advanced_tools:connections
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advanced_tools:connections [2018/01/08 09:25] jakobadmin [Researcher] |
advanced_tools:connections [2018/12/19 11:01] (current) jakobadmin ↷ Links adapted because of a move operation |
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| ====== Connections ====== | ====== Connections ====== | ||
| - | <tabbox Why is it interesting?> | + | //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>Our interest in connections was originally motivated (in | + | <tabbox Intuitive> |
| - | 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 Layman> | + | |
| <blockquote>The phase of a charged particle moving in an electromagnetic field (e.g., | <blockquote>The phase of a charged particle moving in an electromagnetic field (e.g., | ||
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| for any alteration in the ball’s internal spinning.**<cite>page 23 in Topology, Geometry and Gauge Fields: Foundations by Naber</cite></blockquote> | for any alteration in the ball’s internal spinning.**<cite>page 23 in Topology, Geometry and Gauge Fields: Foundations by Naber</cite></blockquote> | ||
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| - | <tabbox Student> | + | <tabbox Concrete> |
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| + | The two most important types of connections are | ||
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| + | * 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 [[models:general_relativity|General Relativity]]. The gravitational field is described by a Levi-Civita connection | ||
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| + | ---- | ||
| * 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 | * 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 Researcher> | + | |
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| + | <tabbox Abstract> | ||
| <blockquote>The wavefunction of the particle takes values in some vector space $V$ (for our purposes, $V$ | <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) | will be some $\mathbb{C}_k$ ). The particle is coupled to (i.e., experiences the effects of) | ||
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| field (selecting these equations is, of course, the business of the physicists).<cite>Topology, Geometry and Gauge Fields: Foundations by Naber</cite></blockquote> | field (selecting these equations is, of course, the business of the physicists).<cite>Topology, Geometry and Gauge Fields: Foundations by Naber</cite></blockquote> | ||
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| - | <tabbox Examples> | + | <tabbox Why is it interesting?> |
| - | --> Example1# | + | <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). | |
| - | --> Example2:# | + | <cite>Topology, Geometry and Gauge Fields: Foundations by Naber</cite></blockquote> |
| - | + | ||
| - | + | ||
| - | <-- | + | |
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| - | <tabbox FAQ> | + | |
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| <tabbox History> | <tabbox History> | ||
advanced_tools/connections.1515399918.txt.gz · Last modified: 2018/01/08 08:25 (external edit)
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