Seminari del Dipartimento di Matematica

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Seminario del 2024

Ottobre

2024

Maria Clara Nucci

Hidden Lie and Noether symmetries in Classical and Quantum Mechanics

nell'ambito della serie: TOPICS IN MATHEMATICS 2023/2024

fisica matematica

The use of Lie symmetries for differential equations has been tremendous, and many textbooks are available. A major drawback of Lie’s method is that it is useless when applied to systems of first-order equations, e.g. Hamiltonian equations, because they admit an infinite number of Lie symmetries, and there is no systematic way to find even one-dimensional Lie symmetry algebra, apart from trivial groups like translations in time admitted by autonomous systems. However, in 1996 I have remarked that any system of first-order equations could be transformed into an equivalent system where at least one of the equations is of second order. Then, the admitted Lie symmetry algebra is no longer infinite dimensional, and hidden symmetries of the original system could be retrieved: consequently I determined hidden symmetries of the Kepler problem. Since then, with my co-authors, I have found hidden symmetries uncovering the linearity of nonlinear superintegrable systems in two and three dimensions, even in the presence of a static electromagnetic field. In the Avertissement to his Mécanique Analitique (1788), Joseph-Louis Lagrange (1736-1813) wrote in French: "Those who love Analysis will, with joy, see mechanics become a new branch of it, and will be grateful to me for thus having extended its field. (Tr. J.R. Maddox). Although it may seem a joke, we show that it has actually dire consequences for the physical reality of Mechanics, in particular by means of the Jacobi last multiplier, and its connection with Lie symmetries. In 1918 Noether published her landmark paper, and since then her namesake theorem has been applied in different areas of Physics, especially classical Lagrangian mechanics and general relativity. In this seminar, I will show the application of Noether symmetries in the quantization of classical mechanics, a quantization method that preserves the Noether point symmetries and consequently gives rise to the Schroedinger equation of various classical problems.

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