Complexification

Complexification-Quantum Mechanics-Cosmology

The quantum mechanical amplitudes are given by Fourier sums or series of the form \( \sum_{n} a_{n} e^{i t} \) where \( a_{n} \) are complex numbers, and \( t \) is time, whereas probabilities in classical statistic descriptions are given by the similar sums with real \( a_{n}, \) and \( i t \) replaced by inverse (also real) temperature \( -1 / T \).  In this sense, quantum mechanics is a complexification of Ptolemy's epicycles. In the currently acceptable picture, our evolving Universe can be dissected into "space sections" corresponding to the values of global cosmological time (e.g. in the so called Bianchi cosmological models) to each of which a specific temperature of background cosmic radiation can be ascribed. Going back in time, our Universe becomes hotter, so that at the moment of the Big Bang (time \( =0 \) ) its temperature becomes infinite. This provides a highly romantic interpretation of the correspondence \( -1 / T \leftrightarrow i t \).

 -Y. Manin

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