Quantization of the theory of half-differentiable strings

The problem of quantizing the space omega(d) of smooth loops taking values in the d-dimensional vector space can be solved in the framework of the standard Dirac approach. But a natural symplectic form on omega d can be extended to the Hilbert completion of omega(d) coinciding with the Sobolev space V-d:= H-0(1/2) (S1), Double-struck capital R-d) of half-differentiable loops with values in Double-struck capital R-d. We regard V-d as the phase space of the theory of half-differentiable strings. This theory can be quantized using ideas from noncommutative geometry.