Noncommuting Operators Research Articles

A process algebra model of QED

The process algebra approach to quantum mechanics posits a finite, discrete, determinate ontology of primitive events which are generated by processes (in the sense of Whitehead). In this ontology, primitive events serve as elements of an emergent space-time and of emergent fundamental particles and fields. Each process generates a set of primitive elements, using only local information, causally propagated as a discrete wave, forming a causal space termed a causal tapestry. Each causal tapestry forms a discrete and finite sampling of an emergent causal manifold (space-time) M and emergent wave function. Interactions between processes are described by a process algebra which possesses 8 commutative operations (sums and products) together with a non-commutative concatenation operator (transitions). The process algebra possesses a representation via nondeterministic combinatorial games. The process algebra connects to quantum mechanics through the set valued process and configuration space covering maps, which associate each causal tapestry with sets of wave functions over M. Probabilities emerge from interactions between processes. The process algebra model has been shown to reproduce many features of the theory of non-relativistic scalar particles to a high degree of accuracy, without paradox or divergences. This paper extends the approach to a semi-classical form of quantum electrodynamics.

, a C++ library for Yukawa decomposition in [formula omitted] models

We present in this paper the ▪ library, which calculates an analytic decomposition of the Yukawa interactions invariant under SO(2N) in terms of an SU(N) basis. We make use of the oscillator expansion formalism, where the SO(2N) spinor representations are expressed in terms of creation and annihilation operators of a Grassmann algebra acting on a vacuum state. These noncommutative operators and their products are simulated in ▪ through the implementation of doubly-linked-list data structures. These data structures were determinant to achieve a higher performance in the simplification of large products of creation and annihilation operators. We illustrate the use of our library with complete examples of how to decompose Yukawa terms invariant under SO(2N) in terms of SU(N) degrees of freedom for N=2 and 5. We further demonstrate, with an example for SO(4), that higher dimensional field-operator terms can also be processed with our library. Finally, we describe the functions available in ▪ that are made to simplify the writing of spinors and their interactions specifically for SO(10) models. Program summaryProgram title: SOSpinCatalogue identifier: AEZM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZM_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 25623No. of bytes in distributed program, including test data, etc.: 200391Distribution format: tar.gzProgramming language: C++.Computer: PC, Apple.Operating system: UNIX (Linux, Mac OS X 11).RAM: >20 MB depending on the number of processes requiredClassification: 4.2, 11.1, 11.6.External routines: In order to make further simplifications on the expressions obtained the library calls the Symbolic Manipulation System FORM program [1].Nature of problem: The decomposition of an Yukawa interaction invariant under SO(2N) in terms of SU (N) fields.Solution method: We make use of the oscillator expansion formalism, where the SO(2N) spinor representations are expressed in terms of creation and annihilation operators of a Grassmann algebra acting on a vacuum state.Running time: It depends on the input expressions, it can take a few seconds or more for very large representations (because of memory exhaustion).

Generalized Uncertainty Relation in the Non-commutative Quantum Mechanics

In this paper the non-commutative quantum mechanics (NCQM) with the generalized uncertainty relations \({\Delta } x_{1} {\Delta } x_{2} \ge \frac {\theta }{2}, {\Delta } p_{1} {\Delta } p_{2} \ge \frac {\bar {\theta }}{2}, {\Delta } x_{i} {\Delta } p_{i} \ge \frac {\hbar _{eff}}{2}\) is discussed. Four each uncertainty relation, wave functions saturating each uncertainty relation are explicitly constructed. The unitary operators relating the non-commutative position and momentum operators to the commutative position and momentum operators are also investigated. We also discuss the uncertainty relation related to the harmonic oscillator.

Signals on graphs: Transforms and tomograms

Development of efficient tools for the representation of large datasets is a precondition for the study of dynamics on networks. Generalizations of the Fourier transform on graphs have been constructed through projections on the eigenvectors of graph matrices. By exploring mappings of the spectrum of these matrices we show how to construct more general transforms, in particular wavelet-like transforms on graphs. For time-series, tomograms, a generalization of the Radon transforms to arbitrary pairs of non-commuting operators, are positive bilinear transforms with a rigorous probabilistic interpretation which provide a full characterization of the signals and are robust in the presence of noise. Here the notion of tomogram is also extended to signals on arbitrary graphs.

The Bell Inequality Is Satisfied by Quantum Correlations Computed Consistently with Quantum Non-Commutation

In constructing his theorem, Bell assumed that correlation functions among non-commuting variables are the same as those among commuting variables. However, in quantum mechanics, multiple data values exist simultaneously for commuting operations while for non-commuting operations data are conditional on prior outcomes, or may be predicted as alternative outcomes of the non-commuting operations. Given these qualitative differences, there is no reason why correlation functions among non-commuting variables should be the same as those among commuting variables, as assumed by Bell. When data for commuting and noncommuting operations are predicted from quantum mechanics, their correlations are different, and they now satisfy the Bell inequality.

The Generalized Diffusion Phenomenon and Applications

We study the asymptotic behavior of solutions to dissipative wave equations involving two noncommuting self-adjoint operators in a Hilbert space. The main result is that the abstract diffusion phenomenon takes place. Thus solutions of such equations approach solutions of diffusion equations at large times. When the diffusion semigroup has the Markov property and satisfies a Nash-type inequality, we obtain precise estimates for the consecutive diffusion approximations and remainders. We present several important applications including sharp decay estimates for dissipative hyperbolic equations with variable coefficients on an exterior domain. In the nonlocal case we obtain the first decay estimates for nonlocal wave equations with damping; the decay rates are sharp.

Structure of the Algebra Generated by a Noncommutative Operator Graph which Demonstrates the Superactivation Phenomenon for Zero-Error Capacity

Structure of the Algebra Generated by a Noncommutative Operator Graph which Demonstrates the Superactivation Phenomenon for Zero-Error Capacity

Real-Time Propagation via Time-Dependent Density Functional Theory Plus the Hubbard U Potential for Electron-Atom Coupled Dynamics Involving Charge Transfer.

We present methods for combining time-dependent density functional theory and the Hubbard U potential in the framework of the real-time propagation of Kohn-Sham orbitals to describe electron-atom coupled dynamics beyond the Born-Oppenheimer approximation. The time evolution of the noncommuting nonlocal operators were realized through Crank-Nicolson's inversion method and Suzuki-Trotter's split exponentiation. The electron dynamics related to the high speed motion of an alkali atom on a conjugated carbon plane is presented. The nonequilibrium charge oscillation between a metal surface and a localized atomic orbital, as modeled with graphene and Ca, is discussed.

Euler polynomials and identities for non-commutative operators

Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt [Phys. Rev. D 54(12), 7710-7723 (1996)], expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, by Pain [J. Phys. A: Math. Theor. 46, 035304 (2013)], links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Figuieira de Morisson and Fring [J. Phys. A: Math. Gen. 39, 9269 (2006)] in the context of non-Hermitian Hamiltonian systems. In each case, we provide several proofs and extensions of these identities that highlight the role of Euler and Bernoulli polynomials.

Readers offer their own magic moments with John Bell

Bertlmann replies: I am pleased at the positive response of readers to my “Magic moments with John Bell.”I enjoyed very much the contributions of Charles Clement and Kerson Huang, who reported about their own “magic moments” with Bell. Their amusing anecdotes help complete the portrait of Bell, who was an outstanding personality indeed.I found the comments of Nicholas Bykovetz very interesting. I remember that Bell strongly sympathized with Lorentzian relativity and Fitzgerald contraction. In my opinion, the ether-based Lorentzian view of relativity is just closer to the heart of the realist that Bell was. I think it was the acceptance of Lorentz’s conception on relativity rather than Einstein’s that led Bell to the correct answer that a string between two equally accelerated spaceships will break, what is known as Bell’s spaceship paradox.11. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge U. Press (2004), p. 67. In Bell’s quote “An ‘ether’ would be the cheapest solution. But the unobservability of this ether would be disturbing,” he did not mean for the “cheapest solution” to be a derogatory phrase, but rather to mean “simple.” The “unobservability of this ether” disturbed Bell, since why should the laws of physics conspire to prevent us from identifying the ether experimentally.Regarding Robert Griffiths’s arguments, I agree with some parts but strongly disagree with others. Griffiths remarks that the specific form of the expectation value used by Bell for the joint measurement of Alice and Bob in the EPR-Bohm-Bell experiment does not make sense in the context of quantum theory with noncommuting operators. That is not the point Bell wanted to make. In his formalism, the quantum states are supplemented by hidden variables, which are governed by a classical probability distribution, in order to predetermine the measurement results.The expectation value, assigned to the local hidden-variable theory, is not applied to quantum mechanics but compared with the corresponding quantum mechanical result. Its specific form is excluded via a Bell inequality, in which a certain combination of expectation values is needed. Concentrating on the expectation value alone is not enough to distinguish local hidden variable theories from quantum mechanics; for example, the predictions of quantum mechanics can also be achieved by working with a local hidden variable theory. See reference 22. R. A. Bertlmann, J. Phys. A 47, 424007 (2014). https://doi.org/10.1088/1751-8113/47/42/424007 for a more explicit discussion.Quantum mechanics as a mathematical formalism is a theory on (mathematical) Hilbert spaces, no matter whether the quantities associated with those spaces correspond to internal degrees of freedom, like spin, or external ones, like the position. In the EPR-Bohm-Bell context, the quantum formalism contains no reference to our three-dimensional space. Nevertheless, experiments are carried out in 3D space. Alice and Bob perform joint measurements of their particles at different, very remote locations. Then this nonlocal feature turns up: A measurement by Alice on the spin of her particle does have an effect— instantaneously—on Bob’s result, in contrast to what Griffiths claims in his comments. Therefore, quantum correlations are locally inexplicable.I sympathize with some of the comments by Michael Nauenberg, who collaborated with Bell long ago on “The moral aspect of quantum mechanics.”33. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics in Honor of V. F. Weisskopf, A. De-Shalit, H. Feshbach, L. Van Hove, eds., North-Holland (1966), p. 279. In particular, Nauenberg’s comment that “experiments have revealed that the nature of reality in the quantum world is different from our experience in the classical world” is, in my opinion, the lesson we have to learn from Bell inequalities.I do not appreciate so much Nauenberg’s example of the helium atom since it distracts from the issue of nonlocality. In fact, it is at macroscopic distances where the “puzzle” arises and not at atomic distances of separation.I have a confession: I am not the realist one might expect after reading Bell’s article “Bertlmann’s socks and the nature of reality”; the world in its very foundations is much more abstract than we think with our ”anschauliche” (intuitive) concepts, to borrow Werner Heisenberg’s term. My personal feeling is that Bell’s theorem, which reveals an apparent nonlocality in nature, points to a more radical conception whose onset we do not yet have.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge U. Press (2004), p. 67. Google ScholarCrossref2. R. A. Bertlmann, J. Phys. A 47, 424007 (2014). https://doi.org/10.1088/1751-8113/47/42/424007, Google ScholarCrossref3. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics in Honor of V. F. Weisskopf, A. De-Shalit, H. Feshbach, L. Van Hove, eds., North-Holland (1966), p. 279. Google Scholar© 2015 American Institute of Physics.

Mental, Behavioural and Physiological Nonlocal Correlations within the Generalized Quantum Theory Framework: A Review

Generalized Quantum Theory (GQT) seeks to explain and predict quantum-like phenomena in areas usually outside the scope of quantum physics, such as biology and psychology. It draws on fundamental theories and uses the algebraic formalism of quantum theory that is used in the study of observable physical matter such as photons, electrons, etc.In contrast to quantum theory proper, GQT is a very generalized form that does not allow for the full application of formalism. For instance neither a commutator, such as Planck’s constant, nor any additive operations are defined, which precludes the usage of a full Hilbert-space formalism. But it is a formalized phenomenological theory that is applicable whenever the core element of a quantum theory needs to be captured, namely in the presence of incompatible or non-commuting operations. As a consequence, it also predicts nonlocal, generalized entanglement correlations in systems other than proper quantum systems.In this review we summarize the specific scientific evidence relating to the quantum-like mental, behavioral and physiological nonlocal correlations. Such non-local, generalized entanglement correlations are expected, both in space and time, between subsystems of a larger system, whenever observables pertaining to the global system are incompatible or complementary to observables pertaining to subsystems, as predicted by GQT.The result is a coherent explanation of a significant amount of controversial and seemingly weird occurrences that cannot be explained by classical physical laws. This review also offers a new perspective of the human mind’s potential.

Jamming Based on an Ephemeral Key to Obtain Everlasting Security in Wireless Environments

Secure communication over a wiretap channel is considered in the disadvantaged wireless environment, where the eavesdropper channel is (possibly much) better than the main channel. We present a method to exploit inherent vulnerabilities of the eavesdropper's receiver to obtain everlasting secrecy. Based on an ephemeral cryptographic key pre-shared between the transmitter Alice and the intended recipient Bob, a random jamming signal is added to each symbol. Bob can subtract the jamming signal before recording the signal, while the eavesdropper Eve is forced to perform these non-commutative operations in the opposite order. Thus, information-theoretic secrecy can be obtained, hence achieving the goal of converting the vulnerable "cheap" cryptographic secret key bits into "valuable" information-theoretic (i.e. everlasting) secure bits. We evaluate the achievable secrecy rates for different settings, and show that, even when the eavesdropper has perfect access to the output of the transmitter (albeit through an imperfect analog-to-digital converter), the method can still achieve a positive secrecy rate. Next we consider a wideband system, where Alice and Bob perform frequency hopping in addition to adding the random jamming to the signal, and we show the utility of such an approach even in the face of substantial eavesdropper hardware capabilities.

Multiple operator integrals in perturbation theory

The purpose of this survey article is to give an introduction to double operator integrals and multiple operator integrals and to discuss various applications of such operator integrals in perturbation theory. We start with the Birman–Solomyak approach to define double operator integrals and consider applications in estimating operator differences $$f(A)-f(B)$$ for self-adjoint operators A and B. Next, we present the Birman–Solomyak approach to the Lifshits–Krein trace formula that is based on double operator integrals. We study the class of operator Lipschitz functions, operator differentiable functions, operator Holder functions, obtain Schatten–von Neumann estimates for operator differences. Finally, we consider in Chapter 1 estimates of functions of normal operators and functions of d-tuples of commuting self-adjoint operators under perturbations. In Chapter 2 we define multiple operator integrals in the case when the integrands belong to the integral projective tensor product of $$L^\infty $$ spaces. We consider applications of such multiple operator integrals to the problem of the existence of higher operator derivatives and to the problem of estimating higher operator differences. We also consider connections with trace formulae for functions of operators under perturbations of class $${\varvec{S}}_m$$ , $$m\ge 2$$ . In the last chapter we define Haagerup-like tensor products of the first kind and of the second kind and we use them to study functions of noncommuting self-adjoint operators under perturbation. We show that for functions f in the Besov class $$B_{\infty ,1}^1({\mathbb R}^2)$$ and for $$p\in [1,2]$$ we have a Lipschitz type estimate in the Schatten–von Neumann norm $${\varvec{S}}_p$$ for functions of pairs of noncommuting self-adjoint operators, but there is no such a Lipschitz type estimate in the norm of $${\varvec{S}}_p$$ with $$p>2$$ as well as in the operator norm. We also use triple operator integrals to estimate the trace norms of commutators of functions of almost commuting self-adjoint operators and extend the Helton–Howe trace formula for arbitrary functions in the Besov space $$B_{\infty ,1}^1({\mathbb R}^2)$$ .

Nonclassical properties of coherent light in a pair of coupled anharmonic oscillators

The Hamiltonian and hence the equations of motion involving the field operators of two anharmonic oscillators coupled through a linear one is framed. It is found that these equations of motion involving the non-commuting field operators are nonlinear and are coupled to each other and hence pose a great problem for getting the solutions. In order to investigate the dynamics and hence the nonclassical properties of the radiation fields, we obtain approximate analytical solutions of these coupled nonlinear differential equations involving the non-commuting field operators up to the second orders in anharmonic and coupling constants. These solutions are found useful for investigating the squeezing of pure and mixed modes, amplitude squared squeezing, principal squeezing, and the photon antibunching of the input coherent radiation field. With the suitable choice of the parameters (photon number in various field modes, anharmonic, and coupling constants, etc.), we calculate the second order variances of field quadratures of various modes and hence the squeezing, amplitude squared, and mixed mode squeezing of the input coherent light. In the absence of anharmonicities, it is found that these nonlinear nonclassical phenomena (squeezing of pure and mixed modes, amplitude squared squeezing and photon antibunching) are completely absent. The percentage of squeezing, mixed mode squeezing, amplitude squared squeezing increase with the increase of photon number and the dimensionless interaction time. The collapse and revival phenomena in squeezing, mixed mode squeezing and amplitude squared squeezing are exhibited. With the increase of the interaction time, the monotonic increasing nature of the squeezing effects reveal the presence of unwanted secular terms. It is established that the mere coupling of two oscillators through a third one does not produces the squeezing effects of input coherent light. However, the pure nonclassical phenomena of antibunching of photons in vacuum field modes are obtained through the mere coupling and hence the transfers of photons from the remaining coupled mode.

An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

Let be a finite measure on whose Laplace transform is analytic in a neighbourhood of zero. An anyon Lévy white noise on is a certain family of noncommuting operators on the anyon Fock space over , where runs over a space of test functions on , while is interpreted as an operator-valued distribution on . Let be the noncommutative -space generated by the algebra of polynomials in the variables , where is the vacuum expectation state. Noncommutative orthogonal polynomials in of the form are constructed, where is a test function on , and are then used to derive a unitary isomorphism between and an extended anyon Fock space over . The usual anyon Fock space over is a subspace of . Furthermore, the equality holds if and only if the measure is concentrated at a single point, that is, in the Gaussian or Poisson case. With use of the unitary isomorphism , the operators are realized as a Jacobi (that is, tridiagonal) field in . A Meixner-type class of anyon Lévy white noise is derived for which the corresponding Jacobi field in has a relatively simple structure. Each anyon Lévy white noise of Meixner type is characterized by two parameters, and . In conclusion, the representation is obtained, where and are the annihilation and creation operators at the point . Bibliography: 57 titles.

Non-Poissonian run-and-turn motions

Swimming bacteria exhibit a variety of motion patterns, in which persistent runs are punctuated by turning events. A simple yet fundamental question is to establish the properties of these random walks. While a complete answer is available when turning events follow a Poisson process, much less is known outside this particular case. We present a generic framework for such non-Poissonian run-and-turn motions. Extending the formalism of continuous time random walks, we obtain the generating function of moments in terms of noncommuting operators. We characterize analytically a bimodal model of persistent motion, which describes all types of swimming pattern, and is also relevant for cell motility.

Generators of quantum Markov semigroups

Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by a non-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to the works of Lindblad [Commun. Math. Phys. 48, 119-130 (1976)], Stinespring [Proc. Am. Math. Soc. 6, 211-216 (1955)], and Kraus [Ann. Phys. 64, 311-335 (1970)] imply that the generator of the semigroup has the form L(A)=∑n=1∞Vn∗AVn+GA+AG∗, where Vn and G are elements of the underlying operator algebra. In the present paper, we investigate the form of the generators of QMSs which are not necessarily uniformly continuous and act on the bounded operators of a Hilbert space. We prove that the generators of such semigroups have forms that reflect the results of Lindblad and Stinespring. We also make some progress towards forms reflecting Kraus’ result. Finally, we look at several examples to clarify our findings and verify that some of the unbounded operators we are using have dense domains.

Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators

We study perturbations of functions f(A,B) of noncommuting self-adjoint operators A and B that can be defined in terms of double operator integrals. We prove that if f belongs to the Besov class B∞,11(R2), then we have the following Lipschitz-type estimate in the Schatten–von Neumann norm Sp, 1≤p≤2: ‖f(A1,B1)−f(A2,B2)‖Sp≤const(‖A1−A2‖Sp+‖B1−B2‖Sp). However, the condition f∈B∞,11(R2) does not imply the Lipschitz-type estimate in Sp with p>2. The main tool is Schatten–von Neumann norm estimates for triple operator integrals.

Norm estimates for functions of two non-commuting operators

Analytic functions of two non-commuting bounded operators in a Banach space are considered. Sharp norm estimates are established. Applications to operator equations and differential equations in a Banach space are discussed.

Banach–Lie algebroids associated to the groupoid of partially invertible elements of a W∗-algebra

In the paper we study the algebroid A of the groupoid of partially invertible elements over the lattice of orthogonal projections of a $W^*$-algebra. In particular the complex analytic manifold structure of these objects is investigated. The expressions on the Lie brackets for A and related algebroids are given in noncommutative operator coordinates in the explicit way. We also prove statements describing structure of the groupoid of partial isometries and the frame groupoid of A as well as the structure of their algebroids.