Geodesic
Formulation of quantum mechanics with dynamical time
Informatics and Automation, Selected topics of mathematical physics and analysis, Tome 285 (2014), pp. 154-165

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We consider quantum mechanics for which the system time is one of generalized coordinates. The generalized Hamiltonian has an unbounded spectrum, which allows us to introduce a Hermitian time operator. In the proposed formulation of quantum mechanics, a system time and observer's time are introduced. The Schrödinger equation in the system time either does not hold or holds only approximately. The wave function is assumed to be square integrable with respect to all coordinates, including the system time. In some limit, this formalism reproduces standard quantum mechanics and the corresponding measurement theory. 
@article{TRSPY_2014_285_a10,
     author = {M. G. Ivanov},
     title = {Formulation of quantum mechanics with dynamical time},
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     pages = {154--165},
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     volume = {285},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2014_285_a10/}
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M. G. Ivanov. Formulation of quantum mechanics with dynamical time. Informatics and Automation, Selected topics of mathematical physics and analysis, Tome 285 (2014), pp. 154-165. http://geodesic.mathdoc.fr/item/TRSPY_2014_285_a10/