Schroedinger's Equation for Conformal Symmetry

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Polyakov’s bootstrap programme aims at solving conformal field theories using 
unitarity and conformal symmetry. Its implementation in two dimensions has been 
highly successful and numerical studies, in particular of the 3-dimensional Ising 
model, have clearly demonstrated the potential for higher dimensional theories. 
Analytical results in higher dimensions, however, require significant insight 
into the conformal group and its representations. Surprisingly little is actually 
known about this important group theory challenge. I  will explain a remarkable 
and unexpected connection with a class of Schroedinger equations that was uncovered 
in recent joint work with M. Isachenkov. The study of the relevant quantum mechanics 
systems has created an entire branch of modern mathematics whose results can now be 
put to use in the conformal bootstrap program.