Non-commutative integration method and its application to certain problems in QFT

In this talk, I plan to review the so-called noncommutative integration method of linear PDEs, which is an alternative to the method of separation of variables. This method considerably uses non-commutative symmetry algebras of PDEs and allows to construct more convenient bases of the respective solution spaces. In the first part of my talk, I give a brief introduction to the essence of the noncommutative integration method. In the second part, I expose a more advanced description of the method for left-invariant PDEs on Lie groups. In conclusion, I discuss the applications of the noncommutative integration method to some QFT problems: 1) constructing exact solutions of the Klein-Gordon equation in external electromagnetic fields; 2) vacuum polarization; 3) the evaluation of heat kernel on non-compact Lie groups.