Reviews in Mathematical Physics

We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr's doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative C*-algebra of observables is empirically accessible only through associated commutative C*-algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of single-case probabilities (which will be the subject of a sequel paper). Our derivation of the Born rule uses ideas from a program begun by Finkelstein [17] and Hartle [21], intending to remove the Born rule as a separate postulate of quantum mechanics. Mathematically speaking, our approach refines previous elaborations of this program — notably the one due to Farhi, Goldstone, and Gutmann [15] as completed by Van Wesep [50] — in replacing infinite tensor products of Hilbert spaces by continuous fields of C*-algebras. Furthermore, instead of relying on the controversial eigenstate-eigenvalue link in quantum theory, our derivation just assumes that pure states in classical physics have the usual interpretation as truthmakers that assign sharp values to observables.

This paper discusses an algorithmic way of constructing integrable evolution equations based on Lie algebraic structure. We derive, in a pedagogical style, a large class of two-component peakon type dual systems from their two-component soliton equations counter part. We study the essential aspects of Hamiltonian flows on coadjoint orbits of the centrally extended semidirect product group to give a systematic derivation of the dual counter parts of various two-component of integrable systems, viz., the dispersive water wave equation, the Kaup–Boussinesq system and the Broer–Kaup system, using moment of inertia operators method and the (frozen) Lie–Poisson structure. This paper essentially gives Lie algebraic explanation of Olver–Rosenau's paper [31].

We present a systematic and detailed review of the application of the method of Hirota and the rational dressing method to abelian Toda systems associated with the untwisted loop groups of complex general linear groups. Emphasizing the rational dressing method, we compare the soliton solutions constructed within these two approaches, and show that the solutions obtained by the Hirota's method are a subset of those obtained by the rational dressing method.

We study the Aharonov–Bohm effect through the semiclassical analysis for the spectral shift function and its derivative in magnetic scattering by two solenoidal fields in two dimensions, assuming that the total magnetic flux vanishes. The corresponding classical system has a trajectory oscillating between the centers of two solenoidal fields. The emphasis is placed on analyzing how the trapping effect is reflected in the semiclassical asymptotic formula. We also make a comment on the case of scattering by a finite number of solenoidal fields and discuss the relation between the Aharonov–Bohm effect from quantum mechanics and the trapping effect from classical mechanics.

We study the relativistic electron-positron field at positive temperature in the Hartree–Fock approximation. We consider both the case with and without exchange terms, and investigate the existence and properties of minimizers. Our approach is non-perturbative in the sense that the relevant electron subspace is determined in a self-consistent way. The present work is an extension of previous work by Hainzl, Lewin, Séré and Solovej where the case of zero temperature was considered.