# Introductory Remarks to Relativistic Quantum Fields

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In Pa¡t I of euantum Leaps and Bounds, we explored consequences of incorporating the Princþles of Special Relativity (SR) in euantum Mechanics (QM). Incorporating SR in eM, that is, setting up the eM of a Lorentz invariant physical system, led to rhe poincare Argebra, which is a set of commutarion rolations fo¡ the Hamiltonian and the thÌee components of the total momentum, total angular momentum and boost operators for the system. These commutation relations aro the basic equations of relativistic eM. The last five chapters ofpart I contain exampres of Lorentz invariant systems. Relativistic quantum fieid theory ßeFT) is not discussed in part I. That is, not discussed is a physical system whose fundamenta-l dynamical variabres are quantum fields. This example does not appear in part I because we had not developed the tools to do this. These tools are developed here in pa¡t II. In developing quantum field theory, we develop a language (second quantiza_ tion) which provides techniques for studying nonrelativisiic many_body sysrems. The Lagrangian method in ReFT receives a very brief introduction in part IL The many excellent books on ReFT should be consulted for further discussion of the Lagrangirn melhod.l The QM of a system of identical paÍic1es is considered in Chapter 2. occupation number representation for systems of fermions and bosons is discussed in Chapter 3. A list of selectcd ¡eferencc books, joumal articles ånd theses follows Chalter 12 of pútl The Fock space description of the QM of a system of fe¡mions is given in Chapters 4 and 5. A derivation of the Hartree-Fock potential is given in Chapter 4. Quantum flelds for fermions are introduced in Chapter 5. Chapter 5 contains expressions for the Poincare generators for a Lorentz invariant system of noninte¡acting fermions. A system of noninte¡acting spin ] fermions and antifermions is considered in Chapter 6. Femions and antifermions are treated on tho same footing; both appear in the theory with positive energy. It is shown how Dirac,s hole theory a¡ises from the finite theory developed. The Dirac fie1d r/(z) is constructed in Chapter 6 and the poincare generators are expressed in terms of a Lorentz invariant Lagrangian density. The Dirac field satisfios the Dirac equation. The Fock space description of the QM of a system of bosons is given in Chapter 7. Quanfum fields fo¡ bosons are introduced and expressions for the Poincare generatots for a Lorentz invariant system of noninteracting bosons are given. 'l.L^ -^^t^- t:.^ta l/, \ :- ^^-^--.-^--i : ,,ru ùLlr¡úù ¡rc¡r¡ ?l¿./ rr LU[5uulrçLt tll \_llaprcf ^r / an(l tne romcale gengrators are expressed in terms of a Lorcntz invariant Lagrangian density. The scalar field satisfies lhe Klein-Gordon equation. A Lorentz invariant system of noninteracting photons is described in Chapter 8. Quantum electric and magnetic fields are defined in terms of transvorse photon creators and annihilato¡s. The quantum electromagnetic field satisfies Maxwell's equations in free space. Each momentum component of the elecfiomagnetic field corresponds to a hansverss wave moving with the speed of light in the tlirection of the photon momentum. A manifestly cova.riant and gauge invâdant theory of elecûomagnetism is developed in Secrion 8.7. The development is accomplishecl tl'ough innocluction of creators and annihilators for fictious longitudinal and timelike photons. The Lagrangian for the free elect¡omagnotic field is also given fur Section g.7. A very brief infoduction to quantum electrodynamics is given in Section g.g. The instant and point forms of dynamics fo¡ a system of interacting fermions ancl bosons a¡e conside¡ed in chapter 9. The dressing hansfbrmation method for expressing the Hamiltonian in terms of c¡eators aniannihilators for physicar particres is given in section gi|. 'ttre yukawa potential for interacting irrysicar fermions is derived using the dressing transformldon method in Topic 9.g.4. . An Appendix containing some useful commùtation relations fo¡ fermion and boson va¡iables foÌlows Chapter 9.