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By Professor Dr. G. Scharf (auth.)
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Next we consider absorption (annihilation) operators. Let us first assume £1 = L 2 (IR\ Then the operator (a(f) 32, 33) now reads 1I1'(X') 't' = ( Uo(OC) 0) U (oc) 1p (R() oc - 1x') . 37) The unitary matrix in front realizes a spinor representation Dl/2 EB Dl/2 of the rotation group. 37) with respect to (Xj at oc = 0, we find the infinitesimal generators J = S +L with !. 1(a 0) SJ = _ 2 ° J aj a) . 1 (x' / \ - . 38) S is the spin and L the orbit angular momentum, such that J is the total angular momentum. We finally go one step further in our discussion of invariant wave equations, namely to the representation D(1/2,1/2) considered as case 4) at the beginning of this section. Field Quantization The preceding chapter, entitled "Relativistic Quantum Mechanics" did not really deal with quantum mechanics. The theory so far has no consistent interpretation because it suffers from two defects: (i) The hamiltonian operators are unbounded from below and, (ii) there is no many-particle formulation of the classical Dirac theory. These defects will now be removed. We concentrate first on the second problem (ii). In the course of its treatment, problem (i) will be solved, too. There is a powerful method, called second quantization in Fock space, which transforms a one-particle theory into a many-particle theory.