Traces,Dispersions of States and Hidden Variables

No Heading The interplay between the tracial property and minimality of dispersions of states on projections of von Neumann algebras and C^*-algebras is investigated. Let be a state on a C^*-algebra A with the projection structure P(A). The dispersion () is defined as () = sup{(p) – (p)² | p P(A)}. It is proved that () 2/9 whenever is a state on a real rank zero C^*-algebra with no nonzero abelian representation. New characterization of traces in terms of dispersions is proved: A state on a von Neumann algebra without abelian and Type I₂ direct summands is a trace if and only if has the minimal dispersion on all 3x3 matrix substructures. A similar characterization of semifinite normal traces on von Neumann algebras is obtained. The connection between unitary invariance of states and minimal dispersion property on C^*-algebras is studied. Besides providing a new characterization of trace in terms of physically relevant properties, the existing results on hidden variables in W^*- and C^*-formalism of quantum mechanics are strengthen.     

States and Structure of von Neumann Algebras

We summarize and deepen recent results on the interplay between properties of states and the structure of von Neumann algebras. We treat Jauch–Piron states and the concept of independence in noncommutative probability theory.     

Congruences and States on Pseudoeffect Algebras

We study congruences on pseudoeffect algebras, which were recently introduced as a non-commutative generalization of effect algebras. We introduce ideals for these algebras and give a sufficient condition for an ideal to determine a congruence. Furthermore, states on pseudoeffect algebras are considered. It is shown that any interval pseudoeffect algebra maps homomorphically into an effect algebra whose states are in a one-to-one correspondence to the states of the original algebra.     

Existence of States on Pseudoeffect Algebras

Pseudoeffect (PE) algebras have been introduced as a noncommutative generalization of effect algebras. We study in this paper PE algebras with the special property of having a nonempty state space. To this end, we consider PE algebras which are po-group intervals and which are, in a certain sense, noncommutative only in the small. Such a PE algebra is shown to possess a nontrivial commutative homomorphic image from which then follows that there exist states. A typical example is given by an interval of the lexicographical product of two po-groups the first of which is abelian.     

Chiral Charged Fermions,One-Dimensional Quantum Field Theory and Vertex Algebras

We give an explicit L ²-representation of chiral charged fermions using the Hardy–Lebesgue octant decomposition. In the pure case such a representation has already been used by M. Sato in holonomic field theory. We study both pure and mixed cases. In the compact case, we rigorously define unsmeared chiral charged fermion operators inside the unit circle. Using chiral fermions, we orient our findings towards a functional analytic study of vertex algebras as one-dimensional quantum field theory.     

Groupoids,von Neumann Algebras and the Integrated Density of States

We study spectral properties of random operators in the general setting of groupoids and von Neumann algebras. In particular, we establish an explicit formula for the canonical trace of the von Neumann algebra of random operators and define an abstract density of states. While the treatment applies to a general framework we lay special emphasis on three particular examples: random Schrödinger operators on manifolds, quantum percolation and quasi–crystal Hamiltonians. For these examples we show that the distribution function of the abstract density of states coincides with the integrated density of states defined via an exhaustion procedure.     

Husimi算符与热真空态的Husimi分布函数

利用Husimi算符作为一个纯态密度算符的事实,我们导出了热场态的Husimi函数及其边缘分布.通过绘制相空间的Husimi分布图形,我们简要讨论了高斯展宽参数以及热真空态的温度对Husimi函数的影响.     

Hidden Variables and Bell Inequalities on Quantum Logics

In the quantum logic approach, Bell inequalities in the sense of Pitowski are related with quasi hidden variables in the sense of Deliyannis. Some properties of hidden variables on effect algebras are discussed.     

Bound States in Quantum Electrodynamics: Theory and Application

The basic methods that have been used for describing bound-state quantum electrodynamics are described and critically discussed. These include the external field approximation, the quasi-potential approaches, the effective potential approach, the Bethe–Salpeter method, and the three-dimensional equations of Lepage and other workers. Other methods less frequently used but of some intrinsic interest such as applications of the Duffin–Kemmer equation are also described. A comparison of the strengths and shortcomings of these various approaches is included.     

Criterion of Quantum Entanglement and the Covariance Correlation Tensor in the Theory of Quantum Network

This article discusses the covariance correlation tensor (CCT) in quantum network theory for four Bell bases in detail. Furthermore, it gives the expression of the density operator in terms of CCT for a quantum network of three nodes, thus gives the criterion of entanglement for this case, i.e. the conditions of complete separability and partial separability for a given quantum state of three bodies. Finally it discusses the general case for the quantum network of m≥3 nodes.     

Prequantum Classical Statistical Field Theory: Complex Representation, Hamilton-Schrödinger Equation, and Interpretation of Stationary States

We develop a prequantum classical statistical model in that the role of hidden variables is played by classical (vector) fields. We call this model Prequantum Classical Statistical Field Theory (PCSFT). The correspondence between classical and quantum quantities is asymptotic, so we call our approach asymptotic dequantization. We construct the complex representation of PCSFT. In particular, the conventional Schrödinger equation is obtained as the complex representation of the system of Hamilton equations on the infinite-dimensional phase space. In this note we pay the main attention to interpretation of so called pure quantum states (wave functions) in PCSFT, especially stationary states. We show, see Theorem 2, that pure states of QM can be considered as labels for Gaussian measures concentrated on one dimensional complex subspaces of phase space that are invariant with respect to the Schrödinger dynamics. “A quantum system in a stationary state ψ” in PCSFT is nothing else than a Gaussian ensemble of classical fields (fluctuations of the vacuum field of a very small magnitude) which is not changed in the process of Schrödinger's evolution. We interpret in this way the problem of stability of hydrogen atom. One of unexpected consequences of PCSFT is the infinite dimension of physical space on the prequantum scale.     

测量相位算符在压缩热场态中的压缩效应

利用Bannett和Pegg提出的测量相位算符讨论了压缩热场态中测量相位算符的压缩效应。结果表明测量相位算符在压缩热场态中存在CS类压缩,压缩特性与相干态参数幅角θ、平均光子数n^-和压缩参量r有密切的关系。随着n^-和r的增大,CS类压缩效应逐渐减弱,直至完全消失。另有计算表明测量相位算符在压缩热场态中不存在CN和SN类压缩。     

Information and Distinguishability of Ensembles of Identical Quantum States

We consider a fixed quantum measurement performed over n identical copies of quantum states. Using a rigorous notion of distinguishability based on Shannon’s 12th theorem, we show that in the case of a single qubit, the number of distinguishable states is , where (α₁,α₂) is the angle interval from which the states are chosen. In the general case of an N-dimensional Hilbert space and an area Ω of the domain on the unit sphere from which the states are chosen, the number of distinguishable states is . The optimal distribution is uniform over the domain in Cartesian coordinates.     

Wigner Function:from Ensemble Average of Density Operator to Its One Matrix Element in Entangled Pure States

We show that the Wigner function W = Tr(△ρ) (an ensemble average of the density operator ρ, △ is theWigner operator) can be expressed as a matrix element of ρ in the entangled pure states. In doing so, converting fromquantum master equations to time-evolution equation of the Wigner functions seems direct and concise. The entangledstates are defined in the enlarged Fock space with a fictitious freedom.     

Automorphisms of Order Structures of Abelian Parts of Operator Algebras and Their Role in Quantum Theory

It is shown that any order isomorphism between the structures of unital associative JB subalgebras of JB algebras is given naturally by a partially linear Jordan isomorphism. The same holds for nonunital subalgebras and order isomorphisms preserving the unital subalgebra. Finally, we recover usual action of time evolution group on a von Neumann factor from group of automorphisms of the structure of Abelian subalgebras.     

Polarization States of Light and Their Quantum Tomography

The notion and main features of polarization states of light are discussed within the framework of classical and quantum optics. This notion is shown to be correctly defined for arbitrary light beams only within quantum optics by using the P-quasispin formalism developed earlier. Polarization states of quantum light are shown to be fully described by a polarization density operator (PDO) obtained via reducing the total field density operator. Theoretical foundations are given for quantum tomography of polarization states of light fields considered as a way of measuring PDO. Herewith, the main attention is paid to a method where proper polarization tomographic observables (PDO “measurers”) are used. The method is shown to be adequately formulated by means of quasi-spectral tomographic expansions of PDO in special operator bases (given by finite sums of partially orthogonal projectors), which determine probability distributions of tomographic observables as expansion coefficients. Matrix versions of such “tomographic” PDO representations are obtained. In particular, projections of these expansions on quasiclassical operator bases, determining polarization quasiprobability functions, are given. An example of experimental implementation of polarization tomography of unpolarized light (biphoton radiation with hidden polarization) is analyzed.     

From Local Perturbation Theory to Hopf and Lie Algebras of Feynman Graphs

We review the algebraic structures imposed on the renormalization procedure in terms of Hopf and Lie algebras of Feynman graphs, and exhibit the connection to diffeomorphisms of physical observables.     

Constructions of Some Quantum Structures and Fuzzy Effect Space

Quantum structures like effect algebras, -effect algebras, orthoalgebras, orthomodular posets, and -orthomodular posets are constructed by use of special fuzzy sets on posets. The concept of fuzzy effect space is introduced and a representation of a lattice effect algebra with a strong order determining system of states by means of fuzzy effect space is established.