Second quantization of a covariant relativistic spacetime string in Steuckelberg–Horwitz–Piron theory

A relativistic 4D string is described in the framework of the covariant quantum theory first introduced by Stueckelberg [Helv. Phys. Acta 14, 588 (1941)], and further developed by Horwitz and Piron [Helv. Phys. Acta 46, 316 (1973)], and discussed at length in the book of Horwitz [Relativistic Quantum Mechanics, Springer (2015)]. We describe the space-time string using the solutions of relativistic harmonic oscillator [J. Math. Phys. 30, 66 (1989)]. We first study the problem of the discrete string, both classically and quantum mechanically, and then turn to a study of the continuum limit, which contains a basically new formalism for the quantization of an extended system. The mass and energy spectrum are derived. Some comparison is made with known string models.