advanced_tools:connections
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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., |
| - | + | a monopole field) is quite like the internal spinning of our ping-pong ball. | |
| - | <note tip> | + | We have seen that a phase change alters the wavefunction of the charge |
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | only by a factor of modulus one and so does not effect the probability of |
| - | </note> | + | 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> | ||
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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 | ||
| + | |||
| + | ---- | ||
| * 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> | + | |
| - | <blockquote>The wavefunction of the particle takes values in some vector space V (for our purposes, V | + | |
| - | will be some C k ). The particle is coupled to (i.e., experiences the effects of) | + | <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. | 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 | 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 | 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. | + | 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” | + | $V$ 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. | + | 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 | 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 | of the particle coupled to the gauge field. Because of the manner in which | ||
| the local wavefunctions respond to a gauge transformation the corresponding | 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 | 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 | + | 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 | connection on the principal bundle representing the gauge field gives rise to | ||
| a natural gauge invariant differentiation process for such wavefunctions. In | a natural gauge invariant differentiation process for such wavefunctions. In | ||
| Line 39: | Line 51: | ||
| 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> | ||
| | | ||
| - | <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 | |
| - | --> Example2:# | + | 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 FAQ> | + | |
| | | ||
| <tabbox History> | <tabbox History> | ||
advanced_tools/connections.1510739754.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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