Massive Klein-Gordon Field Research Articles

Stable Orbit of Massive Klein Gordon Field near Schwarzschild Black Hole and Its Quasi-Normal Modes

The findings of bound states of the massive Klein-Gordon field near the Schwarzschild black hole have provided certain conditions for a stable orbit. Unlike classical methods, I have obtained a specific relation between the mass of the scalar particle and the distance from the black hole. Furthermore, by applying a polynomial condition to the Heun function, more accurate predictions of quasi-normal modes have been made. Based on this, the time of a stable orbit and its relationship with mass, polynomial order, and distance from the Schwarzschild black hole have been calculated.

Localizable Particles in the Classical Limit of Quantum Field Theory

A number of arguments purport to show that quantum field theory cannot be given an interpretation in terms of localizable particles. We show, in light of such arguments, that the classical $\hbar\to 0$ limit can aid our understanding of the particle content of quantum field theories. In particular, we demonstrate that for the massive Klein-Gordon field, the classical limits of number operators can be understood to encode local information about particles in the corresponding classical field theory.

Klein Gordon Field in FLRW Space-Time

As a continuation of solving the equations governing the perturbation of the Friedmann-Lemaitre-Robertson- Walker (FLRW) space-time in Newman-Penrose formalism, the behaviour of the massive Klein-Gordon (KG) field coupled to the FLRW has been investigated. The Equation of Motion has been written and solved separately for radial and temporal parts. The former solution has come to be in terms of the Gegenbauer polynomials and spherical harmonics and the latter being in the WKB approximation. The particle current, energy momentum tensor and potential have also been obtained.

Exact quasinormal frequencies of the Dirac field in a Lifshitz black brane

In the zero momentum limit we exactly calculate the quasinormal frequencies of the massive Dirac field propagating in a Lifshitz black brane. Our results are an extension of those for the massive Klein–Gordon field in the zero momentum limit, but in contrast to the boson field for which only the Dirichlet boundary condition was imposed at the asymptotic region, here we impose two different boundary conditions far from the event horizon and compare the values obtained for the quasinormal frequencies. Furthermore, based on our results we determine some relevant limits, study the classical stability of the quasinormal modes of the Dirac field and determine its relaxation times.

Mode solutions for a Klein-Gordon field in anti–de Sitter spacetime with dynamical boundary conditions of Wentzell type

We study a real, massive Klein-Gordon field in the Poincar\'e fundamental domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime, subject to a particular choice of dynamical boundary conditions of generalized Wentzell type, whereby the boundary data solves a non-homogeneous, boundary Klein-Gordon equation, with the source term fixed by the normal derivative of the scalar field at the boundary. This naturally defines a field in the conformal boundary of the Poincar\'e fundamental domain of AdS. We completely solve the equations for the bulk and boundary fields and investigate the existence of bound state solutions, motivated by the analogous problem with Robin boundary conditions, which are recovered as a limiting case. Finally, we argue that both Robin and generalized Wentzell boundary conditions are distinguished in the sense that they are invariant under the action of the isometry group of the AdS conformal boundary, a condition which ensures in addition that the total flux of energy across the boundary vanishes.

Metric emerging to massive modes in quantum cosmological space-times

We consider a massive quantum test Klein-Gordon field probing a homogeneous isotropic quantum cosmological space-time in the background. In particular, we derive a semiclassical space-time which emerges to a mode of the field. The method consists of a comparison between quantum field theory on a quantum background and quantum field theory on a classical curved space-time, giving rise to an emergent metric tensor (its components being computed from the equation of propagation of the quantum Klein-Gordon field in the test field approximation). If the field is massless the emergent metric is of the Friedmann-Robertson-Walker form, but if a mass term is considered it turns out that the simplest emergent metric that displays the symmetries of the system is of the Bianchi I type, deformed in the direction of propagation of the particle. This anisotropy is of a quantum nature: it is proportional to $\ensuremath{\hbar}$ and ``dresses'' the isotropic classical space-time obtained in the classical limit.

Green’s function of a massless scalar field in curved space-time and superluminal phase velocity of the retarded potential

We study a retarded potential solution of a massless scalar field in curved space-time. In a special ansatz for a particle at rest whose magnitude of the (scalar) charge is changing with time, we found an exact analytic solution. The solution indicates that the phase velocity of the retarded potential of a non-moving scalar charge is position dependent, and may easily be greater than the speed of light at a given point. In the case of the Schwarzschild space-time, at the horizon, the phase velocity becomes infinitely faster than the coordinate speed of light at that point. Superluminal phase velocity is relatively common phenomenon, with the the phase velocity of the massive Klein-Gordon field as the best known example. We discuss why it is possible to have modes with superluminal phase velocity even for a massless field.

Information metric from a linear sigma model

The idea that a space-time metric emerges as a Fisher-Rao "information metric" of instanton moduli space has been examined in several field theories, such as the Yang-Mills theories and nonlinear σ models. In this paper, we report that the flat Euclidean or Minkowskian metric, rather than an anti-de Sitter metric that generically emerges from instanton moduli spaces, can be obtained as the Fisher-Rao metric from a nontrivial solution of the massive Klein-Gordon field (a linear σ model). This realization of the flat space from the simple field theory would be useful to investigate the ideas that relate the space-time geometry with the information geometry.

Entanglement between smeared field operators in the Klein-Gordon vacuum

Quantum field theory is the application of quantum physics to fields. It provides a theoretical framework widely used in particle physics and condensed matter physics. One of the most distinct features of quantum physics with respect to classical physics is entanglement or the existence of strong correlations between subsystems that can even be spacelike separated. In quantum fields, observables restricted to a region of space define a subsystem. While there are proofs on the existence of local observables that would allow a violation of Bell's inequalities in the vacuum states of quantum fields as well as some explicit but technically demanding schemes requiring an extreme fine-tuning of the interaction between the fields and detectors, an experimentally accessible entanglement witness for quantum fields is still missing. Here we introduce smeared field operators which allow reducing the vacuum to a system of two effective bosonic modes. The introduction of such collective observables is motivated by the fact that no physical probe has access to fields in single spatial (mathematical) points but rather smeared over finite volumes. We first give explicit collective observables whose correlations reveal vacuum entanglement in the Klein-Gordon field. We then show that the critical distance between the two regions of space above which two effective bosonic modes become separable is of the order of the Compton wavelength of the particle corresponding to the massive Klein-Gordon field.

Dephasing of a non-relativistic quantum particle due to a conformally fluctuating spacetime

We investigate the dephasing suffered by a non-relativistic quantum particle within a conformally fluctuating spacetime geometry. Starting from a minimally coupled massive Klein–Gordon field, we derive an effective Schrödinger equation in the non-relativistic limit. The wavefunction couples to gravity through an effective nonlinear potential induced by the conformal fluctuations. The quantum evolution is studied through a Dyson expansion scheme up to second order. We show that only the nonlinear part of the potential can induce dephasing. This happens through an exponential decay of the off-diagonal terms of the particle density matrix. The bath of conformal radiation is modeled in three dimensions and its statistical properties are described in terms of a general power spectral density. Vacuum fluctuations at a low energy domain are investigated by introducing an appropriate power spectral density and a general formula describing the loss of coherence is derived. This depends quadratically on the particle mass and on the inverse cube of a particle-dependent typical cutoff scale. Finally, the possibilities for experimental verification are discussed. It is shown that current interferometry experiments cannot detect such an effect. However this conclusion may improve by using high mass entangled quantum states.

Instability of the massive Klein-Gordon field on the Kerr spacetime

We investigate the instability of the massive scalar field in the vicinity of a rotating black hole. The instability arises from amplification caused by the classical superradiance effect. The instability affects bound states: solutions to the massive Klein-Gordon equation which tend to zero at infinity. We calculate the spectrum of bound state frequencies on the Kerr background using a continued-fraction method, adapted from studies of quasinormal modes. We demonstrate that the instability is most significant for the $l=1$, $m=1$ state, for $M\ensuremath{\mu}\ensuremath{\lesssim}0.5$. For a fast rotating hole ($a=0.99$) we find a maximum growth rate of ${\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremath{\approx}1.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}(GM/{c}^{3}{)}^{\ensuremath{-}1}$, at $M\ensuremath{\mu}\ensuremath{\approx}0.42$. The physical implications are discussed.

Fourth order AMR and nonlinear dynamical systems in compactified space

Time evolutions of certain spherically symmetric systems are investigated where simple explicit second order finite difference methods are not applicable. Due to a compactified space coordinate, efficiency and long-term numerical stability require at least fourth order accuracy for both the massive Klein–Gordon field and the SU(2) Yang–Mills–Higgs system. Moreover, adaptive mesh refinement (AMR) has a crucial role in dealing with high frequency oscillations that appear as an initial disturbance is radiated away. The incompatibility of AMR with fully fourth order accuracy is discussed and a solution is presented. Finally, compactification is compared to standard spherical coordinates and truncated grids in terms of efficiency.

TESTING A NEW MESH REFINEMENT CODE IN THE EVOLUTION OF A SPHERICALLY SYMMETRIC KLEIN–GORDON FIELD

Numerical evolution of the spherically symmetric, massive Klein–Gordon field is presented using a new adaptive mesh refinement (AMR) code with fourth order discretization in space and time, along with compactification in space. The system is non-interacting, thus the initial disturbance is entirely radiated away. The main aim is to simulate its propagation until it vanishes near [Formula: see text]. By numerical investigations of the violation of the energy balance relations, the space–time boundaries of "well-behaving" regions are determined for different values of the AMR parameters. An important result is that mesh refinement maintains precision in the central region for a longer time even if the mesh is only refined outside of this region. The speed of the algorithm was also tested; in the case of ten refinement levels the algorithm was two orders of magnitude faster than the extrapolated time of the corresponding unigrid run.

Scalar source in circular motion interacting with massive klein-gordon field in Minkowski spacetime

We analyze the scalar radiation emitted by a source in uniform circular motion in Minkowski spacetime interacting with a massive Klein-Gordon field. We assume the source rotating around a central object due to a Newtonian force. By considering the canonical quantization of this field, we use perturbation theory to compute the radiation emitted at the tree level. Regarding the initial state of the field as being the Minkowski vacuum, we compute the emission amplitude for the rotating source, assuming it as being minimally coupled to the massive Klein-Gordon field. We then compute the power emitted by the swirling source as a function of its angular velocity, as measured by asymptotic static observers.

The massive Klein–Gordon field coupled to a harmonic oscillator at the boundary

We consider the massive Klein-Gordon field on the half line with and without a Robin boundary potential.The field is coupled at the boundary to a harmonic oscillator.We solve the system classically and observe the existence of classical boundary bound states in some regions of the parameter space. The system is then quantized, the quantum reflection matrix and reflection cross section are calculated. Resonances and Ramsauer-Townsend effects are observed in the cross section. The pole structure of the reflection matrix is discussed.

U(1)gauge invariant noncommutative Schrödinger theory and gravity

We consider the complex, massive Klein-Gordon field living in the noncommutative space, and coupled to noncommutative electromagnetic fields. After employing the Seiberg-Witten map to first order, we analyze the noncommutative Klein-Gordon theory as $c$, which corresponds to the velocity of light, goes to infinity. We show that the theory exhibits a regular ``magnetic'' limit only for certain forms of magnetic fields. The resulting theory is nothing but the Schr\"odinger theory in a gravitational background generated by the gauge fields.

Massive fields tend to form highly oscillating self-similarly expanding shells

The time evolution of self-interacting spherically symmetric scalar fields in Minkowski spacetime is investigated based on the use of Green's theorem. It is shown that a massive Klein-Gordon field can be characterized by the formation of certain expanding shell structures where all the shells are built up by very high frequency oscillations. This oscillation is found to be modulated by the product of a simple time decaying factor of the form ${t}^{\ensuremath{-}3/2}$ and of an essentially self-similar expansion. Apart from this self-similar expansion the developed shell structure is preserved by the evolution. In particular, the energy transported by each shell appears to be time independent.

Bose-Einstein condensation on closed Robertson-Walker spacetimes

In this letter we summarize our analysis of Bose-Einstein condensation on closed Robertson-Walker spacetimes. In a previous work we defined an adiabatic KMS state on the Weyl-algebra of the free massive Klein-Gordon field [M. Trucks, M. Keyl, Phys. Lett. B 399 (1997) 223, M. Trucks, Commun. Math. Phys. 197 (1998) 387]. This state describes a free Bose gas on Robertson-Walker spacetimes. We use this state to analyze the possibility of Bose-Einstein condensation on closed Robertson-Walker spacetimes. We take into account the effects due to the finiteness of the spatial volume and show that they are not relevant in the early universe. Furthermore we show that a critical radius can be defined. The condensate disappears above the critical radius.

A KMS-like State of Hadamard Type on Robertson-Walker Spacetimes and its Time Evolution

In this work we define a new state on the Weyl algebra of the free massive scalar Klein-Gordon field on a Robertson-Walker spacetime and prove that it is a Hadamard state. The state is supposed to approximate a thermal equilibrium state on a Robertson-Walker spacetime and we call it an adiabatic KMS state. This opens the possibility to do quantum statistical mechanics on Robertson-Walker spacetimes in the algebraic framework and the analysis of the free Bose gas on Robertson-Walker spacetimes. The state reduces to an adiabatic vacuum state if the temperature is zero and it reduces to the usual KMS state if the scaling factor in the metric of the Robertson-Walker spacetime is constant. In the second part of our work we discuss the time evolution of adiabatic KMS states. The time evolution is described in terms of semigroups. We prove the existence of a propagator on the classical phase space. This defines a time evolution on the one-particle Hilbert space. We use this time evolution to analyze the evolution of the two-point function of the KMS state. The inverse temperature change is proportional to the scale factor in the metric of the Robertson-Walker spacetime, as one expects for a relativistic Bose gas.

The symplectic geometry of a parametrized scalar field on a curved background

We study the real, massive Klein–Gordon field on a C∞ globally-hyperbolic background space–time with compact Cauchy hypersurfaces. In particular, the parametrization of this system as initiated by Dirac and Kuchař is put on a rigorous basis. The discussion is focussed on the structure of the set of spacelike embeddings of the Cauchy manifold into the space–time, and on the associated e-tensor density bundles and their tangent and cotangent bundles. The dynamics of the field is expressed as a set of automorphisms of the space of initial data in which each pair of embeddings defines one such automorphism. Using these results, the extended phase space of the system is shown to be a weak-symplectic manifold, and the Kuchař constraint is shown to define a smooth constraint submanifold which is foliated smoothly by the constraint orbits. The pull-back of the symplectic form to the constraint surface is a presymplectic form which is singular on the tangent spaces to the constraint orbits. Thus, the geometric structure of this infinite-dimensional system is analogous to that of a finite-dimensional, first-class parametrized system, and hence many of the results for the latter can be transferred to the infinite-dimensional case without difficulty.