advanced_tools:fiber_bundles

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advanced_tools:fiber_bundles [2018/05/05 10:37] jakobadmin ↷ Links adapted because of a move operation |
advanced_tools:fiber_bundles [2018/05/06 09:58] jakobadmin [Intuitive] |
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Therefore, as an electron moves through space, we have above each point that it passes a specific point on the unit circle that denotes the specific phase that the electron wave function has at this location. A set of unit circles above each space point is like a notebook that keeps track of the phase of the electron at this location. | Therefore, as an electron moves through space, we have above each point that it passes a specific point on the unit circle that denotes the specific phase that the electron wave function has at this location. A set of unit circles above each space point is like a notebook that keeps track of the phase of the electron at this location. | ||

- | | + | {{ :advanced_tools:fiberbundels1.png?nolink&600 |}} |

- | [{{ :advanced_tools:circlesfibers.png?nolink |[[http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf|Source]]}}] | + | |

It is convenient to cut the circles such that they become lines. We only need to remember that the end points of these lines need to be identified. | It is convenient to cut the circles such that they become lines. We only need to remember that the end points of these lines need to be identified. | ||

- | [{{ :advanced_tools:circlesfibers2.png?nolink |[[http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf|Source]]}}] | + | {{ :advanced_tools:fiberbundles2.png?nolink&600 |}} |

The picture that then emerges, is that we have a line above each point in space. A dot in each such line represents the specific phase our electron has. | The picture that then emerges, is that we have a line above each point in space. A dot in each such line represents the specific phase our electron has. | ||

- | |||

- | [{{ :advanced_tools:circlesfibers3.png?nolink |[[http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf|Source]]}}] | ||

In other words, we now have a bundle of circles above the space in which our electron moves. This is an important tool since it allows us nicely to think about how phase factors evolve as a particle moves through space. It is especially useful, as soon as we are dealing with more than one electron. | In other words, we now have a bundle of circles above the space in which our electron moves. This is an important tool since it allows us nicely to think about how phase factors evolve as a particle moves through space. It is especially useful, as soon as we are dealing with more than one electron. | ||

- | [{{ :advanced_tools:circlesfibers4.png?nolink |[[http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf|Source]]}}] | + | {{ :advanced_tools:fiberbundles3.png?nolink&600 |}} |

In the lower part of the image we have the actual space where our electron moves. In the upper part of the image we have the "internal" space, which is our fiber bundle. As the particles follows a path in our actual space, it also traces out a path on the fiber bundle. | In the lower part of the image we have the actual space where our electron moves. In the upper part of the image we have the "internal" space, which is our fiber bundle. As the particles follows a path in our actual space, it also traces out a path on the fiber bundle. | ||

The actual tool that tells us which path in the fiber bundle our electron will follow is called the connection, and in physics corresponds to the gauge field. We can think of this connection like a family of ramps. Our electron starts at one specific location with one specific phase. Then, as it moves trough space the ramps tell us how the phase changes, i.e. which path in the bundle electron traces out: | The actual tool that tells us which path in the fiber bundle our electron will follow is called the connection, and in physics corresponds to the gauge field. We can think of this connection like a family of ramps. Our electron starts at one specific location with one specific phase. Then, as it moves trough space the ramps tell us how the phase changes, i.e. which path in the bundle electron traces out: | ||

- | |||

- | [{{ :advanced_tools:fiberramps.png?nolink |[[http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf|Source]]}}] | ||

Now imagine two electrons that follow different path trough space, for example, like they do in the [[experiments:double_slit_experiment]] or the [[experiments:aharonov-bohm|Aharonov-Bohm experiment]]. Depending on the physical situation, it can happen that a different path through space also leads to a different path through the fiber. As a result we get a phase difference between the two electrons that is measurable in terms of an interference pattern. | Now imagine two electrons that follow different path trough space, for example, like they do in the [[experiments:double_slit_experiment]] or the [[experiments:aharonov-bohm|Aharonov-Bohm experiment]]. Depending on the physical situation, it can happen that a different path through space also leads to a different path through the fiber. As a result we get a phase difference between the two electrons that is measurable in terms of an interference pattern. | ||

- | [{{ :advanced_tools:fiberramps-aharonovbohm.png?nolink |[[http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf|Source]]}}] | + | {{ :experiments:aharnovbohm.png?nolink&600 |}} |

advanced_tools/fiber_bundles.txt · Last modified: 2018/05/06 09:58 by jakobadmin

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