Generalized quantum mechanics

  title={Generalized quantum mechanics},
  author={Bogdan Mielnik},
  journal={Communications in Mathematical Physics},
  • B. Mielnik
  • Published 1 September 1974
  • Physics
  • Communications in Mathematical Physics
A convex scheme of quantum theory is outlined where the states are not necessarily the density matrices in a Hilbert space. The physical interpretation of the scheme is given in terms of generalized “impossibility principles”. The geometry of the convex set of all pure and mixed states (called a statistical figure) is conditioned by the dynamics of the system. This provides a method of constructing the statistical figures for non-linear variants of quantum mechanics where the superposition… 
Geometrization of quantum mechanics
Quantum mechanics is cast into a classical Hamiltonian form in terms of a symplectic structure, not on the Hilbert space of state-vectors but on the more physically relevant infinite-dimensional
Fundamental principles of quantum theory. II. From a convexity scheme to the DHB theory
Some classical and quantum theories are characterized within the convexity approach to probabilistic physical theories. In particular, the structure of the so-called DHB quantum theory will be
A geometric approach to quantum mechanics
It is argued that quantum mechanics is fundamentally a geometric theory. This is illustrated by means of the connection and symplectic structures associated with the projective Hilbert space, using
Nonlinear description of quantum dynamics: Generalized coherent states
In this work it is shown that there is an inherent nonlinear evolution in the dynamics of the so-called generalized coherent states. To show this, the immersion of a classical manifold into the
Is quantum mechanics exact
We formulate physically motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to quantum mechanics as the only nontrivial consistent
An Axiomatic Basis for Quantum Mechanics
In this paper we use the framework of generalized probabilistic theories to present two sets of basic assumptions, called axioms, for which we show that they lead to the Hilbert space formulation of
Quartic quantum theory: an extension of the standard quantum mechanics
We propose an extended quantum theory, in which the number K of parameters necessary to characterize a quantum state behaves as fourth power of the number N of distinguishable states. As the simplex
A generic approach to the quantum mechanical transition probability
  • G. Niestegge
  • Mathematics
    Proceedings of the Royal Society A
  • 2022
In quantum theory, the modulus-square of the inner product of two normalized Hilbert space elements is to be interpreted as the transition probability between the pure states represented by these
PT-symmetry, indefinite metric, and nonlinear quantum mechanics
If a Hamiltonian of a quantum system is symmetric under space-time reflection, then the associated eigenvalues can be real. A conjugation operation for quantum states can then be defined in terms of
Axioms for quantum theory
The first three of these axioms describe quantum theory and classical mechanics as statistical theories from the very beginning. With these, it can be shown in which sense a more general than the


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It is shown that two quantum theories dealing, respectively, in the Hilbert spaces of state vectors H1 and H2 are physically equivalent whenever we have a faithful representation of the same abstract
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Recent work of Davies and Lewis has suggested a mathematical framework in which the notion of repeated measurements on statistical physical systems can be examined. This paper is concerned with an
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We present a reformulation of the axiomatic basis of quantum mechanics with particular reference to the manner in which the usual algebraic structures arise from certain natural physical
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Postulates for General Quantum Mechanics
We present in this paper a set of postulates for a physical system and deduce from these the main general features of the quantum theory of stationary states. Our theory is strictly operational in
An operational approach to quantum probability
In order to provide a mathmatical framework for the process of making repeated measurements on continuous observables in a statistical system we make a mathematical definition of an instrument, a
Attempt of an axiomatic foundation of quantum mechanics and more general theories V
We continue here the series of papers treated byLudwig in [1–5]. Using some results ofDähn in [6], we point out that each irreducible solution of the axiomatic scheme set up in [5] is represented by
Attempt of an axiomatic foundation of quantum mechanics and more general theories. III
Starting from axioms as physical as possible [1, 2, 3] about “effects” and “ensembles”, we shall investigate further consequences.Concerning part I and II [4, 5] the axioms can be so formulated as to
Quantum stochastic processes
In order to describe rigorously certain measurement procedures, where observations of the arrival of quanta at a counter are made throughout an interval of time, it is necessary to introduce the
Attempt of an axiomatic foundation of quantum mechanics and more general theories. IV
This contribution continues the series of papers on the same subject which has been treated byLudwig in [1–3]. Using the system of axioms as given in [3], we shall succeed in constructing an