We do this symmetrically by making two complex elds. Klein gordon equation relativistic description of quantum theory once upon a time, many physicists tried to nd the equation that is both relativistic and quantum at once. The kleingordon equation for a real scalar field reads in positionspace. Physics 221b academic year 202021 notes 46 introduction. Separation of variables and exact solution of the kleingordon and. On the fluid dynamical approximation to the nonlinear klein. We will see that, unlike the klein gordon equation, the dirac equation, being.
But pretty soon this interpretation run into trouble with bad probabilities negative. In elementary particle physics, especially the dirac equation is of fundamental importance and occurs at a central place in the standard model. The problem of a particle in a box is probably the simplest problem in quantum mechanics which allows for significant insight into the nature of quantum systems and thus is a cornerstone in the teaching of quantum mechanics. Solutions of kleingordon and dirac equations for nonpure. Generalized kleingordon and dirac equations from nonlocal. Interplay between dynamic systems described by the klein. The emphasis is given to those elements of the formalism which can be carried on. The main goal of this paper is to adopt a multivector calculus scheme to study finite difference discretizations of klein gordon and dirac equations for which.
Solutions to common problems with probability interpretation and inde. A novelapproach to the synthesis of nondispersive wave. The dirac equation notes 45 describes the state of a. Relativistic kleingordonmaxwell multistream model for. Dirac was searching for a split of the klein gordon equation into two. As in klein gordon equation, we have negative energy solutions and they can not be discarded. Recognizing that the negative probabilities of the klein gordon equation were related to the fact that the klein gordon equation is second order in time, dirac decided to. Therefore, it is my goal to show how the earlier results of the purely real solutions could be applied when the. The dirac equation, like the kleingordon equation, possesses solutions with negative energy, which. Request pdf on dec 31, 20, jay theodore cremer published the klein gordon and dirac equations find, read and cite all the research you need on researchgate.
Comparison among kleingordon equation, dirac equation and relativistic stationary schr. 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 klein gordon equation klein fock gordon equation or sometimes klein gordon fock equation is a relativistic wave equation, related to the schrodinger equation. In the dirac case, the discrete solution converges to the continuum propagator and it can also be rewritten in the form. Pdf the geometrization of quantum mechanics, the nonlinear.
Its theoretical relevance is similar to that of the dirac equation. In this paper we will consider the nonlinear klein gordon equation 2 2m c 0 1 2 mc2 2vj j 0. Wave klein gordon dirac equations in the standard model of particle physics zoe wyatt migsaa joint with philippe lefloch sorbonne and shijie dong fudan. Generalized dirac and kleingordon equations for spinor. More particularly in quantum field theory, higherorder field equations such as the generalized klein gordon equation were discussed in literature. Following the klein gordon analysis, we may now expres s the radia l equation for the dirac atom in th e form of 4. On the spectrum of the kleingordon equation isaac newton institute. However, we will see that both the klein gordon equation and the dirac equation have solutions corresponding to states with negative energies. Exact bound states of the ddimensional klein gordon equation 1427 the radial and angular parts of the klein gordon equation with this type of potential are solved using the nu method.
It is a quantized version of the relativistic energymomentum relation. This is an historic equation, the klein gordon equation. Comparison among kleingordon equation, dirac equation and. Scattering solutions of the klein gordon equation for a step potential. Veselic 11,12 dealt with spectral convergence and analyticity of eigenfunctions in 1c. Dirac equation from wikipedia, the free encyclopedia in particle physics, the dirac equation is a relativistic wave equation derived by british physicist paul dirac in 1928. On the fluid dynamical approximation to the nonlinear.
Because in order to interpret the wavefunction as a probability we must have a continuity equation. The klein gordon oscillator in a noncommutative spaceit is known that the nonrelativistic harmonic oscillator in a noncommutative space has a similar behaviour to the landau problem in a commuatative space 10,11,14. A stringy defor mation of the w eyl heisenberg algebra x, p i. This was introduced by the mathematical physicist p. Therefore, one has to actually consider the dirac equation to treat the electron with including it. Lecture notes wave equations of relativistic quantum mechanics. The dirac equation dirac in his effort to discover an equation that was free of the problems seen with the klein gordon equation tried an equation that was linear in the energy, mass and momentum. Rylov institute for problems in mechanics, russian academy of sciences, 101, vernadskii ave. Global wellposedness and parametrices for critical maxwell. Therefore, one has to actually consider the dirac equation to treat the electron with including it is spin part use in a mu. For every solution to the dirac equation, its components will be a solution of the klein gordon equation, but the converse isnt true. In sec iv, using the algebraic method of separation of variables, we reduce the dirac equation to a system of rst order coupled di erential equations that we solve in terms of special functions. Accordingly, the klein gordon kg equation and dirac equation are restudied and compared with the relativistic stationary schrodinger equation rsse.
A novelapproach to the synthesis of nondispersive wave packet solutions to the klein gordon and dirac equations. Is the dirac equation equivalent to the kleingordon. Null structure and almost optimal local regularity for the diracklein. The dirac equation can be thought of in terms of a square root of the klein gordon equation. Mapping of the relativistic kinetic balance equations onto. Separation of variables and exact solution of the klein. Solutions for the kleingordon and dirac equations on the.
Then we solve the equations for the nonpure dipole in sect. Is the dirac equation equivalent to the kleingordon equation. For mkg, this work extends to the general case m 2 0 the results of kriegersterbenz. Introduction the problem has originally arisen from the paper 1 and the more later one 2, where the klein gordon dirac equation kgd, i. E p2 m2 this demonstrates that there are both positive and negative solutions. The bound state solutions of the swave dirac equation with equal vector and scalar eckarttype potentials in terms of the basic concepts of the shapeinvariance approach in the susyqm. Iii we separate variables and solve the klein gordon equation. Equations onto the klein gordon and secondorder dirac equations adrianaipesciy,raymondegoldsteinyz,andhermann uysy ydepartmentofphysics,universityofarizona,tucson,az85721 zprograminappliedmathematics,universityofarizona,tucson,az85721 abstract.
We also apply a similar but simpler than in the case of the kleingordon equation argument to prove an analogous result for a class of nonlinear dirac. The dirac equation is more restrictive than the klein gordon equation. The problem, at least on a formal level, has a long history, but rigorous results have a rather recent provenance. Thus dirac factored the the klein gordon equation into 2 components. Pdf a discrete scheme for the dirac and kleingordon. We explore klein gordon equation in the new framework of quasihermitian quantum mechanics. Also, the nonlinear klein gordon equation appears as a relativistic. But he failed, and many other people too, until pauli and weisskopf gave the manyparticle theory in 1934 12. The idea of describing quantum physics by an equation will be abandoned in favour of a description in.
In mathematical physics, the dirac algebra is the clifford algebra c. The obvious way of creating a manifestly invariant wave equation is to start from the well known relativistic energy equation and. Klein gordon dirac equation, plane harmonic waves with di. Pdf stationary solutions of the maxwelldirac and the. Wave kleingordon dirac equations in the standard model of. We solve the klein gordon and dirac equations in an open cosmological universe with a partially horn topology in the presence of a time dependent magnetic eld. The klein gordon maxwell system of equations has been applied to the analysis of parametric scattering instabilities in relativistic laserquantum plasma interactions 6, while recent models have been introduced based on the dirac maxwell equations describing the propagation of light in dirac. The klein gordon equation 21 is completely separable in 3, therefore we look for its solution in the form. Bachelot, global existence of large amplitude solutions for dirac klein gordon systems in minkowski space, lecture notes in math. Global wellposedness and parametrices for critical. The kleingordon equation is nice and relativistically invariant, but the fact that its secondorder in time is awkward. In section 3, the dirac oscillator in a noncommutative space is defined and its relation to the landau problem is clarified.
On the rotating nonlinear kleingordon equation archive ouverte. Quantum field theory i babis anastasiou institute for theoretical physics, eth zurich, 8093 zurich, switzerland email. China weimin zhouy and jun yanz department of physics, new york university, 4 washington place, new york, ny, 3 a particle is always not pure. Relativistic wave equation kleinfockgordon equation. Ic9568 international centre for theoretical physics stationary solutions of the maxwell dirac and the klein gordon dirac equations international maria j. It means that if a given field is a solution of the dirac equation, then its components are automatically solutions of the kleingordon equation. Esteban atomic energy agency vladimir georgiev and eric sere united nations educational, scientific and cultural organization miramaretrieste i l l ic9568 international atomic energy agency and i. The kleingordon equation in curved spacetime from within the proper time formalism of schwinger and dewitt has been treated, at least from the gaussian approximation, by beken stein 2, 3. Lecture notes wave equations of relativistic quantum. Dirac equation, kleingordon equation, null structure, local wellposedness, null form estimates. Pdf the dirac equation in quaternionic format researchgate. Kleingordon and dirac equations with thermodynamic. Originally, the klein gordon equation was thought to be the relativistic version of the schr odinger equation that is, an equation for the wave function x. Bournaveas, a new proof of global existence for the dirac klein gordon equations in one space dimension.
Other fundamental equations of qft are the dirac 6 and proca 7 equations, which describe. Bound states of the kleingordon for exponentialtype. All exact solutions of dirac and klein gordon equations, known in 1ate 1981. The dirac equation, like the klein gordon equation, possesses solutions with negative energy, which. Sec iv, using the algebraic method of separation of variables, we reduce the dirac equation to a system of first order coupled differential equations that we solve. It is used to describe all the elementary fermions. Its solutions include a quantum scalar or pseudoscalar field, a field whose. Were firm unique field like electron is represented by the dirac equation, it is not there are certain features the spins of this is not going to be adequately addressed in klein gordon equation ok.
This is basically how the dirac equation is derived justified, really. Hence for the dirac equation to be consistent with the kg equation require. It is secondorder in space and time and manifestly lorentzcovariant. For qcd, each equation has the same structure as the ordinary onebody dirac equation in the presence of an external field similar to the electromagnetic field and an additional external field given by in terms of a. Since the exact solution cannot be obtained explicitly for arbitrary timedependence of the eld, we discuss the asymptotic behavior of. Esteban atomic energy agency vladimir georgiev and eric sere united nations educational, scientific and cultural organization miramaretrieste i l l ic9568 international atomic energy. For qed, each equation has the same structure as the ordinary onebody dirac equation in the presence of an external electromagnetic field, given by the 4potential. Pdf the kleingordon and the dirac oscillators in a non.
For ultrarelativistic problems most of this course, the \weyl representation is more convenient. This equation is nasty, it is hard to see the relativistic invariance and the square root is difficult to interpret quantum mechanically. The kleingordon equation is a relativistic wave equation, related to the schrodinger equation. Dirac in 1928 in developing the dirac equation for spin. Physics 221b academic year 202021 notes 46 introduction to. Pdf in its original form the dirac equation for the free electron and the. Canonical quantization and particle interpretation. Comparison among kleingordon equation, dirac equation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. We can rewrite the kfg equation in a relativistically symmetric form which clearly exhibits its relativistic invariance. Pdf stationary solutions of the maxwelldirac and the klein. Electromagnetic field in radiation and covariant gauge. The dirac equation automatically includes spin, and thus gives a reason why fermionic particles should have spin. Exact solution of kleingordon and dirac equations with snyder.
Local smooth solutions of the nonlinear kleingordon equation. The klein gordon kg equation 4,5, one of the fundamental poincar. To show this we write the klein gordon equation as. Relativistic wave equation klein fock gordon equation. Conversely, if we started with 3 then we could take 2 as our definition of. Derivation of klein gordon equation from dirac equation.
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