Stability and Related Properties of Vacua and Ground States

TL;DR

This paper investigates the non-relativistic limit of a quantum field theory, revealing how different limiting procedures affect stability, scattering, and bound states, with implications for the low-energy behavior of the system.

Contribution

It compares two methods of taking the non-relativistic limit in :4:_{s+1} theory, showing their impact on stability and scattering properties.

Findings

In case i), the scattering amplitude tends to zero as the cutoff increases.

In case ii), a bound state appears, indicating an attractive potential.

Stability of matter fails for the boson system under the second limit.

Abstract

We consider the formal non relativistc limit (nrl) of the :\phi^4:_{s+1} relativistic quantum field theory (rqft), where s is the space dimension. Following work of R. Jackiw, we show that, for s=2 and a given value of the ultraviolet cutoff \kappa, there are two ways to perform the nrl: i.) fixing the renormalized mass m^2 equal to the bare mass m_0^2; ii.) keeping the renormalized mass fixed and different from the bare mass m_0^2. In the (infinite-volume) two-particle sector the scattering amplitude tends to zero as \kappa -> \infty in case i.) and, in case ii.), there is a bound state, indicating that the interaction potential is attractive. As a consequence, stability of matter fails for our boson system. We discuss why both alternatives do not reproduce the low-energy behaviour of the full rqft. The singular nature of the nrl is also nicely illustrated for s=1 by a rigorous stability/instability result of a different nature.