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--- theorems:noethers_theorems [2018/04/08 16:14]
63.143.42.253 ↷ Links adapted because of a move operation
+++ theorems:noethers_theorems [2021/03/31 18:43] (current)
edi [Concrete]
@@ Line 3: / Line 3: @@
-<tabbox Intuitive?>
+<tabbox Intuitive>
+When an object is at rest it has no momentum. When it has a lot of momentum it changes its location quickly. Hence there is a connection between momentum and the change of location. Similarly, if an object doesn't rotate it has no angular momentum. Hence there is a connection between rotations and angular momentum. In a similar spirit we can say that an object that does not change at all over time has no energy.
+In a somewhat more formal way we say that momentum generates translations (= changes of the position), angular momentum generates rotations and energy generates temporal translations (= movements forward in time). Momentum is responsible that an object changes its location, angular momentum that it rotates and energy that it changes as time passes on.
+Noether's theorem tells us that there is also a connection the other way round. Namely starting from the transformations: change of location, rotations and movements forward in time; we can //derive// the quantities; momentum, angular momentum and energy.
+Whenever we have a physical system, we can transform this system and if the transformed system is indistinguishable from the original one, the transformation we performed is a [[basic_tools:symmetry|symmetry]]. So, for example, if we rotate our system and it is indistinguishable to the unrotated system, we say it has rotational symmetry.
+  * Now, Noether's theorem tells us that whenever a system is rotational symmetric there is a conserved quantity that we can recognize as the usual angular momentum.
+  * Similarly, if a system does not change under translations we get a conserved quantity that looks exactly like the quantity we call momentum.
+  * And if a system does not change as time passes on its energy is conserved.
+-----
 <blockquote>If we assume that the laws of physics are describable by a minimum principle, then we can show
@@ Line 238: / Line 253: @@
 Therefore, if $G$ is a symmetry then $G$ is conserved. Equally, if $G$ is conserved we automatically know that it generates a canonical transformation.
+----
+**Graphical Summary**
+The diagram below shows the relationship between symmetry and conservation in the Lagrangian formalism for the case of a linear point transformation, such as a rotation. For a more detailed explanation see [[https://esackinger.wordpress.com/|Fun with Symmetry]].
+{{:theorems:sym_and_cons_lagrange.jpg?nolink}}
+The diagram below shows the relationship between symmetry and conservation in the Hamiltonian formalism. For a more detailed explanation see [[https://esackinger.wordpress.com/|Fun with Symmetry]].
+{{:theorems:sym_and_cons_hamilton.jpg?nolink}}
@@ Line 277: / Line 305: @@
 </blockquote>
-<blockquote>Noether’s 1918 theorem [41] relating infinitesimal “global” symmetries to conservation laws, is a cherished cornerstone of modern theoretical physics; however, the second theorem (appearing in the same work) applicable to “local” symmetry remains somewhat obscure [43]. Our goal is to use Noether’s second theorem as a starting point for a general approach to [[advanced_notions:ward_indentities|Ward identities]] for gauge symmetry. In particular, we are motivated by recent new Ward identities for large gauge symmetry in gravity and QED [4–6, 8–13], and recent discussions in [37]. (This assertion is based, in part, on informal discussions. An important exception is [[https://arxiv.org/abs/hep-th/0111246|44]], which introduced the authors
+<blockquote>Noether’s 1918 theorem [41] relating infinitesimal “global” symmetries to conservation laws, is a cherished cornerstone of modern theoretical physics; however, the second theorem (appearing in the same work) applicable to “local” symmetry remains somewhat obscure [43]. Our goal is to use Noether’s second theorem as a starting point for a general approach to [[advanced_tools:ward_indentities|Ward identities]] for gauge symmetry. In particular, we are motivated by recent new Ward identities for large gauge symmetry in gravity and QED [4–6, 8–13], and recent discussions in [37]. (This assertion is based, in part, on informal discussions. An important exception is [[https://arxiv.org/abs/hep-th/0111246|44]], which introduced the authors
 to Noether’s second theorem)
@@ Line 357: / Line 385: @@
 <tabbox Why is it interesting?>
+<blockquote>In 1915, Emmy Noether demonstrated that differentiable symmetries give rise to conservation laws. Her work is a foundational document in quantum theory because it verifies the ancient insight that what is most important in any physical system is what remains the same in the system as the system is changing.<cite>http://inference-review.com/article/woits-way</cite></blockquote>
 Noether's most famous first theorem connects each symmetry of a system with a conserved quantity.
@@ Line 433: / Line 463: @@
 </blockquote>
 <--
+-->Is there a symmetry associated to the conservation of information?#
+see https://physics.stackexchange.com/questions/41765/is-there-a-symmetry-associated-to-the-conservation-of-information
+<--
 <tabbox History>
-See Chapter 1 in The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century
+  * [[https://arxiv.org/abs/1902.01989|Colloquium: A Century of Noether's Theorem]] by Chris Quigg
+  * See also Chapter 1 in The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century
 by Yvette Kosmann-Schwarzbach
 </tabbox>

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