  In the Bargmann representation of quantum mechanics, physical states are
mapped into entire functions of a complex variable z*, whereas the creation and
annihilation operators $\hat{a}^\dagger$ and $\hat{a}$ play the role of
multiplication and differentiation with respect to z*, respectively. In this
paper we propose an alternative representation of quantum states, conjugate to
the Bargmann representation, where the roles of $\hat{a}^\dagger$ and $\hat{a}$
are reversed, much like the roles of the position and momentum operators in
their respective representations. We derive expressions for the inner product
that maintain the usual notion of distance between states in the Hilbert space.
Applications to simple systems and to the calculation of semiclassical
propagators are presented.
