ИСТИНА

The classical system associated with the theory of closed smooth strings can be identified with the pair $(\Omega(R_d),\text{Vect}(S^1))$ where the phase space $\Omega(R_d)$ is the space of smooth loops in the $d$-dimensional Minkowski space $R_d$ and the algebra of observables $\text{Vect}(S^1)$ is the Lie algebra of the Lie group $\text{Diff}_+(S^1)$ of diffeomorphisms of the circle. In the case of non-smooth strings we take for the phase space of the theory the Sobolev space $V=H_0^{1/2}(S^1,R_d)$ of half-differentiable vector-functions on the circle. The group $\text{Diff}_+(S^1)$ is replaced by the group $\text{QS}(S^1)$ of quasisymmetric homeomorphisms of the circle. However, the action of this group on the Sobolev space $V$ by reparameterization is not smooth so we cannot associate any classical algebra of observables associated with $\text{QS}(S^1)$. However, we can define a quantized infinitesimal action of $\text{QS}(S^1)$ on the Sobolev space $V$ which allows us to construct a quantum algebra of observables associated with the phase space $V$ provided with the action of the group $\text{QS}(S^1))$.