  We define operators on semistandard shifted tableaux and use Stembridge's
local characterization for regular graphs to prove they define a crystal
structure. This gives a new proof that Schur $P$-polynomials are Schur
positive. We define queer crystal operators (also called odd Kashiwara
operators) to construct a connected queer crystal on semistandard shifted
tableaux of a given shape. Using the tensor rule for queer crystals, this
provides a new proof that products of Schur $P$-polynomials are Schur
$P$-positive. Finally, to facilitate applications of queer crystals in the
context of Schur $P$-positivity, we give local axioms for queer regular graphs,
generalizing Stembridge's axioms, that partially characterize queer crystals.
