The numbers are geometrized
The grammar of Nature is a discourse made up of numbers and operations between numbers.
In this view we can follow two opposite approaches:
1) dry up the numbers as much as possible, leaving them only the property of magnitude, in the form of a real number and making the operations between them very complex. For example, the product of two complex numbers becomes:
2) empower the concept of number by embedding in it geometric properties, which greatly simplifies the operations between them. For example, the complex number as operator is a scaled rotation and therefore the product of two complex numbers is simply the sum of the rotations with scale factors multiplied.
The most striking example of the power of this second approach is undoubtedly the reduction of the four Maxwell equations to one.
Dimension scales smoothly
Gibbs and Heaviside vectors algebra is a bit limp about dimensions: the vector product is not defined in 2D and only works in 3D. The geometric product of GA, on the other hand, is defined in any dimension and all operations defined in the GA scale smoothly for any number of dimensions.
The fact that geometric algebra is not commutative is often seen as a big problem: the expression cannot be simplified to zero, because the two terms are not equal.
Actually this is far from a problem, in fact non-commutativity prove to be a farsighted and faithful representation of the physical world (for example the rotations in 3D) and it’s the basis of the formulation of operators in quantum physics.
Moreover, even matrix algebra is non-commutative and despite this – or perhaps exactly for this reason – it lies as the foundation of linear algebra courses.
In linear algebra and its physical applications, the use of subspaces of dimension 1 (vectors), has historically been established. For some reason the subspaces of dimension 2 (planes) and 3 (volumes) have always been manipulated indirectly, using vectors.
Finally, with geometric algebra, subspaces of any size have an existence of their own and can be added, subtracted, multiplied and even divided.
Complex numbers have nothing mysterious or mystical: they are rotodilation operators or duality transformations
For centuries we have had them staring us in the face – as rotations – they had nothing magical about them, but our mindset was fixed on coordinates, that is, on the points of some space. Even after their discovery, complex numbers have been trapped in the Argand-Gauss plane, but their power is best expressed when considered as rotation operators.
The second aspect that explains the presence of the imaginary unity in the formulas used in physics, especially in quantum mechanics, is the operation of duality. In this case we are not dealing with rotations, but with the transformation of an entity into the perpendicular subspace (orthogonal complement). Therefore the duality in 3D, that is the multiplication by transforms a vector in the perpendicular plane and vice versa (except for the sign).
The circle closes
In the GA the product between two vectors, so badly set in the ordinary approach (split in the two halves of scalar and vector product), is not only fully defined and invertible, but leads directly to complex numbers.
Vectors and complex numbers
GA manages to overcome that horrible mess that is seen in some areas of engineering, for example in alternating currents, where there is a mixture of vectors and complex numbers. For example, the expression used for the calculation of electrical power
has no valid explanation in the context of complex numbers. To justify it, Steinmetz even went so far affirming that, when it comes to electrical power, the imaginary unit squared is +1.
All this, finally, acquires total clarity in the light of the GA.
GA is the most general expression of already known systems
In fact it includes as sub-algebras the systems so far perceived as distinct:
– complex numbers (the even sub-algebra of )
– quaternions (the even sub-algebra of )
– Pauli’s spin matrices.
The right hand rule
In the study of electromagnetism there are apparent deviations from symmetry and they are the right-hand rule (valid for generators) and the left-hand rule (for motors). Actually the equations of electromagnetism have no intrinsic chirality and the GA clearly shows this!