Theory In Momentum Space Research Articles

Extended GUP formulation and the role of momentum cut-off

We analyze the extension of the GUP theory deriving from the modified uncertainty principle in agreement with the string low energy limit, which represents one of the most general formulations satisfying the Jacobi identity, in the context of the associative algebras. After providing some physical insights on the nature of the considered approaches exploiting the cosmological arena, first, we show how a natural formulation of the theory in an infinite momentum space does not lead to the emergence of a nonzero minimal uncertainty in position, then we construct a truncated formulation of the theory in momentum space, proving that only in this case we can recover the desired feature of the presence of a nonzero minimal uncertainty in position, which – as usual in these theories – can be interpreted as a phenomenological manifestation of cut-off physics effects. Both quantization schemes are completely characterized and finally applied to study wave packets’ behavior and their evolution in time. The obtained results can shed light on which generalizations of the GUP theory are more coherent with the string low energy limit, in view of the existence of a minimum length in the form of a minimal uncertainty in position.

Exact correlators from conformal Ward identities in momentum space and the perturbative TJJ vertex

We present a general study of 3-point functions of conformal field theory in momentum space, following a reconstruction method for tensor correlators, based on the solution of the conformal Ward identities (CWI's), introduced in recent works by Bzowski, McFadden and Skenderis (BMS). We investigate and detail the structure of the CWI's, their non-perturbative solutions and the transition to momentum space, comparing them to perturbation theory by taking QED as an example. We then proceed with an analysis of the TJJ correlator, presenting independent and detailed re-derivations of the conformal equations in the reconstruction method of BMS, originally formulated using a minimal tensor basis in the transverse traceless sector. A careful comparison with a second basis introduced in previous studies shows that this correlator is affected by one anomaly pole in the graviton (T) line, induced by renormalization. The result shows that the origin of the anomaly, in this correlator, should be necessarily attributed to the exchange of a massless effective degree of freedom. Our results are then exemplified in massless QED at one-loop in d-dimensions, expressed in terms of perturbative master integrals. An independent analysis of the Fuchsian character of the solutions, which bypasses the 3K integrals, is also presented. We show that the combination of field theories at one-loop – with a specific field content of degenerate massless scalar and fermions – is sufficient to generate the complete non-perturbative solution, in agreement with a previous study in coordinate space. The result shows that free conformal field theories, in specific dimensions, arrested at one-loop, reproduce the general result for the TJJ. Analytical checks of this correspondence are presented in d=3,4 and 5 spacetime dimensions. This implies that the generalized 3K integrals of the BMS solution can be expressed in terms of the two single master integrals B0 and C0 of 2- and 3-point functions, with significant simplifications.

Momentum-space cigar geometry in topological phases

.In this paper, we stress the importance of momentum-space geometry in the understanding of two-dimensional topological phases of matter. We focus, for simplicity, on the gapped boundary of three-dimensional topological insulators in class AII, which are described by a massive Dirac Hamiltonian and characterized by an half-integer Chern number. The gap is induced by introducing a magnetic perturbation, such as an external Zeeman field or a ferromagnet on the surface. The quantum Bures metric acquires a central role in our discussion and identifies a cigar geometry. We first derive the Chern number from the cigar geometry and we then show that the quantum metric can be seen as a solution of two-dimensional non-Abelian BF theory in momentum space. The gauge connection for this model is associated to the Maxwell algebra, which takes into account the Lorentz symmetries related to the Dirac theory and the momentum-space magnetic translations connected to the magnetic perturbation. The Witten black-hole metric is a solution of this gauge theory and coincides with the Bures metric. This allows us to calculate the corresponding momentum-space entanglement entropy that surprisingly carries information about the real-space conformal field theory describing the defect lines that can be created on the gapped boundary.

Emergent phase space description of unitary matrix model

We show that large N phases of a 0 dimensional generic unitary matrix model (UMM) can be described in terms of topologies of two dimensional droplets on a plane spanned by eigenvalue and number of boxes in Young diagram. Information about different phases of UMM is encoded in the geometry of droplets. These droplets are similar to phase space distributions of a unitary matrix quantum mechanics (UMQM) ((0 + 1) dimensional) on constant time slices. We find that for a given UMM, it is possible to construct an effective UMQM such that its phase space distributions match with droplets of UMM on different time slices at large N . Therefore, large N phase transitions in UMM can be understood in terms of dynamics of an effective UMQM. From the geometry of droplets it is also possible to construct Young diagrams corresponding to U(N) representations and hence different large N states of the theory in momentum space. We explicitly consider two examples: single plaquette model with TrU2 terms and Chern-Simons theory on S3. We describe phases of CS theory in terms of eigenvalue distributions of unitary matrices and find dominant Young distributions for them.

Six-point remainder function in multi-Regge-kinematics: an efficient approach in momentum space

Starting from the known all-order expressions for the BFKL eigenvalue and impact factor, we establish a formalism allowing the direct calculation of the six-point remainder function in $$ \mathcal{N} $$ = 4 super-Yang-Mills theory in momentum space to — in principle — all orders in perturbation theory. Based upon identities which relate different integrals contributing to the inverse Fourier-Mellin transform recursively, the formalism allows to easily access the full remainder function in multi-Regge kinematics up to 7 loops and up to 10 loops in the fourth logarithmic order. Using the formalism, we prove the all-loop formula for the leading logarithmic approximation proposed by Pennington and investigate the behavior of several newly calculated functions.

Finite-temperature three-point function in 2D CFT

We calculate the finite temperature three-point correlation function for primary fields in a 2D conformal field theory in momentum space. This result has applications to any strongly coupled field theory with a 2D CFT dual, as well as to Kerr/CFT.

Dynamics of e+ + H(ns) → e+ + H(n′s) with exponential cosine-screened Coulomb potentials

The dynamics of ns → n′s excitations of hydrogen by positron impact with exponential cosine-screened Coulomb potentials (ECSCPs) has been investigated using a distorted-wave theory in momentum space. Making use of a simple variationally determined wave function for the hydrogen atom, it has been possible to obtain the distorted-wave scattering amplitude in a closed analytical form. A detailed study has been made on differential and total cross sections in the energy range 20–250 eV. To the best of our knowledge, such a study on the differential and total cross sections for inelastic positron–hydrogen collisions with ECSCPs is the first reported in the literature.

On the eigenfunctions of the Douglas–Kroll operator

The matrix elements of the quasirelativistic Douglas–Kroll operators up to the fourth order for hydrogen-like ions is constructed with as few additional approximations as possible, to investigate the behaviour of its 1s eigenfunctions in the vicinity of the nucleus. Because Douglas–Kroll is a momentum space theory, we use a basis set of spherical waves which are eigenfuntions of the square of the momentum operator. While this avoids the most serious approximation of the standard Douglas–Kroll–Hess protocol, namely that the basis functions used to construct the Douglas–Kroll operator are eigenfunctions of the (squared) momentum operator, it also makes the convergence of this expansion very slow, because spherical waves are not well suited to represent the (weak) singularities of the eigenfunctions at the position of the point-like nucleus. On the other hand, the convergence is quite monotonic, and information on the behaviour close to the nucleus can be extracted from the convergence rate. Starting with the second-order, the eigenfunctions of the Douglas–Kroll operator are not “more singular” than the Dirac eigenfunctions, and the occurence of an additional error when using regular basis sets, as postulated in the literature, can not be observed. The resolution of the identity, which is involved in any practical approach to construct the matrix elements of the Douglas–Kroll operators beyond the first order, is a minor problem for heavy nuclei.

Bragg Diffraction of X-Rays by Single Crystals with Large Microdefects I. Generalized Dynamical Theory

Basic equations of the dynamical scattering theory in momentum space have been considered in the two-beam approximation for single crystals containing randomly distributed microdefects commensurable with extinction length. The amplitudes of coherent and diffusely scattered waves inside the crystal have been found by using the perturbation theory with average and fluctuating parts of crystal polarizability as small parameters. In the complex dispersion corrections to the wave vectors of coherent and diffuse waves, the imaginary parts of which describe the attenuation of these waves due to diffuse scattering, the dynamical effects in diffuse scattering and their dependences on the incidence angle have been taken into account. These corrections also take account of the influence of any multiple diffuse scattering processes. The coherent component of crystal reflectivity has been calculated in the approximation of semiinfinite crystal for an arbitrary diffraction geometry.

A numerical experiment in DLCQ: microcausality, continuum limit and all that

Issues related with microcausality violation and continuum limit in the context of (1+1) dimensional scalar field theory in discretized light-cone quantization (DLCQ) are addressed in parallel with discretized equal time quantization (DETQ) and the fact that Lorentz invariance and microcausality are restored if one can take the continuum limit properly is emphasized. In the free case, it is shown with numerical evidence that the continuum results can be reproduced from DLCQ results for the Pauli–Jordan function and the real part of Feynman propagator. The contributions coming from k + near zero region in these cases are found to be very small. In the interacting case, aspects related to the continuum limit of DLCQ results in perturbation theory in momentum space are discussed.

Application of Renner-Teller and jahn-teller effect to highT c superconductivity theory with LCG-MG and Covalon models

For combination and extension of physical and chemical approaches, instead of the BCS Theory in momentum space for lowT c superconductors, we propose the position-space theory for highT c superconductors using quantum chemistry of moleculargeminal orbital and valence-bond states. Our spectroscopy related vibronic geminals and vibronic ‘Covalons’ will serve as Boson-state quasi-particles. With emphasis on equal-minima double-well potential, besides the Jahn-Teller effect we consider the Renner-Teller effect because of thegeminals and cyclic crystal boundary. It will have “first-order” vibronic degenerate perturbation in addition to the second-order perturbation energies of BCS and Bipolaron theories.

Relativistic quantum theory in momentum space for weak and strong coupling.

In this paper a relativistic quantum theory introduced by de Groot and Ruijgrok [Nucl. Phys. 101B, 95 (1975)] is reformulated. It is shown that it has the correct nonrelativistic limit and that it gives the same lowest-order relativistic corrections as the Breit equation. In this paper it is applied to bound systems of two charged particles with spin 1/2. For small coupling constants the lowest-order fine structure (up to order ${\mathrm{\ensuremath{\alpha}}}^{4}$) for all states and all mass ratios is calculated. No expansion in the mass ratio is made, so that the full recoil effects are obtained. It embraces as special cases the well-established fine structure of the hydrogen atom and positronium and the Fermi splitting of muonium. For the strong-coupling case the critical couplings are calculated. The convergence to the static limit for couplings in the neighborhood of the critical point is extremely slow. The bound-state masses for states with the same total angular momentum J but opposite parity are not equal at the critical point, so there is no parity doubling. The results are compared with a number of other theories.

On the correlation energy density functional in momentum space

A generalization of the Coulson-March formula has been developed interconnecting the electron density and the momentum denstiy within the framework of Thomas-Fermi theory. This has been employed to derive a correlation energy density functional in momentum space and numerical results are presented. Some remarks on the density functional theory in momentum space are also made.

Atom–diatom processes in helium: The multiple-scattering description of elastic and dissociative collisions

Elastic and dissociative scattering of He on He2 are studied for an incident laboratory energy of 1 K. We apply Faddeev-AGS multiple-scattering theory in momentum space to the analysis. We show that this leads to a simple justification for the importance of the ‘‘complex-formation mechanism’’ in termolecular recombination, the time reverse of collision-induced dissociation. Our work, in which separable expansions of two phenomenological He–He potentials are used, predicts significant magnitude differences in the cross sections derived from the interactions and also verifies the validity of the ‘‘peaking approximation’’ for hyperthermal-energy elastic collisions.

A quantum field theory in momentum space of constant curvature

The model of a polarization operator in the “rainbow” diagram approximation in momentum space of constant curvature is studied. An integral equation is obtained for the absorptive component of the hadron polarization operator in a stereographic parametrization of the de Sitter space. The solution of the integral equation is found for the case of zero mass of the exchange particle. The sections and mean multiplicity of hadron generation in e+e− annihilation are calculated. The infrared divergences which develop in an exact summation of the class of diagrams considered here manifest themselves as a multiplicative renormalization constant.

Momentum-space derivation of the Runge–Lenz vector

Momentum-space theory is used to derive the Runge–Lenz vector. This very simple derivation unites the Runge–Lenz vector theory and the momentum-space theory.

The role of iterative isobar processes in nuclear matter and the effective nucleon-nucleon interaction

A calculation is performed using lowest order Brueckner theory in momentum space, with explicit isobar configurations included through the coupled channel mathod. The effective interaction for the1S0-5D0 channel is extracted from this calculation. Two different transition potentials are used — one due to Green and Niskanen (1976), the other, due to Green and co-workers (1978). The nucleon-nucleon (NN) interaction used is the Reid soft core potential, compensated for the inclusion of the explicit isobar channel. The effective interaction shows marked momentum dependence in the intermediate range. The loss of attraction depends on the transition potential one chooses. The correlation function involving the nucleon-isobar intermediate state is anti-correlated to the NN part.

Momentum-space theory of exciton transport. II. Sensitized luminescence calculations for specific trapping models

Momentum-space theory of exciton transport. II. Sensitized luminescence calculations for specific trapping models

Products of Principal Value Singularities Used in the Formal Theory of Scattering

A mathematically neat derivation is given of the relation between the S matrix and the transformation function for a finite time. It is shown that one can dispense with the adiabatic switching on and off, and yet one reaches the same result as when one employs it. Necessary conditions are discussed for the validity of this statement. Systematic prescriptions are given of handling products of principal value singularities, which is relevant to the scattering theory in momentum space.