitrary and could be switched.
mass conserved real commutative (1-D)
time nonconserved imaginary commutative (1-D)
charge conserved imaginary anticommutative (3-D)
space nonconserved real anticommutative (3-D)
We can represent the structure in a convenient way using the algebraic symbols ±x, ±y, ±z, whose + / – duality is also an expression of the zero totality of Nature:
mass x y z
time –x –y z
charge x –y –z
space –x y –z
This allows a visual presentation in a diagram closely related to those used in category theory;
A more visually-appealing version has been provided by my colleague Vanessa Hill (©):