# Geometrical algebra in theoretical physics

by scottbekerham
Tags: algebra, geometrical, physics, theoretical
 P: 834 I consider it worth learning just for the simplicity it introduces into problems that otherwise would've been extremely tedious or difficult. It's not uncommon that one will have to manipulate a tensor expression to try to get to a simpler result. GA's generality and identities make this much easier to do than laboring through index notation, in my opinion. Example: find ##\epsilon_{ijk} \epsilon^{ljk}##. I can't speak to proving this in index notation, but in GA, you can keep things grounded and simple. The Levi-Civita tensor is just components of the pseudoscalar evaluated on some basis. In this case, we can generalize this problem to an equivalent one: Simplify ##(a \wedge B) i (c \wedge B^{-1}) i = (a \wedge B)(B^{-1} \wedge c)## for vectors ##a,c## and bivector ##B##. This isn't a hard problem to attack, especially with the power of GA. Projection onto grade and associativity make it rather straightforward. Note that ##ac = a B B^{-1} c## and project out some components. $$\langle a B B^{-1} c \rangle_0 = (a \wedge B)\cdot (B^{-1} \wedge c) = a \cdot c$$ This is actually so much easier than most identity problems, I was surprised I was done at this point. Usually you have to consider two grades at least, but since the result must be scalar, we're done here. Yes, for the identity we meant to consider, there's a missing factor of 2. I can't quite find it--probably would if I were more methodical--but it does show that there's some work in converting a tensor expression to a GA one.