Derivation of Dirac equation using Lorentz transform

[krishna mohan]

Hi..I was studying Ryder, Chapter 2[Quantum Field Theory]...he derives the Dirac eq using Lorentz transformations..I found the approach fascinating..but there is one part I do not really understand...

Just a few lines before he writes down the Dirac equation, he identifies [tex]\varphi_{R}(0)[/tex] with [tex]\varphi_{L}(0)[/tex] using the argument that there is no handedness when the particle at rest..

Although I do know this, I am not able to see how this follows from the arguments presented till then...Can someone please clarify this point?

 

[LightHero]

The idea is that when a particle is at rest, it has no handedness. This means that its right- and left-handed components (\varphi_{R}(0) and \varphi_{L}(0)) must be equal. Ryder then uses this fact to derive the Dirac equation. He assumes that there is a relationship between the two components which is invariant under Lorentz transformations. From this assumption, he is able to derive the Dirac equation.

 

[sir-pinski]

Thanks

Sure, I'd be happy to clarify that point for you. The identification of \varphi_{R}(0) with \varphi_{L}(0) is based on the fact that in the rest frame of a particle, there is no preferred direction of spin or handedness. This means that the probability of finding a particle with spin up or spin down is the same, and therefore the wavefunctions for the two states must be equivalent. This is known as the principle of relativity, and it is a fundamental concept in the derivation of the Dirac equation using Lorentz transformations.

To understand this better, let's look at the Lorentz transform for a spinor field, which is given by \psi(x) = S(\Lambda)\psi(\Lambda^{-1}x), where S(\Lambda) is the spinor representation of the Lorentz group and \Lambda is the Lorentz transformation. In the rest frame of a particle, \Lambda is the identity matrix, so the transform simplifies to \psi(x) = S(I)\psi(x) = \psi(x). This means that in the particle's rest frame, the spinor field is unchanged by the Lorentz transformation. This is why we can equate \varphi_{R}(0) with \varphi_{L}(0) in the derivation of the Dirac equation, because both fields are unaffected by the transformation.

I hope this helps clarify the point for you. If you have any further questions, please don't hesitate to ask. The derivation of the Dirac equation using Lorentz transformations is a complex topic, so it's completely understandable that some parts may be difficult to grasp at first. Keep studying and asking questions, and you'll eventually gain a deeper understanding of this fascinating topic. Good luck!