Structural Problems and Foundations of Quantum Mechanics
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". Principal publications (numbers according to B.M. publication list):

[6] B.Mielnik,
"Geometry of Quantum States" CMP 9, 55 (1968).

[8] B. Mielnik,
"Theory of Filters" CMP 15, 1 (1969).

[11] B. Mielnik,
"Generalized Quantum Mechanics" CMP 31, 221 (1974).

[12] B. Mielnik,
"Quantum Logic: is it necessarily orthocomplemented?"
in "Quantum Mechanics, Determinism, Causality and Particles", Eds. Flato et al. Reidel (1976).

[22] B. Mielnik,
"Phenomenon of mobility in nonlinear theories",
Commun. Math. Phys. 101, 323 (1985).

Discussions:
[6] quoted by Pascual Jordan, (CMP 1968), discussed by C.V. Stanojevic, TT. Am. Math. Soc., 183, 441 (1974), J.G. Belinfante, Nat. Am. Math. M 21, A344 (1974), S. Pulmannova, J. Math. Phys., 27, 1791 (1987), in "Theory and Decision", 68, (2010) 2547. The work [11] quoted by R. Penrose in
relation to "nonlinear graviton" Gen. Rel Grav. 7, 171 (1976) discussed also by R. Haag and U. Bannier, CMP 60, 1 (1978), discussed and reprinted
in the Hooker's anthology, "Physical Theory as LogicoOperatorial Structure", Reidel, (1978). The idea supported by T. W. Kibble and R. Daemi, J.
Phys. A 13, 141 (1980) discussed by 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] supported 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) 082003, . The article [12] caused multiple controversies, eg G. Cattaneo, J. Math. Phys 25, 513 (1984), but supported by
R. Greechie and D. Foulis, Int. J. Th. Phys 34, 1369 (1995), [22] discussed by D. Gatarek 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).
Prospects and difficulties:
The progress of nonlinear models in quantum mechanics (QM) faces the phenomenon of excessive mobility (described by Haag and Bannier) and by the
author [22] . It is also blocked by the theorem of N. Gisin about superluminal signals in the nonlinear 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: in 2D 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 timedependent external fields was an open
problem for several decades since the early work of Baker, Campbell and Hausdorff in the years 19031906. In the
fifties approached by W. Magnus who applied a nonlinear iterative algorithm (the series of Magnus). The explicit solution was offered
in 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. Important quantum control operations, squeezing operations, inversion of the free evolution. In B.M. publication list:

[7] I. BialynickiBirula, B. Mielnik, J. Plebanski,
"Explicit Solution of the Continuous BakerCampbellHausdorff Problem and a new Expansion of the phase Operator",
Ann. Phys. 51, 187 (1969).

[10] B. Mielnik, J. Plebanski,
"Combinatorial Approach to BakerCampbellHausdorf formula",
Ann. Inst. H. Poincare, 12, 215 (1970).

[14] B. Mielnik,
"Global Mobility of Schrödinger's particle",
Rep. Math. Phys. 12, 331 (1977).

[25] B. Mielnik,
"Evolution Loops",
J. Math. Phys. 27, 2290 (1986).

[36] D.J. Fernandez and B. Mielnik,
"Controlling quantum motion",
J. Math. Phys. 35, 2083 (1994).

[45] F. Delgado C., B. Mielnik,
Phys. Lett. A 249, 369 (1998).

[65] S. Cruz y Cruz and B. Mielnik,
Phys. Lett. A 352, 3640 (2005).

Discussions:
The algorithms of [7,10] 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 e.q. by the Austin group (A. Emmanouilidou et al. Phys Rev. Lett. 85,
1626, (2000). Historical review by D. Fernandez in proc. of XXX Workshop on Geometric Methods of Physics,...
Prospects and difficulties:
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 potentials equivalent to the supersymmetric harmonic oscillator: the case when the operators step between neighboring levels of energy are
differential operators of third or higher orders. Supersymmetric deformations of periodic potentials. According to B.M. publication list:

[21] B. Mielnik,
J. Math. Phys. 25, 3387 (1984).

[44] D. J. Fernandez C., V. Hussin, B. Mielnik,
Phys. Lett. A 244, 1 (1998).

[55] D. J. Fernandez C., B. Mielnik, O. RosasOrtiz, B. F. Samsonov,
Phys. Lett. A 294, 168 (2002).

[57] D. J. Fernandez C., B. Mielnik, O. RosasOrtiz, B. F. Samsonov,
J. Phys. A 35, 4279 (2002).

[62] B. Mielnik and O. RosasOrtiz,
J. Phys. A 37, 1000710035 (2004).

Prospects and difficulties:
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 socalled "time operator" and the relationship of uncertainty timeenergy 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:
Prospects and difficulties
The problems of "time operator" are still unsolved. An experiment reported by D. Kuznetsov and H. Oberst, Optical Review 12, 363366 (2005) seems
to confirm the possibility of the Zeno effect caused by the measurements of time [37] .
