By Professor Dr. F. Scheck (auth.)
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Extra resources for Electroweak and Strong Interactions: An Introduction to Theoretical Particle Physics
Go to the rest system of the particle where P = (m,O). 85) reduces to (o~ o)(~:i~;) 0 V3(0) v4 (0) =0 , the solutions of which are respectively. Here i r ) is a two-component object which, in fact, must be a Pauli spin or, well-known from nonrelativistic quantum mechanics. 2. The solutions for arbitrary momentum p can then be expressed in terms of the rest-frame solutions by utilizing the relation y/1 P/1 Yvpv =! 75). 85) respectively. 4 Plane Wave Solutions of the Dirac Equation 27 This is a covariant normalization of one-particle states.
Ej'-'(x)AI'(x) is the interaction with the Maxwell field, represented by the potentials AI" In this case we can compute single particle matrix elements of j I'(x) and of the corresponding charge operator Q. 122) are not well-behaved and, strictly speaking, we should smooth them out with appropriate weight functions. 8 Lagrange Density of Dirac Particles 39 that the integral of /(x) over all space is a constant of the motion: Q: = f d 3 x/(x; t), :t Q = O. 92-94). One finds Q= I fd3 ---.!!.. 124b), Q= ±fd P 3 2Ep r= 1 [drlt(p)drJ(p) _ HrJt(p) b(r)(p)].
As for f(x - y) we note that there are precisely two linearly independent, Lorentz invariant solutions of the Klein-Gordon equation for mass m. 111) + e'k Z). 112) 1 e Z), Z - 1 with w k = (k 2 + m2 )l/2 and z = x - y. We note, in particular, that Llo(z; m) is antisymmetric when z is replaced by - z and vanishes for spacelike separation of x and y. Thus, Llo is a causal distribution. Lll (z; m) on the other hand, is symmetric in z and does not vanish for Z2 < O. Thus, Lll cannot be a causal distribution.