Invariant Perturbation Theory Research Articles

Perturbations of a class of locally rotationally symmetric cosmologies with applications to dissipative fluids

A gauge invariant perturbation theory, based on the covariant split of spacetime, is used to study first order perturbations on a class of anisotropic cosmological backgrounds. The perturbations as well as the energy-momentum tensor are kept general, giving a system of equations on which different physical situations may be imposed. Through a harmonic decomposition, the system is then transformed to evolution equations in time and algebraic constraints. This result is then applied to dissipative one-component fluids, and on using the simplified acausal Eckart theory the system is reduced to two closed subsystems, governed by four and eight harmonic coefficients for the odd and even sectors respectively. The system is also seen to close in a simplified causal theory. It is then demonstrated, within the Eckart theory, how vorticity can be generated from viscosity.

Relating dust reference models to conventional systems in manifestly gauge invariant perturbation theory

Models with dust reference fields in relational formalism have proved useful in understanding the construction of gauge-invariant perturbation theory to arbitrary orders in the canonical framework. These reference fields modify the dynamical equations for perturbation equations. However, important questions remain open on the relation with conventional perturbation theories of inflaton coupled to gravity and of multi-fluid systems, and on understanding modifications in terms of physical degrees of freedom. These gaps are filled in this manuscript for Brown-Kucha\v{r} and Gaussian dust models, both of which involve three scalar physical degrees of freedom. We establish a relationship of these models with conventional inflationary and multi-fluid system of inflation and ordinary dust by introducing a set of gauge invariant variables on the reduced phase space of the dust reference models. We find the modifications due to dust clocks to Bardeen equation in the longitudinal gauge and Mukhanov-Sasaki equation in the spatially-flat gauge, in terms of physical degrees of freedom. This results in a closed system of equations for all the degrees of freedom needed to explore the evolution of the scalar perturbations. Our analysis shows for the first time that even for two-fluid systems, there is a natural choice of the set of gauge invariant variables for each chosen gauge which not only offers a direct physical interpretation but also results in simplifications to the dynamical equations.

Gauge invariant perturbation theory via homotopy transfer

We show that the perturbative expansion of general gauge theories can be expressed in terms of gauge invariant variables to all orders in perturbations. In this we generalize techniques developed in gauge invariant cosmological perturbation theory, using Bardeen variables, by interpreting the passing over to gauge invariant fields as a homotopy transfer of the strongly homotopy Lie algebras encoding the gauge theory. This is illustrated for Yang-Mills theory, gravity on flat and cosmological backgrounds and for the massless sector of closed string theory. The perturbation lemma yields an algorithmic procedure to determine the higher corrections of the gauge invariant variables and the action in terms of these.

Peculiarities of a helium interaction with hydrogen and free electrons from the point of view of the exchange perturbation theory solutions

We consider an adiabatic potential of a Hydrogen-Helium interaction by using a new approach based on the exchange perturbation theory (EPT). This method allows to present results analytically. We find the solution as an expansion in an antisymmetric basis thanks to the proved completeness property for a nonorthogonal set of antisymmetric wave functions. We use the developed invariant form with respect to interatomic electron permutations for both the Hamiltonian describing an unperturbed system and the perturbation operator. The invariant form for both operators allows one to obtain properly antisymmetric corrections to the antisymmetric wave function of the zero approximation. The probability of electron scattering with electron capture by the helium atom is discussed. We use an EPT formalism developed for the time-dependent perturbations. A formalism of the Time-dependent EPT has a form of an invariant perturbation theory that accounts for intercentre electron permutations among the overlapping states. The expressions for the matrix elements of an S-matrix and a T-matrix account for interatomic electron permutations. We use the obtained differential cross section formulas for the description of the electron capture during the scattering processes. The differential cross section of the electron capture by Helium atom has a complex shape of peak for the scattering energy around 0.7 atomic units, associated with the adiabatic creation of the negative ion He(-1).

Universal thermodynamics of a trapped Fermi gas in the superfluid regime: the role of the Pauli principle

Understanding the universal behavior of strongly-interacting systems of particles has been a major goal in multiple fields of physics from atomic physics to cosmology. Using a recently introduced manybody perturbation formalism for fermions which uses group theoretic and graphical techniques to solve the N-body problem, I study the thermodynamic behavior of the unitary regime for a trapped Fermi gas and investigate the role of the Pauli principle in the emergence of collective behavior, specifically superfluidity at ultralow temperatures. The method, which is called symmetry invariant perturbation theory, has no adjustable parameters and currently offers a solution to the manybody Schrodinger equation through first order in inverse dimensionality exactly. The solution at first order defines collective coordinates in terms of five types of N-body normal modes, identified as symmetric stretch, symmetric bend, single particle angular excitation, single particle radial excitation and phonon. A correspondence is established ‘on paper’ that enforces the Pauli principle through the assignment of specific normal mode quantum numbers. Applied at ultracold temperatures, this normal mode assignment yields occupation only in an extremely low frequency N-body phonon mode. A single particle radial excitation mode at a much higher frequency creates a gap that stabilizes the superfluidity at low temperatures. This radial excitation involves excitation of a single particle out of the synced motion of the phonon mode, rather than the breaking of a fermion pair. Coupled with the corresponding values for the energies at unitarity obtained by this manybody calculation, I obtain good agreement with experimental thermodynamic results. In particular, the lambda transition in the specific heat is clearly seen. Based on these findings, I propose a mechanism for the emergence of superfluidity at ultralow temperatures in this unitary regime that is driven by quantum statistics and the enforcement of the Pauli principle through the selection of normal modes. My results for the calculated thermodynamic quantities suggest that the emergence of some collective behaviors in macroscopic systems is due to the Pauli principle producing manybody pairing and the resulting macroscopic occupation of an N-body phonon mode.

Second order gauge invariant perturbation theory and conserved charges in cosmological Einstein gravity

Recently a new approach in constructing the conserved charges in cosmological Einstein gravity was given. In this new formulation, instead of using the explicit form of the field equations, a covariantly conserved rank-four tensor was used. In the resulting charge expression, instead of the first derivative of the metric perturbation, the linearized Riemann tensor appears along with the derivative of the background Killing vector fields. Here we give a detailed analysis of the first-order and the second-order perturbation theory in a gauge-invariant form in cosmological Einstein gravity. The linearized Einstein tensor is gauge-invariant at the first order but it is not so at the second order, which complicates the discussion. This method depends on the assumption that the first-order metric perturbation can be decomposed into gauge-variant and gauge-invariant parts and the gauge-variant parts do not contribute to physical quantities.

S-duality invariant perturbation theory improved by holography

We study anomalous dimensions of unprotected low twist operators in the four-dimensional SU(N ) mathcal{N} = 4 supersymmetric Yang-Mills theory. We construct a class of interpolating functions to approximate the dimensions of the leading twist operators for arbitrary gauge coupling τ . The interpolating functions are consistent with previous results on the perturbation theory, holographic computation and full S-duality. We use our interpolating functions to test a recent conjecture by the mathcal{N} = 4 superconformal bootstrap that upper bounds on the dimensions are saturated at one of the duality-invariant points τ = i and τ = eiπ/3. It turns out that our interpolating functions have maximum at τ =eiπ/3, which are close to the conjectural values by the conformal bootstrap. In terms of the interpolating functions, we draw the image of conformal manifold in the space of the dimensions. We find that the image is almost a line despite the conformal manifold is two-dimensional. We also construct interpolating functions for the subleading twist operator and study level crossing phenomenon between the leading and subleading twist operators. Finally we study the dimension of the Konishi operator in the planar limit. We find that our interpolating functions match with numerical result obtained by Thermodynamic Bethe Ansatz very well. It turns out that analytic properties of the interpolating functions reflect can expectation on a radius of convergence of the perturbation theory.

Invariant time-dependent exchange perturbation theory and its application to the particles collision problem

We present the formalism of time-dependent exchange perturbation theory built to all orders of perturbation, for the arbitrary time dependency of perturbation. The theory takes into account the rearrangement of electrons among centers. We show how the formalism can be reduced to the standard form of invariant perturbation theory by “switching off” the re-arrangement of electrons among centers. The elements of the scattering S-matrix and transitions T-matrix and the formula for the electron scattering differential cross section are derived. The application of the theory to scattering and collision problems is discussed. As an example, we apply the theory to proton scattering on a lithium atom, calculating the differential and total cross sections.

The Relativistic Theory of Electromagnetic Interactions

Quantum – mechanical problem on electron motion in Coulomb field plays the fundamental role in the modern physics, because solution of this problem provides the most realistic model of atomic structure. In nowadays the model of meson atom is of the same importance. Indeed, the mesoatom is unstable, hence the study of interaction of mesons with atomic nuclei can provide the valuable information on the nature of hadronic interactions and structure of atomic nucleus. The mesoatoms arise in the process of scattering of the negatively charged pions in matter. The pions captured by the target atoms populate initially the highly excited atomic orbits and then go down to 2p and 1s atomic states via cascade of radiative transitions. As far as the meson mass exceeds the electron mass for the two orders of magnitude then the mesonic atom orbit radius is much smaller than the radius of electron atom orbits. For example, the radius of the lowest mesonic orbits becomes comparable with the atomic nucleus size in the case of nucleus charge 12 Z [1]. As a result, for meson in 2p state the probability to annihilate on atomic nucleus is higher than the probability to decay radiatively in 2 1 p s transition. In the atoms with the nucleus charge 40 Z the radiative transitions disappear before the cascade reaches the 2p state. In the case of atoms with more heavy nuclei even the 3d states can not often be occupied. Thus the study of the cascade x-ray spectra of mesonic atoms provides the information on the efficiency of meson interaction with the different nuclei. The nonrelativistic Schredinger theory produces the atomic spectrum in zero-order approximation which does not include fine and hyperfine structure. The atomic spectrum calculated in the frames of the Dirac relativistic theory includes the fine structure. The hyperfine atomic structure can be calculated with the help of the invariant perturbation theory methods, which are based on the field quantization theory. The smallness parameter of invariant perturbation theory is the product of the hyperfine structure constant and nucleus charge, Z . Hence, the atoms with the nucleus charge of 1 Z are out of the frames of the perturbation theory and therefore they are of special interest. This interest is due to a number of reasons: Firstly, the precise mathematical calculations of hyperfine atomic structure can be based on the non-perturbative theory which is valid for arbitrary value of parameter Z . Secondly, the Dirac theory predicts that at 1 Z the energy eigenvalues become complex. It means that the corresponding atomic states are unstable, i.e. the lifetime of these states is finite [2-4].

Invariant exchange perturbation theory for multicenter systems and its application to the calculation of magnetic chains in manganites

The formalism of exchange perturbation theory is presented with regard to the general principles of constructing an antisymmetric vector with the use of the Young diagrams and tableaux in which the coordinate and spin parts are not separated. The form of the energy and wave function corrections coincides with earlier obtained expressions, which are reduced in the present paper to a simpler form of a symmetry-adapted perturbation operator, which preserves all intercenter exchange contributions. The exchange perturbation theory (EPT) formalism itself is presented in the standard form of invariant perturbation theory that takes into account intercenter electron permutations between overlapping nonorthogonal states. As an example of application of the formalism of invariant perturbation theory, we consider the magnetic properties of perovskite manganites La1/3Ca2/3MnO3 that are associated with the charge and spin ordering in magnetic chains of manganese. We try to interpret the experimental results obtained from the study of the effect of doping the above alloys by the model of superexchange interaction in manganite chains that is constructed on the basis of the exchange perturbation theory (EPT) formalism. The model proposed makes it possible to carry out a quantitative analysis of the effect of substitution of manganese atoms by doping elements with different electron configurations on the electronic structure and short-range order in a magnetic chain of manganites.

GMRES Convergence for Perturbed Coefficient Matrices, with Application to Approximate Deflation Preconditioning

How does GMRES convergence change when the coefficient matrix is perturbed? Using spectral perturbation theory and resolvent estimates, we develop simple, general bounds that quantify the lag in convergence such a perturbation can induce. This analysis is particularly relevant to preconditioned systems, where an ideal preconditioner is only approximately applied in practical computations. To illustrate the utility of this approach, we combine our analysis with Stewart's invariant subspace perturbation theory to develop rigorous bounds on the performance of approximate deflation preconditioning using Ritz vectors.

Gauge invariant cosmological perturbations for the nonminimally coupled inflaton field

We construct the gauge invariant free action for cosmological perturbations for the nonminimally coupled inflaton field in the Jordan frame. For this the phase space formalism is used, which keeps track of all the dynamical and constraint fields. We perform explicit conformal transformations to demonstrate the physical equivalence between the Jordan and Einstein frames at the level of quadratic perturbations. We show how to generalize the formalism to the case of a more complicated scalar sector with an internal symmetry, such as Higgs inflation. This work represents a first step in developing gauge invariant perturbation theory for nonminimally coupled inflationary models.

Gauge invariant perturbation theory and non-critical string models of Yang-Mills theories

We carry out a gauge invariant analysis of certain perturbations of $D-2$-branes solutions of low energy string theories. We get generically a system of second order coupled differential equations, and show that only in very particular cases it is possible to reduce it to just one differential equation. Later, we apply it to a multi-parameter, generically singular family of constant dilaton solutions of non-critical string theories in $D$ dimensions, a generalization of that recently found in arXiv:0709.0471[hep-th]. According to arguments coming from the holographic gauge theory-gravity correspondence, and at least in some region of the parameters space, we obtain glue-ball spectra of Yang-Mills theories in diverse dimensions, putting special emphasis in the scalar metric perturbations not considered previously in the literature in the non critical setup. We compare our numerical results to those studied previously and to lattice results, finding qualitative and in some cases, tuning properly the parameters, quantitative agreement. These results seem to show some kind of universality of the models, as well as an irrelevance of the singular character of the solutions. We also develop the analysis for the T-dual, non trivial dilaton family of solutions, showing perfect agreement between them.

Consistency of Equations in the Second-Order Gauge- Invariant Cosmological Perturbation Theory

Along the general framework of the gauge-invariant perturbation theory developed in the papers [K. Nakamura, Prog. Theor. Phys. {\bf 110} (2003), 723; {\it ibid}, {\bf 113} (2005), 481.], we re-derive the second-order Einstein equations on four-dimensional homogeneous isotropic background universe in gauge-invariant manner without ignoring any mode of perturbations. We consider the perturbations both in the universe dominated by the single perfect fluid and in that dominated by the single scalar field. We also confirmed the consistency of all equations of the second-order Einstein equation and the equations of motion for matter fields which are derived in the paper [K. Nakamura, arXiv:0804.3840 [gr-qc]]. This confirmation implies that the all derived equations of the second order are self-consistent and these equations are correct in this sense.

Quantum adiabatic polarons by translationally invariant perturbation theory

The translationally invariant diagrammatic quantum perturbation theory (TPT) is applied to the polaron problem on the 1D lattice, modeled through the Holstein Hamiltonian with the phonon frequency omega0, the electron hopping t and the electron-phonon coupling constant g. The self-energy diagrams of the fourth-order in g are calculated exactly for an intermittently added electron, in addition to the previously known second-order term. The corresponding quadratic and quartic corrections to the polaron ground state energy become comparable at t/omega0>1 for g/omega0~(t/omega0)^{1/4} when the electron self-trapping and translation become adiabatic. The corresponding non adiabatic/adiabatic crossover occurs while the polaron width is large, i.e. the lattice coarsening negligible. This result is extended to the range (t/omega0)^{1/2}>g/omega0>(t/omega0)^{1/4}>1 by considering the scaling properties of the high-order self-energy diagrams. It is shown that the polaron ground state energy, its width and the effective mass agree with the results found traditionally from the broken symmetry side, kinematic corrections included. The Landau self trapping of the electron in the classic self-consistent, localized displacement potential, the restoration of the translational symmetry by the classic translational Goldstone mode and the quantization of the polaronic translational coordinate are thus all encompassed by a quantum theory which is translationally invariant from the outset.

Second-Order Gauge Invariant Perturbation Theory: -- Perturbative Curvatures in the Two-Parameter Case --

Based on the gauge invariant variables proposed in our previous paper [K. Nakamura, Prog. Theor. Phys. vol.110 (2003), 723.], some formulae of the perturbative curvatures of each order are derived. We follow the general framework of the second order gauge invariant perturbation theory on arbitrary background spacetime to derive these formulae. These perturbative curvatures do have the same form as the definitions of gauge invariant variables for arbitrary perturbative fields which are previously proposed. As a result, we explicitly see that any perturbative Einstein equations are given in terms of gauge invarinat form. We briefly discuss physical situations to which this framework should be applied.

Measurement of αs and the β function with the DELPHI detector at LEP

Event shape distributions in e + e − annihilation are determined from the DELPHI data taken between 183 and 207 GeV. From these the strong coupling a, is extracted with several techniques. Together with the results from other LEP2 energies and at about m Z this allows both, a combined measurement of α s and a test of the scale dependence of the strong interaction. Alternatively the renormalisation group invariant (RGI) perturbation theory is applied to measure the β function of strong interaction. The results are good agreement with the QCD expectation and allow to exclude the existence of light gluinos with a mass below 30 GeV in a model independent way.

On asymptotic perturbation theory for quantum mechanics: Almost invariant subspaces and gauge invariant magnetic perturbation theory

Singular perturbation theory for quantum mechanics is considered in a framework generalizing the spectral concentration theory. Under very general conditions, asymptotic estimations on the Rayleigh–Schrödinger expansions of the perturbed spectral projections are obtained. As a consequence almost invariant subspaces of exponential order are constructed. The results cover practically all singular perturbations considered in nonrelativistic quantum mechanics. In the magnetic field case, under the condition that the magnetic field does not increase at infinity, a gauge invariant perturbation theory leading to convergent series with field-dependent coefficients is developed.

The theory of CMB anisotropies

This is a review of the theory of CMB anisotropies, an updated version of a course given at the troisieme cycle de la Suisse Romande. An introduction to gauge invariant cosmological perturbation theory is given and the theory CMB anisotropies is develiped in this context. Simple analytical approximations for the acoustic peak positions for adiabatic and isocurvature perturbations are derived. Silk damping is discussed by an analytic approximation. A short description of the present status of observations and parameter estimation followed by a critical discussion terminate the review. The full system of differential equations for CMB anisotropies and polarization needed in a numerical treatment is also developed and given in an appendix.

Preheating and the Einstein field equations

We inaugurate a framework for studying preheating and parametric resonance after inflation without resorting to any approximations, either in gravitational perturbation theory or in the classical evolution of the field(s). We do this by numerically solving the Einstein field equations in the post-inflationary universe. In this paper we show how to compare our results to those of gauge invariant perturbation theory. We then verify the analysis of Finelli and Brandenberger (hep-ph/9809490) of super-horizon modes in ${m}^{2}{\ensuremath{\varphi}}^{2}$ inflation, showing that they are not amplified by resonant effects. Lastly, we make a preliminary survey of the nonlinear couplings between modes, which will be important in models where the primordial metric perturbations undergo parametric amplification.