The Mathematical Formulation of Quantum Mechanics

Walter E Thirring

Martin Pieper

BongJu Kim

In mathematical aspect, we introduce quantum algorithm and the mathematical structure of quantum computer. Quantum algorithm is expressed by linear algebra on a finite dimensional complex inner product space. The mathematical formulations of quantum mechanics had been established in around 1930, by von Neumann. The formulation uses functional analysis, linear algebra and probability theory. The knowledge of the mathematical formulation of QM is enough quantum mechanical knowledge for approaching to quantum algorithm and it might be efficient way for mathematicians that starting with mathematical formulations of QM. We explain the mathematical formulations of quantum mechanics briefly, quantum bits, quantum gates, quantum discrete Fourier transformation, Deutsch's algorithm and Shor's algorithm.

Andrés J. Kálnay

Different level sources of possible inconsistencies of Quantum Physics are studied. The point of view is defended that inconsistencies may arise due to improper philosophical foundations of Quantum Physics but that there is a way out to each of the up to now suggested purely mathematical inconsistencies of each of the main mathematical formulations of Quantum mechanics. An example is worked out explicitly. The point of view is also supported that even if the above statement may be correct, i.e., that no purely mathematical inconsistencies exist in each of the main mathematical formulations, it is highly worthwhile to study alternatives as those offered by the Lie-Admissible Formulations of Quantum Mechanics.

Luigi Barletti

The paper is devoted to review, from a mathematical point of view, some fundamental aspects of the Wigner formulation of quantum mechanics. Starting from the axioms of quantum mechanics and of quantum statistics, we justify the introduction of \nthe Wigner transform and eventually deduce the Wigner equation.