| Summary: | 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.
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