Geometrized physics

The physics of the twentieth century, especially after the theory of general relativity, has evolved towards a “geometrization of physics”. Some have even coined the expression geometrodynamics to emphasize the fact that the central equation of general relativity expresses the link between the energy-momentum tensor and the local spacetime curvature.

On the other hand, the course of quantum physics was more abstract, which instead crystallized in mathematical rather than physical spaces (e.g. Hilbert space), with abundant use of operators and imaginary quantities hardly endowed with a geometric interpretation.
Thus the idea that non-commutativity was a peculiarity of quantum physics and the idea that matrices were the necessary tool to master it were established.

Geometric algebra, on the other hand, brings geometry and non-commutativity directly into the DNA of algebraic expressions.
Here are the main advantages of this approach:

  • overcomes the limitation of the cross product beyond 3D space
  • clarifies the ambiguity between vectors and pseudovectors
  • expresses the imaginary unit i both as a geometric object and as a rotation or duality operator, managing to extend the complex analysis beyond 2D
  • expresses Lorentz rotations and transformations with simple algebraic multiplications
  • unifies the language of classical mechanics with that of quantum mechanics (use of spinors and projection operators explained in a geometric sense)
  • brings the Pauli and Dirac matrices back to the bases of geometric algebra respectively G^3 and G^{1,3}

A preliminary note concerns the units of measurement: dimensional analysis in physics is very important and allows you to check the correctness of a calculation or even to derive certain expressions starting from dimensional considerations.
With a joke, physics is a strongly typed language.
Having said this, however, the use of several quantities merged in the same multivector is not prohibited, as long as it is borne in mind that the unit of measurement is part of the coefficient (the quantum ) and not of the base (the thing ).
For example a bivector B = 2.32 \: e_1e_2 can be expressed in square meters, but it does not mean that the bivector unit expresses an area (the how much ), expresses instead a plane (which position ? which orientation ?) and therefore must be understood as B = (2.32 \: m^2) \: e_1e_2.

Speaking of bivectors, we have already said that GA replaces cross products with bivectors, but we must pay attention to units. Take angular velocity, an axial vector defined by the right-hand rule: it would seem natural to obtain the corresponding bivector from the outer product (wedge) of the radius rr and the velocity vv. However, r∧vr \wedge v has dimensions [m]2[s]−1[m]^2[s]^{-1} — these are the dimensions of specific angular momentum, not of an angular velocity [s]−1[s]^{-1}.
To obtain the angular-velocity bivector we must normalize by dividing by r2r^2:

\begin{equation} \Omega = \frac{\pmb{r} \wedge \pmb{v}}{r^2}

In a very general way, to approach physics problems using geometric algebra, one must:

  • use the geometric object suitable for the physical quantity in question (see table)
  • replace the vector product with the wedge
  • for ease of calculation, replace the bivectors with their dual expression, that is j \pmb{n}