Tom Lawrence

We recap the theory of diffeomorphisms and covariances acting on a space and their induced action on fields, and how this leads to their decomposition into Fourier modes and harmonics. We show that a careful consideration of this theory reveals some unexpected results. For a start, harmonics are distinguished by their ‘quantum numbers’, but this analysis is valid for classical fields, so that classical fields can be written as a linear sum of modes with ‘quantum numbers’. Furthermore, certain models of compactification, as described in previous works by the author, are found to exploit a loophole in a key ‘no-go’ theorem, allowing the Poincaré group and an internal O(N ) symmetry group to be embedded in a larger symmetry group. This embedding is achieved without the need for the fermionic generators of supersymmetry and it is manifest in the ‘decompactification limit’. Away from this limit – for a realistic universe – the homogeneous part of this larger symmetry group is non-linearly realised. The translation part of the original symmetry group is realised as additional rotational transformations on the compact factor space.

This paper looks at how changes of coordinates on a pseudo-Riemannian manifold induce homogeneous linear transformations on its tangent spaces. We see that a pseudo-orthonormal frame in a given tangent space is the basis for a set of Riemann normal coordinates. A Lorentz subgroup of the general linear transformations preserves this pseudo-orthonormality. We borrow techniques from the methodology of non-linear realizations to analyze this group-subgroup structure. `Parallel maps' are used to relate tangent space at different points. `Parallelisms' across a finite region of the manifold may be built up from them. These are used to define Weitzenb\"{o}ck connections and Levi-Civita connections. This provides a new formulation of teleparallel gravity, in which the tetrad field is viewed as a field-valued group element relating the coordinate basis to the frame basis used in defining a parallelism. This formulation separates the metric degrees of freedom from those associated with the choice of parallelism. The group element can be combined by matrix multiplication with Lorentz transformations of frame or with other Jacobian matrices. We show how this facilitates a new understanding of inertial forces and local Lorentz transformations. The analysis is also applied to translations of the coordinates. If they are constant across spacetime, this has no effect on the tangent space bases. If the translation parameters become fields, they induce general linear transformations of the coordinate basis; however, the tetrad components can only be expressed in terms of translations on a flat spacetime.

This paper considers the relationship between geometry, symmetry and fundamental interactions — gravity and those mediated by gauge fields. We explore product spacetimes which (a) have the necessary symmetries for gauge interactions and four-dimensional gravity and (b) reduce to an [ ]-dimensional isotropic universe in their flat space limit. The key technique is looking at orbits of the operator form of symmetric rank-two tensors under changes of coordinate system. Orbits containing diagonal matrices are seen to correspond to product manifolds. The [ ] symmetry of the decompactified universe acts nonlinearly on such a product spacetime. We explore the resulting Kaluza–Klein theories, in which the internal symmetries act indirectly on space of the extra dimensions, and give two examples: a six-dimensional model in which the gauge symmetry is [ ] and a seven-dimensional model in which it is [ ]. We identify constraints that can be placed on any rank-two symmetric tensor to obtain such spacetimes: relationships between polynomial invariants. The multiplicities of its eigenvalues determine the dimensionalities of the factor spaces and hence the gauge symmetries. If the tensor in question is the Ricci tensor, other than two-dimensional factor spaces all the factor spaces are Einstein manifolds. This situation represents the classical vacuum of the Kaluza–Klein theory.

This paper looks at how changes of coordinates on a pseudo-Riemannian manifold induce homogeneous linear transformations on its tangent spaces. We see that a pseudo-orthonormal frame in a given tangent space is the basis for a set of Riemann normal coordinates. A Lorentz subgroup of the general linear transformations preserves this pseudo-orthonormality. We borrow techniques from the methodology of non-linear realizations to analyze this group-subgroup structure. “Parallel maps” are used to relate tangent space at different points. “Parallelisms” across a finite region of the manifold may be built up from them. These are used to define Weitzenböck connections and Levi-Civita connections. This provides a new formulation of teleparallel gravity, in which the tetrad field is viewed as a field-valued group element relating the coordinate basis to the frame basis used in defining a parallelism. This formulation separates the metric degrees of freedom from those associated with the choice of parallelism. The group element can be combined by matrix multiplication with Lorentz transformations of frame or with other Jacobian matrices. We show how this facilitates a new understanding of inertial forces and local Lorentz transformations. The analysis is also applied to translations of the coordinates. If they are constant across spacetime, this has no effect on the tangent space bases. If the translation parameters become fields, they induce general linear transformations of the coordinate basis; however, the tetrad components can only be expressed in terms of translations on a flat spacetime.

This paper analyses the geometry of the Lie algebra of SO(6) by making use of its homomorphism with SU(4). We study the vector space of 4×4 traceless, Hermitian matrices from four different viewpoints and examine the connections between them. We review the strata of this space under group transformations using established techniques for su(N) algebras. We focus on orbits of special types of vectors and their interpretation as rotations of SO(6) spinors.