共查询到20条相似文献,搜索用时 15 毫秒
1.
Observables of a quantum system, described by self-adjoint operators in a von Neumann algebra or affiliated with it in the unbounded case, form a conditionally complete lattice when equipped with the spectral order. Using this order-theoretic structure, we develop a new perspective on quantum observables. In this first paper (of two), we show that self-adjoint operators affiliated with a von Neumann algebra ${\mathcal{N}}$ can equivalently be described as certain real-valued functions on the projection lattice ${\mathcal{P}(\mathcal{N}})$ of the algebra, which we call q-observable functions. Bounded self-adjoint operators correspond to q-observable functions with compact image on non-zero projections. These functions, originally defined in a similar form by de Groote (Observables II: quantum observables, 2005), are most naturally seen as adjoints (in the categorical sense) of spectral families. We show how they relate to the daseinisation mapping from the topos approach to quantum theory (Döring and Isham , New Structures for Physics, Springer, Heidelberg, 2011). Moreover, the q-observable functions form a conditionally complete lattice which is shown to be order-isomorphic to the lattice of self-adjoint operators with respect to the spectral order. In a subsequent paper (Döring and Dewitt, 2012, preprint), we will give an interpretation of q-observable functions in terms of quantum probability theory, and using results from the topos approach to quantum theory, we will provide a joint sample space for all quantum observables. 相似文献
2.
Antun Milas 《Communications in Mathematical Physics》2008,277(2):497-529
This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of
Heisenberg and lattice vertex algebras. In this paper we examine logarithmic intertwining operators associated with rank one
Heisenberg vertex operator algebra M(1)
a
, of central charge 1 − 12a
2. We classify these operators in terms of depth and provide explicit constructions in all cases. Our intertwining operators
resemble puncture operators appearing in quantum Liouville field theory. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)0 and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct
explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine
logarithmic L(1, 0)-module. 相似文献
3.
In this paper, the factorizable S matrices with toroidal rapidity values are constructed explicitly and the symmetry operators of which tensor product commutes with S are found. These operators have a structure which is similar to the ordinary Hopf algebra. As a limit of the elliptic case slq(2) quantum algebra with qN = 1 can be obtained from the symmetry operators. 相似文献
4.
Andrzej ?uczak 《International Journal of Theoretical Physics》2010,49(12):3176-3184
We construct quantum stochastic integrals for the integrator being a martingale in a von Neumann algebra, and the integrand—a
suitable process with values in the same algebra, as densely defined operators affiliated with the algebra. In the case of
a finite algebra we allow the integrator to be an L
2-martingale in which case the integrals are L
2-martingales too. 相似文献
5.
In this paper, we study (n-1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator. Finally, we give various constructions of Nijenhuis operators and some examples. 相似文献
6.
Boris Feigin Michio Jimbo Tetsuji Miwa Alexandr Odesskii Yaroslav Pugai 《Communications in Mathematical Physics》1998,191(3):501-541
We construct a family of intertwining operators (screening operators) between various Fock space modules over the deformed
W
n
algebra. They are given as integrals involving a product of screening currents and elliptic theta functions. We derive a
set of quadratic relations among the screening operators, and use them to construct a Felder-type complex in the case of the
deformed W
3 algebra.
Received: 3 March 1997 / Accepted: 20 May 1997 相似文献
7.
Marcel Polakovi? Zdenka Rie?anová 《International Journal of Theoretical Physics》2011,50(4):1167-1174
Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely,
we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the
set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned.
Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the
partial binary operation is defined making this set a generalized effect algebra. 相似文献
8.
Tense Operators on Basic Algebras 总被引:1,自引:0,他引:1
M. Botur I. Chajda R. Halaš M. Kolařík 《International Journal of Theoretical Physics》2011,50(12):3737-3749
The concept of tense operators on a basic algebra is introduced. Since basic algebras can serve as an axiomatization of a
many-valued quantum logic (see e.g. Chajda et al. in Algebra Univer. 60(1):63–90, 2009), these tense operators are considered to quantify time dimension, i.e. one expresses the quantification “it is always going
to be the case that” and the other expresses “it has always been the case that”. We set up the axiomatization and basic properties
of tense operators on basic algebras and involve a certain construction of these operators for left-monotonous basic algebras.
Finally, we relate basic algebras with tense operators with another quantum structures which are the so-called dynamic effect
algebras. 相似文献
9.
We show that considerable sets of positive linear operators namely their extensions as closures, adjoints or Friedrichs positive
self-adjoint extensions form operator (generalized) effect algebras. Moreover, in these cases the partial effect algebraic
operation of two operators coincides with usual sum of operators in complex Hilbert spaces whenever it is defined. These sets
include also unbounded operators which play important role of observables (e.g., momentum and position) in the mathematical
formulation of quantum mechanics. 相似文献
10.
This article is concerned with crossed products and their applications to random operators. We study the von Neumann algebra of a dynamical system using the underlying Hilbert algebra structure. This gives a particularly easy way to introduce a trace on this von Neumann algebra. We review several formulas for this trace, show how it comes as an application of Connes" noncommutative integration theory and discuss Shubin"s trace formula. We then restrict ourselves to the case of an action of a group on a group and include new proofs for some theorems of Bellissard and Testard on an analogue of the classical Plancherel theorem. We show that the integrated density of states is a spectral measure in the periodic case, thereby generalizing a result of Kaminker and Xia. Finally, we discuss duality results and apply a method of Gordon et al. to establish a duality result for crossed products by Z. 相似文献
11.
E. Horozov 《Communications in Mathematical Physics》2002,231(2):287-308
The aim of this paper is to solve the bispectral problem for bispectral operators whose order is a prime number. More precisely
we give a complete list of such bispectral operators. We use systematically the operator approach and in particular – Dixmier
ideas on the first Weyl algebra. When the order is 2 the main theorem is exactly the result of Duistermaat-Grünbaum. On the
other hand our proofs seem to be simpler.
Received: 14 February 2002 / Accepted: 10 June 2002 Published online: 21 October 2002 相似文献
12.
We consider Dirichlet realizations of Pauli-Fierz type operators generating the dynamics of non-relativistic matter particles which are confined to an arbitrary open subset of the Euclidean position space and coupled to quantized radiation fields. We find sufficient conditions under which their domains and a natural class of operator cores are determined by the domains and operator cores of the corresponding Dirichlet-Schrödinger operators and the radiation field energy. Our results also extend previous ones dealing with the entire Euclidean space, since the involved electrostatic potentials might be unbounded at infinity with local singularities that can only be controlled in a quadratic form sense, and since locally square-integrable classical vector potentials are covered as well. We further discuss Neumann realizations of Pauli-Fierz type operators on Lipschitz domains. 相似文献
13.
Yusuke Arike 《Letters in Mathematical Physics》2012,101(1):13-26
We show that the space of logarithmic intertwining operators among logarithmic modules for a vertex operator algebra is isomorphic to the space of 3-point conformal blocks over the projective line. This can be viewed as a generalization of Zhu??s result for ordinary intertwining operators among ordinary modules. 相似文献
14.
《International Journal of Theoretical Physics》2010,49(9):2230-2245
The irreducible tensor operators and their tensor products employing Racah algebra are studied. Transformation procedure of
the coordinate system operators act on are introduced. The rotation matrices and their parametrization by the spherical coordinates
of vector in the fixed and rotated coordinate systems are determined. A new way of calculation of the irreducible coupled
tensor product matrix elements is suggested. As an example, the proposed technique is applied for the matrix element construction
for two electrons in a field of a fixed nucleus. 相似文献
15.
Abdellah Sebbar 《Communications in Mathematical Physics》1998,198(2):283-309
In this paper a close connection is established between certain cohomology spaces of representations of the quantum affine
algebra , and a twisted $q$-de Rham (Jackson–Aomoto) cohomology of configuration spaces using the quantum screening operators.
Received: 24 December 1997 / Accepted: 2 April 1998 相似文献
16.
We consider the universal central extension of the Lie algebra Vect(S
1) C(S
1). The coadjoint representation of thisLie algebra has a natural geometric interpretation by matrix analogues ofthe Sturm –Liouville operators. This approach leads to new Liesuperalgebras generalizing the well-known Neveu –Schwarz algebra. 相似文献
17.
18.
Erik Carlsson 《Communications in Mathematical Physics》2010,300(3):599-613
We approximate the infinite Grassmannian by finite-dimensional cutoffs, and define a family of fermionic vertex operators
as the limit of geometric correspondences on the equivariant cohomology groups, with respect to a one-dimensional torus action.
We prove that in the localization basis, these are the well-known fermionic vertex operators on the infinite wedge representation.
Furthermore, the boson-fermion correspondence, locality, and intertwining properties with the Virasoro algebra are the limits
of relations on the finite-dimensional cutoff spaces, which are true for geometric reasons. We then show that these operators
are also, almost by definition, the vertex operators defined by Okounkov and the author in Carlsson and Okounkov ( [math.AG], 2009), on the equivariant cohomology groups of the Hilbert scheme of points on
\mathbb C2{\mathbb C^2} , with respect to a special torus action. 相似文献
19.
We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain C
*-subalgebra to discuss a Shubin trace formula. 相似文献
20.
Yun Gao 《Communications in Mathematical Physics》2000,211(3):745-777
We use the underlying Fock space for the homogeneous vertex operator representation of the affine Lie algebra to construct a family of vertex operators. As an application, an irreducible module for an extended affine Lie algebra of
type A
N
−1 coordinatized by a quantum torus ℂ
q
of 2 variables (or 3 variables) is obtained. Moreover, this module turns out to be a highest weight module which is an analog
of the basic module for affine Lie algebras.
Received: 16 August 1999 / Accepted: 18 January 2000 相似文献