arXiv subjects

Gabor Zsolt Toth

Publications and source records attributed to Gabor Zsolt Toth.

At least 19 recordsLinked to original sources

arXiv2024

Numerical investigation of the late-time tails of the solutions of the Fackerell-Ipser equation

The late-time behaviour of the solutions of the Fackerell-Ipser equation (which is a wave equation for the spin-zero component of the electromagnetic field strength tensor) on the closure of the domain of outer communication of sub-extremal Kerr spacetime is studied numerically. Within the Kerr family, the case of Schwarzschild background is also considered. Horizon-penetrating compactified hyperboloidal coordinates are used, which allow the behaviour of the solutions to be observed at the event horizon and at future null infinity as well. For the initial data, pure multipole configurations that have compact support and are either stationary or non-stationary are taken. It is found that with such initial data the solutions of the Fackerell-Ipser equation converge at late times either to a known static solution (up to a constant factor) or to zero. As the limit is approached, the solutions exhibit a quasinormal ringdown and finally a power-law decay. The exponents characterizing the power-law decay of the spherical harmonic components of the field variable are extracted from the numerical data for various values of the parameters of the initial data, and based on the results a proposal for a Price's law relevant to the Fackerell-Ipser equation is made. Certain conserved energy and angular momentum currents are used to verify the numerical implementation of the underlying mathematical model. In the construction of these currents a discrete symmetry of the Fackerell-Ipser equation, which is the product of an equatorial reflection and a complex conjugation, is also taken into account.

gr-qc↗

arXiv2022

Energy-momentum tensor and duality symmetry of linearized gravity in the Fierz formalism

A formulation of linearized gravity in flat background, based on the Fierz tensor as a counterpart of the electromagnetic field strength, is discussed in detail and used to study fundamental properties of the linearized gravitational field. In particular, the linearized Einstein equations are written as first order partial differential equations in terms of the Fierz tensor, in analogy with the first order Maxwell equations. An energy-momentum tensor ($T_{\mathrm{lg}}^{ab}$) with favourable properties and exhibiting remarkable similarity to the standard energy-momentum tensor of the electromagnetic field is found for the linearized gravitational field. $T_{\mathrm{lg}}^{ab}$ is quadratic in the Fierz tensor (which is constructed from the first derivatives of the linearized metric), traceless, and satisfies the dominant energy condition in a gauge that contains the transverse traceless gauge. It is further shown that in suitable gauges, including the transverse traceless gauge, linearized gravity in the absence of matter has a duality symmetry that maps the Fierz tensor, which is antisymmetric in its first two indices, into its dual. Conserved currents associated with the gauge and duality symmetries of linearized gravity are also determined. These currents show good analogy with the corresponding currents in electrodynamics.

gr-qc↗

arXiv2020

On the relations between two construction methods for conserved currents of differential equations

The relations between two construction methods (called multiplier and embedding methods) for conserved currents of general systems of ordinary or partial differential equations (DEs) are investigated. Recent studies indicate that the multiplier method, which is a generalization of Noether's theorem, has significant advantages in comparison with the embedding method, which uses adjoint-symmetry/symmetry pairs and is based on embedding the original system of DEs in a larger one that follows from a Lagrangian. In particular, the multiplier method can generally give a wider range of conserved currents than the embedding method. In this paper simple extended forms of general systems of DEs, obtained by treating parameters present in the equations or introduced into them as dependent variables, are studied. A variant of a fundamental result on the connection between the embedding method and the action of symmetries on conserved currents that correspond to a multiplier is derived for the extended systems of DEs. Using this connection and by considering particular extensions that endow the extended DEs with scaling symmetry, it is shown that the embedding method becomes significantly stronger if it is also allowed to be applied to the extended forms of the original DEs. It is also shown that up to equivalence the multipliers of an extended DE system contain the parametric multipliers of the original system together with the derivatives of the corresponding conserved currents with respect to the parameters.

math-ph↗

arXiv2018

Noether currents for the Teukolsky Master Equation

Conserved currents associated with the time translation and axial symmetries of the Kerr spacetime and with scaling symmetry are constructed for the Teukolsky Master Equation (TME). Three partly different approaches are taken, of which the third one applies only to the spacetime symmetries. The results yielded by the three approaches, which correspond to three variants of Noether's theorem, are essentially the same, nevertheless. The construction includes the embedding of the TME into a larger system of equations, which admits a Lagrangian and turns out to consist of two TMEs with opposite spin weight. The currents thus involve two independent solutions of the TME with opposite spin weights. The first approach provides an example of the application of an extension of Noether's theorem to nonvariational differential equations. This extension is also reviewed in general form. The variant of Noether's theorem applied in the third approach is a generalization of the standard construction of conserved currents associated with spacetime symmetries in general relativity, in which the currents are obtained by the contraction of the symmetric energy-momentum tensor with the relevant Killing vector fields. Symmetries and conserved currents related to boundary conditions are introduced as well, and Noether's theorem and its variant for nonvariational differential equations are extended to them. The extension of the latter variant is used to construct conserved currents related to the Sommerfeld boundary condition.

gr-qc↗

arXiv2017

Noether's theorems and conserved currents in gauge theories in the presence of fixed fields

We extend the standard construction of conserved currents for matter fields in general relativity to general gauge theories. In the original construction the conserved current associated with a spacetime symmetry generated by a Killing field $h^μ$ is given by $\sqrt{-g}\,T^{μν}h_ν$, where $T^{μν}$ is the energy-momentum tensor of the matter. We show that if in a Lagrangian field theory that has gauge symmetry in the general Noetherian sense some of the elementary fields are fixed and are invariant under a particular infinitesimal gauge transformation, then there is a current $\mathcal{B}^μ$ that is analogous to $\sqrt{-g}\,T^{μν}h_ν$ and is conserved if the non-fixed fields satisfy their Euler-Lagrange equations. The conservation of $\mathcal{B}^μ$ can be seen as a consequence of an identity that is a generalization of $\nabla_μT^{μν}=0$ and is a consequence of the gauge symmetry of the Lagrangian. This identity holds in any configuration of the fixed fields if the non-fixed fields satisfy their Euler-Lagrange equations. We also show that $\mathcal{B}^μ$ differs from the relevant canonical Noether current by the sum of an identically conserved current and a term that vanishes if the non-fixed fields are on-shell. As example we discuss the case of general, possibly fermionic, matter fields propagating in fixed gravitational and Yang-Mills background. We find that in this case the generalization of $\nabla_μT^{μν}=0$ is the Lorentz law $\nabla_μT^{μν} - F^{aνλ}\mathcal{J}_{aλ} = 0$, which holds as a consequence of the diffeomorphism, local Lorentz and Yang-Mills gauge symmetry of the matter Lagrangian. As a second simple example we consider the case of general fields propagating in a background that consists of a gravitational and a real scalar field.

gr-qc↗

arXiv2016

On the mass-coupling relation of multi-scale quantum integrable models

We determine exactly the mass-coupling relation for the simplest multi-scale quantum integrable model, the homogenous sine-Gordon model with two independent mass-scales. We first reformulate its perturbed coset CFT description in terms of the perturbation of a projected product of minimal models. This representation enables us to identify conserved tensor currents on the UV side. These UV operators are then mapped via form factor perturbation theory to operators on the IR side, which are characterized by their form factors. The relation between the UV and IR operators is given in terms of the sought-for mass-coupling relation. By generalizing the $Θ$ sum rule Ward identity we are able to derive differential equations for the mass-coupling relation, which we solve in terms of hypergeometric functions. We check these results against the data obtained by numerically solving the thermodynamic Bethe Ansatz equations, and find a complete agreement.

hep-th↗

arXiv2016

Weak cosmic censorship, dyonic Kerr-Newman black holes and Dirac fields

It was investigated recently, with the aim of testing the weak cosmic censorship conjecture, whether an extremal Kerr black hole can be converted into a naked singularity by interaction with a massless classical Dirac test field, and it was found that this is possible. We generalize this result to electrically and magnetically charged rotating extremal black holes (i.e. extremal dyonic Kerr-Newman black holes) and massive Dirac test fields, allowing magnetically or electrically uncharged or nonrotating black holes and the massless Dirac field as special cases. We show that the possibility of the conversion is a direct consequence of the fact that the Einstein-Hilbert energy-momentum tensor of the classical Dirac field does not satisfy the null energy condition, and is therefore not in contradiction with the weak cosmic censorship conjecture. We give a derivation of the absence of superradiance of the Dirac field without making use of the complete separability of the Dirac equation in dyonic Kerr-Newman background, and we determine the range of superradiant frequencies of the scalar field. The range of frequencies of the Dirac field that can be used to convert a black hole into a naked singularity partially coincides with the superradiant range of the scalar field. We apply horizon-penetrating coordinates, as our arguments involve calculating quantities at the event horizon. We describe the separation of variables for the Dirac equation in these coordinates, although we mostly avoid using it.

gr-qc↗

arXiv2015

Higher spin fields with reversed spin-statistics relation

A construction of massive free fields with arbitrary spin and reversed spin-statistics relation is presented. The main idea of the construction is to consider fields that transform according to representations of the Lorentz group that are doubled in comparison with the representations according to which normal (physical) fields transform. This allows the definition of opposite commutation properties for these fields, while the spin of the particles they describe remains unchanged. The correspondence established by the construction between fields obeying normal and reversed spin-statistics relation makes it possible to express e.g. the polarization states, (anti)commutators, or Feynman propagators of the latter fields in terms of those of the normal fields to which they correspond. The cases of the scalar and Dirac fields are discussed in additional detail.

hep-th↗

arXiv2015

Projection operator approach to the quantization of higher spin fields

A general method to construct free quantum fields for massive particles of arbitrary definite spin in a canonical Hamiltonian framework is presented. The main idea of the method is as follows: a multicomponent Klein-Gordon field that satisfies canonical (anti)commutation relations and serves as an auxiliary higher spin field is introduced, and the physical higher spin field is obtained by acting on the auxiliary field by a suitable differential operator. This allows the calculation of the (anti)commutation relations, the Green functions and the Feynman propagators of the higher spin fields. In addition, canonical equations of motions, which are expressed in terms of the auxiliary variables, can be obtained also in the presence of interactions, if the interaction Hamiltonian operator is known. The fields considered transform according to the (n/2,m/2) + (m/2,n/2) and (n/2,m/2) representations of the Lorentz group.

hep-th↗

arXiv2013

Reformulating the TBA equations for the quark anti-quark potential and their two loop expansion

The boundary thermodynamic Bethe Ansatz (BTBA) equations introduced in arXiv:1203.1913 and arXiv:1203.1617 to describe the cusp anomalous dimension contain imaginary chemical potentials and singular boundary fugacities, which make its systematic expansion problematic. We propose an alternative formulation based on real chemical potentials and additional source terms. We expand our equations to double wrapping order and find complete agreement with the direct two-loop gauge theory computation of the cusp anomalous dimension.

hep-th↗

arXiv2012

Test of the weak cosmic censorship conjecture with a charged scalar field and dyonic Kerr-Newman black holes

A thought experiment considered recently in the literature, in which it is investigated whether a dyonic Kerr-Newman black hole can be destroyed by overcharging or overspinning it past extremality by a massive complex scalar test field, is revisited. Another derivation of the result that this is not possible, i.e. the weak cosmic censorship is not violated in this thought experiment, is given. The derivation is based on conservation laws, on a null energy condition, and on specific properties of the metric and the electromagnetic field of dyonic Kerr-Newman black holes. The metric is kept fixed, whereas the dynamics of the electromagnetic field is taken into account. A detailed knowledge of the solutions of the equations of motion is not needed. The approximation in which the electromagnetic field is fixed is also considered, and a derivation for this case is also given. In addition, an older version of the thought experiment, in which a pointlike test particle is used, is revisited. The same result, namely the non-violation of the cosmic censorship, is rederived in a way which is simpler than in earlier works.

gr-qc↗

arXiv2012

On some aspects of the definition of scattering states in quantum field theory

The problem of extending quantum-mechanical formal scattering theory to a more general class of models that also includes quantum field theories is discussed, with the aim of clarifying certain aspects of the definition of scattering states. As the strong limit is not suitable for the definition of scattering states in quantum field theory, some other limiting procedure is needed. Two possibilities are considered, the abelian limit and adiabatic switching. Formulas for the scattering states based on both methods are discussed, and it is found that generally there are significant differences between the two approaches. As an illustration of the application and the features of these formulas, S-matrix elements and energy corrections in two quantum field theoretical models are calculated using (generalized) old-fashioned perturbation theory. The two methods are found to give equivalent results.

hep-th↗

arXiv2011

Numerical investigation of the late-time Kerr tails

The late-time behavior of a scalar field on fixed Kerr background is examined in a numerical framework incorporating the techniques of conformal compactification and hyperbolic initial value formulation. The applied code is 1+(1+2) as it is based on the use of the spectral method in the angular directions while in the time-radial section fourth order finite differencing, along with the method of lines, is applied. The evolution of various types of stationary and non-stationary pure multipole initial states are investigated. The asymptotic decay rates are determined not only in the domain of outer communication but along the event horizon and at future null infinity as well. The decay rates are found to be different for stationary and non-stationary initial data, and they also depend on the fall off properties of the initial data toward future null infinity. The energy and angular momentum transfers are found to show significantly different behavior in the initial phase of the time evolution. The quasinormal ringing phase and the tail phase are also investigated. In the tail phase, the decay exponents for the energy and angular momentum losses at future null infinity are found to be smaller than at the horizon which is in accordance with the behavior of the field itself and it means that at late times the energy and angular momentum falling into the black hole become negligible in comparison with the energy and angular momentum radiated toward future null infinity. The energy and angular momentum balances are used as additional verifications of the reliability of our numerical method.

gr-qc↗

arXiv2010

The Rayleigh-Schrödinger perturbation series of quasi-degenerate systems

We present the first representation of the general term of the Rayleigh-Schrödinger series for quasidegenerate systems. Each term of the series is represented by a tree and there is a straightforward relation between the tree and the analytical expression of the corresponding term. The combinatorial and graphical techniques used in the proof of the series expansion allow us to derive various resummation formulas of the series. The relation with several combinatorial objects used for special cases (degenerate or non-degenerate systems) is established.

quant-ph↗

arXiv2010

On Lagrangian and Hamiltonian systems with homogeneous trajectories

Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmetry group G is a solution of the Euler-Lagrange or Hamiltonian equations. In particular, we generalize the `geodesic lemma' known in Riemannian geometry to Lagrangian and Hamiltonian systems. We present results on the existence of homogeneous trajectories of Lagrangian systems. We study Hamiltonian and Lagrangian g.o. spaces, i.e. homogeneous spaces G/H with G-invariant Lagrangian or Hamiltonian functions on which every solution of the equations of motion is homogeneous. We show that the Hamiltonian g.o. spaces are related to the functions that are invariant under the coadjoint action of G. Riemannian g.o. spaces thus correspond to special Ad*(G)-invariant functions. An Ad*(G)-invariant function that is related to a g.o. space also serves as a potential for the mapping called `geodesic graph'. As illustration we discuss the Riemannian g.o. metrics on SU(3)/SU(2).

math-ph↗

arXiv2009

Particle spectrum of the 3-state Potts field theory: a numerical study

The three-state Potts field theory in two dimensions with thermal and magnetic perturbations provides the simplest model of confinement allowing for both mesons and baryons, as well as for an extended phase with deconfined quarks. We study numerically the evolution of the mass spectrum of this model over its whole parameter range, obtaining a pattern of confinement, particle decay and phase transitions which confirms recent predictions.

hep-th↗

arXiv2009

Proposal to improve the behaviour of self-energy contributions to the S-matrix

A simple modification of the definition of the S-matrix is proposed. It is expected that the divergences related to nonzero self-energies are considerably milder with the modified definition than with the usual one. This conjecture is verified in a few examples using perturbation theory. The proposed formula is written in terms of the total Hamiltonian operator and a free Hamiltonian operator and is therefore applicable in any case when these Hamiltonian operators are known.

hep-th↗

arXiv2007

Investigations in Two-Dimensional Quantum Field Theory by the Bootstrap and TCSA Methods

This thesis contains three main parts, which are largely independent. In the first part we deal with the boundary bootstrap in supersymmetric factorized scattering theory. We give a description of supersymmetry in the case when the space is a half-line and present rules for the determination of the representations in which higher level boundary bound states transform, and for the determination of the supersymmetric one-particle reflection matrix factors for the higher level boundary bound states. These rules apply under the condition that the bulk particles transform in the kink or in the boson-fermion representation. Examples for the application of these rules to specific models are also given. In the second part we investigate the problem whether the TCSA spectrum can be approximated by the spectrum of the original Hamiltonian operator in which the coefficients of the terms are suitably changed. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. Another truncation method that preserves the solvability of the model is also considered. The results of perturbative and numerical calculations show that the above approximation is possible and that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method. In the third part we investigate the phase structure of the two- and three-frequency sine-Gordon models using the TCSA. In the case of the three-frequency model the tricritical point, several points of the critical line and a few points of the line of first order transition are found.

hep-th↗