Relativistic Physics in the Clifford Algebra Cℓ(1, 3)
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
There is growing evidence that the Clifford algebra Cℓ(1, 3) is the appropriate mathematical structure to formulate physical theories. The geometries of 3-space and spacetime are naturally reflected in the algebras Cℓ(0, 3) and Cℓ(1, 3) respectively. The choice of metric is important and we give further evidence that only the anti-Euclidean metric allows a proper treatment of rotations. The algebra Cℓ(1, 3) is not a division algebra. The invertibility or non-invertibility of elements in the algebra gives physical insight into the limitations of physical systems and non-invertbility should therefore not be regarded as a weakness of the algebra. The Lorentz force law is shown to arise from energy considerations of the electromagnetic field. This result shows that the Lorentz force is not a necessary addition to Maxwell's equations but rather follows from supplementing the electromagnetic energy density by Hamilton's principle. Maxwell's equations are written as a single geometric equations in Cℓ(1, 3). We review this derivation and other electromagnetic theory in the Clifford algebra framework. Taking the massless limit of Weinberg's spin one field equations results in a set of equations more general than Maxwell's equations, containing extra scalar fields. A derivation of these equations in Cℓ(1, 3) is presented and it is shown that, like the Maxwell equations, this set of equations can also be written as a single geometric equation. It has been suggested that the stabilised Poincaré-Heisenberg algebra gives an algebraic signature of quantum cosmology. It is shown that there exists a limit in which this algebra reduces to the conformal algebra. This limit describes how the present day Poincaré algebraic description relates to the conformal-algebraic description of the universe in the past. Furthermore, the proposed algebra inevitably leads to geometric changes in the underlying physical space and any cosmologically derived quantum effects may carry a strong polarisation and spin dependence. The algebra introduces a new dimensionless parameter, the importance of which has been difficult to pin down in the past. It is shown that this dimensionless parameter is closely related to the geometry of the underlying space and if non-zero will affect some of the quantum relativistic notions. The non-scalar basis elements of Cℓ(1, 3) are shown to generate the stabilised Poincar Heisenberg algebra under the Lie bracket [x, y] = xy − yx. The advantage of the Cℓ(1, 3) approach to the stabilised Poincaré-Heisenberg algebra is that it avoids the traditional stability considerations. It has been previously noted that gravitational effects in quantum measurement necessarily renders spacetime non-commutative and induces modifications to the fundamental commutators. This non-commutativity of spacetime and the corresponding modifications to the fundamental commutators arise naturally from the algebra Cℓ(1, 3). The study of the conformal group in Rp,q usually involves the conformal compactification of Rp,q. This allows the transformations to be represented by linear transformations in Rp+1,q+1. This embedding into a higher dimensional space comes at the expense of the geometric properties of the transformations. We show that this linearization procedure can be achieved with no loss of geometric insight, if, instead of using this compactification, we let the conformal transformations act on two copies of the associated Clifford algebra.