advanced_tools:connections
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advanced_tools:connections [2018/04/14 11:18] aresmarrero ↷ Page moved from advanced_notions:connections to advanced_tools:connections |
advanced_tools:connections [2025/01/11 17:09] (current) 91.243.89.241 |
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| - | ====== Connections ====== | + | Just like you got this message, we can submit your message to millions of contact forms. |
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| - | <blockquote>The phase of a charged particle moving in an electromagnetic field (e.g., | + | Let’s strengthen brand awareness together! |
| - | a monopole field) is quite like the internal spinning of our ping-pong ball. | + | Need more brand visibility? |
| - | 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> | + | |
| - | * 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 | + | Pricing: |
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| - | <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> | ||
| + | |||
| + | If you wish to stop getting subsequent messages from us, simply use https://bit.ly/getdelisted | ||
advanced_tools/connections.1523697511.txt.gz · Last modified: 2018/04/14 09:18 (external edit)
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