Stability and related properties of vacua and ground states

• Published in: Annals of physics 2008-02, Vol.323 (2), p.251-266
• Main Authors: Wreszinski, Walter F., Jäkel, Christian D.
• Format: Article
• Language: English
• Subjects: BOSONS; BOUND STATE; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONSTRUCTIVE FIELD THEORY; GROUND STATES; INSTABILITY; INTERACTIONS; MATTER; Non-relativistic limit; POTENTIALS; SCATTERING AMPLITUDES; Stability of matter; ULTRAVIOLET RADIATION
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2007.02.002

We consider the formal non-relativistic limit (nrl) of the :ϕ4:s+1 relativistic quantum field theory (rqft), where s is the space dimension. Following the work of R. Jackiw [R. Jackiw, in: A. Ali, P. Hoodbhoy (Eds.), Bég Memorial Volume, World Scientific, Singapore, 1991], we show that, for s=2 and a given value of the ultraviolet cutoff κ, there are two ways to perform the nrl: (i) fixing the renormalized mass m2 equal to the bare mass m02; (ii) keeping the renormalized mass fixed and different from the bare mass m02. In the (infinite-volume) two-particle sector the scattering amplitude tends to zero as κ→∞ 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.