Complex probabilities on R(N) as real probabilities on C(N) and an application to path integrals

We establish a necessary and sufficient condition for averages over complex-valued weight functions on R(N) to be represented as statistical averages over real, non-negative probability weights on C(N). Using this result, we show that many path integrals for time-ordered expectation values of bosonic degrees of freedom in real-valued time can be expressed as statistical averages over ensembles of paths with complex-valued coordinates, and then speculate on possible consequences of this result for the relation between quantum and classical mechanics.     

Division Algebras and Quantum Theory

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly ‘complex’ representations), those that are self-dual thanks to a symmetric bilinear pairing (which are ‘real’, in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are ‘quaternionic’, in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds—real, complex and quaternionic—can be seen as Hilbert spaces of the other kinds, equipped with extra structure.     

Quantum mechanics of the quaternionic Hilbert spaces based upon the imprimitivity theorem

We survey the realization of quantum mechanics in quaternionic Hilbert spaces following the methods of Mackey, who examined the complex and real cases exploiting the imprimitivity theorem. We show that there exists a unique unitary skew-adjoint operator which commutes with all the observables. This operator not only plays the role of the imaginary unit in the complex case, but allows a complexification of the Hilbert space by the choice of any quaternionic imaginary unit. Difficulties in the definition of time reversal, however, arise because of the properties of the quaternionic field. The introduction of an extra imaginary unit, commuting with the others, is suggested in order to implement time reversal properly. In the Appendix we give the proof of the imprimitivity theorem, in the quaternionic case, that we use in the paper.     

Einstein,Incompleteness, and the Epistemic View of Quantum States

Does the quantum state represent reality or our knowledge of reality? In making this distinction precise, we are led to a novel classification of hidden variable models of quantum theory. We show that representatives of each class can be found among existing constructions for two-dimensional Hilbert spaces. Our approach also provides a fruitful new perspective on arguments for the nonlocality and incompleteness of quantum theory. Specifically, we show that for models wherein the quantum state has the status of something real, the failure of locality can be established through an argument considerably more straightforward than Bell’s theorem. The historical significance of this result becomes evident when one recognizes that the same reasoning is present in Einstein’s preferred argument for incompleteness, which dates back to 1935. This fact suggests that Einstein was seeking not just any completion of quantum theory, but one wherein quantum states are solely representative of our knowledge. Our hypothesis is supported by an analysis of Einstein’s attempts to clarify his views on quantum theory and the circumstance of his otherwise puzzling abandonment of an even simpler argument for incompleteness from 1927.     

Multistability of delayed complex-valued recurrent neural networks with discontinuous real-imaginarytype activation functions

In this paper, the multistability issue is discussed for delayed complex-valued recurrent neural networks with discontinuous real-imaginary-type activation functions. Based on a fixed theorem and stability definition, sufficient criteria are established for the existence and stability of multiple equilibria of complex-valued recurrent neural networks. The number of stable equilibria is larger than that of real-valued recurrent neural networks, which can be used to achieve high-capacity associative memories. One numerical example is provided to show the effectiveness and superiority of the presented results.     

Synthesization of high-capacity auto-associative memories using complex-valued neural networks

In this paper, a novel design procedure is proposed for synthesizing high-capacity auto-associative memories based on complex-valued neural networks with real-imaginary-type activation functions and constant delays. Stability criteria dependent on external inputs of neural networks are derived. The designed networks can retrieve the stored patterns by external inputs rather than initial conditions. The derivation can memorize the desired patterns with lower-dimensional neural networks than real-valued neural networks, and eliminate spurious equilibria of complex-valued neural networks.One numerical example is provided to show the effectiveness and superiority of the presented results.     

On the duality between Boolean-valued analysis and reduction theory under the assumption of separability

It is well known that the real and complex numbers in the Scott-Solovay universeV ^(B) of ZFC based on a complete Boolean algebraB are represented by the real-valued and complex-valued Borel functions on the Stonean space ofB. The main purpose of this paper is to show that the separable complex Hilbert spaces and the von Neumann algebras acting on them inV ^(B) can be represented by reasonable classes of families of complex Hilbert spaces and of von Neumann algebras over. This could be regarded as the duality between Boolean-valued analysis developed by Ozawa, Takeuti, and others and the traditional reduction theory based not on measure spaces but on Stonean spaces. With due regard to Ozawa, this duality could pass for a sort of reduction theory forAW ^*-modules over commutativeAW ^*-algebras and embeddableAW ^*-algebras. Under the duality we establish several fundamental correspondence theorems, including the type correspondence theorems of factors.     

Finite-time Mittag—Leffler synchronization of fractional-order complex-valued memristive neural networks with time delay

Without dividing the complex-valued systems into two real-valued ones, a class of fractional-order complex-valued memristive neural networks (FCVMNNs) with time delay is investigated. Firstly, based on the complex-valued sign function, a novel complex-valued feedback controller is devised to research such systems. Under the framework of Filippov solution, differential inclusion theory and Lyapunov stability theorem, the finite-time Mittag—Leffler synchronization (FTMLS) of FCVMNNs with time delay can be realized. Meanwhile, the upper bound of the synchronization settling time (SST) is less conservative than previous results. In addition, by adjusting controller parameters, the global asymptotic synchronization of FCVMNNs with time delay can also be realized, which improves and enrich some existing results. Lastly, some simulation examples are designed to verify the validity of conclusions.     

Constructing Exactly Solvable Pseudo-hermitian Many-Particle Quantum Systems by Isospectral Deformation

A class of non-Dirac-hermitian many-particle quantum systems admitting entirely real spectra and unitary time-evolution is presented. These quantum models are isospectral with Dirac-hermitian systems and are exactly solvable. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. Examples of a pseudo-hermitian rational Calogero model and XXZ spin-chain are considered.     

Practical image hiding method using a phase wrapping rule and real-valued decoding key

We propose a practical image hiding method using phase wrapping and real-valued decoding key. A zero-padded original image, multiplied with a random-phase pattern, is Fourier transformed and its real-valued data denotes an encoded image in the embedding process. The encoded image is divided into two phase-encoded random patterns which are generated based on the phase wrapping rule. The imaginary part and the real part of these phase-encoded random patterns are used as a hidden image and a decoding key, respectively. A host image is then made from the linear superposition of the weighted hidden image and a cover image. The original image is simply obtained by the inverse-Fourier transform of the product of the host image and the decoding key in the reconstruction process. The embedding process and the reconstruction process are performed digitally and optically, respectively. Computer simulation and an optical experiment are shown in support of the proposed method.     

An approach to measurement

We present a new approach to measurement theory. Our definition of measurement is motivated by direct laboratory procedures as they are carried out in practice. The theory is developed within the quantum logic framework. This work clarifies an important problem in the quantum logic approach; namely, where the Hilbert space comes from. We consider the relationship between measurements and observables, and present a Hilbert space embedding theorem. We conclude with a discussion of charge systems.     

Importance of reversibility in the quantum formalism

In this Letter I stress the role of causal reversibility (time symmetry), together with causality and locality, in the justification of the quantum formalism. First, in the algebraic quantum formalism, I show that the assumption of reversibility implies that the observables of a quantum theory form an abstract real C^{?} algebra, and can be represented as an algebra of operators on a real Hilbert space. Second, in the quantum logic formalism, I emphasize which axioms for the lattice of propositions (the existence of an orthocomplementation and the covering property) derive from reversibility. A new argument based on locality and Soler's theorem is used to derive the representation as projectors on a regular Hilbert space from the general quantum logic formalism. In both cases it is recalled that the restriction to complex algebras and Hilbert spaces comes from the constraints of locality and separability.     

有限维希尔伯特空间q广义相干态振幅平方压缩

构造了有限维希尔伯特空间q变形非简谐振子广义相干态,研究了它的振幅平方压缩效应,结果表明,该量子态存在振幅平方压缩效应,并且给出了压缩条件与参数s,r,q之间的关系。     

Amplitude-Squared Squeezing of Generalized q -Deformation Coherent States of a Non-harmonic Oscillator in a Finite-Dimensional Hilbert Space

Generalized q-deformation coherent states of a non-harmonic oscillator in a finite-dimensional Hilbert Space are constructed. Their amplitude-squared squeezing are discussed. The result shows that there are squeezing effects in these quantum states. The relationship between condition of squeezing and parameters S, r, q is found.     

From quantum mechanics to classical statistical physics: Generalized Rokhsar-Kivelson Hamiltonians and the “Stochastic Matrix Form” decomposition

Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar-Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with stochastic classical systems described by a Master equation of the matrix type, hence their name. It then follows that the equilibrium partition function of the stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.     

Homogeneous Decoherence Functionals¶in Standard and History Quantum Mechanics

General history quantum theories are quantum theories without a globally defined notion of time. Decoherence functionals represent the states in the history approach and are defined as certain bivariate complex-valued functionals on the space of all histories. However, in practical situations – for instance in the history formulation of standard quantum mechanics – there often is a global time direction and the homogeneous decoherence functionals are specified by their values on the subspace of homogeneous histories. In this work we study the analytic properties of (i) the standard decoherence functional in the history version of standard quantum mechanics and (ii) homogeneous decoherence functionals in general history theories. We restrict ourselves to the situation where the space of histories is given by the lattice of projections on some Hilbert space ℋ. Among other things we prove the non-existence of a finitely valued extension for the standard decoherence functional to the space of all histories, derive a representation for the standard decoherence functional as an unbounded quadratic form with a natural representation on a Hilbert space and prove the existence of an Isham–Linden–Schreckenberg (ILS) type representation for the standard decoherence functional. Received: 26 November 1998 / Accepted: 2 December 1998     

Complex Classical Fields: A Framework for Reflection Positivity

We explore a framework for complex classical fields, appropriate for describing quantum field theories. Our fields are linear transformations on a Hilbert space, so they are more general than random variables for a probability measure. Our method generalizes Osterwalder and Schrader’s construction of Euclidean fields. We allow complex-valued classical fields in the case of quantum field theories that describe neutral particles. From an analytic point-of-view, the key to using our method is reflection positivity. We investigate conditions on the Fourier representation of the fields to ensure that reflection positivity holds. We also show how reflection positivity is preserved by various spacetime compactifications of ${\mathbb{R}^{d}}$ in different coordinate directions.     

Dilations of dynamical semi-groups

We prove the existence of isometric and unitary dilations of a class of semi-groups of completely positive maps on an algebra of operators on a Hilbert space. The result has relevance to the problem of embedding an open quantum mechanical system in a closed one.