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Abstract

We consider the Wheeler-DeWitt operator associated with the bosonic part of
the Hamiltonian of D=11 supergravity in a formulation with only the spatial
components of the three-form and six-form fields, and compare it with the E10
Casimir operator at low levels, to show that these two operators precisely
match modulo spatial gradients up to and including gl(10) level two. The
uniqueness of the E10 Casimir operator eliminates all ordering ambiguities in
the quantum Hamiltonian, at least up to the level considered. Beyond level
three the two operators are expected to start to differ from each other, as
they do so for the classical expressions. We then consider truncations of the
E10 Wheeler-DeWitt operator for various finite-dimensional subgroups of E10 in
order to exhibit the automorphic properties of the associated wave functions
and to show that physically sensible wave functions generically vanish at the
cosmological singularity, thus providing new and more sophisticated examples of
DeWitt's proposed mechanism for singularity resolution in quantum gravity. Our
construction provides novel perspectives on several unresolved conceptual
issues with the Wheeler-DeWitt equation, such as the question of observables in
quantum gravity, or the issue of emergent space and time in a purely algebraic
framework. We also highlight remaining open questions of the E10 framework.