Gauge theory

Description

See the key example Time zones example.
Gauge theory is a term that refers to a quite general type of theories in physics which share some key concepts. In this note we are going to name them, relating them with their mathematical counterpart.
In a gauge theory we start with a manifold $M$ that represents, usually, space, time or spacetime (see relativistic spacetime), and a G-bundle over it

with typical fibre $S$. The physical situation that we are dealing with determines an atlas $\{(U_{\alpha },{\varphi }_{\alpha })\}$. The fibre $E_m$ is called the internal space for $m\in M$ and $G$ is called the gauge group of the theory. A section $\sigma :M\to E$ is called a matter field and represents an observable quantity.
A local (global) section of the associated principal bundle $P$,

is called a local (global) gauge, and the choice of such a section is called gauge fixing. As it is said here, it let us to speak of the components of the matter field respect to this gauge fixing. In a sense, fixing a gauge is like specifying a reference frame in $E_m$ for every point $m$ of the manifold.
Given any other gauge $p^{\prime }$ defined on $U$ we can define a map

such that $p(m)=p^{\prime }(m)\cdot \gamma (m)$, and we call it a gauge transformation. The set of all gauge transformations constitutes a group $\mathcal{G}$, which some physicists call also gauge group.

In addition, a gauge transformation can also be described (@baez1994gauge page 222) as a transformation $T\in \text{End}(E)$, where $T(p)$ for every $p\in M$ is a map from $E_p$ to $E_p$. In local trivializations, this map $T(p)$ can be expressed as an element of $G$, the gauge group.

Observe that given a trivialization $(U_{\alpha },{\varphi }_{\alpha })$ of $E$ we have the associated local $G$-frame

and then, any local matter field

can be expressed by

given by the components of $\sigma$ respect to the $G$-frame $p_{\alpha }$:

See the note principal bundle for more details.

A principal connection in $P$, given by an horizontal distribution $H$ or a global $\mathfrak{g}$-valued 1-form $\omega$, both related by $H_{u}P=Ker(\omega (u))$, is called a gauge connection. It provides us with a mechanism to identify $G$-bases at different points of $M$ as being the same.

We can express $\omega$ respect to a chosen local gauge $p$, obtaining the 1-form $p^{\ast }(\omega )$ called the gauge potential (or Yang-Mills field, following [Schuller 2013] page 185). The 2-form obtained by exterior differentiation is called the gauge field or gauge strength tensor.

Coming from Classical field theories

We have classical field theories, which are specified by the following data:

• A $n$-dimensional manifold (space-time) $M$,
• A target space $V\cong {\mathbb{R}}^m$, to construct the space of matter fields $\mathcal{F}=\text{Maps}(M,V)$.
• A Lagrangian, which is a polynomial $L(\varphi ,{\partial }_{\mu }\varphi )$, where $\varphi \in \mathcal{F}$.
• If there is an "external element" affecting the dynamic of the matter field $\varphi$, this is codified through specific terms in the Lagrangian.

Then, they are generalized to gauge theories, which are the same stuff but:

• Instead of $\mathcal{F}=\text{Maps}(M,V)$ we use sections of a fiber bundle $E\to M$ as matter fields.
• Instead of partial derivatives ${\partial }_{\mu }$ we use a connection $D_{\mu }$. The connection codify, within itself, the presence of a "external element" affecting the dynamic of the matter field $\varphi$. Different choices of connections lead to different external "forces".
• The Lagrangians should now be simpler. For example, the Lagrangian for the combined system of the electromagnetic field and a charged particle field is, in a classical field theory,

If we define a covariant derivative $D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }$, where $q$ is the charge of the field $\varphi$, we can rewrite the total Lagrangian as:

• The Lagrangians are now invariant by gauge transformations. We know that different $A$'s could define the same EM field, so we had different Lagrangians describing the same system. With the new approach, all these Lagrangians correspond to the same, and the different $A$s correspond to different expressions in a local frame of the same connection.
• The external forces depends, indeed, on the curvature of the connection. So several connections with the same curvature yield equivalent Lagrangians.
• Fermions (particles) are related to sections of a complex vector bundle which is associated to a principal bundle over space-time. This complex vector bundle is typically obtained via a representation of the structure group of the principal bundle.
• Bosons (forces) are related to connections on the principal bundle, which determine how the associated vector bundles are parallel transported.
  Related: quantum particle, Standard Model.

Examples

1. The time zone model. See me paper attempt.
2. See the meteor tracking problem at @sharpe2000differential and [xournal 130]

Enfoque Gauge on a point