Position Space Representation Research Articles

Genus-one open string amplitudes on AdS5×S3 from CFT

We bootstrap one-loop string corrections to the four-point function of half-BPS operators in a 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFT with flavour group SO(8), dual to gluon scattering at genus one on AdS5×S3. We identify an 8-dimensional organising principle which governs the spectrum of double-trace anomalous dimensions, valid to all orders in the string length. This has precise implications for the structure of one-loop Mellin amplitudes, which we explicitly compute for the first three orders beyond the field-theory limit. We also consider the corresponding position space representation, which is entirely determined by the square of a certain differential operator acting on a simpler “pre-correlator”. Finally, we show that the flat-space limit of the Mellin amplitudes exactly matches the logarithmic terms of the genus-one amplitude in 8-dimensional flat space, which we compute via a partial-wave analysis.

Path integral in position-deformed Heisenberg algebra with maximal length uncertainty

In this work, we study the path integral in a position-deformed Heisenberg algebra with quadratic deformation which implements both minimal momentum and maximal length uncertainties. We construct propagators of path integrals within this deformed algebra using the position space representation on the one hand and the Fourier transform and its inverse representations on the other. The result is remarkably similar to the one obtained by Pramanik (2022) from the Perivolaropoulos’s deformed algebra (Perivolaropoulos, 2017). Then, the propagators and the corresponding actions of a free particle and a simple harmonic oscillator are discussed as examples. We also show that the actions which describe the classical trajectories of both systems are bounded by the ordinary ones of classical mechanics due to the existence of this maximal length. Consequently, particles of these systems travel faster from one point to another with low kinetic and mechanical energies.

Isomorphism between the Białynicki-Birula and the Landau-Peierls Fock space quantization of the electromagnetic field in position representation

We first present a summary of the quantization of the electromagnetic field in position space representation, using two main approaches: the Landau-Peierls approach in the Coulomb gauge and the Białynicki-Birula (BB) approach, based on the Riemann–Silberstein vector. We describe both in a framework that starts with a classical Hamiltonian structure and builds the quantum model in a bosonic Fock space by a precisely defined principle of correspondence. We show that the two approaches are completely equivalent. This is formulated by showing that there is a unitary map between the Fock spaces that makes them isomorphic. Since all the physically measurable quantities can be expressed in terms of scalar products, this implies that the two quantizations lead to exactly the same physical properties. We show furthemore that the isomorphism is preserved in the time evolutions. To show the equivalence, we use the concepts of helicity and frequency operators. The combination of these two operators provides a formulation that allows one to make the link between these two methods of quantization in a precise way. We also show that the construction in the BB quantization that avoids the presence of negative eigenvalues in the Hamiltonian, in analogy with the one for the Dirac equation for electrons and positrons, can be performed through an alternative choice of the canonical variables for Maxwell’s equations.

Momentum Gauge Fields and Non-Commutative Space–Time

In this work, we present a gauge principle that starts with the momentum space representation of the position operator (x^i=iℏ∂∂pi), rather than starting with the position space representation of the momentum operator (p^i=−iℏ∂∂xi). This extension of the gauge principle can be seen as a dynamical version of Born’s reciprocity theory, which exchanges position and momentum. We discuss some simple examples with this new type of gauge theory: (i) analog solutions from ordinary gauge theory in this momentum gauge theory, (ii) Landau levels using momentum gauge fields, and (iii) the emergence of non-commutative space–times from the momentum gauge fields. We find that the non-commutative space–time parameter can be momentum dependent, and one can construct a model where space–time is commutative at low momentum, but becomes non-commutative at high momentum.

Position-space representation of charged particles\u2019 propagators in a constant magnetic field as an expansion over Landau levels

We have obtained propagators in the position space as an expansion over Landau levels for the charged scalar particle, fermion, and massive vector boson in a constant external magnetic field. The summation terms in the resulting expressions consisted of two factors, one being rotationally invariant in the 2-dimensional Euclidean space perpendicular to the direction of the field, and the other being Lorentz-invariant in the 1+1-dimensional space-time. The obtained representations are unique in the sense that they allow for the simultaneous study of the propagator from both space-time and energetic perspectives which are implicitly connected. These results contribute to the development of position-space techniques in QFT and are expected to be of use in the calculations of loop diagrams.

The spin-one DKP equation with a nonminimal vector interaction in the presence of minimal uncertainty in momentum

In this work, we consider the relativistic Duffin–Kemmer–Petiau equation for spin-one particles with a nonminimal vector interaction in the presence of minimal uncertainty in momentum. By using the position space representation, we exactly determine the bound-states spectrum and the corresponding eigenfunctions. We discuss the effects of the deformation and nonminimal vector coupling parameters on the energy spectrum analytically and numerically.

The Duffin-Kemmer-Petiau oscillator in the presence of minimal uncertainty in momentum

In the anti-de Sitter background, the position and momentum operators obey the extended commutation relations leading to the emergence of a minimal uncertainty in momentum. Here, by using the position space representation we exactly determine the energy eigenvalues and the corresponding eigenfunctions for a (3+1) dimensional Duffin-Kemmer-Petiau oscillator of spin-zero and one in the framework of extended commutation relations.

Two-loop evolution equations for flavor-singlet light-ray operators

QCD in non-integer d = 4 − 2ϵ space-time dimensions enjoys conformal invariance at the special fine-tuned value of the coupling. Counterterms for composite operators in minimal subtraction schemes do not depend on ϵ by construction, and therefore the renormalization group equations for composite operators in physical (integer) dimensions inherit conformal symmetry. This observation can be used to restore the complete evolution kernels that take into account mixing with the operators containing total derivatives from their eigenvalues (anomalous dimensions). Using this approach we calculate the two-loop (NLO) evolution kernels for the leading twist flavor-singlet operators in the position space (light-ray operator) representation. As the main result of phenomenological relevance, in this way we are able to confirm the evolution equations of flavor-singlet generalized hadron parton distributions derived earlier by Belitsky and Müller using a different approach.

Anomalous dimensions at finite conformal spin from OPE inversion

We compute anomalous dimensions of higher spin operators in Conformal Field Theory at arbitrary space-time dimension by using the OPE inversion formula of [1], both from the position space representation as well as from the integral viz. Mellin representation of the conformal blocks. The Mellin space is advantageous over the position space not only in allowing to write expressions agnostic to the space-time dimension, but also in that it replaces tedious recursion relations in terms of simple sums which are easy to perform. We evaluate the contributions of scalar and spin exchanges in the t-channel exactly, in terms of higher order Hypergeometric functions. These relate to a particular exchange of conformal spin β = Δ + J in the s-channel through the inversion formula. Our results reproduce the special cases for large spin anomalous dimension and OPE coefficients obtained previously in the literature.

On the quantum-mechanics of a single photon

It is shown that a Dirac(-type) equation for a rank-two bi-spinor field ψph on Minkowski (configuration) spacetime furnishes a Lorentz-covariant quantum-mechanical wave equation in position-space representation for a single free photon. This equation does not encounter any of the roadblocks that have obstructed previous attempts (by various authors) to formulate a quantum-mechanical photon wave equation. In particular, it implies that the photon wave function ψph yields conserved non-negative Born-rule-type quantum probabilities and that its probability current density four-vector transforms properly under Lorentz transformations. Moreover, the eigenvalues of the pertinent photon Dirac Hamiltonian and the vector eigenvalues of the photon momentum operator yield the familiar Einstein relations E = ℏω and p = ℏk, respectively. Furthermore, these spin-1 wave modes are automatically transversal without the need of an additional constraint on the initial data. Some comments on other proposals to set up a photon wave equation are supplied as well.

Optical potential for light nuclei and momentum-space eikonal phase function

The space radiation environment comprises all of the nuclei in the periodic table with energies that extend from a fraction of an MeV/n to TeV/n. The vast range of projectile–target and energy combinations necessitates highly efficient and accurate cross section codes for use in radiation transport codes. As particles in the space radiation environment impinge on shielding materials, nuclear reactions, such as nuclear fragmentation, occur. One way of estimating nuclear fragmentation cross sections is to use an abrasion–ablation model, which describes how nucleons are dislodged from the nuclei as a result of nuclear collisions and the mechanism by which excited pre-fragments decay via particle emission to more stable states. The well-known partial wave solution method cannot be used directly for the computation of abrasion cross sections. Instead, abrasion cross sections may be computed by slightly altering the Eikonal solution method, which is a high energy (small scattering angle) approximation that depends on the nucleus–nucleus optical potential. The aim of the current work is to present two efficient methods for the computation of the Eikonal phase shift function. Analytic formulas of the optical potential are presented in the position-space representation for nuclei that are well-represented by harmonic-well nuclear matter densities (A < 20), which reduces the Eikonal phase factor to an integration over a single dimension. Next, the Eikonal phase function is presented in the momentum-space representation, which is particularly useful when the Fourier transform of the position-space optical potential is known. These new methods increase the computational efficiency by three orders of magnitude and allow for rapid prediction of elastic differential, total, elastic, and reaction cross sections in the Eikonal approximation.

Dirac and Klein-Gordon oscillators on anti-de Sitter space

The Dirac and Klein-Gordon oscillators on anti-de Sitter space were considered in a space with deformed commutation relations. Anti-de Sitter commutation relations give rise to the appearance of minimal uncertainty in the momentum. Using the position space representation, we determine the energy eigenvalues and the eigenfunctions for both cases. The wave functions can be given in terms of Gegenbauer polynomials. The high-temperature thermodynamic properties of the relativistic harmonic oscillators are then analyzed.

Heteropolar bonding and a position-space representation of the 8 - N rule.

Chemical bonding models are one of the most powerful tools in chemistry and provide essential guidance in the understanding of composition and structure of chemical compounds, as well as in the development of new preparation routes. Facing the tremendous diversity of crystal structures and properties of intermetallic compounds, it is highly desirable to make the predictive power of chemical bonding models also available for this field of inorganic chemistry. Within the framework of quantum-chemical position-space analysis the concept of the 8 - N rule is recovered and extended for a consistent and quantitative treatment of heteropolar bonding situations as in compounds of the MgAgAs type and their relatives. A first evaluation of the predictive capabilities of the position-space view is obtained in the analysis of 51 zinc-blende, wurtzite and rock-salt-type compounds. An outlook on future investigations is given and modifications of main-group elements and (pseudo) main-group compound families are classified within the presented model framework.

Position-space cuts for Wilson line correlators

We further develop the formalism for taking position-space cuts of eikonal diagrams introduced in [Phys.Rev.Lett. 114 (2015), no. 18 181602, arXiv:1410.5681]. These cuts are applied directly to the position-space representation of any such diagram and compute its discontinuity to the leading order in the dimensional regulator. We provide algorithms for computing the position-space cuts and apply them to several two- and three-loop eikonal diagrams, finding agreement with results previously obtained in the literature. We discuss a non-trivial interplay between the cutting prescription and non-Abelian exponentiation. We furthermore discuss the relation of the imaginary part of the cusp anomalous dimension to the static interquark potential.

Imaginary Parts and Discontinuities of Wilson Line Correlators

We introduce a notion of position-space cuts of eikonal diagrams, the set of diagrams appearing in the perturbative expansion of the correlator of a set of straight semi-infinite Wilson lines. The cuts are applied directly to the position-space representation of any such diagram and compute its imaginary part to the leading order in the dimensional regulator. Our cutting prescription thus defines a position-space analog of the standard momentum-space Cutkosky rules. Unlike momentum-space cuts which put internal lines on shell, position-space cuts constrain a number of the gauge bosons exchanged between the energetic partons to be lightlike, leading to a vanishing and a non-vanishing imaginary part for space- and timelike kinematics, respectively.

Schrödinger equation and resonant scattering in the presence of a minimal length

In this work we have studied the consequences of the minimal length, which arises in many theories of quantum gravity, on the Scattering of a point particle by a spherically symmetric potential. The modified Schrödinger equation is factorized to be of second order in position space representation. For the square well potential analytic expressions for the scattering states are obtained. Then the phase shifts are deduced. It is shown that the minimal length has two effects on the resonant scattering. The first one is that the minimal length increases slightly the resonant cross section and the second is the shift of the position of the resonances.

Some consequences of spacetime fuzziness

Abstract Finite resolution of spacetime at Planck scale gives it a fuzzy structure (the so-called foamy or fractal spacetime). This fuzzy structure of spacetime is a consequence of quantum fluctuation of geometry itself and can be described within non-commutative geometry and some alternative approaches to quantum gravity. In this paper, some consequences of spacetime fuzziness are studied. Due to this fuzzy structure, some basic notions of ordinary quantum mechanics such as position space representation, wave packet broadening during its propagation and coherent states of quantum mechanical systems should be re-examined.

The Topological Analysis of the Electron Localization Function. A Key for a Position Space Representation of Chemical Bonds

The purpose of the topological theory of Chemical Bonding is to provide a mathematical bridge between chemical concepts, such as bonds and lone pair, and rigorous Quantum Mechanics. The theory of dynamical systems enables to achieve a partition of the geometrical molecular space into basins of attractors bearing a chemical significance. A one to one correspondence is established between these basins and the chemical objects used to describe the bonding. The definition of population operators gives access to quantitative information and provides a firm basis for the concept of mesomery.

Algebraic tools for dealing with the atomic shell model. I. Wavefunctions and integrals for hydrogen-like ions

Today, the ‘hydrogen atom model’ is known to play its role not only in teaching the basic elements of quantum mechanics but also for building up effective theories in atomic and molecular physics, quantum optics, plasma physics, or even in the design of semiconductor devices. Therefore, the analytical as well as numerical solutions of the hydrogen-like ions are frequently required both, for analyzing experimental data and for carrying out quite advanced theoretical studies. In order to support a fast and consistent access to these (Coulomb-field) solutions, here we present the D IRAC program which has been developed originally for studying the properties and dynamical behavior of the (hydrogen-like) ions. In the present version, a set of MAPLE procedures is provided for the Coulomb wave and Green’s functions by applying the (wave) equations from both, the nonrelativistic and relativistic theory. Apart from the interactive access to these functions, moreover, a number of radial integrals are also implemented in the DIRAC program which may help the user to construct transition amplitudes and cross sections as they occur frequently in the theory of ion–atom and ion–photon collisions.

The calculated photon: Visualization of a quantum field

We calculate spatial localization and phase properties of spontaneously emitted photons. Our model is simple yet fully quantized: the emitting atom is a two-level atom located in a one-dimensional multimode optical cavity. Although the photon state vector does not have a position-space representation, the expectation value of the square of the electric field operator (intensity) is spatially localized and this pattern shifts at the speed of light. The emitted photon exhibits classical-like phase properties in the intensity expectation value when it “interferes with itself” after reflection. The phase properties of the emitted radiation are also evident at times sufficiently long for the radiation to have returned to the emitting atom before the decay of excited atomic state is complete.