Quantum Gravity.

t’Hooft builds a Quantum Field Theory based on this step of the discretization process. He builds a wavefunction ¦Ψ_N>.  In this next picture, we are generalising the idea of a divisor N.

We suggest that it is possible to define N as an element of a set of such numbers, each of which is defined by some observer. Each element in that set being a positive integer value. That is;

t_i = {m,r,t, … }

t_k = {N, S, V, …}

t_l = {j, z, w, …}

We have arbitrary groups of observers, of which N is an observation by a single observer.  K is an invariant amongst all observers. We take its value to be the same in every frame. This is possible for quantities like the Planck length which is a constant. For example, we imagine that K = l_p = Planck length. P_i becomes some measure that depends on the values t_i and e_i so that it is considered an invariant quantity by all observers.

For example, suppose that t_i = t_1 for i = 1 and that t_1 = N. That is, at some position 1 in the set t_i we locate the quantity (N).

P_1 = K/t_1 + e_1

P_1 = K/N + e_1

Another way of thinking about this is to realise that N is inversely proportional to P_1. We can allow K to vary along with P_1 and hold e_1 as a constant of the theory.

For example, we can allow

P_1(t) = K(t)/t_1 + e_1

Which by differentation in the the time variable (t) we write;

dP_1(t)/dt = dK(t)dt (1/t_1) + de_1/dt

dP_1(t)/dt = dK(t)dt (1/t_1)

\dot{P _1} = \dot{K} (1/t_1)

P_1 is one of many theories possible. What we will discover is that it will be necessary for us to work in the general space

*dP_i = dK (1/t_i)

* There is an interesting physics associated with this p-form as we will see.

To connect with t’Hooft, we may tentatively suggest a function ¦Ψ_t_i>.

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