  Although local Hamiltonians exhibit local time dynamics, this locality is not
explicit in the Schr\"{o}dinger picture in the sense that the wavefunction
amplitudes do not obey a local equation of motion. We show that geometric
locality can be achieved explicitly in the equations of motion by "gauging" the
global unitary invariance of quantum mechanics into a local gauge invariance.
That is, expectation values $\langle \psi|A|\psi \rangle$ are invariant under a
global unitary transformation acting on the wavefunction $|\psi\rangle \to U
|\psi\rangle$ and operators $A \to U A U^\dagger$, and we show that it is
possible to gauge this global invariance into a local gauge invariance. To do
this, we replace the wavefunction with a collection of local wavefunctions
$|\psi_J\rangle$, one for each patch of space $J$. The collection of spatial
patches is chosen to cover the space; e.g. we could choose the patches to be
single qubits or nearest-neighbor sites on a lattice. Local wavefunctions
associated with neighboring pairs of spatial patches $I$ and $J$ are related to
each other by dynamical unitary transformations $U_{IJ}$. The local
wavefunctions are local in the sense that their dynamics are local. That is,
the equations of motion for the local wavefunctions $|\psi_J\rangle$ and
connections $U_{IJ}$ are explicitly local in space and only depend on nearby
Hamiltonian terms. (The local wavefunctions are many-body wavefunctions and
have the same Hilbert space dimension as the usual wavefunction.) We call this
picture of quantum dynamics the gauge picture since it exhibits a local gauge
invariance. The local dynamics of a single spatial patch is related to the
interaction picture, where the interaction Hamiltonian consists of only nearby
Hamiltonian terms. We can also generalize the explicit locality to include
locality in local charge and energy densities.
