Covariant Expansion Research Articles

Covariant Jacobi-Legendre expansion for total energy calculations within the projector augmented wave formalism

Covariant Jacobi-Legendre expansion for total energy calculations within the projector augmented wave formalism

Pole skipping in holographic theories with gauge and fermionic fields

Using covariant expansions, recent work showed that pole skipping happens in general holographic theories with bosonic fields at frequencies i(lb − s)2πT, where lb is the highest integer spin in the theory and s takes all positive integer values. We revisit this formalism in theories with gauge symmetry and upgrade the pole-skipping condition so that it works without having to remove the gauge redundancy. We also extend the formalism by incorporating fermions with general spins and interactions and show that their presence generally leads to a separate tower of pole-skipping points at frequencies i(lf − s)2πT, lf being the highest half-integer spin in the theory and s again taking all positive integer values. We also demonstrate the practical value of this formalism using a selection of examples with spins 0,12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}, 1,32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{3}{2} $$\\end{document}, 2.

Non-Abelian chiral kinetic equations in the Cartan-Weyl basis

<sec>Non-Abelian gauge field is the fundamental element of the standard model. Non-Abelian chiral kinetic theory can be used to describe how the chiral fermions in standard model transport in a non-equilibrium system. </sec><sec>In our previous work, we decomposed the non-Abelian chiral kinetic equations into color singlet and multiplet in the <inline-formula><tex-math id="M1">\begin{document}$SU(N)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M1.png"/></alternatives></inline-formula> color space. In this formalism, the chiral kinetic equations preserve the gauge symmetry in a very apparent way. However, sometimes we need to describe the microscopic process of the specific color degree, e.g. the color connection in the hadronization stage. In order to describe such a process, it will be more convenient to decompose the non-Abelian chiral kinetic equations in the Cartan-Weyl basis. </sec><sec>In this work, we choose the matrix elements of the Wigner function in fundamental representation of color space as the direct variables and decompose the gauge field or strength tensor field in the Cartan-Weyl basis. By using the covariant gradient expansion, we decompose the non-Abelian chiral kinetic equations into the coupled kinetic equations for diagonal distribution function and non-diagonal distribution function up to the first order. When only diagonal elements exist in the gauge field with non-diagonal elements and diagonal elements decoupled, the non-Ableian chiral kinetic equation will be reduced to the form in the Abelian case. When the non-diagonal elements of the gauge field are present, the kinetic equations are totally tangled between diagonal distribution function and non-diagonal distribution function. Especially, the <inline-formula><tex-math id="M2">\begin{document}$0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M2.png"/></alternatives></inline-formula>th-order non-diagonal distribution function could induce the <inline-formula><tex-math id="M3">\begin{document}$1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M3.png"/></alternatives></inline-formula>st-order diagonal Wigner function, and the <inline-formula><tex-math id="M4">\begin{document}$0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M4.png"/></alternatives></inline-formula>th-order diagonal distribution function could also induce the <inline-formula><tex-math id="M5">\begin{document}$1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20222471_M5.png"/></alternatives></inline-formula>st-order non-diagonal Wigner function. </sec>

Model two-particle kinetic equation for pairs of quasiparticles

Because the main process of interaction in the Boltzmann model of real gases is the binary collision of molecules, it is convenient to use the two-particle kinetic equation to describe the dynamics of a rarefied gas. This equation was written using the same physical assumptions as those used by Ludwig Boltzmann. The right-hand side of this equation contains the product of the linear scattering operator and chaos projector. The Boltzmann equation follows from this equation without any additional approximations after simple integration of the velocities and positions of the second particle. Using the divergence form of the scattering operator, this equation can be represented as the Liouville equation, which implies that real molecules can be replaced by quasiparticles whose distribution function is the same as that of real molecules but whose dynamics are completely different. Pairs of quasiparticles do not collide but move along continuous trajectories in the phase space. The relative velocities in pairs of quasiparticles slowly rotate with an angular velocity vector depending on the distribution function. We provide an explicit approximate expression for the angular velocity through the first few velocity moments, using a special covariant Grad expansion for the velocity distribution function, which reduces to the exact Bobylev–Kruk–Wu solution in the isotropic case. We simulated the relaxation of distribution function to equilibrium and compared results with the existing exact solutions. The described algorithm will be effective for modeling flow regions with low Knudsen numbers, where the standard Direct Monte Carlo Simulation (DSMC) method encounters significant difficulties.

Heat kernel expansion for higher order minimal and nonminimal operators

We build a systematic calculational method for the covariant expansion of the two-point heat kernel $\hat K(\tau|x,x')$ for generic minimal and non-minimal differential operators of any order. This is the expansion in powers of dimensional background field objects -- the coefficients of the operator and the corresponding spacetime and vector bundle curvatures, suitable in renormalization and effective field theory applications. For minimal operators whose principal symbol is given by an arbitrary power of the covariant Laplacian $(-\Box)^M$, $M>1$, this result generalizes the well-known Schwinger--DeWitt (or Seeley--Gilkey) expansion to the infinite series of positive and negative fractional powers of the proper time $\tau^{1/M}$, weighted by the generalized exponential functions of the dimensionless argument $-\sigma(x,x')/2\tau^{1/M}$ depending on the Synge world function $\sigma(x,x')$. The coefficients of this series are determined by the chain of auxiliary differential operators acting on the two-point parallel transport tensor, which in their turn follow from the solution of special recursive equations. The derivation of these operators and their recursive equations are based on the covariant Fourier transform in curved spacetime. The series of negative fractional powers of $\tau$ vanishes in the coincidence limit $x'=x$, which makes the proposed method consistent with the heat kernel theory of Seeley--Gilkey and generalizes it beyond the heat kernel diagonal in the form of the asymptotic expansion in the domain $\sigma(x,x')\ll\tau^{1/M}$, $\tau\to 0$.$\nabla^a\sigma(x,x')\ll\tau^{1/2M}$, $\tau\to 0$.

Lagrangians for nonrelativistic gravity

We study the covariant expansion of Einstein-Hilbert action in powers of $1/c^2$ with arbitrary spacetime foliation where $c$ is the speed of light. This is done firstly by suitable parametrization of geometry which is called 'pre-non-relativistic' parametrization. This allows us to rewrite the general relativity in a form suitable for the analytical $1/c^2$ expansion. Consequently, we can study the expansion of Einstein-Hilbert action up to the next-to-next-to-leading order.

Covariant chiral kinetic equation in non-Abelian gauge field from \u201ccovariant gradient expansion\u201d

We derive the chiral kinetic equation in 8 dimensional phase space in non- Abelian SU(N) gauge field within the Wigner function formalism. By using the “covariant gradient expansion”, we disentangle the Wigner equations in four-vector space up to the first order and find that only the time-like component of the chiral Wigner function is independent while other components can be explicit derivative. After further decomposing the Wigner function or equations in color space, we present the non-Abelian covariant chiral kinetic equation for the color singlet and multiplet phase-space distribution functions. These phase-space distribution functions have non-trivial Lorentz transformation rules when we define them in different reference frames. The chiral anomaly from non-Abelian gauge field arises naturally from the Berry monopole in Euclidian momentum space in the vacuum or Dirac sea contribution. The anomalous currents as non-Abelian counterparts of chiral magnetic effect and chiral vortical effect have also been derived from the non-Abelian chiral kinetic equation.

Covariant series in the normal neighborhood of a submanifold

We consider the covariant series in a some normal neighborhood of a submanifold. Such a neighborhood is a generalization of the normal neighborhood of a point. We discuss how the coefficients of the covariant Taylor series of an arbitrary tensor field can be expressed in terms covariant derivatives of the torsion, Riemann curvature and the field under consideration. We also discuss the algorithm of calculating coefficients of a pseudo-Riemannian metric with respect to the corresponding metric connection without torsion. As an example, we calculate the covariant expansion of the Schwarzschild metric in the normal tubular neighborhood of a circular orbit up to fifth order using the Fermi coordinate system.

Gradient effects on false vacuum decay in gauge theory

We study false vacuum decay for a gauged complex scalar field in a polynomial potential with nearly degenerate minima. Radiative corrections to the profile of the nucleated bubble as well as the full decay rate are computed in the planar thin-wall approximation using the effective action. This allows to account for the inhomogeneity of the bounce background and the radiative corrections in a self-consistent manner. In contrast to scalar or fermion loops, for gauge fields one must deal with a coupled system that mixes the Goldstone boson and the gauge fields, which considerably complicates the numerical calculation of Green's functions. In addition to the renormalization of couplings, we employ a covariant gradient expansion in order to systematically construct the counterterm for the wave-function renormalization. The result for the full decay rate however does not rely on such an expansion and accounts for all gradient corrections at the chosen truncation of the loop expansion. The ensuing gradient effects are shown to be of the same order of magnitude as non-derivative one-loop corrections.

Oddity in nonrelativistic, strong gravity

We consider the presence of odd powers of the speed of light c in the covariant nonrelativistic expansion of General Relativity (GR). The term of order c in the relativistic metric is a vector potential that contributes at leading order in this expansion and describes strong gravitational effects outside the (post-)Newtonian regime. The nonrelativistic theory of the leading order potentials contains the full non-linear dynamics of the stationary sector of GR.

Covariant multipole expansion of local currents for massive states of any spin

We study the structure of scalar, vector, and tensor currents for on-shell massive particles of any spin. When considering higher values for the spin of the particle, the number of form factors (FFs) involved in the decomposition of the matrix elements associated with these local currents increases. We identify all the fundamental structures that give rise to the independent FFs, systematically for any spin value. These structures can be conveniently organised using an expansion in covariant multipoles, built solely from the Lorentz generators. This approach allows one to uniquely identify the terms which are universal and those that arise because of spin. We derive counting rules which relate the number of FFs to the total spin $j$ of the state, showing explicitly that these rules match all the well-known cases up to spin 2.

Vacuum Effective Actions and Mass-Dependent Renormalization in Curved Space

We review past and present results on the non-local form-factors of the effective action of semiclassical gravity in two and four dimensions computed by means of a covariant expansion of the heat kernel up to the second order in the curvatures. We discuss the importance of these form-factors in the construction of mass-dependent beta functions for the Newton’s constant and the other gravitational couplings.

Towards apparent convergence in asymptotically safe quantum gravity

The asymptotic safety scenario in gravity is accessed within the systematic vertex expansion scheme for functional renormalisation group flows put forward in Christiansen et al. (Phys Lett B 728:114, 2014), Christiansen et al. (Phy Rev D 93:044036, 2016), and implemented in Christiansen et al. (Phys Rev D 92:121501, 2015) for propagators and three-point functions. In the present work this expansion scheme is extended to the dynamical graviton four-point function. For the first time, this provides us with a closed flow equation for the graviton propagator: all vertices and propagators involved are computed from their own flows. In terms of a covariant operator expansion the current approximation gives access to Lambda , R, R^2 as well as R_{mu nu }^2 and higher derivative operators. We find a UV fixed point with three attractive and two repulsive directions, thus confirming previous studies on the relevance of the first three operators. In the infrared we find trajectories that correspond to classical general relativity and further show non-classical behaviour in some fluctuation couplings. We also find signatures for the apparent convergence of the systematic vertex expansion. This opens a promising path towards establishing asymptotically safe gravity in terms of apparent convergence.

Computational Algorithm for Covariant Series Expansions in General Relativity

We present a new algorithm for computing covariant power expansions of tensor fields in generalized Riemannian normal coordinates, introduced in some neighborhood of a parallelized k-dimensional submanifold (k = 0, 1, . . .< n; the case k = 0 corresponds to a point), by transforming the expansions to the corresponding Taylor series. For an arbitrary real analytic tensor field, the coefficients of such series are expressed in terms of its covariant derivatives and covariant derivatives of the curvature and the torsion. The algorithm computes the corresponding Taylor polynomials of arbitrary orders for the field components and is applicable to connections that are, in general, nonmetric and not torsion-free. We show that this computational problem belongs to the complexity class LEXP.

Hyperscaling violating Lifshitz holography

We present an overview of the construction of the general holographic dictionary for asymptotically locally Lifshitz and hyperscaling violating Lifshitz backgrounds with arbitrary dynamical exponents z and θ, compatible with the null energy condition, which was recently developed in [W. Chemissany, I. Papadimitriou, Generalized dilatation operator method for non-relativistic holography, arXiv:1405.3965, W. Chemissany, I. Papadimitriou, Lifshitz holography: The whole shebang, arXiv:1408.0795]. A concrete definition of asymptotically locally Lifshitz and hyperscaling violating Lifshitz backgrounds is provided in the context of a generic bottom-up Einstein-Proca-Dilaton theory, and a systematic procedure for solving the radial Hamilton-Jacobi equation via a covariant expansion in eigenfunctions of two commuting operators is presented. The resulting asymptotic solution of the Hamilton-Jacobi equation is subsequently used to derive the full holographic dictionary, including the Fefferman-Graham asymptotic expansions and the non-relativistic holographic Ward identities.

Covariant Perturbation Expansion of Off-Diagonal Heat Kernel

Covariant perturbation expansion is an important method in quantum field theory. In this paper an expansion up to arbitrary order for off-diagonal heat kernels in flat space based on the covariant perturbation expansion is given. In literature, only diagonal heat kernels are calculated based on the covariant perturbation expansion.

Background field method in the gradient flow

In perturbative consideration of the Yang--Mills gradient flow, it is useful to introduce a gauge non-covariant term ("gauge-fixing term") to the flow equation that gives rise to a Gaussian damping factor also for gauge degrees of freedom. In the present paper, we consider a modified form of the gauge-fixing term that manifestly preserves covariance under the background gauge transformation. It is shown that our gauge-fixing term does not affect gauge-invariant quantities as the conventional gauge-fixing term. The formulation thus allows a background gauge covariant perturbative expansion of the flow equation that provides, in particular, a very efficient computational method of expansion coefficients in the small flow time expansion. The formulation can be generalized to systems containing fermions.

Editorial Note: Covariant expansion of the gravitational Stückelberg trick [Phys. Rev. D91, 094001 (2015)

Editorial Note: Covariant expansion of the gravitational Stückelberg trick [Phys. Rev. D<b>91</b>, 094001 (2015)

Covariant expansion of the gravitational Stückelberg trick

A new approach to expanding the "St\"{u}ckelbergized" fiducial metric in a covariant manner is developed. The idea is to consider the curved 4-dimensional space as a codimension-one hypersurface embedded in a 5-dimensional Minkowski bulk, in which the 5-dimensional Goldstone modes can be defined as usual. After solving one constraint among the five 5-dimensional Goldstone modes and projecting onto the 4-dimensional hypersurface, we are able to express the "St\"{u}ckelbergized" fiducial metric in terms of the 4-dimensional Goldstone modes as well as 4-dimensional curvature quantities. We also compared the results with expressions got using the Riemann Normal Coordinates (RNC) in Gao et al [Phys. Rev. D90, 124073 (2014)] and find that, after a simple field redefinition, results got in two approaches exactly coincide.

Geodesic deviation at higher orders via covariant bitensors

We review a simple but instructive application of the formalism of covariant bitensors, to use a deviation vector field along a fiducial geodesic to describe a neighboring worldline, in an exact and manifestly covariant manner, via the exponential map. Requiring the neighboring worldline to be a geodesic leads to the usual linear geodesic deviation equation for the deviation vector, plus corrections at higher order in the deviation and relative velocity. We show how these corrections can be efficiently computed to arbitrary orders via covariant bitensor expansions, deriving a form of the geodesic deviation equation valid to all orders, and producing its explicit expanded form through fourth order. We also discuss the generalized Jacobi equation, action principles for the higher-order geodesic deviation equations, results useful for describing accelerated neighboring worldlines, and the formal general solution to the geodesic deviation equation through second order.