Abstract
The problem of quantizing the space Ωd 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 Ωd can be extended to the Hilbert completion of Ωd coinciding with the Sobolev space Vd:= H01/2 (\(\mathbb{S}^1\), ℝd) of half-differentiable loops with values in ℝd. We regard Vd as the phase space of the theory of half-differentiable strings. This theory can be quantized using ideas from noncommutative geometry.
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To Andrei Alekseevich Slavnov on his 80th birthday.
This research was supported in part by the Russian Foundation for Basic Research (Grant Nos. 18-51-05009 and 19-01-00474) and the Presidium of the Russian Academy of Sciences (Program “Nonlinear dynamics”).
Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 203, No. 2, pp. 220–230, May, 2020.
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Sergeev, A.G. Quantization of the theory of half-differentiable strings. Theor Math Phys 203, 621–630 (2020). https://doi.org/10.1134/S0040577920050050
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DOI: https://doi.org/10.1134/S0040577920050050