## Sunday, August 20, 2006

Fractal Cartan Forms are Anyons

is contracted over both the GCT indices u and the tangent space indices a.
Therefore it's locally frame invariant both under GCT Diff(4) for COINCIDENT non-geodesic LNIFs at fixed physical event E as well as for O(1,3) COINCIDENT LIF transformations at same E.

Cartan's whole idea of differential forms is that they are local frame invariants - local coordinate independent.

A general 1 form is

1 = 1udx^u

that's scalar invariant

1 = 1udx^u = 1u'dx^u'

A general 2-form is

2 = 2uvdx^u/\dx^v

2uv = - 2vu

etc.

d(p/\q) = dp/\q + (-1)^|p|p/\dq

d^2 = 0

If p is a zero form scalar O

d(O/\q) = dO/\g + O/\dq is a q + 1 form

If p is a 1-form

d(1/\q) = d1/\q - 1/\dq

etc.

p/\q = (-1)^|pq|q/\p

If p & q are both 1-forms then they anti-commute sort of like fermion operators.

ckck' + ck'ck = 0

when k = k' that's the Pauli exclusion principle.

If p & q are both 0-forms then they commute like bosons.

bkbk' - bk'bk = 0

You get Heisenberg uncertainty principle by taking canonical conjugates

e.g. c*k = d/dck is conjugate to ck, one must assume c*K is also a 1-form?

c*kck + ckc*k = 1

So now let p & q be rational numbers, i.e. fractal forms.

This gives fractional quantum statistics & fractional charges!