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The Holographic Quantum

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Abstract

We present a map of standard quantum mechanics onto a dual theory, that of the classical thermodynamics of irreversible processes. While no gravity is present in our construction, our map exhibits features that are reminiscent of the holographic principle of quantum gravity.

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Notes

  1. We will henceforth omit all normalistion factors, bearing in mind that all probabilites are to be normalised at the end.

  2. What quantum theorists call the Feynman path integral was independently developed in Ref. [23] by Onsager and collaborators, who appear to have arrived at the notion of a path integral all by themselves, without previous knowledge of Feynman’s earlier work [14].

  3. Implicit in the replacements (23) is the assumption that the thermodynamical extensive variable x, and the mechanical variable x, both have units of length. A dimensionful conversion factor is to be understood in case the dimensions do not match.

  4. In case more than just one normal coordinate is needed, this statement is to be understood as meaning the sum of all the lengths so obtained.

  5. We should remark that the assumption of compactness of the leaves \(\mathbb {L}_n\) can be lifted without altering our conclusions. A noncompact leaf encloses an infinite (yet countable) number of volume quanta \(Q_{D-1}\). Upon multiplication by an infinite (yet countable) number of momentum-space quanta \(P_{D-1}\), the dimension of the tangent Hilbert space \(\mathcal{H}_t\) remains denumerably infinite. This form of holography in which the leaves are noncompact replaces the notion of inside vs. outside the leaf with the equivalent notion of one side of the leaf vs. the other side. One should not dismiss this possibility as unphysical: the constant potential, for example, can be regarded as having either compact or noncompact equipotential submanifolds.

  6. Unless, of course, one is willing to allow for pair creation out of the vacuum, thus quitting quantum mechanics and entering field theory.

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Authors and Affiliations

  1. Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, 46022, Valencia, Spain

    P. Fernández de Córdoba, J. M. Isidro & J. Vazquez Molina

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  1. P. Fernández de Córdoba
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  2. J. M. Isidro
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  3. J. Vazquez Molina
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Correspondence to J. M. Isidro.

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Fernández de Córdoba, P., Isidro, J.M. & Vazquez Molina, J. The Holographic Quantum. Found Phys 46, 787–803 (2016). https://doi.org/10.1007/s10701-015-9986-2

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