## I. INTRODUCTION 

One of the most important physical systems for both classical and quantum mechanics is the harmonic oscillator. In contrast to the nonrelativistic harmonic oscillator that is dis- cussed in most textbooks, the theory of relativistic harmonic oscillator is far from complete. The reason for this is the complexity of the problem related to the nonlinearity of differen- tial equations of motion for the classical relativistic oscillator. No wonder that even in the simple case of the massive one-dimensional relativistic oscillator there are problems with identification of periodic solutions to equations of motion [1]. The problem of a quantum relativistic harmonic oscillator is usually formulated in one of three different frameworks: the Klein-Gordon, Dirac or Salpeter equations. The first one uses the spinless Klein-Gordon equation with a Lorentz invariant oscillatory potential [2]. However, the solutions of that equation are blamed by pathologies such as the appearance of ghost states. The second approach, referred to by Moshinsky [3] as the “Dirac oscillator” and describing spin one-half particles utilizes the Dirac equation with an appropriate combination of the scalar, vector and tensor couplings with an external field [4]. It can be successively applied to analysis of relativistic symmetries which recently were recognized experimentally in both nuclear and hadron spectroscopy [5]. Unfortunately, this approach has no classical relativistic counter- part. Finally, the third approach follows from the relativistic Hamiltonian dynamics for a scalar particle and on the quantum level it is based on the spinless Salpeter equation [6]. The Salpeter equation [6–16] is a ”square root” of the Klein-Gordon equation [17] and can be regarded as its alternative [16]. The serious advantages of the Salpeter scheme are the lack of problems with probabilistic interpretation on the quantum level as well as the classi- cally well-defined physical content of this theory. This last framework is frequently used as a phenomenological description of the quark-antiquark-gluon system as a hadron model. 

Surprisingly, to our best knowledge, the simplest case of the massless relativistic harmonic oscillator was not discussed in the literature. In this work we perform a detailed analysis of the massless relativistic harmonic oscillator. In particular we find the exact solutions to the classical Hamilton equations as well as to the corresponding quantum Salpeter equation and discuss their basic properties. The article is organized as follows. In Sec. II, by integrating the corresponding Hamilton system we identify all kinds of possible motion of the oscillator as well as find its quantative characteristics. For an easy illustration of the dynamics of the 

