✎ Connections [The Physics Travel Guide]

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====== Connections ======

//also known as a path lifting rules in the context of [[advanced_tools:fiber_bundles|fiber bundles]] and as gauge fields in particle physics //

<blockquote>The phase of a charged particle moving in an electromagnetic field (e.g.,
a monopole field) is quite like the internal spinning of our ping-pong ball.
We have seen that a phase change alters the wavefunction of the charge
only by a factor of modulus one and so does not effect the probability of
finding the particle at any particular location, i.e., does not effect its motion
through space. Nevertheless, when two charges interact (in, for example, the
Aharonov-Bohm experiment), phase differences are of crucial significance to
the outcome. **The gauge field (connection), which mediates phase changes
in the charge along various paths through the electromagnetic field, is the
analogue of the room’s atmosphere, which is the agency (“force”) responsible
for any alteration in the ball’s internal spinning.**<cite>page 23 in Topology, Geometry and Gauge Fields: Foundations by Naber</cite></blockquote>
  
<tabbox Concrete> 
The two most important types of connections are

* The [[advanced_tools:connections:ehresmann_connection]], which is appropriate tool to describe parallel transport in gauge theories. The parallel transport here happens on the fiber bundle, i.e. from one fiber to the next. Each fiber is one copy of our gauge group, e.g. $U(1)$. Our gauge fields, like the electromagnetic field is described by an Ehresmann connection.
  * The [[advanced_tools:connections:levi_civita_connection]], which is the appropriate tool to describe parallel transport in [[theories:general_relativity|General Relativity]]. The gravitational field is described by a Levi-Civita connection

----

* For a nice explanation of connections with pictures, see page 26 and 27 here:http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf

<tabbox Abstract> 
<blockquote>The wavefunction of the particle takes values in some vector space $V$ (for our purposes, $V$
will be some $\mathbb{C}_k$ ). The particle is coupled to (i.e., experiences the effects of)
a gauge field which is represented by a connection on a principal G-bundle.
The connection describes (via Theorem 6.1.4) the evolution of the particle’s
internal state. The response of the wavefunction at each point to a gauge
transformation will be specified by a left action (representation) of $G$ on $V$.
$V$ and this left action of $G$ on $V$ determine an “associated vector bundle”
obtained by replacing the $G$-fibers of the principal bundle with copies of $V$.
The local cross-sections of this bundle then represent local wavefunctions
of the particle coupled to the gauge field. Because of the manner in which
the local wavefunctions respond to a gauge transformation the corresponding
local cross-sections piece together to give a global cross-section of the associated vector bundle and this, we will find, can be identified with a certain
type of $V$-valued function on the original principal bundle space. Finally, the
connection on the principal bundle representing the gauge field gives rise to
a natural gauge invariant differentiation process for such wavefunctions. In
terms of this derivative one can then postulate differential equations (field
equations) that describe the quantitative response of the particle to the gauge
field (selecting these equations is, of course, the business of the physicists).<cite>Topology, Geometry and Gauge Fields: Foundations by Naber</cite></blockquote>
  
<tabbox Why is it interesting?>

<blockquote>Our interest in connections was originally motivated (in
Chapter 0) by the suggestion that such a structure would provide the unique
path lifting procedure whereby one might keep track of the evolution of a
particle’s internal state (e.g., phase) as it traverses the field established by
some other particle (e.g., the electromagnetic field of a magnetic monopole).
<cite>Topology, Geometry and Gauge Fields: Foundations by Naber</cite></blockquote>
  
<tabbox History>

<blockquote>The historical evolution of our definition of the curvature form
from more familiar notions of curvature (e.g., for curves and surfaces) is not
easily related in a few words. Happily, Volume II of [Sp2] is a leisurely and
entertaining account of this very story which we heartily recommend to the
reader in search of motivation.
<cite>Topology, Geometry and Gauge Fields: Foundations by Naber</cite>
</blockquote>
[Sp2] is Spivak, M., A Comprehensive Introduction to Differential Geometry, Volumes I–V, Publish or Perish, Inc., Boston, 1979.
</tabbox>

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