The Models Of Non-perturbative Quantum Field Theory

In theoretical physics, quantum field theory (QFT) is a theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics, by treating a particle as an excited state of an underlying physical field. As the basic physics theory of microphenomenon, quan-tum field theory has been widely used in various field in physics. The development of particle physics stimulates quantum field theory to study new topics, and make impres-sive progresses, such as composed particle field theory, spontaneous symmetry break-ing, non-Abelian gauge field theroy, vacuum theory and so on, they are all connected to each other through QFT. When study these topics, a widely used tool is path integral and functional analysis. Although the Lagrangians for these theories are always con-cise and exquisite, but when we try to solve physics problems by using path integral to deal with the Lagrangian, the physicists are always inevitably facing complicated calculations, and these calculations are hardly to simplify, hence it is hard to reveal the laws beneath these formulas. For solving this problem, many methods are proposed to calculate the path integral, from perturbative expansion to lattice QCD, all of them serve this purpose. When we study weak coupling theory,such as QED, it is appropriate to use perturbative expansion, most of QED’s experiments’data can be perfectly fitted through the leading orders of perturbative expansion, but it is not a good tool to study strong coupling theory, firstly, the large coupling constants in strong coupling theories lead higher orders of the expansion to strongly affect the results, which could be even stronger than leading orders, and in perturbative expansion, the expanded terms are infi-nite, hence it is impossible to judge the influence from infinite higher orders. Secondly, some theories’non-perturbative phenomenons are invisible to perturbative expansion, such as dynamical chiral symmetry breaking(DCSB), many non-perturbative methods reveal that DCSB produces most of quark’s mass, but it can not be deduced from per-turbative method. Therefore, non-perturbative methods are needed for the explaining of real world. In my dissertation, I have studied various models that were produced in the process of comprehending the path integral and non-perturbative phenomenons. Although some of these models are not suitable to the real world, but they are still useful for us to understand field theory.The first chapter of this dissertation gives a brief introduction to quantum field theory, and then some basic concepts and topics that are important in the context.The second chapter of this dissertation is about Chern-Simons theory in1+2dimension. In this chapter, I have focused on the induced Chern-Simons term, which implies Chern-Simons theory has nature origin. I have also studied the relation be-tween irreducible and reducible representations of QED in1+2dimension, found that reducible representation has U(1)×UA(1) gauge symmetry, and after the localization of UA(1) symmetry, for maintaining the invariance of Lagrangian, we have to introduce new gauge field, this causes we find the connection between reducible representation and two flavors fermion’s irreducible representation. More than that, we realize Chern-Simons-like term in the reducible representation.The third chapter is about Yukawa coupling. On the bases of Yukawa coupling and Higgs mechanism, we build there types of Yukawa couplings that are not included in Standard Model:LY1,LY2and LY3. These there Yukawa couplings are studied in different space-time, because LY1and LY2are not renormalizable in1+3dimension by renormalization rules, hence we have to put them in lower dimension, while LY3is renormalizable in1+3dimension space-time. After the spontaneous symmetry breaking has happened, we find that the produced mass of fermion from these there couplings are different, especially for the3coupling, the matrix boson field Φ in this coupling presents unique properties when Higgs mechanism has been considered. Beside these, we have studied the dynamical properties of LY1and LY2by a non- perturbative method, the large N expansion, and in this method, we have introduced new auxiliary fields that are different from original large N expansion.In the fourth chapter, I have studied chiral anomaly at finite temperature and finite chemical potential, and prove that chiral anomaly is irrelevant to temperature and chem-ical potential. Especially for the problem with chemical potential, we have used two methods to deduce consistent results, one is with the real-time propagator of fermion of imaginary-time finite temperature field theory, the other is with the perturbation in the order of chemical potential. In the meanwhile, we have introduced chiral chemical potential, and found out that in the chiral anomaly mechanism, chiral chemical poten-tial will induce Chern-Simons-like term in the radiative correction, beside that, we have also considered a space-dependent chemical potential which will cause a correction to the original chiral anomaly.The last chapter presents a summary of this dissertation, and then gives some outlook for the investigation.