The so-called Ultimate Question is the question whose answer is a Theory-of-Everything. The trouble is that as far as I can tell theoretical physics are not certain what the question is they're trying to solve.
In quantum mechanics and quantum field theory, which has a separate notion of time, it seems like everything is contained in a function $\Delta_t(A,B)$ where $\Delta$ maps an initial and final state and a duration, $t$ to a complex number (called an amplitude). And according to the rules of quantum mechanics this function must also satisfy:
$$\int \Delta_t(A,B)\Delta_{t'}(B,C)dB = \Delta_{t+t'}(A,C)$$
In the case of quantum gravity where time is not independent but a property of the states themselves. It seems like everything is contained within the function $\Delta(A,B)$, where this time:
$$\int_\Sigma \Delta(A,B)\Delta(B,C)dB = \Delta(A,C)$$
Where $\Sigma$ is a smooth surface in the space of physical states separating the states $A$ and $C$. i.e. any smooth path from $A$ to $C$ must pass through one of the states $B$. (Similar to how Cauchy's theorem works). Which is just another way of saying you sum over (a complete set of) intermediate states.
So it seems like if we had a set of states, an amplitude function $\Delta$, and an initial condition $O$. Then this would specify a complete set of laws of nature.
The second equation seems to suggest that for sufficiently complicated states, a notion of diffeomorphism symmetry would help satisfy this equation, but other then that it doesn't seem very restrictive. (At least not restrictive enough that the answer should be Supergravity or Superstrings).
What other axioms must be satisfied. In other words, how can one formulate the Ultimate Question in a mathematically precise way?
