ABSTRACT We show that a result of symplectic topology, Gromov's non-squeezing theorem, al... more ABSTRACT We show that a result of symplectic topology, Gromov's non-squeezing theorem, also known as the `principle of the symplectic camel', can be used to quantize phase space in cells. That quantization scheme leads to the correct energy levels for integrable systems and to Maslov quantization of Lagrangian manifolds by purely topological arguments. We finally show that the argument leading to the proof of the non-squeezing theorem leads to a classical form of Heisenberg's inequalities.
The emergence of quantum mechanics from classical world mechanics is now a well-established theme... more The emergence of quantum mechanics from classical world mechanics is now a well-established theme in mathematical physics. This book demonstrates that quantum mechanics can indeed be viewed as a refinement of Hamiltonian mechanics, and builds on the work of George Mackey in relation to their mathematical foundations. Additionally when looking at the differences with classical mechanics, quantum mechanics crucially depends on the value of Planck's constant h. Recent cosmological observations tend to indicate that not only the fine structure constant α but also h might have varied in both time and space since the Big Bang. We explore the mathematical and physical consequences of a variation of h; surprisingly we see that a decrease of h leads to transitions from the quantum to the classical. Emergence of the Quantum from the Classical provides help to undergraduate and graduate students of mathematics, physics and quantum theory looking to advance into research in the field. Readership: Undergraduate and graduate students of mathematics and physics, interested in analysis and differential equations, probability and statistics, geometry and topology and mathematical physics
There are known obstructions to a full quantization in the spirit of Dirac's approach, the most k... more There are known obstructions to a full quantization in the spirit of Dirac's approach, the most known being the Groenewold-van Hove no-go result. We show, following a suggestion of S. K. Kauffmann, that it is possible to construct a well-defined quantization procedure by weakening the usual requirement that commutators should correspond to Poisson brackets. The weaker requirement consists in demanding that this correspondence should only hold for Hamiltonian functions of the type T(p)+V(q). This reformulation leads to a non-injective quantization of all observables which, when restricted to polynomials, is the rule proposed by Born and Jordan in the early days of quantum mechanics.
ABSTRACT We show that a result of symplectic topology, Gromov's non-squeezing theorem, al... more ABSTRACT We show that a result of symplectic topology, Gromov's non-squeezing theorem, also known as the `principle of the symplectic camel', can be used to quantize phase space in cells. That quantization scheme leads to the correct energy levels for integrable systems and to Maslov quantization of Lagrangian manifolds by purely topological arguments. We finally show that the argument leading to the proof of the non-squeezing theorem leads to a classical form of Heisenberg's inequalities.
The emergence of quantum mechanics from classical world mechanics is now a well-established theme... more The emergence of quantum mechanics from classical world mechanics is now a well-established theme in mathematical physics. This book demonstrates that quantum mechanics can indeed be viewed as a refinement of Hamiltonian mechanics, and builds on the work of George Mackey in relation to their mathematical foundations. Additionally when looking at the differences with classical mechanics, quantum mechanics crucially depends on the value of Planck's constant h. Recent cosmological observations tend to indicate that not only the fine structure constant α but also h might have varied in both time and space since the Big Bang. We explore the mathematical and physical consequences of a variation of h; surprisingly we see that a decrease of h leads to transitions from the quantum to the classical. Emergence of the Quantum from the Classical provides help to undergraduate and graduate students of mathematics, physics and quantum theory looking to advance into research in the field. Readership: Undergraduate and graduate students of mathematics and physics, interested in analysis and differential equations, probability and statistics, geometry and topology and mathematical physics
There are known obstructions to a full quantization in the spirit of Dirac's approach, the most k... more There are known obstructions to a full quantization in the spirit of Dirac's approach, the most known being the Groenewold-van Hove no-go result. We show, following a suggestion of S. K. Kauffmann, that it is possible to construct a well-defined quantization procedure by weakening the usual requirement that commutators should correspond to Poisson brackets. The weaker requirement consists in demanding that this correspondence should only hold for Hamiltonian functions of the type T(p)+V(q). This reformulation leads to a non-injective quantization of all observables which, when restricted to polynomials, is the rule proposed by Born and Jordan in the early days of quantum mechanics.
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Papers by Maurice de Gosson