Advanced Quantum Field Theory: Intuition With Path Integrals

Master Quantum Field Theory and Renormalization group with Path Integrals: Intuitive Insights & Practical Applications

What you'll learn

Master Path Integrals: Understand the concept of path integrals in Quantum Field Theory, and learn how they offer a unique perspective on the subject

Derive Feynman Rules: Gain the ability to derive Feynman rules naturally from the path integral formulation

Dive into Renormalization: Delve into the essential concept of renormalization, with a particular focus on the renormalization group.

Comprehend Non-Commutativity: Explore the non-commutative nature of Quantum Field Theory by examining how path integrals incorporate non-differentiable paths

Derive the Yukawa potential: while discussing renormalization, we will see how some theories give rise to long-range potentials and some others short-range ones

Discover the partition function: essential tool to the definition of path integrals

Learn how to use correlation functions, whose interpretation is related to Feynman diagrams and particle interactions

Learn how to use perturbation theory in QFT

Learn the orbit-stabilizer theorem, another key concept related to the interpretation of Feynman diagrams

Discover the effective action: this tool is key to understanding renormalization

Discover the Callan Symanzik equation, which appears in the theory of the renormalization group

Learn why "anomalous" dimensions arise in QFT

Requirements

Schrödinger equation

Operators, states, eigenstates, eigenvalues

familiarity with bra-ket notation

Classical theory of Fields (Lagrangian, action, etc)

Multivariable Calculus

Complex calculus (in particular, the Residue Theorem)

Familiarity with QFT and second quantization will enhance your learning experience

Special Relativity (and tensors)

Description

Welcome to "Advanced Quantum Field Theory: Intuition with Path Integrals." In this course, we take a unique approach to delve deeper into the fascinating world of Quantum Field Theory (QFT). The foundations of this course are based on the notes of Professor David Skinner, although an original perspective will be given, which emphasizes intuition and the power of path integrals.What You Will Learn:Path Integrals Demystified: Explore Quantum Field Theory from a different angle, using path integrals as our guiding tool. Unlike the traditional "second quantization" approach, we won't begin with a classical field and then transform it into an operator, but rather, we'll start directly from the action and Lagrangian, offering a more intuitive understanding.Summing Over Infinite Paths: In classical field theory, a single trajectory minimizes the action. Path integrals take us beyond this limitation. You'll grasp the essence of QFT by summing over countless possible paths, gaining insight into the fundamental role of uncertainty.Zero-Dimensional QFT: We'll begin with simpler mathematics in a zero-dimensional QFT, paving the way for a natural derivation of Feynman rules directly from the path integral formulation.Exploring Non-Commutativity: Delve into the concept of non-commutativity in Quantum Field Theory. Discover how path integrals naturally encompass non-commutative behaviors due to the summation over non-differentiable paths.Renormalization Insights: Gain a deep understanding of renormalization, a crucial concept often overlooked in basic QFT courses. Explore the intricacies of the renormalization group, a fundamental aspect of Quantum Field Theory.Course Content: the current course content covers path integrals, zero-dimensional QFT, one-dimensional QFT, and renormalization. The material may be expanded in the future to include additional sections.Prerequisites: To fully benefit from this course, it's essential to have a grasp of:Schrödinger equationoperatorsbra-ket notationmultivariable calculus and complex calculusClassical Theory of fieldsSpecial Relativity and tensorsFamiliarity with QFT and second quantization will enhance your learning experience.Enroll today and embark on a captivating journey into the heart of Quantum Field Theory. Discover the power of path integrals and develop a deep, intuitive understanding of this fascinating field. Join this course to reshape your perspective on Advanced Quantum Field Theory!

Overview

Section 1: Path integrals and Quantum Mechanics

Lecture 1 Introduction to the course

Lecture 2 Course prerequisites

Lecture 3 Path integral derivation

Lecture 4 Some intuition behind the path integral

Section 2: QFT in zero dimensions

Lecture 5 Free field theory in zero dimensions

Lecture 6 The current in the Free field theory in zero dimensions

Lecture 7 Wick theorem as a probabilistic theorem in QFT

Lecture 8 Integrals in perturbation theory

Lecture 9 Example of perturbation theory

Lecture 10 Feynman diagrams from a combinatoric perspective

Lecture 11 Feynman diagrams and orbit stabilizer theorem

Lecture 12 Wilsonian effective action

Lecture 13 Feynman rules with two fields and 4-valent vertices

Lecture 14 Example of action with two fields

Lecture 15 Direct calculation of the effective action without Feynman rules

Lecture 16 Direct calculation of the correlation function without Feynman rules

Lecture 17 How renormalization comes forth

Section 3: QFT in 1 dimension

Lecture 18 QFT in one dimension: action with two fields

Lecture 19 Logarithm of a determinant

Lecture 20 Non-locality of interactions in a one-dimensional Quantum Field Theory

Lecture 21 Non commutativity and path integrals

Lecture 22 Correlation functions and time ordered products

Section 4: Theory of renormalization in physics

Lecture 23 Theory of the renormalization group

Lecture 24 Running of couplings and Callan Symanzik equation

Lecture 25 Anomalous dimensions - part 1

Lecture 26 Anomalous dimensions - part 2

Lecture 27 Scale invariant theories

Lecture 28 Massive scalar field and the Yukawa potential

Lecture 29 Derivation of the Yukawa potential from the Klein Gordon field in 3 dimension

Lecture 30 Renormalization group flow

Lecture 31 The local potential approximation: example on the running of couplings

Section 5: Appendix

Lecture 32 Complex Gaussian integrals

Lecture 33 Lagrange duplication formula

Lecture 34 Relation between Beta and Gamma function

Lecture 35 Derivation of a green function from a differential equation

Advanced (Master-level) Students,Physicists and Researchers: Professionals in the field of theoretical physics, including physicists, researchers, and academics, who wish to enhance their expertise in Quantum Field Theory.,Mathematics Enthusiasts, Mathematicians, interested in the intersection of advanced mathematics and theoretical physics, looking to explore the beauty of Quantum Field Theory from a mathematical perspective.,Physics Enthusiasts, passionate about the world of quantum physics and eager to deepen their understanding of Quantum Field Theory.