Fundamental Problems in Quantum Field Theory

This chapter discusses the basic equations in quantum field theory. First, we clarify some important properties of Maxwell equation so that the main part of the electromagnetisms can be easily understood. Then, we present some useful properties of the Dirac equation and its free wave solution. These two equations are the basic ingredients in understanding quantum field theory. We also give the exact energy spectrum of Dirac equation with Coulomb plus gravity potential in hydrogenlike atom

In this chapter, we discuss the S-matrix theory in quantum field theory. Here, we first treat the non-relativistic scattering theory and its relation to the Tmatrix. In particular, we discuss the scattering problem in terms of the Lippmann- Schwinger equation. Then we discuss the S-matrix theory in quantum field theory. This is based on the perturbation theory and we present the example of the S-matrix evaluation. In particular, we discuss some basic problems in the Feynman propagator of photon and show a possible physical difference between Feynman and correct propagators of photon.

Here, we present the calculations of the self-energy of photon and electron. First the renormalization scheme of photon is discussed in connection with the triangle diagrams which have no divergence at all, and therefore the self-energy of photon is not related to any physical observables. Then, we discuss the renormalization scheme of fermion self-energy term which is still connected with the vertex correction in QED. Also, we discuss briefly the physics of Lamb shifts which should be treated in detail in chapter 7. Then, we present the calculation of the photonphoton scattering cross section and some possible experiments on the photon-photon cross section. Finally, the problem of the chiral anomaly equation is discussed, and we see that there are neither anomaly equation nor the violation of the axial current conservation in physical world.

Quantum chromodynamics is the theoretical frame work in which one can treat the physics of the strong interactions. This is the non-abelian gauge field theory, and it cannot be solved in the perturbation theory since the free Lagrangian densities of quarks and gluons are not gauge invariant. In the perturbation theory, we describe all the physical observables in terms of the properties of quarks and gluons, and if they are not related to physical observables, then there is no point of employing the perturbation theory. Here, we also discuss the nucleon-nucleon interactions based on the meson exchange processes and present the calculation of the two pion exchange potential. Further, we discuss some physical observables in connection with quarks, that is, the magnetic moments of nucleons in terms of quark model and the total cross section ratio between σe+e-→ hadrons and σe+e-→μ+μ-. These physical quantities are related to the quark degrees of freedom.

In this chapter, we first present a brief review of the weak interaction theory. In particular, we discuss why the conserved vector current model had to be modified to a new theory. After that, we clarify the physics of the spontaneous symmetry breaking and then discuss the intrinsic problem of the Higgs mechanism in the Weinberg-Salam model. In addition, we present the calculation of the vertex correction due to the weak vector bosons and show that there is no logarithmic divergence in this vertex corrections. Therefore, there is no need of the renormalization procedure in the weak interaction models with massive vector bosons.

In this chapter, we present the new model of the quantum field theory of gravitation which is, by now, properly included into the Lagrangian density of quantum electrodynamics. In this model of QED plus gravity, Dirac fields couple to the electromagnetic field A^μ as well as the gravitational field G. The gravity appears in the mass term as m(1 + gG)ψψ with the coupling constant of g, thus keeping the local gauge invariance of the total Lagrangian of QED plus gravity. Here, after a brief review of the new gravity model, we present the discussion of applying the new gravity model to the time shifts of various kinds of planet motions.

In this chapter, we discuss some of the problems which are not yet fully understood in the calculations of quantum field theory. First, we discuss the neutron EDM in terms of the vertex corrections from the W^± bosons. Then, we present the calculation of the weak charge which may arise from the parity violating interaction. Also, we discuss the Lamb shifts in muonic hydrogen as well as in muonium, and present the calculation of the center of mass corrections in these exotic atoms. It is shown that the effect of the center of mass corrections destroys the close agreement between theory and experiment in muonic hydrogen. In addition, the effect in muonium is so large that it destroys the excellent agreement between Bethe’s calculation and experiment of the Lamb shifts, and this may well affect on the understanding of the fundamental mechanism of the Lamb shifts.