Abstract
Three Clifford algebras are sufficient to describe all interactions of modern physics: The Clifford algebra of the usual space is enough to describe all aspects of electromagnetism, including the quantum wave of the electron. The Clifford algebra of space-time is enough for electro-weak interactions. To get the gauge group of the standard model, with electro-weak and strong interactions, a third algebra is sufficient, with only two more dimensions of space. The Clifford algebra of space allows us to include also gravitation. We discuss the advantages of our approach.
Highlights
Why Clifford algebras are necessary in physics? Physics uses waves and the Fourier theorem says that any periodic function may be decomposed in a sum of sin and cos functions
The second reason is the spin 1/2 of all fundamental fermions, which uses SL (2, ), that is a subgroup of Cl3* = GL (2, ). This greater group is the group of form invariance of electromagnetism, wave of the electron included [2]-[6]
The use of the three Clifford algebras Cl3, Cl1,3 and Cl1,5 presents many advantages: We reunite the frame of classical physics, which was the space-time and vectors or tensors built on space-time, with the frame of quantum mechanics, since all interactions are described with real Clifford algebras
Summary
Why Clifford algebras are necessary in physics? Physics uses waves and the Fourier theorem says that any periodic function may be decomposed in a sum of sin and cos functions. The second reason is the spin 1/2 of all fundamental fermions, which uses SL (2, ) , that is a subgroup of Cl3* = GL (2, ) This greater group is the group of form invariance of electromagnetism, wave of the electron included [2]-[6]. Space-time is the auto-adjoint part of the space algebra [1]: This allows us to read the Dirac wave of the electron in Cl3. The greater group of invariance has induced the invariant form of the wave equation, and this will bring a new understanding of the existence of the Lagrangian density, and so on. The space algebra Cl3 is 8-dimensional on , the invariant wave Equation (6) is equivalent to a system of 8 numeric equations with partial derivatives. Under the dilation R defined in (7), the A and B are contravariant vectors, moving with the source, while the j and k currents are covariant vectors and the field F satisfies: F=′ MFM −1; A′ + iB=′ M ( A + iB) M †; j + i=k M ( j′ + ik′) M
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