Infinitesimal Multimode Bargmann-State Representation*

In the Hilbert space of a light mode (harmonic oscillator), we construct a representation, in which an arbitrary state vector is expanded using Bargmann states ‖α〉 with real parameters α being in an infinitesimal vicinity of zero. The complete Hilbert-space structure is represented in the one- and multimode cases as well, making the representation able to deal with problems of continuous-variable quantum information processing.     

An Alternative to the Gauge Theoretic Setting

The standard formulation of quantum gauge theories results from the Lagrangian (functional integral) quantization of classical gauge theories. A more intrinsic quantum theoretical access in the spirit of Wigner’s representation theory shows that there is a fundamental clash between the pointlike localization of zero mass (vector, tensor) potentials and the Hilbert space (positivity, unitarity) structure of QT. The quantization approach has no other way than to stay with pointlike localization and sacrifice the Hilbert space whereas the approach built on the intrinsic quantum concept of modular localization keeps the Hilbert space and trades the conflict creating pointlike generation with the tightest consistent localization: semiinfinite spacelike string localization. Whereas these potentials in the presence of interactions stay quite close to associated pointlike field strengths, the interacting matter fields to which they are coupled bear the brunt of the nonlocal aspect in that they are string-generated in a way which cannot be undone by any differentiation.     

Non-standard aspects of Minkowski causal logic

The Minkowski causal logic, which is already known to be a complete orthomodular lattice, is found to be also an atomistic and irreducible logic, but to have no other essential properties to be represented in terms of all the subspace of some Hilbert space. Alternative representation of the logic in terms of subspaces of a real vector space or of the states in terms of probability measures are suggested.     

A Hilbert Space Realization of Nonlinear Quantum Mechanics as Classical Extension of Its Linear Counterpart

This paper studies the state-effect-probability structure associated with thequantum mechanics of nonlinear (homogeneous, in general nonadditive) operatorson a Hilbert space. Its aim is twofold: to provide a concrete representation ofthe features of nonlinear quantum mechanics on a Hilbert space, and to showthat the properties of the nonlinear version of quantum mechanics here describedhave the structure of a classical logic.     

A Vanishing Theorem for Operators in Fock Space

We consider the bosonic Fock space over the Hilbert space of transversal vector fields in three dimensions. This space carries a canonical representation of the group of rotations. For a certain class of operators in Fock space, we show that rotation invariance implies the absence of terms which either create or annihilate only a single particle. We outline an application of this result in an operator theoretic renormalization analysis of Hamilton operators, which occur in non-relativistic qed.     

Algebraic aspects of the wigner distribution in quantum mechanics

The algebraic structure underlying the method of the Wigner distribution in quantum mechanics and the Weyl correspondence between classical and quantum dynamical variables is analysed. The basic idea is to treat the operators acting on a Hilbert space as forming a second Hilbert space, and to make use of certain linear operators on them. The Wigner distribution is also related to the diagonal coherent state representation of quantum optics by this method.     

On the algebra of test functions for field operators

It is shown that every continuous linear functional on the field algebra can be defined by a vector in the Hilbert space of some representation of the algebra. The functionals which can be written as a difference of positive ones are characterized. By an example it is shown that a positive functional on a subalgebra does not always have an extension to a positive functional on the whole algebra.     

Operator structure of a nonquantum and nonclassical system

There exists a connection between the vectors of the Poincaré-sphere and the elements of the complex Hilbert space C². This latter space is used to describe spin-1/2 measurements. We use this connection to study the intermediate cases of a more general spin-1/2 measurement model which has no representation in a Hilbert space. We construct the set of operators of this general model and investigate under which circumstances it is possible to define linear operators. Because no Hilbert space structure is possible for these intermediate cases, it can be expected that no linear operators are possible and it is shown that under very plausible assumptions this is indeed the case.     

混合纠缠态的几何描述

给出了Hilbert-Schmidt(H-S)空间中密度矩阵的向量表示,建立了完整的H-S空间中的度规 ,由此将混合纠缠态的判据纳入到直观的几何图像中,讨论了最大混合态附近可分离态的紧 致邻域,得到了关于该邻域体积测度的更强的结果. 关键词： 纠缠 混合态 Hilbert-Schmidt空间     

Probabilistic Fuzzy Sets and a Related Operator Algebra

The rules of union and intersection of probabilistic fuzzy sets guided us to construct a related operator algebra. In a Hilbert space, where each fuzzy set is represented by an orthonormal vector, the union and the intersection operators generate a well-defined algebra with a unique representation. PACS NUMBER: 02.10.-v     

Stochastic Dynamics of Reduced Wave Functions and Continuous Measurement in Quantum Optics

The stochastic dynamics of open quantum systems interacting with a zero temperature environment is investigated by employing a formulation of quantum statistical ensembles in terms of probability distributions on projective Hilbert space. It is demonstrated that the open system dynamics can consistently be described by a stochastic process on the reduced state space. The physical meaning of reduced probability distributions on projective Hilbert space is derived from a complete, orthogonal measurement of the environment. The elimination of the variables of the environment is shown to lead to a piecewise deterministic process in Hilbert space defined by a differential Chapman-Kolmogorov equation. A Hilbert space path integral representation of the stochastic process is constructed. The general theory is illustrated by means of three examples from quantum optics. For these examples the microscopic derivation of the stochastic process is given and the general solution of the differential Chapman-Kolmogorov equation is constructed by means of the path integral representation.     

On representation spaces in geometric quantization

The structure of the space of wave functions in the representation given by a complete strongly admissible polarization of the phase space is investigated. The conditions that the wave functions should be covariant constant along the real part of the polarization define the Bohr-Sommerfeld set of the representation containing the supports of all wave functions. There is a natural scalar product for the wave functions defined on the Bohr-Sommerfeld set. It is shown, for a real polarization, that the resulting Hilbert space of wave functions is not trivial if and only if the Bohr-Sommerfeld set is not empty.Partially supported by the National Research Council, Grant No. A8091.     

Natural star-products on symplectic manifolds and related quantum mechanical operators

In this paper is considered a problem of defining natural star-products on symplectic manifolds, admissible for quantization of classical Hamiltonian systems. First, a construction of a star-product on a cotangent bundle to an Euclidean configuration space is given with the use of a sequence of pair-wise commuting vector fields. The connection with a covariant representation of such a star-product is also presented. Then, an extension of the construction to symplectic manifolds over flat and non-flat pseudo-Riemannian configuration spaces is discussed. Finally, a coordinate free construction of related quantum mechanical operators from Hilbert space over respective configuration space is presented.     

Quantization, group contraction and zero point energy

We study algebraic structures underlying 't Hooft's construction relating classical systems with the quantum harmonic oscillator. The role of group contraction is discussed. We propose the use of SU(1,1) for two reasons: because of the isomorphism between its representation Hilbert space and that of the harmonic oscillator and because zero point energy is implied by the representation structure. Finally, we also comment on the relation between dissipation and quantization.     

Multi-matrix vector coherent states

A class of vector coherent states is derived with multiple of matrices as vectors in a Hilbert space, where the Hilbert space is taken to be the tensor product of several other Hilbert spaces. As examples vector coherent states with multiple of quaternions and octonions are given. The resulting generalized oscillator algebra is briefly discussed. Further, vector coherent states for a tensored Hamiltonian system are obtained by the same method. As particular cases, coherent states are obtained for tensored Jaynes-Cummings type Hamiltonians and for a two-level two-mode generalization of the Jaynes-Cummings model.     

Measures on Hilbert space and quantum states

A representation of quantum mechanics of one particle by means of integration in Hilbert space H is embedded into a multi-particle theory on boson Fock space F₊. We prove that the grand canonical statistical operator on F₊ can be represented by means of a gaussian measure on the one-particle Hilbert space H.     

Fermi-Dirac quantization of linear systems

After discussing the Fermion analogues of classical mechanics, we show that in finite degrees of freedom, the Segal-Weinless construction of the vacuum representation is always possible. This amounts to an explicit construction of a complex structure J which extends real Euclidean space with orthogonal dynamics to a complex Hilbert space with unitary dynamics. Also, we solve the inverse problem, deducing the class of classical Hamiltonians, given the complex structure J.     

The cauchy problem for non-linear Klein-Gordon equations

We consider in ⁿ⁺¹,n2, the non-linear Klein-Gordon equation. We prove for such an equation that there is a neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincaré covariant then the non-linear representation of the Poincaré Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincaré group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincaré group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra.     

Geometry of quantum states

In the first part of this work, an attempt of a realistic interpretation ofquantum logic is presented. Propositions of quantum logic are interpreted as corresponding to certain macroscopic objects called filters; these objects are used to select beams of particles. The problem of representing the propositions as projectors in a Hilbert space is considered and the classical approach to this question due to Birkhoff and von Neumann is criticized as neglecting certain physically important properties of filters. A new approach to this problem is proposed.The second part of the paper contains a revision of the concept of a state in quantum mechanics. The set of all states of a physical system is considered as an abstract space with a geometry determined by the transition probabilities. The existence of a representation of states by vectors in a Hilbert space is shown to impose strong limitations on the geometric structure of the space of states. Spaces for which this representation does not exist are called non-Hilbertian. Simple examples of non-Hilbertian spaces are given and their possible physical meaning is discussed. The difference between Hilbertian and non-Hilbertian spaces is characterized in terms of measurable quantities.