Let us assume that there exists a Hamiltonian that (together with the initial state) describes the whole universe.
Then my question is : What is the spectrum of this Hamiltonian and what are the multiplicities of the spectrum ? More precisely, what is the representation of the Hamiltonian in Spectral multiplicity theory (see http://en.wikipedia.org/wiki/Self-adjoint_operator, section "Spectral multiplicity theory").
Assume that the universe is asymptotically flat and has finite total energy. (These appear to be the minimal assumptions under which talking about the Hamiltonian of the universe make sense.)
Then the following description holds independent of any other unknown details about its modeling. The latter are relevant only if you want to predict the set of bound states and their masses, and want to know something about other observables than Poincare symmetries.
Because of translation invariance of space-time, the spectrum is the positve halfline, including zero.
The appropriate spectral representation is in terms of an asymptotic representation where the Hamiltonian and the momentum are diagonal. In such a representation, assuming asymptotic completeness and disregarding infrared complications, a basis is given by states $|C;p^1,...,p^{|C|}\rangle$, where $C$ runs over all possible sets of bound states, and $p^j=(p^j_0,{\bf p}^j)$ is the asymptotic 4-momentum of the $j$-th entry of $C$, with $p^j_0=\sqrt{({\bf p}^j)^2+(m^jc)^2}$, where $m^j$ is the mass of the $j$-th entry of $C$ and $c$ is the speed of light.
On these states, 4-momentum $p$ acts by multiplication with $\sum_{j=1}^{|C|} p^j$, and $H=p_0c$ is the Hamiltonian. The representation is the direct sum of the corresponding Fock representations of the Poincare group (with statistics corresponding to the spin of the bound states). This gives a complete description of the spectrum in the $C^*$-algebraic sense. Clearly, every eigenvalue has infinite multiplicity.
A correct infrared description also needs to account for clouds of massless particles accompanying these states, defined formally by very weak limits that are not mathematically well-defined in a separable Hilbert space setting. It leads to the formation of uncountably many superselection sectors modifying the above description, but lacking a clear mathematical basis, the details are poorly understood.
The above assumes that the universe is asymptotically flat and has finite total energy. (These appear to be the minimal assumptions under which talking about the Hamiltonian of the universe make sense.) The description holds independent of any other unknown details about its modeling. The latter are relevant only if you want to predict the set of bound states and their masses, and want to know something about other observables than Poincare symmetries.
