Sine-Gordon Solitons vs. Relativistic Calogero-Moser Particles

Abstract

Abstract. We discuss the connection between the N-soliton subspace of the sine-Gordon field theory and (a specialization of) the relativistic Calogero-Moser N-particle systems. At the classical level this soliton-particle relation is well understood, and we summarize its ma.in features. At the quantum level we expect a physical equivalence of the pertinent particle systems and the sine-Gordon/massive Thirring field theory. We survey the evidence for the 2-body case in some detail.

Effective potential for relativistic scattering

“…Its spectrum and S-matrix is exactly known from its bootstrap solution [18]. Moreover, an equivalent relativistic quantum mechanical description exists, the Ruijsenaars-Schneider model [12,13]. The SG field theory Lagrangian is ¶…”

“…The Ruijsenaars-Schneider (RS) model [12,13] is an integrable relativistic quantum mechanical model whose dynamics and S-matrix is completely equivalent to that of the SG field theory. From the RS description it is possible to read off the corresponding zero-momentum potential [13,11]. In our conventions it reads (after restoring the constants , c)…”

“…In the semiclassical limit an energy-independent effective potential was constructed, which exactly reproduces the semiclassical time delays for all energies. This could be compared to the zero-momentum potential, which is explicitly known in this model from its equivalent Ruijsenaars-Schneider (RS) formulation [12,13].…”

Spatially Extended Relativistic Particles Out of Traveling Front Solutions of Sine-Gordon Equation in (1+2) Dimensions

“…This idea has been discussed extensively in the High Energy literature (see, e.g., [ 6 – 16 ].) In particular, the connection between solutions of the (1+1)-dimensional Sine-Gordon equation and particles within a field-theoretic context has been studied (see, e.g., [ 14 ]).…”

Effective potential from zero-momentum potential

“…Luckily, an alternative description of the SG model exists since it is known that for any fixed particle number subspace of the SG field theory Hilbert space there is a corresponding Ruijsenaars-Schneider (RS) type relativistic quantum mechanical description [14]. The RS wave function is known [15] for both soliton-soliton and soliton-antisoliton scattering and exactly reproduces the scattering phase shifts of SG field theory. Moreover, also the soliton-antisoliton bound state spectrum is calculable and exactly match the SG results.…”