Bogdan Mielnik

Structural Problems and Quantum Mechanics Foundations

The possibility of atypical geometries where the quantum states do not have to be vectors in Hilbert spaces. Convexity and generalized geometry of quantum theories. Alternatives nonlinear quantum equations and the "phenomenon of mobility". In Bogdan Mielnik publication list:

[6] is quoted by a sequence of authors starting from Pascual Jordan, (CMP 1968), ending up by economist in "Theory and Decision", 68, (2010) 25-47. The work [11] quoted by R. Penrose in relation to "non-linear graviton" Gen. Rel Grav. 7, 171 (1976) also discussed by R. Haag and U. Bannier, CMP 60, 1 (1978), discussed and reprinted in the Hooker's anthology, "Physical Theory as Logico-Operatorial Structure", Reidel, (1978). The idea adopted by T. W. Kibble and R. Daemi, J. Phys. A 13, 141 (1980) discussed between R. Penrose and A. Shimony in "The Large, the Small and the Human Mind", CUP (1997), pp. 144, 176. Fragments of [6] and [11] reprinted in the "Encyclopedia of Math. And Its Appl." Ed Vol XV Gian Carlo Rota, Addison Wesley (1981), Ch 18, 19. The ideas of [8] are adopted and developed by John Bell and M. Hallett, Philos. Sci. 49, 335 (1982) recently also by quantum information groups, eg D. C. Brody and L. P. Hughston, J. Phys. A 43, (2010) 8, [12] resulted in multiple controversies, eg G. Cattaneo, J. Math. Phys 25, 513 (1984), but supported by R. Greechie and D. In Int J. Foulis Th Phys 34, 1369 (1995), [22] discussed by D. Gatare and N. Gisin, J. Math. Phys 32, 2152 (1991), R. Cirelli et al. J. Geom. Phys 29, 64 (1999), D. C. Brody and L. P. Hughston, J. Geom. Ph. 38, 2597 (2001).

The progress of nonlinear models in quantum mechanics (QM) faces the phenomenon of excessive mobility (described by Haag and Bannier and the author [22] ) is also blocked by the theorem of N. Gisin about superluminal signals in the non-linear QM entangled states. However, it appears that the idea of ​​infinite tensor products in quantum field theories (with endless repetitions of the same basic cell: the area in 2-D Hilbert) are just a quantum analogue of Ptolemy's epicycles theory. The problem is essentially open.

Quantum Control Problems

The explicit series for the exponent (operatorial phase) of the unitary operator generated by time-dependent external fields was an open problem for several decades since the early work of Baker, Campbell and Hausdorff in the years 1903-1906, in the fifties approached by W. Magnus who applied a nonlinear iterative algorithm (the series of Magnus) but without the explicit solution finally offered in the two papers [7,10] . Related topics: manuals of exact dynamic manipulation of quantum systems. Closed cycles of evolution ("evolution loops"), as the antithesis of chaos (Ref. 44-36). Its importance as starting points for monitoring progress. Squeezing operations, dynamic transformation of Fourier inversion of the free evolution, possible uses and fundamental implications. In B.M. publication list:

The algorithms of [7,10] were discussed in the monograph of J. Czyz, "Paradoxes of Measures and Dimensions in Felix Hausdorff Originated ideas", World Sci Singapore (1994), cited by I. M. Gelfand, Adv. Math. 112, 218 (1995) reexamined the Ph. D. L. Saenz (R. Suarez, L. Saenz, JM Ph. 47, 4582 (2001). The method of "evolution loops" cited as a key for quantum manipulation by the Austin group (A. Emmanouilidou et al. Phys Rev. Lett. 85, 1626, (2000).

The methods of "perturbative calculation in the exponent" [7,10] is a topic barely open, one of the ideas is that they can reduce the infinities of quantum field theories (the commutators of the fields vanish outside the cones of light, reducing the areas of integration in the perturbative series). The exact solutions derived from the cycles of evolution ("evolution loops") suggest new techniques of quantum control.

Construction of exactly soluble spectral models

Study of potential equivalent to the supersymmetric harmonic oscillator: the case when the operators step between neighboring levels of energy are differential operators of third or higher degrees. Supersymmetric deformations of periodic potentials. According to B.M. publication list:

The exactly soluble quantum wells may arise as defects of periodic potential, injected by supersymmetric transformations. One might use fragments exact supersymmetric solutions as interpolation functions in order to facilitate numerical methods in multiple problems.

Problems of Time

The so-called "time operator" and the relationship of uncertainty time-energy in quantum theories. The problem of radiation of a quantum system in the presence of an environment variable: depends on the photon energy emitted from past events? In B.M. publication list:

The problems of "time operator" are still unresolved. An experiment reported by D. Kuznetsov and H. Oberst, Optical Review 12, 363-366 (2005) seems to confirm the possibility of the Zeno effect caused by the measurements of time [37] .