Heisenberg proof of the Balian-Low theorem

M. A. Seifi Mirjafarlou, Arash Jafarizadeh, M. A. Rajabpour

Abstract Fermionic Gaussian states have garnered considerable attention due to their intriguing properties, most notably Wick’s theorem. Expanding upon the work of Balian and Brezin, who generalized properties of fermionic Gaussian operators and states, we further extend their findings to incorporate Gaussian operators with a linear component. Leveraging a technique introduced by Colpa, we streamline the analysis and present a comprehensive extension of the Balian–Brezin decomposition to encompass exponentials involving linear terms. Furthermore, we introduce Gaussian states featuring a linear part and derive corresponding overlap formulas. Additionally, we generalize Wick’s theorem to encompass scenarios involving linear terms, facilitating the expression of generic expectation values in relation to one and two-point correlation functions. We also provide a brief commentary on the applicability of the BB decomposition in addressing the BCH (Zassenhaus) formulas within the so ( N ) Lie algebra.

D. ter Haar, William Menzies Nicol

Colin James

J. J. Halliwell

Any account of the emergence of classicality from quantum theory must address the fact that the quantum operators representing positions and momenta do not commute, whereas their classical counterparts suffer no such restrictions. To address this, we revive an old idea of von Neumann, and seek a pair of commuting operators X, P which are, in a specific sense, "close" to the canonical non-commuting position and momentum operators, x,p. The construction of such operators is related to the problem of finding complete sets of orthonormal phase space localized states, a problem severely limited by the Balian-Low theorem. Here these limitations are avoided by restricting attention to situations in which the density matrix is reasonably decohered (i.e., spread out in phase space).

Shue-Kwing Chan, Kiang-Chuen Young

J. M. Luttinger

It is shown that the expressions for the thermodynamic potential and temperature Green's functions in terms of the exact mean occupation number of the unperturbed single-particle states may be obtained without the elaborate diagrammatic analysis of Balian and his coworkers. The proof is only carried out for fermions, but a similar situation also holds for bosons.