On the one hand, it is known that a real hypersphere of an 8-dimensional neutral space, that is, a space with signature of the metric (+ 4, -4), is homeomorphic to the space R4 × S3. On the other hand, the algebra of linear tangent vector fields of hyperspheres of an 8-dimensional neutral space and hyperspheres of an 8-dimensional Euclidean space is isomorphic to the Lie algebra of the Dirac matrix algebra. At the same time, at the intersection of the hypersphere of the neutral and Euclidean space is the product of the spheres S3 × S3.
The answer is Yes.
See, for example,
Principles of a Unified Theory of Spacetime and Physical Interactions
https://arxiv.org/pdf/hep-th/0111021.pdf
Thank you, but I'm closer to Kassandrov's algebra-dynamic approach.
Vladimir Vsevolodovich Kassandrov
Usually, in a quantum field theory, first one thinks of a vector-space and then looks at various symmetry operations which can be performed within this space. Rotations translations etc. For example if you consider the real line, you can only think of translations and reflections; if you consider 2D real space you can think of translations, reflections ans also rotations. These are so called symmetry operations.
Lorentz group is a group of symmetry operations in 4-D Minkowski space. Group has it's representations and then it also has it's generators. Physical objects are labeled by representations, such as scalar, vector, tensor etc.
The generators on the other hand, satisfy the group algebra. Generators can obviously generate elements of the group. This is a group of transformations, which act on representations.
If you know the group algebra to begin with then there is a problem. I can explain with an example. SU(2) and O(3) has the same Lie algebra. In the first case group elements are 2x2 complex matrices which act on a 2D complex vector space whereas in the second case group elements are 3x3 real matrices which act on 3D real world. Then the geometry on which symmetry transformations act upon, is not fixed uniquely by knowing (only) Lie algebra.
I agree that the geometry of the representation space of the algebra is not fixed. For example, the same Lie algebra su(2) can be represented either as the algebra of linear tangent vector fields of 3-dimensional spheres in R4 or the algebra of linear tangent vector fields of circles in R3. However, if we are talking about the geometry of the microworld, then we have to make a choice. My choice in the Article Applications of the local algebras of vector fields to the m...
By the way, on the one hand, the Lie algebra su(3) can be represented by the algebra of tangent vector fields of 3-spheres in R6, and on the other hand, the group SU(3) can be represented by the motions of a 3-dimensional torus spanned by a 3-dimensional sphere with punctured poles.
In the framework of this topic, I would like to touch upon the question of the geometry of elementary particles. If the vacuum is interpreted as a linear vector velocity field of particles of a continuous medium, then the elementary particles could be interpreted as topological singularities of an arbitrary vector field in which it is closed. In this case, the phrase "the electron has the geometry of a torus spanned by the product S2 × S1" would have a certain meaning ...
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