Spectral representation

B. A. Veklenko

Daniel M. Greenberger, Klaus Hentschel, Friedel Weinert

Kacheong R. Yeung, Lex Yeung

Many familiar quantum puzzles may begin not with what the quantum object is taken to be, but with what we demand of its representations. This paper develops a representability framework in which an admissible representation may be subjected to an additional representational requirement whose admissibility must then be tested. When that stronger demand cannot be satisfied while preserving the relations the representation is required to maintain, a representability obstruction arises. Spectral incompatibility, entanglement, interference, and Bell-violating correlations are examined through this common methodological pattern. Their mathematical structures remain distinct; what they share is the question of whether an already admissible representation can satisfy the additional representational requirement being imposed. The framework also separates representability obstruction from outcome-conditioned refinement, the operation on the represented state conventionally described as “wavefunction collapse.” Given an already admissible spectral structure and a recorded outcome, the represented state undergoes outcome-conditioned refinement: the corresponding spectral alternative is selected and the state is refined relative to it. This operation is defined at the Hilbert-space and spectral level before any coordinate wavefunction is introduced. A coordinate wavefunction expresses the refined state only after a coordinate representation has been chosen. The framework therefore treats recorded outcomes as empirical givens and does not attempt to explain why one particular outcome rather than another is physically produced. At the structural level, the framework is organized by the symmetry group, the observable algebra, and the symmetry action. Their Hilbert-space realization supplies the spectral and symmetry-compatible representational structures within which admissibility and invariant comparison are formulated. Within this setting, the Born weight is treated as an invariant quadratic comparison between the represented state and a spectral alternative and is not, by itself, identified with ignorance over pre-existing jointly defined values. An individual Born measure therefore does not by itself supply the joint classical representation required by classical ignorance. The resulting picture places structure before wave description, distinguishes representational impossibility from conditional refinement, and isolates precisely where stronger representational demands cease to be admissible. This is version 1.0 of the manuscript.

Stephen J. Gustafson, Israel Michael Sigal

Götz Silvester Uhrig, Mona H. Kalthoff, James K. Freericks

Periodically driven nonequilibrium many-body systems are interesting because they have quasi-energy spectra, which can be tailored by controlling the external driving fields. We derive the general spectral representation of retarded Green functions in the Floquet regime, thereby generalizing the well-known Lehmann representation from equilibrium many-body physics. The derived spectral Floquet representation allows us to prove the non-negativity of spectral densities and to determine exact spectral sum rules, which can be employed to benchmark the accuracy of approximations to the exact Floquet many-body Green functions.

Shen Jun, Junde Wu

In 2006, Gudder [Math. Slovaca 56, 573 (2006)] introduced a logic order ≼ on bounded quantum observable set S(H). In 2007, Pulmannova and Vincekova [Math Slovaca 57, 589 (2007)] proved that for each subset D of S(H), the infimum of D exists with respect to the logic order ≼. In 2008, Liu and Wu [J. Math. Phys. 49, 073521 (2008)] found a representation of the infimum A∧B for A,B∊S(H), and by using the limit methods, they gave out a representation for the infimum of D. But, that representation is complicated. In this paper, we present a simpler spectral representation for the infimum of D with respect to the logic order ≼.