The dirac equation under lorentz and parity transformations. Representation of the symmetry group of spacetime in special relativity hendrik antoon lorentz right after whom the lorentz group is named and albert einstein whose special theory of relativity is the main source of application. To extend to parity the spacetime transformations, we combine these onedimensional spinors into bidimensional column vectors. In section 2, we introduce the quaternionic lorentz group 3 by using leftright operators 6,7. In this paper we demonstrate that both of them can be considered as a special cases of the more general equation. Relativistic wave equations and their derivation 5. In pure mathematics, the dirac operator also plays an important role in di erential geometry and index. First we analyze the full group of lorentz transformations and its four distinct, connected components. Contents 1 lorentz group national tsing hua university. It remains to find a transformation matrix that satisfies.
Contents minkowski space and lorentz transformation generators of lorentz group. A comparison of this equation with reveals that the dirac equation takes the same form in frames and. In this chapter, we will study the dirac equation which describes spin12 particles. The dirac equation and the lorentz group part i classical approach 1 derivation of the dirac equation the basic idea is to use the standard quantum mechanical substitutions p. It appears necessary that mechanics in general have the symmetry of the lorentz group, and that requirement corresponds to the general applicability of special relativity. Introduction before we examine how the dirac equation and dirac wave function transform under lorentz transformations we present some material on the lorentz transformations themselves. A real version of the dirac equation and its coupling to. Quaternionic lorentz group and dirac equation springerlink. The lorentz group, its representations and generators. The development of quantum mechanics is presented from a historical perspective. Spinorial representation of lorentz group for solution to dirac equation.
Incidentally, it is clear from and that the matrices are the same in all inertial frames. The lorentz group is a collection of linear transformations of spacetime coordinates x. The dirac equation university of california, san diego. Professor at leiden, founded in 1895 the theory of electrons, with which he explained the zeeman effect as well as the rotation of the plane of polarization of light in a magnetic field. Substituting this into the second equation we obtain x0 l01.
Quaternions, lorentz group and the dirac theory progress. Lorentz symmetry, weyl spinors, chirality and dirac equation. Dirac matrices and lorentz spinors university of texas. Using the lorentz transformation, the other worldline is found at t0 t. From the transformation properties of the twocomponent spinors, we derive a quaternionic chiral representation for the spacetime algebra. The abrahamlorentz force is the source of the radiation resistance of a radio antenna radiating radio waves.
In other words, the dirac equation is lorentz invariant. Quaternionic lorentz group and dirac equation core. Lorentz symmetry, weyl spinors, chirality and dirac equation kow lung chang physics department, national taiwan university may 2, 20 kow lung chang lorentz symmetry, weyl spinors, chirality and dirac equation. Representation theory of the lorentz group wikipedia. To motivate the dirac equation, we will start by studying the appropriate representation of the lorentz group. Fermions and the dirac equation in 1928 dirac proposed the following form for the electron wave equation.
Incidentally, you may be aware that there are two ways to derive dirac s electron equation. Similarly, dirac equation is lorentz covariant, but the wavefunction will change when we make a lorentz transformation. It is described by the dirac equation, and as a eld with halfinteger spin it should obey fermi statistics. If we also require the local lorentz invariance of the dirac lagrangian, the quaternionic nature of the spinor quaternion introduces a coupling of it to a yangmills type gauge field a 18aij bibj bj bi, which has 6 components for each, since the lorentz group is a 6parameter group. Dirac attempted to overcome some of the problems of. Lorentz transformations in quantum mechanics and the covariance of the dirac equation these notes supplement chapter 2 of bjorken and drell, which concerns the covariance. Physics 221ab spring 1997 notes 36 lorentz transformations in quantum mechanics and the covariance of the dirac equation these notes supplement chapter 2 of bjorken and drell, which concerns the covariance. How dirac and majorana equations are related murod abdukhakimov murod. In these notes we study rotations in r3 and lorentz transformations in r4. Consider a frame fwith an observer o and coordinates x. On the basis of the dirac theory using the spinor group spin2. The dirac equation should be invariant under lorentz boosts and under rotations, both of which are just changes in the definition of an inertial coordinate system. In this chapter we will describe the dirac equation, whose quantization gives rise to fermionic spin 12 particles.
In these notes we will work at the level of classical special relativity, without reference to quantum. Tomotivatethediracequation,wewillstart by studying the appropriate representation of the lorentz group. An introduction to the lorentzdirac equation eric poisson department of physics, university of guelph, guelph, ontario, canada n1g 2w1 these notes provide two derivations of the lorentz dirac equation. Representations of the discrete inhomogeneous lorentz group and. The second is patterned after dirac, and is based upon considerations of energymomentum conservation. Sl2, c group, which is a double covering of the lorentz group. Let u be a dirac spinor and suppose that u obeys the equation. The schrodinger equation is not relativistically invariant. Let u be a dirac spinor and suppose that u obeys the equation p. One could say that a spinor is the most basic sort of mathematical object that can be lorentz transformed. Lorentz transformation, chirality, and the spinor minkowski. The abraham lorentz force is the source of the radiation resistance of a radio antenna radiating radio waves. Quaternions, lorentz group and the dirac theory progress of. Lorentz transformations, rotations, and boosts arthur jaffe november 23, 20 abstract.
The lorentz group is a lie group of symmetries of the spacetime of special relativity. This has led to a whole area of research in theoretical and mathematical physics. Quantum field theory i babis anastasiou institute for theoretical physics, eth zurich, 8093 zurich, switzerland email. This material is intended to supplement chapter 1, for those students who are not as familiar with the lorentz group and dirac equation as they. Lorentz covariance of the dirac equation springerlink. Quaternionic lorentz group and dirac equation request pdf. Where the lorentzabrahamdirac equation for the radiation reactionforce fails, andwhy the proofs break down dieter gromes institut fu.
The dirac theory is treated by noting that the lorentz group is realized by a subset of sl2,h, each element being characterized by a pair of unit quatenion rotation and pure quaterion boost. The generators of the lorentz group will later play a critical role in. Quaternions, lorentz group and the dirac theory pdf. Spinorial representation of lorentz group for solution to. C of the homogeneous lorentz group rather than the lorentz group itself.
It is shown that the lorentzdirac equation contradicts the differential consequence of the centreofmass conserved quantity which arises from the invariance of the system under lorentz. The same is true for the relativistic dirac equation of the electron, realized by. A classical form of the dirac equation is obtained, and the quantum prediction. Lorentz covariance of the dirac equation means that the matrices are the same in both frames. An introduction to the lorentzdirac equation eric poisson department of physics, university of guelph, guelph, ontario, canada n1g 2w1 these notes provide two derivations of the lorentzdirac equation. It turns out that they are related to representations of lorentz group. In this section we consider the natural representation of the lorentz group l, i.
Thus in the primed frame at a given instant the two. From the transformation properties of the twocomponent spinors. Verify explicitly the commutation relation between the gamma matrices and the generators of the lorentz group in the dirac representation. Lorentz group and the dirac equation lie groups and lie. Where the lorentzabrahamdirac equation for the radiation. Under lorentz boosts, transforms like a 4vector but the matrices are constant. We will provide in the following a new formulation of the dirac equation in the chiral. We formulate lorentz group representations in which ordinary complex numbers are replaced by linear functions of real quaternions and introduce dotted and undotted quaternionic onedimensional spinors. Lecture notes wave equations of relativistic quantum. The dirac equation we will try to find a relativistic quantum mechanical description of the electron. Lecture 22 relativistic quantum mechanics tcm group. Appendix c lorentz group and the dirac algebra mcgill physics.
The notes conclude with a discussion of the difficulties associated with the interpretation of the lorentzdirac equation as an equation of motion for a. It is shown that the lorentz dirac equation contradicts the differential consequence of the centreofmass conserved quantity which arises from the invariance of the system under lorentz. C will be discussed in some detail laterfor now we just mention that it acts most directly on twocomponent, complex vectors, the basic spinors that are analogous to fourvectors for the lorentz group. From the dirac representation of the lorentz group. The notes conclude with a discussion of the difficulties associated with the interpretation of the lorentz dirac equation as an equation of motion for a point charge. This summary is placed here rather than at the end of the chapter in order to help the reader follow the main thread of the argument. Appendix c lorentz group and the dirac algebra theory groups. The weyl spinors will then fall out automatically from this analysis.
There are pathological solutions of the abraham lorentz dirac equation in which a particle accelerates in advance of the application of a force, socalled preacceleration solutions. The lorentz group is the symmetry group of electrodynamics, of the electroweak gauge theory, and of the strong interactions described by quantum chromodynamics. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly. Also we would like to have a consistent description of the spin of the electron that in the nonrelativistic theory has to be added by hand. Relativistic quantum mechanics is developed, including the kleingordon equation and up to the dirac equation. Consider an infinitesimal lorentz transformation, for which. In 3d, the spinor j 1 2 representation of the spin3 rotation group is constructed from the pauli matrices.
In this section we will describe the dirac equation, whose quantization gives rise to fermionic spin 12particles. The dirac equation under lorentz and parity transformations in the last lecture we studied the solutions of the dirac equation which in a covariant form is given by. The dirac algebra for any spin university of michigan. Incidentally, you may be aware that there are two ways to derive diracs electron equation. The oneparticle states are nonlocal in the spacetime. There are pathological solutions of the abrahamlorentzdirac equation in which a particle accelerates in advance of the application of a force, socalled preacceleration solutions. Spinor formulation of relativistic quantum mechanics. Moreover, the dirac equation leads to nontrivial representations of the lorentz group, so called spinor eprresentations. Presented at the lorentz group symposium, institute for theoretical physics, university of colorado, summer, 1964.