Quantum Mechanics beyond Hilbert Space
نویسنده
چکیده
When describing a quantum mechanical system, it is convenient to consider state vectors that do not belong to the Hilbert space. In the rst part of this paper, we survey the various formalisms have been introduced for giving a rigorous mathematical justiication to this procedure: rigged Hilbert spaces (RHS), scales or lattices of Hilbert spaces (LHS), nested Hilbert spaces, partial inner product spaces. Then we present three types of applications in quantum mechanics, all of them involving spaces of analytic functions. First we present a LHS built around the Bargmann space, thus giving a natural frame for the Fock-Bargmann (or phase space) representation. Then we review the RHS approach to scattering theory (resonances, Gamow vectors, etc.). Finally, we reformulate the Weinberg-van Winter integral equation approach to scattering in the LHS language, and this allows us to prove that it is in fact a particular case of the familiar complex scaling method.
منابع مشابه
کوانتش گرانش و بررسی هندسی مکانیک کوانتمی
We elaborate on some recent results on a solution of the Hilbert-space problem in minisuperspace quantum cosmology and discuss the consequences of making the (geometry of the) Hilbert space of ordinary nonrelativistic quantum systems time-dependent. The latter reveals a remarkable similarity between Quantum Mechanics and General Relativity.
متن کاملLocal quantum measurement and no-signaling imply quantum correlations.
We show that, assuming that quantum mechanics holds locally, the finite speed of information is the principle that limits all possible correlations between distant parties to be quantum mechanical as well. Local quantum mechanics means that a Hilbert space is assigned to each party, and then all local positive-operator-valued measurements are (in principle) available; however, the joint system ...
متن کاملAlgebraic quantum mechanics
Algebraic quantum mechanics is an abstraction and generalization of the Hilbert space formulation of quantum mechanics due to von Neumann [5]. In fact, von Neumann himself played a major role in developing the algebraic approach. Firstly, his joint paper [3] with Jordan and Wigner was one of the first attempts to go beyond Hilbert space (though it is now mainly of historical value). Secondly, h...
متن کاملHow to Differentiate a Quantum Stochastic Cocycle
Two new approaches to the infinitesimal characterisation of quantum stochastic cocycles are reviewed. The first concerns mapping cocycles on an operator space and demonstrates the role of Hölder continuity; the second concerns contraction operator cocycles on a Hilbert space and shows how holomorphic assumptions yield cocycles enjoying an infinitesimal characterisation which goes beyond the sco...
متن کاملThe role of the rigged Hilbert space in Quantum Mechanics
There is compelling evidence that, when continuous spectrum is present, the natural mathematical setting for Quantum Mechanics is the rigged Hilbert space rather than just the Hilbert space. In particular, Dirac’s bra-ket formalism is fully implemented by the rigged Hilbert space rather than just by the Hilbert space. In this paper, we provide a pedestrian introduction to the role the rigged Hi...
متن کامل