Schr\"odinger quantization of infinite-dimensional Hamiltonian systems with a nonquadratic Hamiltonian function

O. G. Smolyanov ; N. N. Shamarov

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 492 (2020), pp. 65-69

Résumé

According to a theorem of Andre Weil, there does not exist a standard Lebesgue measure on any infinite-dimensional locally convex space. Because of that, Schrödinger quantization of an infinite-dimensional Hamiltonian system is often defined using a $\sigma$-additive measure, which is not translation-invariant. In the present paper, a completely different approach is applied: we use the generalized Lebesgue measure, which is translation-invariant. In implicit form, such a measure was used in the first paper published by Feynman (1948). In this situation, pseudodifferential operators whose symbols are classical Hamiltonian functions are formally defined as in the finite-dimensional case. In particular, they use unitary Fourier transforms which map functions (on a finite-dimensional space) into functions. Such a definition of the infinite-dimensional unitary Fourier transforms has not been used in the literature.

Détail

Keywords: quantization, Schrödinger quantization, generalized Lebesgue measure, infinite-dimensional Hamiltonian systems, Heisenberg algebra, infinite-dimensional pseudodifferential operators.