23,166 research outputs found
The Two-fold Role of Observables in Classical and Quantum Kinematics
Observables have a dual nature in both classical and quantum kinematics: they
are at the same time \emph{quantities}, allowing to separate states by means of
their numerical values, and \emph{generators of transformations}, establishing
relations between different states. In this work, we show how this two-fold
role of observables constitutes a key feature in the conceptual analysis of
classical and quantum kinematics, shedding a new light on the distinguishing
feature of the quantum at the kinematical level. We first take a look at the
algebraic description of both classical and quantum observables in terms of
Jordan-Lie algebras and show how the two algebraic structures are the precise
mathematical manifestation of the two-fold role of observables. Then, we turn
to the geometric reformulation of quantum kinematics in terms of K\"ahler
manifolds. A key achievement of this reformulation is to show that the two-fold
role of observables is the constitutive ingredient defining what an observable
is. Moreover, it points to the fact that, from the restricted point of view of
the transformational role of observables, classical and quantum kinematics
behave in exactly the same way. Finally, we present Landsman's general
framework of Poisson spaces with transition probability, which highlights with
unmatched clarity that the crucial difference between the two kinematics lies
in the way the two roles of observables are related to each other.Comment: Corrected typos; revised final arguments of section 2.2 and added a
figure at the end of this sectio
Model theoretic reformulation of the Baum-Connes and Farrell-Jones conjectures
The Isomorphism Conjectures are translated into the language of homotopical
algebra, where they resemble Thomason's descent theorems.Comment: 4 page
Quantum logic and decohering histories
An introduction is given to an algebraic formulation and generalisation of
the consistent histories approach to quantum theory. The main technical tool in
this theory is an orthoalgebra of history propositions that serves as a
generalised temporal analogue of the lattice of propositions of standard
quantum logic. Particular emphasis is placed on those cases in which the
history propositions can be represented by projection operators in a Hilbert
space, and on the associated concept of a `history group'.Comment: 14 pages LaTeX; Writeup of lecture given at conference ``Theories of
fundamental interactions'', Maynooth Eire 24--26 May 1995
Vertex operator algebras and operads
Vertex operator algebras are mathematically rigorous objects corresponding to
chiral algebras in conformal field theory. Operads are mathematical devices to
describe operations, that is, -ary operations for all greater than or
equal to , not just binary products. In this paper, a reformulation of the
notion of vertex operator algebra in terms of operads is presented. This
reformulation shows that the rich geometric structure revealed in the study of
conformal field theory and the rich algebraic structure of the theory of vertex
operator algebras share a precise common foundation in basic operations
associated with a certain kind of (two-dimensional) ``complex'' geometric
object, in the sense in which classical algebraic structures (groups, algebras,
Lie algebras and the like) are always implicitly based on (one-dimensional)
``real'' geometric objects. In effect, the standard analogy between
point-particle theory and string theory is being shown to manifest itself at a
more fundamental mathematical level.Comment: 16 pages. Only the definitions of "partial operad" and of "rescaling
group" have been improve
On Large Games with a Bio-Social Typology
We present a comprehensive theory of large non-anonymous games in which agents have a name and a determinate social-type and/or biological trait to resolve the dissonance of a (matching-pennies type) game with an exact pure-strategy Nash equilibrium with finite agents, but without one when modeled on the Lebesgue unit interval. We (i) establish saturated player spaces as both necessary and sufficient for an existence result for Nash equilibrium in pure strategies, (ii) clarify the relationship between pure, mixed and behavioral strategies via the exact law of large numbers in a framework of Fubini extension, (iii) illustrate corresponding asymptotic results.
Culture and Cognitive Theory: Toward a Reformulation
In a provocative and important recent article Anthony Marsella (1998) makes an eloquent plea for the forging of a new metadiscipline of psychology that he labels global-community psychology. Marsella argues that we need a radical rethinking of the fundamental premises of psychology, rooted as they are in Western cultural traditions. Features of an emergent global-community psychology include an emphasis on multicultural and multidisciplinary approaches to human behavior that draw attention to the importance of context and meaning in human lives. Marsella's call for a global-community psychology reflects, in part, a growing body of literature that demonstrates the importance of cultural factors in a diver-sity of psychological domains such as cognition, emotion, social behavior, and psychopathology
Three-Hilbert-Space Formulation of Quantum Mechanics
In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages,
arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian)
formulation of Quantum Mechanics has been revisited. In the present
continuation of this study (with the spaces in question denoted as and ) we spot a weak point of
the 2HS formalism which lies in the double role played by . As long as this confluence of roles may (and did!) lead to
confusion in the literature, we propose an amended, three-Hilbert-space (3HS)
reformulation of the same theory. As a byproduct of our analysis of the
formalism we offer an amendment of the Dirac's bra-ket notation and we also
show how its use clarifies the concept of covariance in time-dependent cases.
Via an elementary example we finally explain why in certain quantum systems the
generator of the time-evolution of the wave functions may
differ from their Hamiltonian
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