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Heinz von Foerster 100
Andrei Khrennikov

Quantum Mechanics as Theory of Classical Random Signals: the problem of (anti-) bunching

International Center for Mathematical Modeling
in Physics, Engineering, Economics, and Cognitive Science
Linnaeus University, Växjö-Kalmar, Sweden

Prequantum classical statistical field theory [1 -11] (PCSFT) is a model which provides a possibility to represent averages of quantum observables, including correlations of observables on subsystems of a composite system, as averages with respect to fluctuations of classical random fields. PCSFT is a classical model of the wave type. For example, electron'' is described by electronic field. In contrast to QM, this field is a real physical field and not a field of probabilities. An important point is that the prequantum field of e.g. electron contains the irreducible contribution of the background field, vacuum fluctuations. In principle, the traditional QM-formalism can be considered as a special regularization procedure: subtraction of averages with respect to vacuum fluctuations. In this paper we derive a classical analog of the Heisenberg-Robertson inequality for dispersions of functionals of classical (prequantum) fields. PCSFT Robertson-like inequality provides a restriction on the product of classical dispersions. However, this restriction is not so rigid as in QM. The quantum dispersion corresponds to the difference between e.g. the electron field dispersion and the dispersion of vacuum fluctuations. Classical Robertson-like inequality contains these differences. Hence, it does not imply such a rigid estimate from below for dispersions as it was done in QM.

Bunching and anti-bunching are considered as fundamentally quantum phenomena which could not be described in the classical field framework. In this talk we show that, opposite to this very common opinion, bosonic and fermionic (as well as anyonic) correlations can be described with the aid of classical random fields. We present a model of bunching and anti-bunching of classical random (Gaussian) bi-signals. Thus quantum and classical signal theories (and, in particular, classical and quantum optics) are much closer than it is typically assumed.

[1] Khrennikov A 2005 {\it J. Phys. A: Math. Gen.} {\bf 38} 9051.

[2] Khrennikov A 2006 {\it Found. Phys. Letters} {\bf 18} 637.

[3] Khrennikov A 2006 {\it Physics Letters A} {\bf 357} 171.

[4] Khrennikov A 2006 {\it Found. Phys. Lett.} {\bf 19} 299.

[5] Khrennikov A 2006 {\it Nuovo Cimento} B {\bf 121} 505.

[6] Khrennikov A 2010 {\it Physica E: Low-dimensional Systems and Nanostructures} {\bf 42}, 287-292.

[7] Khrennikov A 2006 {\it Nuovo Cimento} B, {\bf 121} (9): 1005-1021.

[8] Ohya M and Watanabe N 1986 {\it Jap. J. Appl. Math.}, {\bf 3}, 197-206.

[9] Khrennikov A 2008 {\it. J. Modern Optics}, {\bf 55}, N 14, 2257-2267.

[10] Khrennikov A 2009 {\it Europhysics Letters}, {\bf 88}, 40005.1.

[11] Khrennikov A 2010 {\it Europhysics Lett.} {\bf 90}, art-number 40004.