Is the Adiabatic approximation Inconsistent? – Solomon Duki

Date: Mon. October 23rd, 2006, 12:30 pm-1:30 pm
Location: Rockefeller 221

The adiabatic theorem is the basis of an approximation scheme that was discovered at the dawn of quantum mechanics and that has been in widespread and continuous use ever since. Applications range from two-level systems (such as nuclei undergoing magnetic resonance or atoms interacting resonantly with a laser field) to quantum field theory (where a low-energy effective theory is derived by integrating out fast, high-energy degrees of freedom). Two decades ago, Berry uncovered the beautiful geometric structure underlying the adiabatic approximation, leading to a resurgence of interest in the subject and to new applications. More recently, it has been proposed that Berry phase effects lead to quantum phase transitions that lie outside the usual Landau-Ginzburg-Wilson paradigm. The adiabatic theorem is also the basis of a newly proposed quantum computing scheme. Considering the significance of the adiabatic approximation to quantum physics, the discovery of an inconsistency would be most disturbing. In recent Letter Marzlin and Sanders[1] ask whether such an inconsistency might exist, at least for a class of Hamiltonians. That question has been further studied by Tong et. al.[2]. We argue that there is no inconsistency and the apparent paradox found in refs [1] and [2] results from the application of an incorrect condition for the validity of the adiabatic approximation [3].