The usual statement is that a lattice formulation severely mutilates lorentz invariance at the outset 1 and that continuous symmetries are only recovered in the continuous limit of a lattice theory. In this paper we present a rigorous derivation of the lorentz invariant solutions of the klein gordon equation. So we have shown the invariance of the dalembert operator under lorentz transformations. Lecture 3 energy momentum tensor, the lorentz force, noether theorem duration. In this paper we consider the classical scalar klein gordon equation coupled noncovariantly. In the classical mechanics, when the action s is invariant under some coordinate transformation, lagrangian gives the equation of motions through eulerlagrange equation.
Van wijngaarden at the meeting of february 23, 1963 introduction the solutions, invariant under a proper lorentz transformation, of. We have found that the existence of this type of solution leads us directly to the klein gordon equation as a. Mod01 lec01 introduction, the kleingordon equation nptelhrd. The energy inner product is nonpositive on a subspace of infinite dimension, not consisting entirely of eigenvectors. The initial condtions to solve the newtons equation of motion are the initial positions and initial velocities. In this paper, we investigate the lorentz invariance of the square root klein gordon equation. As a lorentz transformed sinus remains a sinus, the equation still remains a sinegordon equation, thus it is lorentz invariant. There is a vast array of further quantum field theory texts, many of them with. The lorentzinvariant solutions of the kleingordon equation.
Substituting this into the second equation we obtain x0 l01. In fact the lorentz invariance of the square root kleingordon equation was already shown. Thus in the primed frame at a given instant the two. Lorentzinvariance of solution of kleingordon equation closed. It is argued that this equation can be considered a natural relativistic extension of the schrodinger equation for energies smaller than the energy associated to the particles mass. The klein gordon equation is a secondorder differential equation in t and thus the initial values of.
We would like to show you a description here but the site wont allow us. Since we want the lagrangian density to be invariant under lorentz transformations a linear wave equation, the lagrangian density can contain only the terms. Em lorentz force derived from klein gordons equation 11. Lorentz invariance of positive energy solutions to the klein. There are three fundamental problems arising with the formulation of. The magnetic eld binside the long solenoid and the total ux through the inside surface, the integral of bover the surface, is. If you substitute the free particle solution into the klein gordon equation you get, unsurpisingly, the relation e 22p m. Because of its high derivatives, checking lorentz invariance of this equation is so complicated and cannot be speci. As we can see now, the apparent problems were manifestations of the new physics. It is secondorder in space and time and manifestly lorentzcovariant. Quantum field theory i babis anastasiou institute for theoretical physics, eth zurich, 8093 zurich, switzerland email. In fact the lorentz invariance of the square root klein gordon equation was already shown. The scalar product is lorentz invariant and therefore it has the same value on each spacelike hypersurface. Much use is made of graphical arguments to back up the mathematical results.
Invariance of the kleingordon equation under timereversal transformation. In its free form, or including electromagnetic interactions, it describes all spin1 2 massive particles such as electrons and quarks for which parity is a symmetry. The kleingordon equation in the spacetime representation 9 the quantity q2 is represented by the differential operator dalembert operator from the mass shell condition 6 this results in the kleingordon equation as the basic field equation of the scalar field. Dec 22, 2004 although the usual operator invariance requirements and corresponding commutation conditions encountered in the study of the invariance of relativistic wave equations and other equations of physics are sufficient conditions for invariance, they are by no means. Periodic solitary waves for two coupled nonlinear klein.
Lorentz invariance of kleingordon lagrangian physics forums. Kleingordon equation alternatively, apply quantization to energymomentum invariant. Scattering theory for the kleingordon equation with. Lorentz invariance of positive energy solutions to the klein gordon equation. The klein gordon equation klein fock gordon equation or sometimes klein gordon fock equation is a relativistic wave equation, related to the schrodinger equation. Also and refer to two different points with coordinates in the old and new system, respectively since the klein gordon operator does not change under continuous lorentz transformations, we can reason that the wave function is. Introduction to relativistic quantum mechanics 247 contents 1. Canonical quantization lorentz invariant integration measure. Lorentz invariance, galilei invariance, conformal invariance, etc.
The interaction of pseudo scalar particles with the electromagnetic field is described by the minimal substitution. Em lorentz force derived from klein gordon s equation this means that somewhere b must di er from zero. On the lorentz invariance of the square root kleingordon. Lorentzinvariance of solution of kleingordon equation. As a lorentz transformed sinus remains a sinus, the equation still remains a sine gordon equation, thus it is lorentz invariant. Mod01 lec01 introduction, the kleingordon equation youtube. Relativistic introduction to the kleingordon equation. So, none of the objections that led to discarding the klein equation held up with time. Note that our approach was used in the past in relation to the gravitational theories.
Van wijngaarden at the meeting of february 23, 1963 4. Klein gordon equation, we need all the initial conditions i. But scalar wave function is inconsistent with born interpretation because. Maxwell klein gordon equations with small sobolev data by cristian dan gavrus doctor of philosophy in mathematics university of california, berkeley professor daniel tataru, chair in this thesis we prove global wellposedness and modi ed scattering for the massive maxwell klein gordon mkg and for the massless maxwelldirac md equations, in the. Natural extension of the schrodinger equation to quasi. It is a quantized version of the relativistic energymomentum relation.
Substitution of this solution into the equation for the probability density yields. These lecture notes are based on an introductory course on quantum field theory, aimed at part iii i. Invariance of the equations under the spacereversal transformation. Kleingordon equation mathematics and physics wiki fandom. This arises not only in the context of eld theories as e. Unlike the klein gordon equation which is invariant under lorentz transformation, dirac equation is not. Notice that the kg equation admits every solution of 6 as a solution. Em lorentz force derived from klein gordon s equation 11.
Of course, as klein gordon equation is lorentz invariant, this lagrandian needs to be lorentz invariant scalar, too. The dirac equation the klein gordon equation write quantum mechanics as a quantum field theory slides. In other words, this is not invariant under lorentz transformations. Chapter 3 the lorentz transformation in the wonderful world and appendix 1, the reasoning is kept as direct as possible. The kleingordon equation kleinfockgordon equation or sometimes kleingordonfock equation is a relativistic wave equation, related to the schrodinger equation. Scattering of solitons of the kleingordon equation. The klein gordon equation in the spacetime representation 9 the quantity q2 is represented by the differential operator dalembert operator from the mass shell condition 6 this results in the klein gordon equation as the basic field equation of the scalar field.
Galilean invariance, probability density, and relation to the kleingordon equation are discussed. Relativisticquantum mechanics 1 university of oxford. Schroeder, an introduction to quantum field theory, perseus books, 1995 chapter 2. If one keeps first order derivative of time, then to preserve lorentz invariance, the space coordinate derivatives must be of the first order as well.
Relativistic quantum mechanics janos polonyi university of strasbourg dated. Lorentz invariance of positive energy solutions to the. P r o bl em s w it h k lei n g o r do n e q ua t io n 1. Later, the equation was revived and is now known as kleingordon equation. Request pdf relativistic introduction to the klein gordon equation during schroedingers development of the quantum mechanical wave equation for spin0 particles, he had sought a. The solutions to kge are invariant under relativistic lorentz transforms and subjected to the causality principle. Lorentz group and lorentz invariance in studying lorentz invariant wave equations, it is essential that we put our understanding of the lorentz group on. Lorentzinvariance of solution of kleingor don equation closed ask question asked 5 years, 6 months ago. Lorentz invariance lorentz invariance of the laws of physics is satisfied if the laws are cast in terms of four. Lorentz covariance has two distinct, but closely related meanings.
The invariance in form of maxwells equations under lorentz. There are many reasons to consider quantum physics on a lattice. Using the lorentz transformation, the other worldline is found at t0 t. In a relativistic theory of physics, a lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any lorentz transformation. A physical quantity is said to be lorentz covariant if it transforms under a given representation of the lorentz group.
The point is that the kleingordon equation is second order in time derivative, similarly to the newtons equation of motion in mechanics. It is secondorder in space and time and manifestly lorentz covariant. Mod01 lec01 introduction, the kleingordon equation. True wavefunctions and antiparticles of kleingordon equation. The plane waves 10 are basic solutions and the field 9 is constructed by. Klein gordon equation 249 fourmomentum 250 maintaining lorentz invariance 251 algebra 253 2component form 254 currents and densities 256 charged currents 258 antiparticle 259 klein gordon field 260 example 1. To leave a comment or report an error, please use the auxiliary blog. Since and refer to the same spacetime point and is a scalar, the klein gordon equation is lorentz invariant. Since we want the lagrangian density to be invariant under lorentz transformations a linear wave equation, the lagrangian density can contain only the. However, the klein gordon equation has issues when interpreted as a standard wave equation.
In the relativistic field theories, we must use lorentz scalars to express the iiinteractions. A natural class of solutions are those given by the lagrangian submanifolds of the phase space when it is endowed with the symplectic structure modified by the electromagnetic field. On the other hand,the lorentz invariance of this equation has often been under discussion. The dirac equation was an attempt to solve this issue.
Jun 19, 2014 mod01 lec01 introduction, the kleingordon equation nptelhrd. Proof of lorentz invariance of kleingordon equation. Chapter 11 em lorentz force derived from klein gordons. The lorentz group propriety and orthochronaity unitary operators generators slides. In particle physics, the dirac equation is a relativistic wave equation derived by british physicist paul dirac in 1928.
Since it is secondorder in time like the classical wave equation from classical lorentz invariant em, probability density is not explicitly always conserved. Notes on relativistic quantum mechanics module 4 of refresher. Lorentz invariance of klein gordon lagrangian physics forums. Four vector dot products are said to be lorentz scalars. Relativistic quantum mechanics kleingordon equation dirac. Lorentz invariance in klein gordon momentum states link to. Special attention is paid to various ways that lead to analytical solutions to kge. The klein gordon equation is considered in the case of nondecreasing potentials.
To claim so in the context of solutions to the actual secondorder in time klein gordon equation, we need all the initial conditions i. Lorentz covariance, a related concept, is a property of the underlying spacetime manifold. On the lorentz invariance of maxwells equations vesselin c. Global wellposedness and parametrices for critical. Because of its high derivatives, checking lorentz invariance of this equation is so complicated and cannot be speci ed easily. The lorentz invariant theory of gravitation ligt is the conditional name of the proposed theory of gravity, since lorentz invariance is a very important, although not the only feature of this theory.
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