Catchy title. But seriously, I came across this thing which behaves as an inverse to the p-form.
χ•d = a∂/∂a + (b/a)∂/∂b
where we allow χ = 1 + 1/a to be a 2×2 diagonal matrix. We allow this operator to act on a scalar Φ as…
(χ•d)(Φ) = a∂Φ/∂a + (b/a)∂Φ/∂b
(χ•d)(Φ) = χ•dΦ ≠ d•χΦ
This formula, I think, possibly opens up exciting possibilities. For example, we can imagine that there is some sort of anti-symmetry that can be captured by the Lie bracket so that
[χ•d , Φ] = (χ•d)(Φ) – (Φ)(χ•d)
It is interesting to note some of the properties, if at all, of this rendition of the Lie bracket such as its operation on another scalar field given by φ.
[χ•d , Φ] (φ) = (χ•dΦ)(φ) – ((Φ)(χ•d) )(φ)
[χ•d , Φ] φ = (χ•dΦ)φ – Φ(χ•dφ)
Perhaps the usefulness of the Lie bracket is that it measures the extent to which symmetry is lost, and is itself an operator on scalars. There are many interesting properties to this.
Am I right thta this object
is just a particulary chose vector field? What I mean is that the general form of a vector field is
And, hence, the object in question is just some vector field on a two–dimensional manifold.
Wow. Am so sorry about the ‘delayed’ reply. Yes, it is a vector field. The commutator defines for us a new operator, a new vector field.
Now, I intended to provide a kind of trivial proof that one can indeed formulate an inverse…but, since then I have been focused on applying that commutator to the more interesting problem of devising a structure from which one can derive Newtonian mechanics. I have succeeded. With quite spectacular results, I am sure and hopeful.
So, I apologise to have to end the discussion there, for now, perhaps I can come back to it when I’m salaried to do these things.
Hope remains!
However, I hope you will join me on this blog as I update you – the reader – on progress I have made since then. Again, I do apologise for the long stretch without a response.