resources:books:baez_muniain
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resources:books:baez_muniain [2017/12/27 12:21] jakobadmin [Solutions to Exercises] |
resources:books:baez_muniain [2018/05/05 17:17] (current) jakobadmin ↷ Page moved from resources:baez_muniain to resources:books:baez_muniain |
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| ====== Baez, Muniain: Gauge Fields, Knots and Gravity ====== | ====== Baez, Muniain: Gauge Fields, Knots and Gravity ====== | ||
| + | |||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| <WRAP imageshadow> | <WRAP imageshadow> | ||
| Line 11: | Line 11: | ||
| | | ||
| - | <tabbox Reading Notes> | + | <tabbox Reading Notes/Summaries> |
| - | * http://michaelnielsen.org/blog/yang_mills.pdf | + | * A great summary of Gauge, Fields, Knots and Gravity was [[http://michaelnielsen.org/blog/yang_mills.pdf|published by Michael Nielsen]] |
| Line 21: | Line 21: | ||
| <tabbox Solutions to Exercises> | <tabbox Solutions to Exercises> | ||
| - | * [[https://people.phys.ethz.ch/~renes/gfkg.html|Solutions by Joseph M. Renes]]. ("Solutions to nearly all of the exercises in Part I, many of them in Part II, and none in Part III") | ||
| - | * https://ncatlab.org/brucebartlett/published/BaezMunian. (Solutions to all exercises in Part I and Part II.) | ||
| - | --> 1# | + | **Part I:** |
| - | Q: Let $\vec{k}\in\R^3$ and lt $\omega=|\vec{k}|$. Fix $\vec{E}\in\mathbb{C}^3$ with | + | |
| + | --> Exercise 1# | ||
| + | **Q:** Let $\vec{k}\in\mathbb R^3$ and lt $\omega=|\vec{k}|$. Fix $\vec{E}\in\mathbb{C}^3$ with | ||
| $\vec{k}\cdot\vec{E}=0$ and $\vec{k}\times\vec{E}=i\omega\vec{E}$. Show that | $\vec{k}\cdot\vec{E}=0$ and $\vec{k}\times\vec{E}=i\omega\vec{E}$. Show that | ||
| $\vec{\mathcal{E}}(t,\vec{x})=\vec{E}e^{-i(\omega t-\vec{k}\cdot\vec{x})}$ satisfies the | $\vec{\mathcal{E}}(t,\vec{x})=\vec{E}e^{-i(\omega t-\vec{k}\cdot\vec{x})}$ satisfies the | ||
| vacuum Maxwell equations. | vacuum Maxwell equations. | ||
| - | A: | + | **A:** |
| Line 38: | Line 38: | ||
| <-- | <-- | ||
| - | -->2# | + | -->Exercise 2# |
| - | Q: Given a Lie group $G$, define its identity component $G_0$ to be the connected component | + | **Q:** Given a Lie group $G$, define its identity component $G_0$ to be the connected component |
| containing the identity element. Show that the identity component of any Lie group is a subgroup, and a Lie | containing the identity element. Show that the identity component of any Lie group is a subgroup, and a Lie | ||
| group in its own right. | group in its own right. | ||
| - | A: Suppose we have a path from the identity to $g\in G_0$. Now map this | + | |
| + | **A:** Suppose we have a path from the identity to $g\in G_0$. Now map this | ||
| + | path to a new path by multiplying each element by $h\in G_0$. This path starts at $h$ and since the mapping is continuous, must remain in $G_0$. (Otherwise, smoothly mapping the group manifold to $\mathbb R^n$ would show a | ||
| + | discontinuity at some point.) Thus $hg\in G_0$ for all $h,g\in G_0$. There's a certain tension between having a smooth manifold with disconnected pieces --- given a map from the | ||
| + | the manifold to itself, one must take care that it does not have a discontinuous action, mapping some points in one component to another component. When this map is an element of the group, this requirement makes | ||
| + | $G_0$ into a subgroup. | ||
| + | |||
| + | <-- | ||
| + | |||
| + | **Part II:** | ||
| + | |||
| + | --> Exercise 1# | ||
| + | **Q:**Given a Lie group $G$, define its identity component $G_0$ to be the connected component | ||
| + | containing the identity element. Show that the identity component of any Lie group is a subgroup, and a Lie | ||
| + | group in its own right. | ||
| + | |||
| + | **A:** Suppose we have a path from the identity to $g\in G_0$. Now map this | ||
| path to a new path by multiplying each element by $h\in G_0$. This path starts at $h$ and since the mapping is continuous, must remain in $G_0$. (Otherwise, smoothly mapping the group manifold to $\R^n$ would show a | path to a new path by multiplying each element by $h\in G_0$. This path starts at $h$ and since the mapping is continuous, must remain in $G_0$. (Otherwise, smoothly mapping the group manifold to $\R^n$ would show a | ||
| discontinuity at some point.) Thus $hg\in G_0$ for all $h,g\in G_0$. | discontinuity at some point.) Thus $hg\in G_0$ for all $h,g\in G_0$. | ||
| Line 51: | Line 67: | ||
| the manifold to itself, one must take care that it does not have a discontinuous action, mapping some points in one component to another component. When this map is an element of the group, this requirement makes | the manifold to itself, one must take care that it does not have a discontinuous action, mapping some points in one component to another component. When this map is an element of the group, this requirement makes | ||
| $G_0$ into a subgroup. | $G_0$ into a subgroup. | ||
| - | |||
| <-- | <-- | ||
| - | | + | |
| + | //Source: [[https://github.com/joerenes/Baez-Muniain-solutions|Solutions to Exercises in Gauge Fields, Knots and Gravity]] by Joseph M. Renes licensed under a Creative Commons Attribution 4.0 Licence.// | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | See also: | ||
| + | |||
| + | * [[https://people.phys.ethz.ch/~renes/gfkg.html|Solutions by Joseph M. Renes]]. ("Solutions to nearly all of the exercises in Part I, many of them in Part II, and none in Part III") | ||
| + | * https://ncatlab.org/brucebartlett/published/BaezMunian. (Solutions to all exercises in Part I and Part II.) | ||
| <tabbox Discussions> | <tabbox Discussions> | ||
resources/books/baez_muniain.1514373687.txt.gz · Last modified: 2017/12/27 11:21 (external edit)
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