Ebook Relativistic quantum physics - From advanced quantum mechanics to introductory quantum field theory: Part 2

Part 2 book "Relativistic quantum physics - From advanced quantum mechanics to introductory quantum field theory" includes content: Quantization of the dirac field, Maxwell's equations and quantization of the electromagnetic field, the electromagnetic lagrangian and introduction to Yang-Mills theory, asymptotic fields and the LSZ formalism, perturbation theory,.... and other contents.
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Ebook Relativistic quantum physics - From advanced quantum mechanics to introductory quantum field theory: Part 2 7 Quantization of the Dirac field In this chapter, we will quantize the Dirac field using canonical quantization, i.e. with a similiar method as was used for the Klein-Gordon field (cf. Chapter 6). However, we will observe that we need to replace the canonical commutation rela- tions with canonical anticommutation relations in order to obtain positive energy for the quantized Dirac field and to obey the Pauli exclusion principle. In addition, we will study the transformations of parity, time reversal, and charge conjugation as well as the CPT symmetry for this field. Especially, we will investigate the Majo- rana field, which is a special case of the Dirac field. We will also derive Green's functions and propagators and briefly investigate interactions. The Dirac equation (cf. Chapter 3) establishes a single-particle theory (usually known as the Dirac theory), since it cannot take into account creation and anni- hilation of particles. Therefore, the Dirac equation for wave functions has to be replaced by the Dirac equation for quantum fields, which means that the problems of Dirac theory are circumvented by introducing a quantum field theory reformu- lation of this theory; thus abandoning wave functions in favour of quantum fields. In fact, adding to the theory of the Dirac equation for quantum fields the quantized version of the electromagnetic field (see Chapter 8), we will end up with the theory of QED, which will be investigated in detail in Chapters 11-13. 7.1 The free Dirac field The Lagrangian density for a free Dirac field is given by -(i++ -m]_4 ) 1/f=- 1-(-i3+m]_4)1/f- 1 (iaJI1jfy1I+mo/)1/f, Ln=o/ 3 1/f - - 2 2 2 (7.1) ++ where a!Jb = a!Jb- (!Ja)b. Note that the fields 1/f and {r should be independently varied, since they are treated as dynamically independent fields. Using the Euler- Lagrange field equations for {r, i.e. varying {r, leads to the Dirac equation www.pdfgrip.com 7.1 The free Dirac field 139 (7.2) Similarly, one can use the Euler-Lagrange field equations for 1/J to obtain the cor- responding Dirac equation for 1f. Exercise 7.1 Using Euler-Lagrange field equations for the Lagrangian (7.1) and the field 1f, show that the Dirac equation (7 .2) is obtained. In the case of the free Dirac field, again using Noether's theorem (5.69) and remembering that quantities for a Dirac field do not necessarily commute, the energy-momentum tensor is defined as Tf.lV = a£o av,,, + av,/, a£o f.lV £ (7.3) acaf.11/f) 'f' 'f' acaf.11/f) - g o and inserting Eq. (7.1) into Eq. (7.3), we find that r 11 v = ~ [ {fy 11 av1j; _ (av{f) yl11j;], (7.4) which is not symmetric, but could be made symmetric. However, similar to the case of the free neutral Klein-Gordon field, the energy-momentum tensor T 11 v is conserved, i.e. o T 11 v = 0. Now, one can compute 11 p11 = J Toll d3x = ~ J[ {fyo()l11j;- (all{f) yo1/J] d3x = i J {fyoal11j; d3x, (7.5) where we have used integration by parts in the last step as well as the fact that a 4-divergence does not contribute to the integral, and especially, using the defini- tions of derivatives (2.8) and the Dirac equation (7.2), the total Hamiltonian and 3-momentum operator become H=P 0 = J {f(-iy-V+m]A)1/fd3 x= J 1/lt(-ia·V+f3m)1/fd3 x, (7.6) P = -i j {fy 0V1/f d3x = -i j 1/J tv1j; d3x, (7.7) respectively. In addition, one has the operators for the components of the total angular momentum and the boosts M 11 v = J 1/Jt [ -i(x 11 av -xvo 11 ) - ~a 11 v] 1/fd3x. (7.8) Note that in the case of the Dirac field, the total angular momentum contains spin, since Dirac particles are spin-1/2 particles, which is represented by the term a 11 v /2 in Eq. (7.8). Furthermore, the conjugate momenta are given by a£o i t - a£o i 0 :rr = - . = -1/1 and :rr = - . = --y 1/J. (7.9) o1/f 2 a{f 2 www.pdfgrip.com 140 Quantization of the Dirac field However, from this result, we observe that the conjugate momenta rr and ir are not independent, i.e. they are dependent. Thus, instead of Eq. (7 .I), we consider the Lagrangian density (7.10) which differs from Eq. (7.1) by the fact that we now impose that the derivative operator solely acts on 1/f (and not on lfr) and we are only varying the field lfr [which again leads to the Dirac equation (7.2)]. Therefore, in this case, we obtain the conjugate momenta rr = iljfi' and ir = 0. (7.11) Note that the difference between the Lagrangian densi ...