The quantum Mellin transform

We uncover a new type of unitary operation for quantum mechanics on the half-line which yields a transformation to 'hyperbolic phase space' (η, p_η). We show that this new unitary change of basis from the position x on the half line to the hyperbolic momentum p_η, transforms the wavef unction via a Mellin transform on to the critical line s = 1 /2 -ip_η. We utilize this new transform to find quantum wavefunctions whose hyperbolic momentum representation approximate a class of higher transcendental functions, and in particular, approximate the Riemann-Zeta function. We finally give possible physical realizations to perform an indirect measurement of the hyperbolic momentum of a quantum system on the half-line.