Existence of the Fock representation for current algebras of the Galilei algebra (Q2888938)

The investigation of the 3 problems stated below has led, in the past 10 years, to a multiplicity of new results and to the discovery of several unexpected connections between different fields at mathematics and physics:NEWLINENEWLINE Problem 1: construct a continuous analogue of the \(*\)-Lie algebra (and associative \(*\)-algebra) of differential operators in \(d\) variables with polynomial coefficients NEWLINE\[NEWLINEDOPC(\mathbb{R}^d):= \Biggl\{\sum_{n\in \mathbb{N}^d} P_n(x) \sigma^n_x;\;x\in\mathbb{R}^d,\;P_n\text{ complex polynomials in \(d\) real variables}\Biggl\},NEWLINE\]NEWLINE where continuous means that the space NEWLINE\[NEWLINE\mathbb{R}^d\equiv\{\text{functions} \{1,\dots,d\}\to\mathbb{R}\}NEWLINE\]NEWLINE is replaced by some function space NEWLINE\[NEWLINE\text{functions }\{\mathbb{R}\to\mathbb{R}\}.NEWLINE\]NEWLINE Since, for \(d+1\), this algebra can be canonically identified to the universal enveloping algebra of the one-mode Heisenberg algebra \(\text{Heis\,}\mathbb{C}(1)\), this problem is equivalen to the old standing problem of constructing a theory of nonlinear quantum (boson) fields: hence its connections with the renormalization problem.NEWLINENEWLINE Problem II: construct \(*\)-representations of this algebra (typically a generalization of the Fock representation) as operators acting on some domain in Hilbert space \({\mathcal H}\).NEWLINENEWLINE Problem III: prove the unitarity of these representations, i.e. that the skew symetric elemets of this \(*\)-algebra can be exponentiated; leading to strongly continuous 1-parameter unitery groups.NEWLINENEWLINEFor the entire collection see [Zbl 1238.81004].