Baez, Muniain: Gauge Fields, Knots and Gravity [The Physics Travel Guide]

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 ====== Baez, Muniain: Gauge Fields, Knots and Gravity ======
 <tabbox Why is it interesting?>
 <WRAP imageshadow>
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-<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>
-  * [[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.)
+**Part I:**
+--> 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
+$\vec{\mathcal{E}}(t,\vec{x})=\vec{E}e^{-i(\omega t-\vec{k}\cdot\vec{x})}$ satisfies the
+vacuum Maxwell equations.
+**A:**
+<--
+-->Exercise 2#
+**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 $\mathbb R^n$ 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
+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.
+<--
+//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>

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