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The Quantum Mellin transform

J. Twamley·G. J. Milburn
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Abstract

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 $p_η$, transforms the wavefunction via a Mellin transform on to the critial line $s=1/2-ip_η$. 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.

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J. Twamley, G. J. Milburn. 2007-02-12. The Quantum Mellin transform. https://doi.org/10.1088/1367-2630%2F8%2F12%2F328

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