Hello,
Continuing with our theme; we can more explicitly write an equation
(1/χ) • (χ•d) = d
where 1/χ is the inverse to χ
and χ is a brilliant little symmetric (0,2) tensor that comes in useful if we imagine that somehow a and b define a 2-d manifold. It is a useful tool because we can use it to imagine that we can define a group based upon the multiplication rules of the new exterior derivative defined above.
By this I mean that…
(1/χ) • (χ•d) ≠ (χ•d)(1/χ)
This is part of the reason why we discussed the Lie algebra because that algebra allows us to define, in algebraic terms, the loss of symmetry shown above. Also, the Lie algebra opens up opportunity to describe group elements such as exp(χ•d).
It is by using group theoretic-ish methods that we will try to define the inverse of the p-form. By inverse of the p-form we mean to understand that
(χ•d)Φ = χ•dΦ = Φ
where dΦ is the p-form
The idea is to prove, using physics, that the above equation is correct. My own view is that a mathematical proof is not so far off.
