Quantum sets and clifford algebras

The mathematical language presently used for quantum physics is a high-level language. As a lowest-level or basic language I construct a quantum set theory in three stages: (1) Classical set theory, formulated as a Clifford algebra of “S numbers” generated by a single monadic operation, “bracing,” Br = {…}. (2) Indefinite set theory, a modification of set theory dealing with the modal logical concept of possibility. (3) Quantum set theory. The quantum set is constructed from the null set by the familiar quantum techniques of tensor product and antisymmetrization. There are both a Clifford and a Grassmann algebra with sets as basis elements. Rank and cardinality operators are analogous to Schroedinger coordinates of the theory, in that they are multiplication or “Q-type” operators. “P-type” operators analogous to Schroedinger momenta, in that they transform theQ-type quantities, are bracing (Br), Clifford multiplication by a setX, and the creator ofX, represented by Grassmann multiplicationc(X) by the setX. Br and its adjoint Br* form a Bose-Einstein canonical pair, andc(X) and its adjointc(X)* form a Fermi-Dirac or anticanonical pair. Many coefficient number systems can be employed in this quantization. I use the integers for a discrete quantum theory, with the usual complex quantum theory as limit. Quantum set theory may be applied to a quantum time space and a quantum automaton. This material is based upon work supported in part by NSF Grant No. PHY8007921.     

Shocks in nonlocal media

We investigate the formation of collisionless shocks along the spatial profile of a Gaussian laser beam propagating in nonlocal nonlinear media. For defocusing nonlinearity the shock survives the smoothing effect of the nonlocal response, though its dynamics is qualitatively affected by the latter, whereas for focusing nonlinearity it dominates over filamentation. The patterns observed in a thermal defocusing medium are interpreted in the framework of our theory.     

Catalysis in nonlocal quantum operations

We show how entanglement can be used, without being consumed, to accomplish unitary operations that could not be performed without it. When applied to infinitesimal transformations, our method makes equivalent, in the sense of Hamiltonian simulation, a whole class of otherwise inequivalent two-qubit interactions. The new catalysis effect also implies the asymptotic equivalence of all such interactions.     

Soliton control in inhomogeneous nonlocal media

In this paper, we have addressed a new way to achieve soliton control, such as compression (or broadening) and fission. More concretely, nonlocality management technique has been adopted, and different control purposes can be realized by controlling corresponding nonlocal parameter. Essentially, nonlocality management should be classified into nonlinearity management.     

Electron self-energy in nonlocal field theory

In the framework of quantum electrodynamics with the nonlocal interaction it is shown that the correspondence principle holds in the problem on the self-mass of an electron and a particle with arbitrary spin. It appears that the second order of perturbation theory in the limit ? → 0 gives just the classical expression for the electron self-energy, and all higher order corrections are zero.     

Zero modes in nonlocal domain walls

We generalize the Callan–Harvey mechanism to the case of actions with a nonlocal mass term for the fermions. Using a (2+1)-dimensional model as a concrete example, we show that both the existence and properties of localized zero modes can also be consistently studied when the mass is nonlocal. We derive some general properties from a study of the resulting integral equations, and consider their realization in a concrete example.     

Two-color vector solitons in nonlocal media

We investigate the interaction between two beams differing in wavelength and the properties of dual-frequency spatial solitons in nonlocal birefringent reorientational media. We report the first experimental observations of anisotropic nonlocal vector solitons in unbiased nematic liquid crystals. Model and simulations, based on the paraxiality along the Poynting vectors, include joint walk-off and breathing.     

Soliton mobility in nonlocal optical lattices

We address the impact of nonlocality in the physical features exhibited by solitons supported by Kerr-type nonlinear media with an imprinted optical lattice. We discover that the nonlocality of the nonlinear response can profoundly affect the soliton mobility, hence all the related phenomena. Such behavior manifests itself in significant reductions of the Peierls-Nabarro potential with an increase in the degree of nonlocality, a result that opens the rare possibility in nature of almost radiationless propagation of highly localized solitons across the lattice.     

Anderson localization in nonlocal nonlinear media

We theoretically and numerically investigate the effect of focusing and defocusing nonlinearities on Anderson localization in highly nonlocal media. A perturbative approach is developed to solve the nonlocal nonlinear Schr?dinger equation in the presence of a random potential, showing that nonlocality stabilizes Anderson states.     

Tracking azimuthons in nonlocal nonlinear media

We study the formation of azimuthons, i.e., rotating spatial solitons, in media with nonlocal focusing nonlinearity. We show that whole families of these solutions can be found by considering internal modes of classical non-rotating stationary solutions, namely vortex solitons. This offers an exhaustive method to identify azimuthons in a given nonlocal medium. We demonstrate formation of azimuthons of different vorticities and explain their properties by considering the strongly nonlocal limit of accessible solitons.     

Periodic solitons in nonlocal nonlinear media

We identify periodic solitons in nonlocal nonlinear media: multi-hump soliton solutions propagating in a fully periodic fashion. We also demonstrate recurrences and breathers whose evolution is statistically periodic and discuss why some systems support periodic solitons while others do not.     

Soliton switching in inhomogeneous nonlocal media

We address a simple way to achieve routing of optical spatial solitons via soliton interactions in the inhomogeneous nonlocal media. We reveal that the variation of the nonlocality disturbs the solitons pairs and splits them into two individual solitons which have the same escape angle but opposite deflection directions. In particular, the escape angle monotonically increases with the increase of the nonlocality variation rate. We demonstrate that the soliton pairs could form self-consistent waveguides that are able to controllably guide a weak signal by any output positions.     

Diffusion-path approximation in nonlocal electron kinetics

For the positive column of a discharge, nonlocal distribution functions obtained by averaging over radial diffusion paths are compared with the exact solution to the kinetic elliptic equation. For a discharge in argon, as an example, the limits of applicability of the approximate solution for various macroscopic characteristics of the plasma were identified. It was previously believed that the approximation based on averaging has the limits of applicability determined by the condition that the plasma inhomogeneity size be smaller than the energy relaxation length. This condition restricts the applicability of the approximation to low pressures. In the present work, it is shown that, for determining a number of macroscopic parameters, such as the concentration, mean energy, mobility, diffusion coefficient, and thermal conductivity of electrons, the pathaveraging approximation works well over a pressure range of up to a few Torr. A number of subtle characteristics, such as the excitation rate, ionization rate, and others, largely influenced by fast electrons, cannot be calculated from the averaged distribution functions at pressures above a few tenths of a Torr.     

Spin response in nonlocal nonlinear optics

A novel effect of the electron spin in nonlinear optics is predicted. Starting from the Pauli Hamiltonian, we present the influence of the electron spin on the nonlinear optical response of a homogeneous electron gas in the low temperature limit. It is shown that the longitudinal second-harmonic response which results from a transverse incoming field differs significantly from the usual paramagnetic response when spin effects are included. The calculation is done both with and without screening.     

Bound on nonlocal scale from g − 2 of muon in a nonlocal W-S model

We consider the nonlocal version of the Weinberg Salam model (following Kleppe et al.) with a finite parameter Λ signifying a fundamental length scale. We calculate the extra contributions to the anomalous magnetic moment of the muon coming from the nonlocal structure in this model. We find that the nonlocal contribution can be comparable to weak contributions and goes to increase theoretical estimates. We use this calculation to determine the limit on the scale of nonlocality. We obtain the result 1/Λ ? 3 × 10^?16cm, which could be improved when present experimental errors narrow down.     

Localization of nonlocal theories

We show that a certain class of nonlocal scalar models, with a kinetic operator inspired by string field theory, is equivalent to a system which is local in the coordinates but nonlocal in an auxiliary evolution variable. This system admits both Lagrangian and Hamiltonian formulations, and its Cauchy problem and quantization are well-defined. We classify exact nonperturbative solutions of the localized model which can be found via the diffusion equation governing the fields.     

Collapse-induced quantum nonlocal effect

In this article we attempt to bring out some significant general aspects of what we call collapse-induced quantum nonlocal effects resulting from the use of the hypothesis of wave function collapse.     

Weakly nonlocal irreversible thermodynamics

Weakly nonlocal thermodynamic theories are critically revisited. A relocalized, irreversible thermodynamic theory of nonlocal phenomena is given, based on a modified form of the entropy current and new kind of internal variables, the so called current multipliers. The treatment is restricted to deal with nonlocality connected to dynamic thermodynamic variables. Several classical equations are derived, including Guyer‐Krumhansl, Ginzburg‐Landau and Cahn‐Hilliard type equations.