| Summary: | It appears that the fundamental interactions of nature which are electromagnetic, strong and weak interactions are gauge type. The following analysis basically concerns quantum chromodynamics(QCD) which is a non-abelian gauge theory that respects the gauge symmetry of the SU(3) group. Although in Quantum Field Theory most of the phenomena are treated as perturbative effects, here we will present some non-perturbative ones that happens to give big contribution in QCD. As we shall see, those effects give a nice description for the structure of the vacuum of QCD which remains unknown.
We are starting our analysis with some analogs in Quantum mechanics that appear to be simple for understanding and give us a nice illustration of these effects which are called instantons. Instantons are tunneling effects that appear between minimum energy states of our system and give us some contribution to the energy of the vacuum.
What concerns us the most is how instantons appear in any Yang-Mills theory and as a result, in QCD as well. We will see that there are some topological issues that separate the classical minimas of the energy in a Yang-Mills theory into distinct classes and therefore instanton occur as tunneling trajectories between these minimas. We will also introduce the θ vacuum which is a way of describing the vacuum of QCD as a superposition of these minimum states which are connected through instantons.
We will examine a special category of instantons called BPST instanton as it turns out that this is the one that gives the most contribution in QCD phenomena and thus it is the only one worth analyzing. Therefore, we find some explicit solutions for the BPST instanton which later will be used in some calculations. These calculations will give us in detail the contribution of instantons in the vacuum of any Yang-Mills theory and especially in QCD.
The last chapter focuses mainly in solution of a big problem in QCD which comes with the insertion of instantons in Yang-Mills theory. This problem is known as the U(1) problem and appears when massless fermions are included in our theory and the analysis describes how instantons appear to solve this problem.
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