  I propose that physical theories defined over finite places (including
$p$-adic fields) can be used to construct conventional theories over the reals,
or conversely, that certain theories over the reals "decompose" over the finite
places, and that this decomposition applies to quantum mechanics, field theory,
gravity, and string theory, in both Euclidean and Lorentzian signatures. I
present two examples of the decomposition: quantum mechanics of a free
particle, and Euclidean two-dimensional Einstein gravity. For the free
particle, the finite place theory is the usual free particle $p$-adic quantum
mechanics, with the Hamiltonian obtained from the real one by replacing the
usual derivatives with Vladimirov derivatives, and numerical coefficients with
$p$-adic norms. For Euclidean two-dimensional gravity, the finite place objects
mimicking the role of the spacetime are $\mathrm{SL}(\mathbb{Q}_p)$ Bruhat-Tits
trees. I furthermore propose quadratic extension Bruhat-Tits trees as the
finite place objects into which Lorentzian $\mathrm{AdS}_2$ decomposes, and
Bruhat-Tits buildings as a natural generalization to higher dimensions, with
the same symmetry group on the finite and real sides for the manifolds and
buildings corresponding to the vacuum state.
