Introduction to Quantum Field Theory

• The language used to describe fundamental physics is quantum field theory (QFT)

Introduction to Quantum Field Theory

• QFT combines quantum mechanics with special relativity
• Nature is described in terms of fields, not particles
• Quantum mechanics describes nature in terms of particles with wave-like properties, in contrast to QFT.
• It's the most accurate natural theory ever created
• Quantum Electrodynamics (QED), Quantum Chromodynamics (QCD), and the Standard Model are examples of QFTs

From Quantum Mechanics to Quantum Field Theory

• Quantum mechanics (QM) describes many phenomena successfully, but has issues being combined with special relativity
• QM uses the Schrödinger equation to describe how a particle's wave function evolves over time
• Special relativity is inconsistent with the Schrödinger equation because it treats time and space differently
• There are issues with the Klein-Gordon equation, a relativistic version of the Schrödinger equation, such as predicting negative probabilities
• QM struggles to describe systems with many particles
• A wave function in QM depends on 3N spatial coordinates when there are N particles, this becomes very complicated
• QFT addresses these issues by treating particles as excitations of a field
• A field exists at every point in space and time
• Instead of each particle having a wave function, there is a single field that describes all particles of a given type
• The electron field, for example, describes all electrons in the universe
• Particles are viewed as quantized field excitations
• This automatically incorporates special relativity, enabling particle creation and annihilation

Fields

• A field is a physical quantity possessing a value at each point in space and time
• Temperature, wind velocity, and the electromagnetic field are a few examples
• The fundamental objects in QFT are fields
• Particles are viewed as emergent properties arising from field quantization

Quantum Electrodynamics (QED)

• Describes the interaction between light and matter
• One of the most successful QFTs
• Experimental results are predicted to an astonishing degree of accuracy (e.g., the anomalous magnetic dipole moment of the electron)
• Deals with the electromagnetic force

Structure of QFT

• Free Field Theory: Describes particles that do not interact with each other
• Interacting Field Theory: Describes particles that interact with each other; this is closer to reality
• Interaction can be represented using Feynman diagrams

Feynman Diagrams

• Pictorial representation of particle interactions
• A powerful tool for calculating the probabilities of different processes
• Each line represents a particle, and each vertex represents an interaction

Renormalization

• A procedure to remove infinities that arise in calculations in QFT
• Calculations in QFT sometimes yield infinite results, which are unphysical
• The parameters of the theory (mass and charge) are redefined in renormalization in order to eliminate these infinities
• One of QFT's trickiest aspects

Quantum Chromodynamics (QCD)

• Describes the strong force, which binds quarks together to form protons and neutrons
• It is more complicated than QED, because the force-carrying particles (gluons) also interact with each other

The Standard Model

• Combines QED and QCD with the weak force
• Describes all known fundamental particles and their interactions, gravity excluded
• Although highly successful, it is incomplete
• Dark matter, dark energy, and gravity are not included

Challenges and Open Questions

• Finding a theory of quantum gravity
• Understanding dark matter and dark energy
• Explaining the masses of the neutrinos
• The hierarchy problem (why is gravity so much weaker than the other forces?)