Gleaned from Particles, Sources & FIelds by Julian Sxhwinger pp.225-6
Consider an infinitesimal isotropic dilation.
We are doing global special relativity
The elastic strain is then
dX(x)u,v + dX(x)v,u = nuv(x)dphi(x)
Schwinger proves dphi is a linear form
dphi(x) = 2da + 4dbvx^v
The scalar dilation da
plus 4 special conformal translations bv in
dX(x)^u = dax^u + dbv(x^ux^v - n^u^vx^2)
The special conformal translations are nonlinear in x^u.
added to Poincare group is the 15 parameter Conformal Group beloved of
Roger Penrose in his Twistor theory.
Conformal Group is isomorphic to O(2,4) i.e. RIGID ROTATIONS in 2 times
and 4 space dimensions.
Special relativity is from semi-direct product of T4 with O(1,3).
Einstein's gravity is compensating local gauge field from T4 only.
So add one space dimension y5 and one time dimension y6.
This is like adding a "string space" fiber to each point of 4D space-time.
Define homogeneous coordinates
x^u = y^u/(y5 + y6)
u = 0,1,2,3
y^2 = yuy^u
Define the null 5D hyper-sphere in O(2,4) 6D space
y^2 + y5^2 - y6^2 = 0
10 parameter Poincare group of global Special Relativity is the subgroup
of homogeneous transformations of the null 5D hypersphere that leaves
advanced light cone string coordinate y5 + y6 invariant.
The "Wavelet Transform" scale changes ZOOM IN and ZOOM OUT in 3+1
space-time are ROTATIONS in the STRING SPACE FIBER!
dy5 = -day6
dy6 = -day5
dy^u = 0
The special conformal translations keep y5 - y6 fixed, i.e. retarded
light cone string coordinates fixed.
The 5D null hyper-sphere admits a discrete string space parity mirror
reflection
y5 -> -y5
that INDUCES A DUALITY
x^u --> x^u/x^2
i.e. x^2 -> x^2/x^4
analogous to
r' = Lp^2/r
INVERSION THROUGH ORIGIN in ordinary 4D space-time.
