Gauge theory

A gauge theory is a special kind of field theory used in physics. It describes how particles and forces behave when certain rules stay the same even when you change how you look at things in tiny, local areas of space and time. These unchanging rules are called symmetries, and they help scientists understand the hidden patterns in nature.

In gauge theories, the way things change under these local symmetries is very important. Each symmetry group has special fields called gauge fields. When these theories are studied using quantum ideas, the tiny particles that carry these fields are called gauge bosons. For example, the photon, which carries the force of electricity and magnetism, is a gauge boson.

Gauge theories are very powerful because they explain many of the forces and particles in the universe. They are the foundation of the Standard Model, which describes all the known particles and three of the four basic forces. They also help scientists think about gravity in new ways, offering ideas for how gravity might fit into the same picture. Today, gauge theories are used in many areas of physics, from the study of materials to the search for new particles and forces.

History

The idea of gauge theory started with the work of Hermann Weyl in 1918. Weyl tried to combine ideas from general relativity with electromagnetism. After quantum mechanics was developed, Weyl and others changed their approach, turning it into a study of how certain properties change in a way that keeps the overall system balanced.

Later, Chen Ning Yang and Robert Mills built on this idea in 1954, creating a new theory for understanding atomic nuclei. Their work helped explain the forces that hold atoms together and influenced many areas of physics. Today, gauge theories are important in describing the fundamental forces of nature, such as electromagnetism and the forces inside atoms.

Description

Global and local symmetries

Global symmetry

In physics, the way we describe things often has extra details that don’t change what we see. For example, when we describe motion, changing where we stand to watch still shows the same thing happening. These changes form a group of “symmetries,” meaning different ways to describe the same situation.

This idea can also work when the changes depend on where you are, leading to more complex theories.

Example of global symmetry

Sometimes, numbers we use to describe things have meaning beyond just being numbers, like speed or direction. When we turn our view, these numbers change in a matching way. If we describe fluid moving one way and then turn our view, the fluid still moves the same, just described from a new angle.

Local symmetry

Use of fiber bundles to describe local symmetries

To describe more complex situations, we sometimes need extra details at each point that don’t match simple space-time points. These extra details can change in ways that still keep the physics the same, called gauge transformations.

In many gauge theories, these changes follow a specific pattern, like the simplest one used in modern theories of electricity and magnetism.

Gauge fields

To handle these local changes, gauge theories add something called a gauge field. This field is important because it shows how things change from place to place.

When we study how these theories behave, the gauge field acts like other parts of the physics, adding its own energy.

Physical experiments

Gauge theories help us understand experiments by limiting what can happen based on how we set up the test and then calculating what we might see.

Continuum theories

The theories we use for electricity, magnetism, and gravity work well for everyday things but can fail for very small or very large scales.

Quantum field theories

Main article: Quantum field theory

Quantum field theories, like those used to understand tiny particles, start similarly but handle extra details differently. Some use simpler methods, while others use more advanced steps to deal with complex patterns.

Classical gauge theory

Classical electromagnetism

In electrostatics, we can talk about either the electric field, E, or its related electric potential, V. Knowing one helps us find the other, except when potentials differ by a constant amount. This is because the electric field depends on how the potential changes from one place to another, and a constant difference disappears when we look at these changes. In terms of vector calculus, the electric field is the gradient of the potential, E = −∇V.

When we move from static electricity to electromagnetism, we add a second potential, the vector potential A, with special relationships:

E = −∇V − ∂A/∂t
B = ∇ × A

The general gauge transformations now include not just V → V + C but also A → A + ∇f and V → V − ∂f/∂t, where f is a smooth function of position and time. The electromagnetic fields stay the same under these transformations.

Example: scalar O(n) gauge theory

This section needs knowledge of classical or quantum field theory and Lagrangians.

Imagine a set of n non-interacting real scalar fields, all with the same mass m. This system can be described by adding up actions for each scalar field φ_i.

The Lagrangian can be written compactly by grouping the fields into a vector Φ.

The Lagrangian stays the same under a transformation where Φ is multiplied by a constant matrix from the n-by-n orthogonal group O(n). This is called the gauge group.

To make the Lagrangian work for changes that depend on location, we need to adjust the derivative. This leads to a new "derivative" called a (gauge) covariant derivative, which includes an extra term involving a gauge field A. This gauge field transforms in a specific way to keep the Lagrangian unchanged.

Finally, the full Lagrangian now includes an interaction Lagrangian. This term creates interactions between the scalar fields because of the need for local gauge invariance. In quantum field theory, the particles of the gauge field are called gauge bosons, and they mediate these interactions.

Yang–Mills Lagrangian for the gauge field

Main article: Yang–Mills theory

To fully describe a classical gauge theory, we need to specify the gauge field A at every point in space and time. One way to do this is by using a Lagrangian that depends on the field strength F, derived from A. This Lagrangian is called a Yang–Mills action.

The complete Lagrangian for the gauge theory combines the local Lagrangian, the interaction Lagrangian, and the gauge field Lagrangian.

Example: electrodynamics

As a simple use of these ideas, consider electrodynamics with just the electron field. The basic action for the electron field leads to the Dirac equation.

The global symmetry here is a rotation of the field's phase angle. Making this symmetry local leads to a covariant derivative that includes the electromagnetic vector potential. This naturally introduces the minimal coupling of the electromagnetic field to the electron field.

Adding a Lagrangian for the gauge field gives us the starting point for quantum electrodynamics.

See also: Dirac equation, Maxwell's equations, and Quantum electrodynamics

Mathematical formalism

See also: Gauge theory (mathematics)

Gauge theories are often studied using ideas from a branch of math called differential geometry. In simple terms, a "gauge" is a way to pick a specific view of certain mathematical objects. A "gauge transformation" is a change between two of these views.

These theories mostly focus on special structures called connections, especially when studied by scientists who work with very small particles like high-energy physicists. But connections are not the only important idea in gauge theory. There are also other types of transformations, like those seen in BF theory and nonlinear sigma models.

When space or spacetime is used as a background, and a special kind of math group is chosen, the different views form a structured system. These views help define important tools like the covariant derivative and curvature form, which describe how things change and twist in space.

One key idea in gauge theory is the Yang–Mills action, which helps describe how these structures behave together. Another important concept is the Wilson loop, which stays the same no matter how we change our view, making it very useful in studying these theories.

Quantization of gauge theories

Gauge theories can be studied using special math tools that work for any kind of quantum field theory. However, gauge theories have extra rules that make this more challenging. Even so, these rules can sometimes make calculations easier. For example, special math connections called Ward identities link different parts of the calculations, such as renormalization.

The first gauge theory studied this way was quantum electrodynamics (QED). Scientists developed methods to handle these special rules, like the Gupta–Bleuler method. Today, more complex theories are studied using different approaches. The goal is to calculate how likely different events are in the theory, which relates to special math descriptions called correlation functions in a base state called the vacuum state. This often needs a process called renormalization.

When a key number in the theory, called the running coupling, is small, scientists can use simpler math methods called perturbation theory. Some methods, like canonical quantization, are designed to make these easier and help test the theories with experiments. However, many important questions need more powerful computers and special methods like lattice gauge theory, which are still being developed.

Sometimes, certain symmetry rules that work in simple math descriptions do not hold when studied with quantum tools. This is called an anomaly. Well-known examples include:

• The scale anomaly, which changes how a key number behaves. In QED, this leads to the Landau pole. In quantum chromodynamics (QCD), it causes asymptotic freedom.
• The chiral anomaly, which connects to special math ideas through instantons. In QCD, this anomaly explains why a pion can break down into two photons.
• The gauge anomaly, which must balance out in any proper theory. In the electroweak theory, this balance needs the same number of quarks and leptons.

Pure gauge

A pure gauge is a special set of field setups made by changing a zero field setup through a gauge transformation. This creates a specific path in the space of field setups.

In simpler cases, a pure gauge is made by changing the field setups using a simple math rule. This rule helps us understand how these setups can look different but still work the same way.