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We gave a pedagogical introduction to the Dirac equation in background curved spacetime and particle creation. II. curved spacetime, emphasizing the relation between free-field unitarity and the preservation of the inner prod-uct on the one-particle Hilbert space (for scalar fields this is the Klein-Gordon inner product). More recently, a similar approach has been pursued to study the dynamics of Klein-Gordon ï¬elds in â¦ Introduction We examine the global in time solutions of some class of semililear hyperbolic equations, such as the KleinâGordon equation, which includes the Higgs boson equation in the Minkowski spacetime, de Sitter spacetime, and Einstein & de Sitter spacetime. Exact solutions of the KleinâGordonâFock (KGF) general relativistic equation that describe the dynamics of a massive, electrically charged scalar particle in the curved spacetime geometry of an electrically charged, rotating KerrâNewmanâ(anti) de Sitter black hole are investigated. A more general wave equation, which is satisï¬ed by matter ï¬elds on spacetime is the so-called Klein-Gordon equation. It is well known that the Klein-Gordon equation in curved spacetime is conformally noninvariant, both with and without a mass term. In particular, we give applications to the KleinâGordon and wave equations in the curved spacetimes such as the de Sitter universe. The gravitational interaction is prescribed by the metric of a spherically symmetric space-time. An evolution equation approach to the KleinâGordon operator on curved spacetime. We present the ar-gument in a way that does not rely on global hyperbolici-ty of the spacetime. curved spacetime backgrounds. Klein-Gordon Fields in Curved Spacetime We are now ready to embark on the first major topic of this thesis. It follows that on a curved space-time there seems to be no natural notion of â¦ MATRIX OPERATOR ALGEBRA: DIRAC & KLEIN-GORDON EQUATIONS The covariant generalization of the Dirac equation in a curved space-time of dimension 1n for free spin 1 2 particle in the units c 1 gives im . Keywords Scalar Field, Curved Spacetime, Klein-Gordon Equation, Relativistic Trajectories, Charged Particles 1. These simplifications take the form of the familiar wave equations one meets in everyday life, for example those used in describing ripples of water on the surface of a pond, albeit on a curved spacetime. [F89] S. A. FULLING, Aspects of quantum ï¬eld theory in curved space-time (Cam-bridge University Press, 1989). Among the reasons for studying the theory are: â¢ QFT experiments at CERN, DESY etc are performed in a lightly curved spacetime background. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We show that there exists a duality between the local coordinates and the solutions of the Klein-Gordon equation in curved spacetime in the same sense as in the Minkowski spacetime. Summary. E-mail:jan.derezinski@fuw.edu.pl;daniel.siemssen@fuw.edu.pl. It follows a review of J. DIMOCKâs quantization of the Maxwell ï¬eld on curved spacetimes and then we come to our main result: We show â¦ On a more conceptual side, a curved space-time does not have the large group of isometries (the Poincaré group) of the Minkowski space. The wave equation on a curved space-time, Cambridge Monographs on Mathematical Physics 2, Cambridge University Press, 1975. More recently, however, Atanasov and Saxena have pointed out that, to be consistent with the Heisenberg uncertainty principle, these approaches should not overlook the fact that the carriers are intrinsically three-dimensional â¦ A precise de nition of the adiabatic approximation is The Klein-Gordon equation of a complex scalar ï¬eld in metric (3) reads m2Ï â2â vâ xÏ â f â xÏ â fâ2 x Ï = 0. We derive L1 decay estimates for the solution to the linear Klein-Gordon equation in de Sitter spacetime with and without source term. So we are ignoring half of Wheelerâs famous slogan that âmatter tells spacetime how to curve; spacetime tells matter how to moveâ. As we discuss in the following paper, In the latter, it is assumed that the dynamics of carriers in graphene in low energies is described by Dirac equation in (2 + 1) dimensional curved spaceâtime. ... A free, massive scalar ï¬eld satisï¬es the Klein-Gordon equation3 Then the wave equation and quantization of the Maxwell ï¬eld in ï¬at spacetimes is discussed. Curved Spacetime Ali Mostafazadeh ... deï¬nite, the Klein-Gordon equation may be written as a pair of equations which are linear in the time-derivative of the ï¬eld. to reproduce the usual Klein-Gordon equation. concepts of Differential Geometry to yield the Klein-Gordon equation and the Lagrange equations of Relativistic Mechanics without using the standard postulates of Quantum Mechanics, Special Relativity or even General Relativity. References curved spacetime. We report the methods and results of a computational physics project on the solution of the relativistic Klein-Gordon equation for a light particle gravitationally bound to a heavy central mass. The Klein-Gordon equation is the linear partial differential equation which is the equation of motion of a free scalar field of possibly non-vanishing mass m m on some (possibly curved) spacetime (Lorentzian manifold): it is the relativistic wave equation with inhomogenety the mass m 2 m^2. (1) Here, however, we intend to formulate the Dirac equation in curved space â¦ ... Schrödinger equation and Heisenberg picture. Keywords Klein-Gordon equation Curved spacetime â¦ DECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION IN CURVED SPACETIME MUHAMMET YAZICI Abstract. 6. The Klein-Gordon equation In the space-time representation (9) the quantity -q2 is represented by the differential operator (dâAlembert operator) From the mass shell condition (6) this results in the Klein-Gordon equation as the basic field equation of the scalar field. batic geometrical phase for a classical relativistic charged scalar eld in a curved back-ground spacetime which is minimally coupled to electromagnetism and an arbitrary (non-electromagnetic) scalar potential. Exact solutions of the Klein-Gordon-Fock (KGF) general relativistic equation that describe the dynamics of a massive, electrically charged scalar particle in the curved spacetime geometry of an electrically charged, rotating Kerr-Newman-(anti) de Sitter black hole are investigated. It involves a two-component formulation of the corre-sponding Klein-Gordon equation. However, the duality in curved spacetime does not have the same generality as in flat spacetime and it holds only if the system â¦ The present chapter will be devoted to the formulation of the Klein-Gordon equation in a curved spacetime background. We consider the initial-value problem for the Klein-Gordon equa-tion in de Sitter spacetime. 2.Semi-classical gravity: still treat the background as classical, but now take the back-reaction into account. 1. Klein-Gordon Fields in Curved Spacetimes and Other Backgrounds : In General > s.a. causality violation; geometric phase; huygens' principle; klein-gordon fields [Hamiltonian]; quantum klein-gordon fields; Superradiance. spacetime ï¬eld, but the system does not modify the spacetime itself. A more general wave equation, which is satisfied by matter fields on spacetime is the so-called Klein-Gordon equation. Conclusion. The normal procedure for turning a theory in flat spacetime into one in curved spacetime involves replacing $\partial_\mu$ by $\nabla_\mu$, yes, but the Euler-Lagrange equations don't change . Next we will state its form in the FRW metric, and then solve the equation â¦ 1.Quantum ï¬eld theory in curved space-time: the background space-time is classical, meaning we work in zeroth order in ~. The correspondence you state in $(3)$ is not correct. Using the fact that the Dirac equation in the FRW metric is equivalent to the flat-spacetime Dirac equation with a time-dependent mass term, we demonstrated that a single-particle analog of particle creation can be observed in the dynamical evolution of spinor wave â¦ This paper develops the mathematical foundations for a companion paper on âBlack-Hole Electrodynamicsâ. The positive- and negative-frequency bisolutions, as well as the Feynman and KleinâGordon Operator on Curved Spacetime Jan Derezi´nski, Daniel Siemssen Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093, Warszawa, Poland. We ignore the back-reaction of the matter on space-time. e.g. In recent time (as of 2000) solutions of the KleinâGordon equation on Lorentzian manifolds have attracted increasing attention in connection with the theory of quantized fields on curved space-time, cf. (4) By introducing the variable Ï, mÏ = 2â vÏ + fâ xÏ, (5) Eq. equations one meets in everyday life, for example those used in describing ripples of water on the surface of a pond, albeit on a curved spacetime. the Klein-Gordon equation has then constant coeï¬cients. Abstract. AN EVOLUTION EQUATION APPROACH TO THE KLEINâGORDON OPERATOR ON CURVED SPACETIME 217 On the physical side, on a stationary spacetime with a positive Hamiltonian, it is clear how to deï¬ne the nonclassical propagators. . (4) can be rewritten into two coupled ï¬rst-order equations â vÏ =â f 2 â¦ It More specifically, it re-expresses the equations of curved-spacetime electrodynamics in terms of a 3 + 1 (space + time) split, in which the key quantities are three-dimensional vectors (electric field á¸, magnetic field |$^B_\sim$|â , etc.) Two coupled ï¬rst-order equations â vÏ =â f 2 â¦ II ] S. A. FULLING Aspects! In ï¬at spacetimes is discussed ( 5 ) Eq matter Fields on spacetime is the so-called equation. Â vÏ =â f 2 â¦ II spacetimes is discussed Field, curved spacetime background ï¬eld theory in curved (! Is satisfied by matter ï¬elds on spacetime is the so-called Klein-Gordon equation, 1975 positive- and negative-frequency bisolutions as! 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