1,721,021 research outputs found
Quantum Tomography twenty years later
No full textWe show a sample of some relevant developments in classical and quantum tomography that
have taken place over the last twenty years. We will present a general conceptual framework that
provides a simple unifying mathematical picture for them and, as an effective use of it, three
subjects have been chosen that offer a wide panorama of the scope of classical and quantum
tomography: tomography along lines and submanifolds, coherent state tomography and
tomography in the abstract algebraic setting of quantum system
Alternative linear structures for classical and quantum systems
Full text link26 pages, 2 figures.-- ArXiv pre-print available at: http://arxiv.org/abs/0706.1619The possibility of deforming the (associative or Lie) product to obtain alternative descriptions for a given classical or quantum system has been considered in many papers. Here we discuss the possibility of obtaining some novel alternative descriptions by changing the linear structure instead. In particular we show how it is possible to construct alternative linear structures on the tangent bundle TQ of some classical configuration space Q that can be considered as "adapted" to the given dynamical system. This fact opens the possibility to use the Weyl scheme to quantize the system in different nonequivalent ways, "evading," so to speak, the von Neumann uniqueness theorem
Covariant quantum fields on noncommutative spacetimes
Full text linkA spinless covariant field on Minkowski spacetime \M^{d+1} obeys the relation where is an element of the Poincar\\u27e group \Pg and is its unitary representation on quantum vector states. It expresses the fact that Poincar\\u27e transformations are being unitary implemented. It has a classical analogy where field covariance shows that Poincar\\u27e transformations are canonically implemented. Covariance is self-reproducing: products of covariant fields are covariant. We recall these properties and use them to formulate the notion of covariant quantum fields on noncommutative spacetimes. In this way all our earlier results on dressing, statistics, etc. for Moyal spacetimes are derived transparently. For the Voros algebra, covariance and the *-operation are in conflict so that there are no covariant Voros fields compatible with *, a result we found earlier. The notion of Drinfel\u27d twist underlying much of the preceding discussion is extended to discrete abelian and nonabelian groups such as the mapping class groups of topological geons. For twists involving nonabelian groups the emergent spacetimes are nonassociative
Inequivalence of quantum field theories on noncommutative spacetimes: moyal versus Wick-Voros
No full textIn this paper, we further develop the analysis started in an earlier paper on the inequivalence of certain quantum field theories on noncommutative spacetimes constructed using twisted fields. The issue is of physical importance. Thus it is well known that the commutation relations among spacetime coordinates, which define a noncommutative spacetime, do not constrain the deformation induced on the algebra of functions uniquely. Such deformations are all mathematically equivalent in a very precise sense. Here we show how this freedom at the level of deformations of the algebra of functions can fail on the quantum field theory side. In particular, quantum field theory on the Wick-Voros and Moyal planes are shown to be inequivalent in a few different ways. Thus quantum field theory calculations on these planes will lead to different physics even though the classical theories are equivalent. This result is reminiscent of chiral anomaly in gauge theories and has obvious physical consequences. The construction of quantum field theories on the Wick-Voros plane has new features not encountered for quantum field theories on the Moyal plane. In fact it seems impossible to construct a quantum field theory on the Wick-Voros plane which satisfies all the properties needed of field theories on noncommutative spaces. The Moyal twist seems to have unique features which make it a preferred choice for the construction of a quantum field theory on a noncommutative spacetime
A pedagogical presentation of a C*-algebraic approach toquantum tomography
No full textIt is now well established that quantum tomography provides an alternative picture of quantum mechanics. It is common to introduce tomographic concepts starting with the Schroedinger-Dirac picture of quantum mechanics on Hilbert spaces. In this picture states are a primary concept and observables are
derived from them. On the other hand, the Heisenberg picture, which has evolved in the C*-algebraic approach to quantum mechanics, starts with the algebra of observables and introduce states as a derived concept.
The equivalence between these two pictures amounts essentially, to the Gelfand-Naimark-Segal construction. In this construction, the abstract C*-algebra is realized as an algebra of operators acting on a constructed Hilbert space. The representation one defines may be reducible or irreducible, but in either case it allows to identify an unitary group associated with the C*-algebra by means of its invertible elements.
In this picture both states and observables are appropriate functions on the group, it follows that also quantum tomograms are strictly related with appropriate functions (positive-type)on the group. In this paper we present, by means of very simple examples, the tomographic description emerging from
the set of ideas connected with the C*-algebra picture of quantum
mechanics. In particular, the tomographic probability distributions are introduced for finite and compact groups and an autonomous criterion to recognize a given probability distribution as a tomogram of quantum state is formulated
Geometry from dynamics, classical and quantum
No full textThis book describes, by using elementary techniques, how some geometrical structures widely used today in many areas of physics, like symplectic, Poisson, Lagrangian, Hermitian, etc., emerge from dynamics. It is assumed that what can be accessed in actual experiences when studying a given system is just its dynamical behavior that is described by using a family of variables ("observables" of the system). The book departs from the principle that ''dynamics is first'', and then tries to answer in what sense the sole dynamics determines the geometrical structures that have proved so useful to describe the dynamics in so many important instances. In this vein it is shown that most of the geometrical structures that are used in the standard presentations of classical dynamics (Jacobi, Poisson, symplectic, Hamiltonian, Lagrangian) are determined, though in general not uniquely, by the dynamics alone. The same program is accomplished for the geometrical structures relevant to describe quantum dynamics. Finally, it is shown that further properties that allow the explicit description of the dynamics of certain dynamical systems, like integrability and superintegrability, are deeply related to the previous development and will be covered in the last part of the book. The mathematical framework used to present the previous program is kept to an elementary level throughout the text, indicating where more advanced notions will be needed to proceed further. A family of relevant examples is discussed at length and the necessary ideas from geometry are elaborated along the text. However no effort is made to present an ''all-inclusive'' introduction to differential geometry as many other books already exist on the market doing exactly that. However, the development of the previous program, considered as the posing and solution of a generalized inverse problem for geometry, leads to new ways of thinking and relating some of the most conspicuous geometrical structures appearing in Mathematical and Theoretical Physics
- …
