"The energy representation has no non-zero fixed vectors," by B. Driver and Brian Hall.in {\it Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999)}, 143--155, Amer. Math. Soc., Providence, RI, 2000We consider the ``energy representation'' $W$ of the group $\mathcal{G}$ of smooth mappings of a Riemannian manifold $M$ into a compact Lie group $G.$
Our main result is that if $W\left( g\right) f=f$ for all $g\in \mathcal{G},$ then $f=0.$ In the language of quantum field theory this says that there are
no ``states.'' Our result follows from the irreducibility of the energy representation whenever the irreducibility theorems of
Ismagilov, Gelfand--Graev--Ver\v{s}ik, Albeverio--Hoegh-Krohn--Testard, or Wallach
apply. Our result, however, applies in general, even in cases where the energy representation is known to be reducible. |
08/15/2016 02:53 PM |