Wheeler de witt gravitons

Let us take a wave function of the metric at time T=0. Expanded around a flat space g  = n + h.

Ψ[h] = a + Ψ(x)h(x,0) + Ψ(x,y)h(x,0)h(y,0) + .....

The Wheeler-de-Witt equation should give a relation between the Ψs. Because g is a linear function of h, d/dh=d/dg.

When expanded in terms of h, this has an infinite number of terms. This imposes very special constraints on Ψ to give it the right symmetries. (A similar thing happens in string theory).

((gg+gg-gg) d/dh(x,0) d/dh(x,0) + det(g)R[g] ) Ψ[h] = 0

This equation says how the wave function amplitudes are connected  be when 2 particles are in the same place. 

If h^2 is approximately 0. Then h behaves like a gauge field.

Part 2
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Lets take a different expansion g=exp(h) , det(g) = exp(Tr(h)) , delta h = da + db?

Then this timer d/dg = g^-1 d/dh

( d/dh d/dh +  exp(Tr[h]+h+h-h)*(dh)^2 ) Ψ[h] = 0 ?

Ψ(x) = exp(-x^2) = 1 - x^2 + x^4/2! +...
int exp(-x^2) dx = sqrt(pi)

 exp(-ax^2)(1-x^2+x^4/2! +...)

Ψ[h] = limit(a-->0) exp(-a h(x,0)^2 )

INTERPRETATION

Fields are 4 dimensional entities that cannot be measured. They are mathematical things used for calculation. What we can have knowledge of is the wave function and it's coefficients which are the particle amplitudes. These are 3-dimensional and are on a casual surface.

What is a functional for which it is peaked around a certain field f? This might be:

Ψ[h] = exp( - int[  (h(x)-f(x)) exp(-(x-y)^2) (h(y)-f(y)) ]dx^3 dy^3  )

Which can be expanded with coefficients:

Ψ(x) = int[exp(-(x-x')^2)f(x')]dx'
Ψ(x,y) = int[ exp(-(x-x')^2-(y-y')^2)f(x')f(y')]dx'dy' + exp(-(x-y)^2)