Algebraic invariants in electrodynamics, relativistic quantum mechanics and particle physics


Introduction to the algebraic method

Clifford algebras [see Lounesto reference below] are used to provide phenomenological descriptions of physical systems as sums of products of elements of the algebra, which describe measuring scales, with the corresponding of measurable quantities, expressed as (real) numbers. These descriptions form algebraic invariants under specific sets of transformations, such as the Lorentz transformation. In analogy with the closely related concept of tensors, they are referred to as `structors' herein. The starting point of the algebraic method is to find an appropriate algebra, which can incorporate transformations that correspond to changes in the observer's reference frame.

A simple example is provided by elementary vector algebra, which uses 3-vector invariants to describe the spatial characteristics of physical quantities. In this case the appropriate Clifford algebra is Cl(3), which can be used to express 3-vectors, and the spatial rotations of the reference frame which leave them invariant.

A more interesting example is the so-called space-time algebra, which is usually taken to be one of the Clifford algebras Cl(1,3) or Cl(3,1). Either of these Clifford algebras can be used to describe algebraic Lorentz invariants in Minkowski space-time, but Cl(1,3) turns out to be the most appropriate. It is also found that the description of physical systems by algebraic invariants provides more information than their description by tensors. This information is interpreted as arising from the internal structure of space-time. If we want to go beyond space-time geometry, it is necessary to consider Clifford algebras, such as Cl(3,3), that contain the space-time algebra, but can also describe charged particles and the fields that act upon them.

Clifford algebras are restricted to the description of macroscopic systems. Relating these algebras to the properties of elementary particles requires the introduction of an appropriate matrix representation, which provides an explicit description of the internal structure of space-time. In addition to being conceived as existing at a point in space-time, elementary particles have properties (e.g. spin and charge) associated with their a location with respect to this internal structure. The papers on this site show the real 8x8 matrix representation of the algebra Cl(3,3) to provide a description of the properties of all the leptons and electroweak bosons. Furthermore, it has been shown that gauge potentials reflect the space-time dependence of the internal structure.

One aim of this work is to determine the 'fundamental' Clifford algebra and its representation that describes all the quarks and leptons and their interactions, yet contains no elements without a physical interpretation. This 'Unification Algebra' has been identified as Cl(5,5), which has a 32x32 real matrix representation. The physical objects described by this algebra, but not Cl(3,3), i.e. quarks and gluons, are restricted to the (high energy density) interiors of hadrons.

Essays related to the concepts underlying this work and a series of research papers developing its mathematical aspects are listed on the Essays page. These will be continuously updated.

Essays on the algebraic theory:

How many leptons are there in a given generation

An independent argument showing that there are eight distinct leptons and, if the neutrino is assumed to massless, that this is consistent with there being no R-chirality neutrinos and no L-chirality antineutrinos.

New concepts in particle physics

This essay extends the remarks given above and provides a general description of the new, often controversial, concepts that are produced the use of representations of Clifford algebras to provide a parametrized description of the elementary particles and their interactions.

Research Papers:

The papers listed below are in various stages of development as shown. All are subject to modification and last modification dates are shown in their headings. (References to material on this site should give this date.) None of this work has currently been published elsewhere.

Iv6: Electrodynamics: an algebraic invariant reformulation


The Clifford algebra Cl(1,3) is employed to construct dimensionless algebraic Lorentz invariants that describe fields and particles. This imposes stronger constraints on the form of physical laws than Lorentz covariance alone. Maxwell's equations in vacuo are shown to be closely related to the Dirac equation, and to have a simple geometrical interpretation. The Cl(3,3) algebra also accommodates an `iso-Lorentz' group which is isomorphic to the Lorentz group and commutes with it. Radiative fields are described by the Dirac equation for real eight component photon wave-functions. Charge sensitive expressions are incorporated in the algebraic description of the interaction of charged particles with the electromagnetic field. The real 8x8 matrix representation of Cl(3,3) suggests that the additional degrees of freedom, which go beyond Cl(1,3), describe an `inner structure' of space-time.

Appendices give details of the `canonical' real 8x8 matrix representation of Cl(3,3) that is used in the whole series of papers. [complete, on-site, last updated, 3 October 2014]


IAv2: On the algebra of gauge fields


Lorentz and gauge field invariant spinors are defined. Gauge fields are expressed in terms of the space-time dependence of the internal structure of space-time. The Cl(3,3) algebra is shown to define four independent gauge fields, which can be identified with the electro-weak interaction. A new type of scalar field is described. This paper is not affected by recent changes in the interpretation of the Cl(3,3) algebra. [complete, on-site, last updated April 2014]


IIv10: An 8-spinor reformulation of the Dirac one-lepton theory


The success of Dirac's theory of electrons as the foundation of relativistic quantum field theory has led to various difficulties with the interpretation of single particle properties being put to one side. Here these are addressed by making three changes in the formalism: (a) the properties of physical systems are described by algebraic Lorentz invariants, (b) Dirac's 4x4 gamma-matrices are replaced by 8x8 real matrices and are no longer regarded as invariants, (c) some square roots of minus one that originate in the geometrical and quantum-mechanical aspects of the Dirac equation are algebraically identified. Eight distinct complex 8-spinor solutions are obtained that span the states of electrons, positrons, neutrinos and antineutrinos. This version has undergone a radical revision, and is now fully consistent with the standard formulation of Quantum Electrodynamics. [Last updated 3 October 2014]


IIA: Lepton generations and neutrino mixing


This paper provides an extension of the interpretation of the leptonic states described by Cl(3,3) given in Paper II by showing that the states of different generations, or families, of leptons can be distinguished within this algebra. A phenomenological model of neutrino mixing is developed that is independent of possible mechanisms. [incomplete, under construction]


III: A new look at time reversal and charge conjugation

Undergoing a complete rewriting to accomodate the revised interpretation


IV: Algebraic description of electro-weak interactions


The Cl(3,3) algebra is shown to include a sub-algebra that is isomorphic to the generators of the SU(2) gauge symmetry associated with weak interactions, providing a new, one-particle, formulation of electro-weak theory for leptons. [currently under revision]


V: Clifford algebras, orthogonal groups and their interpretation

Overview of the relationship between the algebraic and group theoretical approaches. [pending]


VI: Bi-structors and general relativity


It is shown that general relativity can be described using the tensor product of Clifford algebras Cl(1,3) X Cl(1,3) and with elements that vary continuously in space-time. The consequences of extending the theory to Cl(3,3) X Cl(3,3) are investigated.

[currently under construction]


VIIIv3: SO(16,16) unification and quark confinement


The Clifford algebra Cl(5,5), and its associated Lie group SO(16,16), are shown to provide a complete algebraic description of all the elementary fermions in the first generation, together with their electro-weak and strong interactions. New quantum numbers are introduced that describe quark colour, and distinguish leptons and quarks. Fermion states are described by five factor projection operators that act on 32 component spinors. 32-spinors separate into a quark 24-spinors and previously described lepton 8-spinors. u and d quarks of a given colour are also described by 8-component spinors. The SU(3) strong interactions are expressed in terms of 32x32 matrices. The lepton 8-spinor and the lepton/quark 32-spinor are interpreted as existing in two distinct space-time structures: leptonic space-time, described by the Cl(3,3) algebra, and hadronic space-time described by the Cl(5,5) algebra. It is argued that quarks can only exist in the energy dense environment of hadronic space-time. Hypothetical SU(4) gauge bosons are identified that could stabilize the boundaries separating these two types of space-time, providing a new explanation of quark confinement. [on site, but remains subject to extension and revision, last updated 20 January 2014]




Lounesto, Pertti, 1997 Clifford Algebras and Spinors (Cambridge University Press)

Page last updated October 2014

Back to home page.