The Witt construction in characteristic one and Quantization

We develop the analogue of the Witt construction in characteristic one. We construct a functor from pairs of a perfect semi-ring of characteristic one and an element strictly larger than one, to real Banach algebras. We find that the entropy function familiar in thermodynamics, ergodic theory and information theory occurs uniquely as the analogue of the Teichmuller polynomials in characteristic one. We then apply the construction to the semi-field of positive real numbers with max as addition, which plays a central role in idempotent analysis and tropical geometry. Our construction gives the inverse process of the ``dequantization" and provides a first hint towards an extension of the field of real numbers relevant both in number theory and quantum physics.