Real Klein-Gordon Field Research Articles

Physical significance of generalized boundary conditions: An Unruh-DeWitt detector viewpoint on [formula omitted

On PAdS2×S2, we construct the two-point correlation functions for the ground and thermal states of a real Klein-Gordon field admitting generalized (γ,v)-boundary conditions. We follow the prescription recently outlined in [1] for two different choices of secondary solutions. For each of them, we obtain a family of admissible boundary conditions parametrized by γ∈[0,π2]. We study how they affect the response of a static Unruh-DeWitt detector. The latter not only perceives variations of γ, but also distinguishes between the two families of secondary solutions in a qualitatively different, and rather bizarre, fashion. Our results highlight once more the existence of a freedom in choosing boundary conditions at a timelike boundary which is often neglected, though it has a notable associated physical significance.

Ground state for the Klein-Gordon field in anti–de Sitter spacetime with dynamical Wentzell boundary conditions

We consider a real Klein-Gordon field in the Poincar\'e patch of $(d+1)$-dimensional anti-de Sitter spacetime, PAdS$_{d+1}$, and impose dynamical boundary condition on the asymptotic boundary of PAdS$_{d+1}$ that depend explicitly on the second time derivative of the field at the boundary. These boundary conditions are of generalized Wentzell type. We construct the Wightman two-point function for the ground state of the Klein-Gordon theory whenever the parameters of the theory (the field mass, curvature coupling and boundary condition parameters) render such ground state admissible. In the cases in which the mass of the Klein-Gordon field and the curvature coupling term yield an effectively massless theory, we can define a boundary field whose dynamics are ruled by the dynamical boundary condition and construct, in addition to the Wightman function for the Klein-Gordon field, boundary-to-boundary, boundary-to-bulk and bulk-to-boundary propagators.

Role of boundary conditions on Lifshitz spacetimes

On a class of four-dimensional Lifshitz spacetimes with critical exponent $z=2$, including a hyperbolic and a spherical Lifshitz topological black hole, we consider a real Klein-Gordon field. Using a mode decomposition, we split the equation of motion into a radial and into an angular component. As first step, we discuss under which conditions on the underlying parameters we can impose to the radial equation boundary conditions of Robin type and whether bound state solutions do occur. Subsequently, we show that, whenever bound states are absent, one can associate to each admissible boundary condition a ground and a KMS state whose associated two-point correlation function is of local Hadamard form.

On the Inner Topological Pressure within the Topological Insulators

AbstractTopological insulators (TIs) have axion electrodynamics instead of the usual Maxwell. Then, there is an additional energy density, whose origin is topological, generating a pressure that is found analytically. It is shown that the topological electric and magnetic fields act only on the surface, while the scalar axion does as a real Klein–Gordon field in all the TI, giving rise to a behavior quite similar to the Casimir effect without the intervention of the virtual photons of the vacuum. This opens the possibility to experimentally test the electrodynamics of axions for the TIs.

Quantum thermodynamic properties of a vector field coupled to a moving environment

In this paper, the quantum thermodynamical properties of a vector field in the presence of an environment in uniform motion are investigated. The environment is modeled by a continuum of real Klein–Gordon fields, and the coupling function is introduced by the tensorial form which includes the environment velocity components. The covariant Lagrangian of the total system is canonically quantized. The Hamiltonian of the system is diagonalized using the Fano technique. It was observed that the memory function of the system depends on the speed of the environment. Using the diagonal Hamiltonian of the total system, the thermal correlation function of the dynamical variables, the thermal expectation value of the moving environment, mean force thermal energy and free energy are calculated.

A Generally Covariant Theory of Quantized Real Klein-Gordon Field in de Sitter Spacetime

We propose in this paper a quantization scheme for the real Klein-Gordon field in de Sitter spacetime.Our scheme is generally covariant with the help of vierbein, which is necessary usually for the spinorfield in curved spacetime. We first present a Hamiltonian structure and then quantize the field followingthe standard approach. For the free field, the time-dependent quantized Hamiltonian is diagonalized byBogliubov transformation, and the eigenstates at each instant are interpreted as the observed particle states at that instant. The interpretation is supported by the known cosmological redshift formula and the on-shell condition of 4-momentum for a free field. Though mathematics is carried out in terms of conformal coordinates for the sake of convenience, the whole theory can be transformed into any other coordinatesbased on general covariance. It is concluded that particle states, such as vacuum states in particular, aretime-dependent and vacuum states at one time evolve into nonvacuum states at later times. The formalismof perturbation is provided with en extended Dirac picture.

Optimal error estimates of finite difference time domain methods for the Klein–Gordon–Dirac system

AbstractWe propose and analyze finite difference methods for solving the Klein–Gordon–Dirac (KGD) system. Due to the nonlinear coupling between the complex Dirac ‘wave function’ and the real Klein–Gordon field, it is a great challenge to design and analyze numerical methods for KGD. To overcome the difficulty induced by the nonlinearity, four implicit/semi-implicit/explicit finite difference time domain (FDTD) methods are presented, which are time symmetric or time reversible. By rigorous error estimates, the FDTD methods converge with second-order accuracy in both spatial and temporal discretizations, and numerical results in one dimension are reported to support our conclusion. The error analysis relies on the energy method, the special nonlinear structure in KGD and the mathematical induction. Thanks to tensor grids and discrete Sobolev inequalities, our approach and convergence results are valid in higher dimensions under minor modifications.

Waiting for Unruh

How long does a uniformly accelerated observer need to interact with a quantum field in order to record thermality in the Unruh temperature? We address this question for a pointlike Unruh–DeWitt detector, coupled linearly to a real Klein–Gordon field of mass and treated within first order perturbation theory, in the limit of large detector energy gap . We first show that when the interaction duration is fixed, thermality in the sense of detailed balance cannot hold as , and this property generalises from the Unruh effect to any Kubo–Martin–Schwinger state satisfying certain technical conditions. We then specialise to a massless field in four spacetime dimensions and show that detailed balance does hold when grows as a power-law in as , provided the switch-on and switch-off intervals are stretched proportionally to and the switching function has sufficiently strong Fourier decay. By contrast, if grows by stretching a plateau in which the interaction remains at constant strength but keeping the duration of the switch-on and switch-off intervals fixed, detailed balance at requires to grow faster than any polynomial in , under mild technical conditions. The results also hold for a static detector in a Minkowski heat bath. The results limit the utility of the large regime as a probe of thermality in time-dependent versions of the Hawking and Unruh effects, such as an observer falling into a radiating black hole. They may also have implications on the design of prospective experimental tests of the Unruh effect.

The Casimir Effect from the Point of View of Algebraic Quantum Field Theory

We consider a region of Minkowski spacetime bounded either by one or by two parallel, infinitely extended plates orthogonal to a spatial direction and a real Klein-Gordon field satisfying Dirichlet boundary conditions. We quantize these two systems within the algebraic approach to quantum field theory using the so-called functional formalism. As a first step we construct a suitable unital ∗-algebra of observables whose generating functionals are characterized by a labelling space which is at the same time optimal and separating and fulfils the F-locality property. Subsequently we give a definition for these systems of Hadamard states and we investigate explicit examples. In the case of a single plate, it turns out that one can build algebraic states via a pull-back of those on the whole Minkowski spacetime, moreover inheriting from them the Hadamard property. When we consider instead two plates, algebraic states can be put in correspondence with those on flat spacetime via the so-called method of images, which we translate to the algebraic setting. For a massless scalar field we show that this procedure works perfectly for a large class of quasi-free states including the Poincare vacuum and KMS states. Eventually Wick polynomials are introduced. Contrary to the Minkowski case, the extended algebras, built in globally hyperbolic subregions can be collected in a global counterpart only after a suitable deformation which is expressed locally in terms of a *-isomorphism. As a last step, we construct explicitly the two-point function and the regularized energy density, showing, moreover, that the outcome is consistent with the standard results of the Casimir effect.

Dissipative Scalar Field Theory: A Covariant Formulation

Caldeira-Leggett model of reservoir is generalized to a reservoir modeled by a continuum of real Klein-Gordon fields, instead of harmonic oscillators. A quantum Langevin type dissipative equation is obtained for the scalar field. The susceptibility of the medium is defined in terms of the reservoir Green’s function and the coupling function satisfying causality condition. The connection between the coupling function and the susceptibility of the medium is found to be a Hankel transform from which the coupling function can be determined in terms of the susceptibility of the medium. Noise currents and their fluctuation-dissipation relation are obtained. In a homogeneous medium or reservoir, explicit form of the quantum scalar field, and its large-time limit, are found.

Quantum Gravitational Corrections to the Real Klein-Gordon Field in the Presence of a Minimal Length

The (D+1)-dimensional (β,β′)-two-parameter Lorentz-covariant deformed algebra introduced by Quesne and Tkachuk (J. Phys., A Math. Gen. 39, 10909, 2006), leads to a nonzero minimal uncertainty in position (minimal length). The Klein-Gordon equation in a (3+1)-dimensional space-time described by Quesne-Tkachuk Lorentz-covariant deformed algebra is studied in the case where β′=2β up to first order over deformation parameter β. It is shown that the modified Klein-Gordon equation which contains fourth-order derivative of the wave function describes two massive particles with different masses. We have shown that physically acceptable mass states can only exist for $\beta<\frac{1}{8m^{2}c^{2}}$ which leads to an isotropic minimal length in the interval 10−17 m<(ΔX i )0<10−15 m. Finally, we have shown that the above estimation of minimal length is in good agreement with the results obtained in previous investigations.

On some meaningful inner product for real Klein—Gordon fields with positive semi-definite norm

A simple derivation of a meaningful, manifestly covariant inner product for real Klein—Gordon (KG) fields with positive semi-definite norm is provided, which turns out — assuming a symmetric bilinear form — to be the real-KG-field limit of the inner product for complex KG fields reviewed by A. Mostafazadeh and F. Zamani in December 2003, and February 2006 (quant-ph/0312078, quant-ph/0602151, quant-ph/0602161). It is explicitly shown that the positive semi-definite norm associated with the derived inner product for real KG fields measures the number of active positive and negative energy Fourier-modes of the real KG field on the relativistic mass shell. The very existence of an inner product with positive semi-definite norm for the considered real, i.e. neutral, KG fields shows that the metric operator entering the inner product does not contain the charge-conjugation operator. This observation sheds some additional light on the meaning of the C operator in the CPT inner product of PT-symmetric quantum mechanics defined by C.M. Bender, D.C. Brody and H.F. Jones.

Quantum mechanics of Klein–Gordon fields I: Hilbert Space, localized states, and chiral symmetry

We derive an explicit manifestly covariant expression for the most general positive-definite and Lorentz-invariant inner product on the space of solutions of the Klein–Gordon equation. This expression involves a one-parameter family of conserved current densities J a μ , with a ∈ (−1, 1), that are analogous to the chiral current density for spin half fields. The conservation of J a μ is related to a global gauge symmetry of the Klein–Gordon fields whose gauge group is U (1) for rational a and the multiplicative group of positive real numbers for irrational a. We show that the associated gauge symmetry is responsible for the conservation of the total probability of the localization of the field in space. This provides a simple resolution of the paradoxical situation resulting from the fact that the probability current density for free scalar fields is neither covariant nor conserved. Furthermore, we discuss the implications of our approach for free real scalar fields offering a direct proof of the uniqueness of the relativistically invariant positive-definite inner product on the space of real Klein–Gordon fields. We also explore an extension of our results to scalar fields minimally coupled to an electromagnetic field.

A CANONICAL DECOMPOSITION IN COLLECTIVE AND RELATIVE VARIABLES OF A KLEIN–GORDON FIELD IN THE REST-FRAME WIGNER-COVARIANT INSTANT FORM

The canonical decomposition of a real Klein–Gordon field in collective and relative variables proposed by Longhi and Materassi is reformulated on spacelike hypersurfaces. This allows us to obtain the complete canonical reduction of the system on Wigner hyperplanes, namely in the rest-frame Wigner-covariant instant form of dynamics. From the study of Dixon's multipoles for the energy–momentum tensor on the Wigner hyperplanes we derive the definition of the canonical center-of-mass variable for a Klein–Gordon field configuration: it turns out that the Longhi–Materassi global variable should be interpreted as a center of phase of the field configuration. A detailed study of the kinematical "external" and "internal" properties of the field configuration on the Wigner hyperplanes is done. The construction is then extended to charged Klein–Gordon fields: the centers of phase of the two real components can be combined to define a global center of phase and a collective relative variable describing the action–reaction between the two Feshbach–Villars components of the field with definite sign of energy and charge. The Dixon multipoles for both the energy–momentum and the electromagnetic current are given. Also the coupling of the Klein–Gordon field to scalar relativistic particles is studied and it is shown that in the reduced phase space, besides the particle and field relative variables, there is also a collective relative variable describing the relative motion of the particle subsystem with respect to the field one.

A Canonical Decomposition in Collective and Relative Variables of a Klein-Gordon Field in the Rest-Frame Wigner-Covariant Instant Form

The canonical decomposition of a real Klein-Gordon field in collective and relative variables proposed by Longhi and Materassi is reformulated on spacelike hypersurfaces. This allows to obtain the complete canonical reduction of the system on Wigner hyperplanes, namely in the rest-frame Wigner-covariant instant form of dynamics. From the study of Dixon's multipoles for the energy-momentum tensor on the Wigner hyperplanes we derive the definition of the canonical center-of-mass variable for a Klein-Gordon field configuration: it turns out that the Longhi-Materassi global variable should be interpreted as a center of phase of the field configuration. A detailed study of the kinematical external and internal properties of the field configuration on the Wigner hyperplanes is done. The construction is then extended to charged Klein-Gordon fields: the centers of phase of the two real components can be combined to define a global center of phase and a collective relative variable describing the action-reaction between the two Feshbach-Villars components of the field with definite sign of energy and charge. The Dixon multipoles for both the energy-momentum and the electromagnetic current are given. Also the coupling of the Klein-Gordon field to scalar relativistic particles is studied and it is shown that in the reduced phase space, besides the particle and field relative variables, there is also a collective relative variable describing the relative motion of the particle subsytem with respect to the field one.

A canonical realization of the BMS algebra

A canonical realization of the BMS (Bondi–Metzner–Sachs) algebra is given on the phase space of the classical real Klein–Gordon field. By assuming the finiteness of the generators of the Poincaré group, it is shown that a countable set of conserved quantities exists (supertranslations); this set transforms under a particular Lorentz representation, which is uniquely determined by the requirement of having an invariant four-dimensional subspace, which corresponds to the Poincaré translations. This Lorentz representation is infinite-dimensional, nonunitary, reducible and indecomposable. Its representation space is studied in some detail. It determines the structure constants of the infinite-dimensional canonical algebra of the Poincaré generators together with the infinite set of the new conserved quantities. It is shown that this algebra is isomorphic with that of the BMS group.

Pair creation and decay of a massive particle near and far away from a cosmic string.

We study the total transition probabilities of the tree-level processes of the pair creation and decay of a massive particle for real Klein-Gordon fields in the spacetime of an infinite straight static cosmic string. Basing the discussion on cylindrical modes characterized by an approximate radius of closest approach ${\mathit{r}}_{\mathrm{min}}$, it is possible to approximately localize the non-Minkowskian processes to cylindrical effective interaction regions around the cosmic string. A physical understanding of the space dependence of the transition probabilities is obtained on the basis of analytic expressions for different energy domains referring to regions close to and far away from the string. For pair creation the Compton wavelength ${\ensuremath{\lambda}}_{\mathit{C}}$ of the created particles proves to be a crucial length scale. For ${\mathit{r}}_{\mathrm{min}}$\ensuremath{\ll}${\ensuremath{\lambda}}_{\mathit{C}}$ the creation probability is insensitive to a variation of ${\mathit{r}}_{\mathrm{min}}$. For large ${\mathit{r}}_{\mathrm{min}}$ it falls off at least exponentially with ${\mathit{r}}_{\mathrm{min}}$. This agrees with an alternative ``integrated'' approach to localization: the cross section around the cosmic string is proportional to the Compton wavelength ${\ensuremath{\lambda}}_{\mathit{C}}$. The decay of the massive particle on the other hand contains processes allowed in Minkowski spacetime and leads to another type of local behavior.

Quantum states of a field partitioned by an accelerated frame.

The discrete Minkowski Bessel mode technology is developed. This development includes (i) the eigenvalue spectrum, (ii) the phase-space integral formulation of the quantized Klein-Gordon (KG) field in accelerated frames with a finite bottom ($b&gt;0$), and (iii) the procedure for making the transition to bottomless ($b=0$) frames. This technology is used to show that the Rindler vacuum state of an acceleration-partitioned real KG field is perpendicular not only to the Minkowski vacuum state, but also to every quantum state associated with a "detector" accelerating through what initially was the Minkowski vacuum. In fact, every element of the Hilbert space containing the Rindler vacuum is perpendicular to the Hilbert space containing the Minkowski vacuum.

Spectral method for solving the system of equations of Schrödinger—Klein—Gordon field

In this paper, we construct the conservative spectral scheme for the periodic initial-value problem for a system of equations of the complex Schrödinger field, interacting with the real Klein—Gordon field and estimate the error which is \\xV;\\GF;( nk) − Φ n N \\xV; + \\xV;χ( nk) − iX n n\\xV; 1 = O( k 2 + N −(γ−1)).

Integrable theories of nonlinearly coupled Klein-Gordon fields

Exactly integrable systems consisting of one complex and one real Klein-Gordon field in 1+1 dimensions are presented. The coupling terms in the field equations are of second order in the derivatives of the fields.