On certain unitary representations of an infinite group of transformations. Thesis, FU Brussels, 1951. Transl. from the French by Marcus Berg and Cécile DeWitt-Morette (Q2763507)

This publication is the English translation of the mathematical thesis presented by Léon Van Hove at the Free University of Brussels in 1951. It deals with the passage from classical to quantum mechanics, a problem known nowaday as Geometrical quantization. Basis for investigating this correspondence is a Hilbert space description of classical dynamic systems of \(2n+1\) variables \((s,p_1, \dots,q_1, \dots,q_n)\) with the Pfaff form \(\overline\omega= ds-\sum_i p_id q_i\), which is the canonical form of what is ordinarily written \(\omega= \sum_j p_jdq_j -hdt\). In this last expression \(t\) denotes time, \(p_j,q_j\) are canonical conjugated variables and \(h=h(p,q)\) the Hamiltonian of the system.NEWLINENEWLINENEWLINEVan Hove considers the group \(\Gamma\) of bijective infinitely differentiable transformations of the space \((s,p,q)\) onto itself which leave the form \(\overline \omega\) invariant. This group is an infinite-dimensional continuous group. The group of canonical transformations, normally considered in classical mechanics, is the quotient group \(\Gamma/C\) of \(\Gamma\) by its center \(C\). The group \(\Gamma/C\) has not the fundamental property of \(\Gamma\): infinitesimal transformations in bijective correspondence with first integrals of motion, Lie brackets corresponding to Poisson brackets.NEWLINENEWLINENEWLINEA representation \({\mathcal R}\) of \(\Gamma\) is defined by unitary transformations in the Hilbert space of measurable and square integral functions, which has irreducible components \({\mathcal R}^{(\alpha)}\), also unitary and depending on an arbitrary real parameter \(\alpha\). For \(\alpha=1/ \hbar\), the self-adjoint operators \(H^{(\alpha)}[f]\), representing infinitesimal transformations corresponding to functions \(f(p,q)\), have strong analogy with the self-adjoint operators describing observables in quantum mechanics.NEWLINENEWLINENEWLINEFor quantities represented in classical mechanics by quadratic polynomials in \(p,q\) there is a bijective correspondence between Poisson brackets and quantum mechanical commutators. The main result obtained by Van Hove is that this bijective correspondence between classical mechanics and quantum mechanics does not extend beyond the quadratic case.