Algebraic invariants in electrodynamics, relativistic quantum mechanics and particle physics


Introduction to the algebraic method

Clifford algebras [see Lounesto reference] 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 expressions 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 thatbcorrespond to changes in the observer's reference frame.

The simplest example is 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.

Another 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. If we want to go beyond space-time geometry, it is necessary to consider Clifford algebras that contain space-time algebra, but can also describe charged particles and the fields that act on them. The papers on this site show the algebra Cl(3,3) to be suitable for this purpose. The description of physical systems by algebraic invariants provides additional information as compared with their description by tensors. This information can be interpreted in terms of an internal structure of space-time.

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. In particular, the real 8x8 matrix representation of Cl(3,3) has been shown to provide a description of the properties of leptons, possibly including the distinction between lepton generations. Gauge potentials describe the space-time dependence of the variations of its internal structure.

One aim of this work is to determine the 'fundamental' Clifford algebra and its representation that that can describe 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), interpreted in terms of a specific 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 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 equivalent to the Dirac equation, which also provides a simple geometrical interpretation. Unidirectional time is imposed on phenomenological grounds.The Cl(3,3) algebra also accommodates an `iso-Lorentz' group which is isomorphic to the Lorentz group, commutes with it, and provides additional constraints on the algebraic formulation of physical laws. 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, 19 March 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 as the electro-weak interaction. A new type of scalar field is described. [complete, on-site, last updated April 2014]


IIv8: 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 four 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, (c) square roots of minus one that originate in the geometrical and quantum-mechanical aspects of the Dirac equation are algebraically identified and (d) measured time is unidirectional. Together, these changes produce an important modification to the Dirac equation and eliminate the `negative energy' problem. Eight distinct complex 8-spinor solutions span the states of electrons, positrons, neutrinos and antineutrinos. Chiral symmetry breaking is shown to be a consequence of the algebra and does not require a special mechanism. A change of the matrix representation of the space-time reference frame shows that neutrinos can have non-zero mass. Dirac's original 4-spinors are interpreted as an overlay of the electron/positron and neutrino/antineutrino amplitudes. [complete, last updated 14 January 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]


IIB: Klein paradox

This paper uses the formalism developed in Paper II to show that a sufficiently high potential barrier stops penetration and tunnelling by electrons. Pair production does not occur. [currently rewriting]


III: Discrete symmetries in quantum mechanics


Paper I began with the observation that the time we measure on clocks is unidirectional, which is equivalent to saying that all measured time intervals have the same sign. How does one reconcile the statements made in the literature about the time reversal symmetry of physical processes with this observation? Certainly some of the fundamental equations of physics, such as Newton's Laws and Maxwell's electromagnetic field equations are time-reversal invariant, but physically significant solutions of these equations are not. This paper focusses on the discrete symmetries of the solutions of the Dirac equation obtained in Paper II. Revised interpretations of the parity P, charge conjugation C and time-reversal T transformations are obtained. The peculiar algebraic expressions used for the charge conjugation and time-reversal transformations in the original Dirac formalism are explained, and it is shown that both involve charge conjugation. It is concluded that relativistic quantum mechanics is not time-reversal invariant. The commonly accepted idea that non-relativistic quantum mechanics is time reversal invariant is then re-examined. [currently under revision]


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. The chiral property of the weak interaction is associated explicitly with the form of the interaction, making the formal separation of electron and positron states into left and right chiral parts unecessary. [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.

[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. Additive quantum numbers 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 spanning 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. Quarks can only exist in hadronic space-time. Hypothetical SU(4) gauge bosons are identified that could stabilize the boundaries separating these two types of space-time, providing an 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 April 2014

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