Even-dimensional Spacetimes Research Articles

Asymptotic structure of higher dimensional Yang-Mills theory

Using the covariant phase space formalism, we construct the phase space for non-Abelian gauge theories in (d+2)-dimensional Minkowski spacetime for any d ≥ 2, including the edge modes that symplectically pair to the low energy degrees of freedom of the gauge field. Despite the fact that the symplectic form in odd and even-dimensional spacetimes appear ostensibly different, we demonstrate that both cases can be treated in a unified manner by utilizing the shadow transform. Upon quantization, we recover the algebra of the vacuum sector of the Hilbert space and derive a Ward identity that implies the leading soft gluon theorem in (d+2)-dimensional spacetime.

The Nielsen–Ninomiya theorem, -invariant non-Hermiticity and single 8-shaped Dirac cone

The Nielsen–Ninomiya theorem implies that any local, Hermitian and translationally invariant lattice action in even-dimensional spacetime possesses an equal number of left- and right-handed chiral fermions. We argue that if one sacrifices the property of Hermiticity while keeping the locality and translation invariance, and imposing invariance of the action under the space-time () reversal symmetry, then the excitation spectrum of the theory may contain a non-equal number of left- and right-handed massless fermions with real-valued dispersion. We illustrate our statement in a simple 1 + 1 dimensional lattice model which exhibits a skewed 8-figure pattern in its energy spectrum. A drawback of the model is that the symmetry of the Hamiltonian is spontaneously broken implying that the energy spectrum contains complex branches. We also demonstrate that the Dirac cone is robust against local disorder so that the massless excitations in this invariant model are not gapped by random space-dependent perturbations in the couplings.

A Chern–Simons gravity action in [formula omitted

Recently, Antoniadis, Konitopoulos and Savvidy have introduced in Refs. [1–4] a procedure to construct background-free gauge invariants, using non-abelian gauge potentials described by forms of higher degree. Their construction is particularly useful because it can be used in both, odd- and even-dimensional spacetimes. Using their technique, we generalize the Chern–Weil theorem and construct a gauge-invariant, (2n+2)-dimensional transgression form, and study its relationship with the generalized Chern–Simons forms introduced in Refs. [1,2].Using the methods for FDA manipulation and decomposition in 1-forms developed in Ref. [5] and applied in Refs. [6] and [7], we construct a four-dimensional Chern–Simons gravity action, which is off-shell gauge invariant under the Maxwell algebra.

Riemann zeta zeros and zero-point energy

The sequence of nontrivial zeros of the Riemann zeta function is zeta regularizable. Therefore, systems with countably infinite number of degrees of freedom described by self-adjoint operators whose spectra is given by this sequence admit a functional integral formulation. We discuss the consequences of the existence of such self-adjoint operators in field theory framework. We assume that they act on a massive scalar field coupled to a background field in a (d+1)-dimensional flat space–time where the scalar field is confined to the interval [0, a] in one of its dimensions and there are no restrictions in the other dimensions. The renormalized zero-point energy of this system is presented using techniques of dimensional and analytic regularization. In even-dimensional space–time, the series that defines the regularized vacuum energy is finite. For the odd-dimensional case, to obtain a finite vacuum energy per unit area, we are forced to introduce mass counterterms. A Riemann mass appears, which is the correction to the mass of the field generated by the nontrivial zeros of the Riemann zeta function.

Self-energy of a scalar charge near higher-dimensional black holes

We study the problem of self-energy of charges in higher-dimensional static spacetimes. Application of regularization methods of quantum field theory to calculation of the classical self-energy of charges leads to model-independent results. The correction to the self-energy of a scalar charge due to the gravitational field of black holes of the higher-dimensional Majumdar-Papapetrou spacetime is calculated exactly. It proves to be zero in even dimensions, but it acquires nonzero value in odd-dimensional spacetimes. The origin of the self-energy correction in odd dimensions is similar to the origin of the conformal anomalies in quantum field theory in even-dimensional spacetimes.

Area Spectra of Schwarzschild-Anti de Sitter Black Holes from Highly Real Quasinormal Modes

Motivated by the new physical interpretation of quasinormal modes proposed by Maggiore [Phys. Rev. Lett. 100 (2008) 141301], we investigate the quantization of large Schwarzschild-Anti de Sitter black holes in even-dimensional spacetimes, from the interesting highly real quasinormal modes found recently. Following Maggiore's treatment and Kunstatter's method, we derive the area and entropy spectra of the black holes. It is found that the results from both approaches are in full consistency. This implies that one can quantize a black hole via different asymptotic quasinormal modes besides the high damping ones that are usually adopted in the literature. Furthermore, we find that the area and entropy spectra are equidistant and independent of the cosmological constant. However, the spacings depend on the black hole dimension.

DeWitt-Schwinger renormalization and vacuum polarization inddimensions

Calculation of the vacuum polarization, $⟨{\ensuremath{\phi}}^{2}(x)⟩$, and expectation value of the stress tensor, $⟨{T}_{\ensuremath{\mu}\ensuremath{\nu}}(x)⟩$, has seen a recent resurgence, notably for black hole spacetimes. To date, most calculations of this type have been done only in four dimensions. Extending these calculations to $d$ dimensions includes $d$-dimensional renormalization. Typically, the renormalizing terms are found from Christensen's covariant point splitting method for the DeWitt-Schwinger expansion. However, some manipulation is required to put the correct terms into a form that is compatible with problems of the vacuum polarization type. Here, after a review of the current state of affairs for $⟨{\ensuremath{\phi}}^{2}(x)⟩$ and $⟨{T}_{\ensuremath{\mu}\ensuremath{\nu}}(x)⟩$ calculations and a thorough introduction to the method of calculating $⟨{\ensuremath{\phi}}^{2}(x)⟩$, a compact expression for the DeWitt-Schwinger renormalization terms suitable for use in even-dimensional spacetimes is derived. This formula should be useful for calculations of $⟨{\ensuremath{\phi}}^{2}(x)⟩$ and $⟨{T}_{\ensuremath{\mu}\ensuremath{\nu}}(x)⟩$ in even dimensions, and the renormalization terms are shown explicitly for four and six dimensions. Furthermore, use of the finite terms of the DeWitt-Schwinger expansion as an approximation to $⟨{\ensuremath{\phi}}^{2}(x)⟩$ for certain spacetimes is discussed, with application to four and five dimensions.

Late-time tails of a self-gravitating massless scalar field, revisited

We discuss the nonlinear origin of the power-law tail in the long-time evolution of a spherically symmetric self-gravitating massless scalar field in even-dimensional spacetimes. Using the third-order perturbation method, we derive explicit expressions for the tail (the decay rate and the amplitude) for solutions starting from small initial data, and we verify this prediction via numerical integration of the Einstein-scalar field equations in four and six dimensions. Our results show that the coincidence of decay rates of linear and nonlinear tails in four dimensions (which has misguided some tail hunters in the past) is in a sense accidental and does not hold in higher dimensions.

Taub–NUT black holes in third order Lovelock gravity

We consider the existence of Taub–NUT solutions in third order Lovelock gravity with cosmological constant, and obtain the general form of these solutions in eight dimensions. We find that, as in the case of Gauss–Bonnet gravity and in contrast with the Taub–NUT solutions of Einstein gravity, the metric function depends on the specific form of the base factors on which one constructs the circle fibration. Thus, one may say that the independence of the NUT solutions on the geometry of the base space is not a robust feature of all generally covariant theories of gravity and is peculiar to Einstein gravity. We find that when Einstein gravity admits non-extremal NUT solutions with no curvature singularity at r=N, then there exists a non-extremal NUT solution in third order Lovelock gravity. In 8-dimensional spacetime, this happens when the metric of the base space is chosen to be CP3. Indeed, third order Lovelock gravity does not admit non-extreme NUT solutions with any other base space. This is another property which is peculiar to Einstein gravity. We also find that the third order Lovelock gravity admits extremal NUT solution when the base space is T2×T2×T2 or S2×T2×T2. We have extended these observations to two conjectures about the existence of NUT solutions in Lovelock gravity in any even-dimensional spacetime.

Self-interaction and regularization of classical electrodynamics in higher dimensions

The classical electrodynamic system of a field and a single point-like source is considered in even-dimensional spacetime. The problem of self-interaction is discussed. It is manifestly shown that all singular terms appearing in these equations can be regularized. Relations between formulas for radiation and radiation friction are discussed.

New tests and applications of the worldline path integral in the first order formalism

We present different non-perturbative calculations within the context of Migdal's representation for the propagator and effective action of quantum particles. We first calculate the exact propagators and effective actions for Dirac, scalar and Proca fields in the presence of constant electromagnetic fields, for an even-dimensional spacetime. Then we derive the propagator for a charged scalar field in a spacelike vortex (i.e., instanton) background, in a long-distance expansion, and the exact propagator for a massless Dirac field in 1+1 dimensions in an arbitrary background. Finally, we present an interpretation of the chiral anomaly in the present context, finding a condition that the paths must fulfil in order to have a non-vanishing anomaly.

On the dimensional dependence of duality groups for massive p-forms

We study the soldering formalism in the context of Abelian p-form theories. We develop further the fusion process of massless antisymmetric tensors of different ranks into a massive p-form and establish its duality properties. To illustrate the formalism we consider two situations. First the soldering mass generation mechanism is compared with the Higgs and Julia–Toulouse mechanisms for mass generation due to condensation of electric and magnetic topological defects. We show that the soldering mechanism interpolates between them for even-dimensional spacetimes, in this way confirming the Higgs/Julia–Toulouse duality proposed by Quevedo and Trugenberger [Nucl. Phys. B 501 (1997) 143] a few years ago. Next, soldering is applied to the study of duality group classification of the massive forms. We show a dichotomy controlled by the parity of the operator defining the symplectic structure of the theory and find their explicit actions.

Vilkovisky-DeWitt effective action for Einstein gravity on Kaluza-Klein spacetimesM4×SN

We evaluate the divergent part of the Vilkovisky-DeWitt effective action for Einstein gravity on even-dimensional Kaluza-Klein spacetimes of the form $M^{4}\times S^{N}$. Explicit results are given for $N$=2, 4, and 6. Trace anomalies for gravitons are also given for these cases. Stable Kaluza-Klein configurations are sought, unsuccessfully, assuming the divergent part of the effective action dominates the dynamics.

A simple algebraic derivation of the covariant anomaly and Schwinger term

An expression for the curvature of the “covariant” determinant line bundle is given in even-dimensional space–time. The usefulness is guaranteed by its prediction of the covariant anomaly and Schwinger term. It allows a parallel derivation of the consistent anomaly and Schwinger term, and their covariant counterparts, which clarifies the similarities and differences between them.

On chiral symmetry breaking in a constant magnetic field in higher dimension

Chiral symmetry breaking in the Nambu–Jona-Lasinio model in a constant magnetic field is studied in spacetimes of dimension D>4. It is shown that a constant magnetic field can be characterized by [ D−1 2 ] parameters. For the maximal number of nonzero field parameters, we show that there is an effective reduction of the spacetime dimension for fermions in the infrared region D→1+1 for even-dimensional spacetimes and D→0+1 for odd-dimensional spacetimes. Explicit solutions of the gap equation confirm our conclusions.

Exact solutions of classical electrodynamics and the Yang-Mills-Wong theory in even-dimensional space-time

Exact solutions of classical gauge theories in even-dimensional (D=2n) space-time are discussed. Common and specific properties of these solutions are analyzed for the particular dimensions D=2, D=4, and D=6. The consistent formulation of classical gauge field theories with pointlike charged or colored particles is proposed for D=6. The particle Lagrangian must then depend on the acceleration. In D=2, radiation is absent and all processes are invertible w.r.t. time. In D=6, the expression for the radiation intensity, as well as the equation of motion of a self-interacting particle, is obtained; trembling always leads to radiation. Non-Abelian solutions are absent for any D≠4, and only Coulomb-like solutions, which correspond to the Abelian limit of the D-dimensional Yang-Mills-Wong theory, are admitted.

Plane symmetric metrics associated with semi-plane symmetric electromagnetic fields in higher dimensions

Electromagnetic fields yielding plane symmetric metrics in higher-dimensional spacetimes are exhausted and classified. It is shown that these EM fields must fall into one of the following two cases: (i)F it =F iz =0,i=1,...,n; (ii)Ftz=0. We give the general solution to the Einstein-Maxwell equations in higher dimensions corresponding to electromagnetic fields of case (ii) withF it =F iz , which covers all even-dimensional spacetimes as well as a subcase of odd-dimensional spacetimes.

Induced Chern-Simons Terms and Parity Anomalies

Stochastic quantization is used to investigate the parity violating anomaly of (2n + 1)-dimensional fermions interacting with classical Yang-Mills and gravitational fields. The anomaly manifests itself by a dynamically generated Chern-Simons term in the effective action, and by a parity-breaking contribution to the induced vacuum current. This anomaly is the odd-dimensional analogue of the chiral anomaly in even-dimensional spacetimes. In the stochastic approach, both types of anomalies can be derived from certain Ito terms, whose presence reflects the non-differentiable nature of the quantum paths.

Fibre bundle generated theory of higher dimensional gravity

With the help of Weil polynomials on the Lie algebra of the Poincare group in D dimensions, we construct a Lagrangian form for higher dimensional gravity on a principal fibre bundle whose base space is an even-dimensional Riemann-Cartan spacetime and whose structure group is the Poincare group.

Extended Skyrme models in even dimensions and higher-dimensional solitons

We propose a higher-dimensional soliton model — an extended Skyrme model — in an even-dimensional spacetime. A hedgehog ansatz is also generalized to higher-dimensional cases and it is found that the soliton generally has the fermion number, i.e., the topological soliton.