Dirac Equation Derivation with Inhomogeneous Lorentz Group in QFT Book

[nklohit]

I've seen the derivation of Dirac Equation using Inhomogeneous Lorentz Group in L H Ryder's QFT book.Can anybody give some comprehensible descriptions of this method?

 

[Avodyne]

You could take a look at Srednicki's book (draft copy free online, google to find it), which starts with reps of the Lorentz group and slowly builds up to the Dirac lagrangian.

Incidentally, the Dirac equation can't really be derived, it is just postulated as following from the simplest lagrangian (that is, terms with the fewest derivatives) that can be written down for a field corresponding to spin-1/2 particles.

For the extreme version of this point of view, see Weinberg's book (which is thorough and extremely detailed, and therefore comprehensible, but only with a lot of effort).

 

[Fra]

It's also interesting to note the relation between KG and Dirac eqs. You can by a special simple change of variables, transform the second order (KG equation) into a system of first order equations (Dirac), out pops tha pauli matrices.

I'm not sure what it prooves, but it's at least when coming from the classical path, an interesting insight about a possible mathematical relation between the spin ½ system and the spinless KG. It sort of allows for a kind of mathematical "interpretation" of what spin ½ is in terms of a "transformation" of a spinless system.

when I took the QM courses I don't recall this beeing the way it was shown in class but I just noted this myself when playing around, and found it to be an interesting curiosity.

/Fredrik

 

[massless]

We can simply derive KG equation from Dirac equation. More generally ,even the Bargmann-Wigner equation which describes the higher spins can also lead to KG equation, but the fai function in the equation are totally different which result in the corresponding spin quantum numbers.

 

[nklohit]

Thanks to everyone. I am trying Weinberg's and Srednicki's besides Ryder's.