A General Theorem on the Transition Probabilities of a Quantized System with Spatial Degeneracy

H.A. Tolhoek, S.R. DE GROOT

Zohar Nussinov, Gerardo Ortíz

We establish, in the spirit of the Lieb-Schultz-Mattis theorem, lower bounds on the spectral degeneracy of quantum systems with higher (gaugelike) symmetries with rather generic physical boundary conditions in an arbitrary number of spatial dimensions. Contrary to applying twists or equivalent adiabatic operations, we exploit the effects of modified boundary conditions. When a general choice of boundary geometry is immaterial in determining spectral degeneracies while approaching the thermodynamic limit, systems that exhibit rigid noncommuting gaugelike symmetries, such as the orbital compass model, must have an exponential (in the size of the boundary) degeneracy of each of their spectral levels.

A. N. Kushnirenko

A. N. Kushnirenko

Zohar Nussinov, Gerardo Ortíz

We establish, in the spirit of the Lieb-Schultz-Mattis theorem, lower bounds on the spectral degeneracy of quantum systems with higher (Gauge Like) symmetries with rather generic physical boundary conditions in an arbitrary number of spatial dimensions. Contrary to applying twists or equivalent adiabatic operations, we exploit the effects of modified boundary conditions. When a general choice of boundary geometry is immaterial in approaching the thermodynamic limit, systems that exhibit non-commuting Gauge Like symmetries, such as the orbital compass model, must have an exponential (in the size of the boundary) degeneracy of each of their spectral levels. We briefly discuss why, in spite of the proven large degeneracy associated with infrared-ultraviolet mixing, some systems may still exhibit conventional physical behaviors, i.e., of those of systems with non-extensive degeneracies, due to entropic "order by disorder" type effects.

H. Gingold

A principle in quantum mechanics is proposed: ‘‘In general, the less degeneracy the less transition.’’ Mathematical support of this principle is given in a setting of a slowly varying time-dependent Schrödinger equation via a theorem of asymptotic decomposition. Formulas that quantitatively relate transition and degeneracy are developed. Ramifications of those formulas are discussed.