Perturbative Quantum Field Theory

OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), 2018

2018

Physical review, 2016

Schwinger's formalism in quantum field theory can be easily implemented in the case of scalar theories in D dimension with exponential interactions, such as µ D exp(αφ). In particular, we use the relation exp α δ δJ(x) exp(−Z 0 [J]) = exp(−Z 0 [J + α x ]) with J the external source, and α x (y) = αδ(y − x). Such a shift is strictly related to the normal ordering of exp(αφ) and to a scaling relation which follows by renormalizing µ. Next, we derive a new formulation of perturbation theory for the potentials V (φ) = λ n! : φ n :, using the generating functional associated to : exp(αφ) :. The ∆(0)-terms related to the normal ordering are absorbed at once. The functional derivatives with respect to J to compute the generating functional are replaced by ordinary derivatives with respect to auxiliary parameters. We focus on scalar theories, but the method is general and similar investigations extend to other theories.

We show that Schwinger's trick in quantum field theory can be extended to obtain the expression of the partition functions of a class of scalar theories in arbitrary dimensions. These theories correspond to the ones with linear combinations of exponential interactions, such as the potential $\mu^D\exp(\alpha\phi)$. The key point is to note that the exponential of the variation with respect to the external current corresponds to the translation operator, so that $$\exp\big(\alpha{\delta\over \delta J(x)}\big) \exp(-Z_0[J]) = \exp(-Z_0[J+\alpha_x])$$ We derive the scaling relations coming from the renormalization of $\mu$ and compute $\langle \phi(x)\rangle$, suggesting a possible role in a non-perturbative framework for the Higgs mechanism. It turns out that $\mu^D\exp(\alpha\phi)$ can be considered as master potential to investigate other potentials, such as $\lambda\phi^n$.

Lecture Notes in Physics, 2007

We comment on the present status, the concepts and their limitations, and the successes and open problems of the various approaches to a relativistic quantum theory of elementary particles, with a hindsight to questions concerning quantum gravity and string theory.

This report is an attempt to develop and explore the principles underlying the venerable theory that is formulated through the amalgamation of special relativity and quantum theory known as quantum field theory, which is by far the most successful theory that provides the best description of the world as we know it. A thorough quantitative investigation of the canonical quantization approach has been taken into account, in particular with regard to scalar, Dirac and interacting field theories within a Minkowski space-time. The most profound consequences and subtleties upon quantization for the respective theories are delineated and discussed. In addition, we discuss an application for the machinery that is implemented by field theories, in the context of the early universe. We study the history and development of the inflationary paradigm. Furthermore, we attempt to understand the link between elementary particle theory and cosmology, through the phenomenon of particle production by understanding the behaviour of an oscillating inflaton field within a non-expanding background.

2005

An approximate procedure for performing nonperturbative calculations in quantum field theories is presented. The focus will be quantum non-Abelian gauge theories with the goal of understanding some of the open questions of these theories such as the confinement phenomenon and glueballs. One aspect of this nonperturbative method is the breaking down of the non-Abelian gauge group into smaller pieces. For example SU(2)→ U(1) + coset or SU(3)→ SU(2) + coset. The procedure also uses some aspects of an old method by Heisenberg to calculate the n-point Green’s function of a strongly interacting, non-linear theory. Using these ideas we will give approximate calculations of the 2 and 4-points Green’s function of the theories considered.

1995

Even the uninitiated will know that Quantum Field Theory cannot be introduced systematically in just four lectures. I try to give a reasonably connected outline of part of it, from second quantization to the path-integral technique in Euclidean space, where there is an immediate connection with the rules for Feynman diagrams and the partition function of Statistical Mechanics.

Quantum Field Theory II, 2017