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Description
Discussing quantum theory foundations, von Neumann noted that the measurement process should not be regarded in terms of a time evolution [1]. The reason for such a claim is the insurmountability of a gap between reversibly and irreversibly evolving systems in physics. The time operator formalism that goes beyond the gap is an adequate framework for elucudation of the measurement problem [2]. It is a straightforward generalization of multiresolution representing the identity through a direct sum of projectors onto subspaces of the signal space [3]. The wavelet base $ψ_{j,k}$ whose elements span a multiresolution is constituted according to the structure of real numbers that concern commensuration of magnitudes in the Euclidean algorithm [4]. These elements are both states and devices of the measurement process, which is an indication of the signal space autoduality. The statistical model of a measurement requires ensembles whose density operators are $ρ=FF^*$, whereby the root $F=f(T)$ is a normed function $‖f‖=1$ of the time operator $T$. Its coefficients $|D_{j,k}⟩=F|ψ_{j,k}⟩$ are considered to be random variables, as well as each energy $|D_{j,k}|^2$ which is a distribution density. Due to the measurement process, it is reduced to an expected value $|d_{j,k}|^2=E|D_{j,k}|^2$ that is the probability of a state $ψ_{j,k}$. Respecting that, the density operator $ρ=∑_{j,k} |d_{j,k}|^2 |ψ_{j,k}⟩⟨ψ_{j,k}|$ has become diagonal in the base.
In order to regard distributions $|D_{j,k}|^2$, one considers an alternative density $F^* F=∑_{j,k}|D_{j,k}⟩⟨D_{j,k}|$ whose component $ϕ_{j,k}=|D_{j,k}⟩⟨D_{j,k}|$ corresponds to the Markov variable $S_{j,k}$. The measurement problem concerns statistical causality that operates through causal variables $S_{j,k}$ which evoke a stochastic computation [5]. It represents the time evolution of a complex system, which is related to the measurement process.
References
[1] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955, 351-354.
[2] I. Prigogine, Time and Complexity in the Physical Science, W.H. Freeman Co., New York, 1980.
[3] I. Antoniou, B. Misra, Z. Suchanecki, Time Operator: Innovation and Complexity, John Wiley & Sons, New York, 2003, 107.
[4] M. Milovanović, S. Vukmirović, The Time Operator of Reals, In: Proceedings of the 4th Conference on Complexity, Future Information Systems and Risk – COMPLEXIS 2019, Heraklion, 2-4 May 2019, SCITERPRESS – Science and Technology Publications, 75-84.
[5] J. P. Crutchfield, K. Young, Computation on the Onest of Chaos, In: Complexity, Entropy, and the Physics of Information, Addison-Wesley, 1990, 223-269.
Details
Miloš Milovanović
Mathematical institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia
http://www.mi.sanu.ac.rs/
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