Reparameterization-Invariant Reduction in the Hamiltonian Description of a Relativistic String

We study the time-reparameterization-invariant dynamics of an open relativistic string using the generalized Dirac–Hamilton theory and resolving the constraints of the first kind. The reparameterization-invariant evolution variable is the time coordinate of the string center of mass. Using a transformation that preserves the diffeomorphism group of the generalized Hamiltonian and the Poincaré covariance of the local constraints, we segregate the center-of-mass coordinates from the local degrees of freedom of the string. We identify the time coordinate of the string center of mass and the proper time measured in the string frame of reference using the Levi-Civita–Shanmugadhasan canonical transformation, which transforms the global constraint (the mass shell) in the new momentum such that the Hamiltonian reduction does not require the corresponding gauge condition. Resolving the local constraints, we obtain an equivalent reduced system whose Hamiltonian describes the evolution w.r.t. the proper time of the string center of mass. The Röhrlich quantum relativistic string theory, which includes the Virasoro operators $L_n$ only with $n>0$, is used to quantize this system. In our approach, the standard problems that appear in the traditional quantization scheme, including the space-time dimension $D=26$ and the tachyon emergence, arise only in the case of a massless string, $M^2=0$.