We introduce a ``prepotential'' $F$ in quantum mechanics and show that the coordinate $x$ is proportional to the Legendre transform of $F$ with respect to the probability density. Inversion of the Schr\"odinger equation leads us to consider an $x$ -- $\ensuremath{\psi}$ duality related to a modular symmetry. The scaling of $x$ is determined by the ``beta function,'' suggesting that in quantum mechanics the space coordinate is a macroscopic variable of a statistical system with $\ensuremath{\Elzxh}$ playing the role of scale. The formalism is extended to higher dimensions and to the Klein-Gordon equation.We introduce a ``prepotential'' $F$ in quantum mechanics and show that the coordinate $x$ is proportional to the Legendre transform of $F$ with respect to the probability density. Inversion of the Schr\"odinger equation leads us to consider an $x$-- $\ensuremath{\psi}$ duality related to a modular symmetry. The scaling of $x$ is determined by the ``beta function,'' suggesting that in quantum mechanics the space coordinate is a macroscopic variable of a statistical system with $\ensuremath{\Elzxh}$ playing the role of scale. The formalism is extended to higher dimensions and to the Klein-Gordon equation.

arXiv:hep-th/9606063Physical Review LettersJan 13, 199751 citations

Quantum Mechanics and Dualism

Jeffrey Alan Barrett

De Gruyter eBooksAug 6, 20143 citations

Wave-particle duality: tribute to the memory of Yves Couder

Yves Pomeau A

The story of wave-particle duality in quantum mechanics has been told a thousand times. This modest contribution is to get back to a few points, including memories of a discussion we had on a possible experimental evidence of quantum effects at macroscopic scale, but in a way totally unrelated to the experiments he and his group made quite later to exhibit a model of wave-particle duality. The more general question of interpreting/understanding quantum mechanics will be also scrutinized once more.The story of wave-particle duality in quantum mechanics has been told a thousand times. This modest contribution is to get back to a few points, including memories of a discussion we had on a possible experimental evidence of quantum effects at macroscopic scale, but in a way totally unrelated to the experiments he and his group made quite later to exhibit a model of wave-particle duality. The more general question of interpreting/understanding quantum mechanics will be also scrutinized once more.

Comptes Rendus MécaniqueNov 16, 20200 citations

From Dualism to Unity in Quantum Mechanics

Alfred Landé

Springer eBooksJan 1, 19888 citations

We introduce a ``prepotential''

F

in quantum mechanics and show that the coordinate

x

is proportional to the Legendre transform of

F

with respect to the probability density. Inversion of the Schr\"odinger equation leads us to consider an

x

-- $\ensuremath{\psi}$ duality related to a modular symmetry. The scaling of

x

is determined by the ``beta function,'' suggesting that in quantum mechanics the space coordinate is a macroscopic variable of a statistical system with $\ensuremath{\Elzxh}$ playing the role of scale. The formalism is extended to higher dimensions and to the Klein-Gordon equation.