By Peskin and Schroeder
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21 = , with all other components of ! equal to zero. Then Eq. B (3:20) @ 00 1 1 00 CA V 0 0 0 1 which is just an in nitesimal rotation in the xy-plane. B (3:21) @ 0 10 01 00 CA V 0 0 0 1 an in nitesimal boost in the x-direction. The other components of ! generate the remaining boosts and rotations in a similar manner. 2 The Dirac Equation Now that we have seen one nite-dimensional representation of the Lorentz group, the logical next step would be to develop the formalism for nding all other representations.
Incidentally, we are now ready to understand the origin of the notation L and R for Weyl spinors. The solutions of the Weyl equations are states of de nite helicity, corresponding to left- and right-handed particles, respectively. The Lorentz invariance of helicity (for a massless particle) is manifest in the notation of Weyl spinors, since L and R live in di erent representations of the Lorentz group. It is convenient to write the normalization condition for u(p) in a Lorentzinvariant way. We saw above that y is not Lorentz invariant.
2 The complex scalar eld. Consider the eld theory of a complex-valued scalar eld obeying the Klein-Gordon equation. The action of this theory is S= Z d4 x (@ @ ; m2 ): 34 Chapter 2 The Klein-Gordon Field It is easiest to analyze this theory by considering (x) and (x), rather than the real and imaginary parts of (x), as the basic dynamical variables. (a) Find the conjugate momenta to (x) and (x) and the canonical commutation relations. Show that the Hamiltonian is Z H = d3 x ( +r r + m2 ): Compute the Heisenberg equation of motion for (x) and show that it is indeed the Klein-Gordon equation.
An Introduction to Quantum Field Theory by Peskin and Schroeder
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