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''. We show that this new unitary change of basis from the position x on the half line to the Hyperbolic momentum $p_\eta$, transforms the wavefunction via a Mellin transform on to the critial line $s=1/2-ip_\eta$. We utilise 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 realisations to perform an indirect measurement of the Hyperbolic momentum of a quantum system on the half-line.Comment: 23 pages, 6 Figure

The technique of inverse Mellin transform for processes occurring in a background magnetic field

We develop the technique of inverse Mellin transform for processes occurring in a background magnetic field. We show by analyticity that the energy (momentum) derivatives of a field theory amplitude at the zero energy (momentum) is equal to the Mellin transform of the absorptive part of the amplitude. By inverting the transform, the absorptive part of the amplitude can be easily calculated. We apply this technique to calculate the photon polarization function in a background magnetic field.Comment: 3 pages, LATEX; talk presented at ICHEP02, Amsterdam, The Netherlands, 24-31 July 200

The distribution of Mahler's measures of reciprocal polynomials

We study the distribution of Mahler's measures of reciprocal polynomials with complex coefficients and bounded even degree. We discover that the distribution function associated to Mahler's measure restricted to monic reciprocal polynomials is a reciprocal (or anti-reciprocal) Laurent polynomial on [1,\infty) and identically zero on [0,1). Moreover, the coefficients of this Laurent polynomial are rational numbers times a power of \pi. We are led to this discovery by the computation of the Mellin transform of the distribution function. This Mellin transform is an even (or odd) rational function with poles at small integers and residues that are rational numbers times a power of \pi. We also use this Mellin transform to show that the volume of the set of reciprocal polynomials with complex coefficients, bounded degree and Mahler's measure less than or equal to one is a rational number times a power of \pi.Comment: 13 pages. To be published in Int. J. Math. Math. Sc