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theorems:noethers_theorems [2018/04/08 13:21]
jakobadmin [Abstract]
theorems:noethers_theorems [2021/03/31 18:43] (current)
edi [Concrete]
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-<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 <​blockquote>​If we assume that the laws of physics are describable by a minimum principle, then we can show
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 **Derivation** **Derivation**
  
-We start with a [[frameworks:​lagrangian_formalism|Lagrangian]] $L=L(q,​\dot{q})$ and consider an infinitesimal (= tiny tiny tiny) transformation+We start with a [[formalisms:​lagrangian_formalism|Lagrangian]] $L=L(q,​\dot{q})$ and consider an infinitesimal (= tiny tiny tiny) transformation
 \begin{eqnarray} \begin{eqnarray}
 q \rightarrow q+\epsilon \delta q \, . q \rightarrow q+\epsilon \delta q \, .
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 \end{eqnarray} \end{eqnarray}
  
-In general the [[frameworks:​lagrangian_formalism|Lagrangian]] for a non-relativistic system reads $L=T-V$ and therefore we have ${\partial L \over+In general the [[formalisms:​lagrangian_formalism|Lagrangian]] for a non-relativistic system reads $L=T-V$ and therefore we have ${\partial L \over
 \partial \dot{q}} = {\partial T \over \partial \dot{q}}$ because the potential $V$ is \partial \dot{q}} = {\partial T \over \partial \dot{q}}$ because the potential $V$ is
 usually a function of $q$ only. In most cases we have usually a function of $q$ only. In most cases we have
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 which is the total energy of the system.  ​ which is the total energy of the system.  ​
  
-Here in this context the total energy $T+V \equiv H$ is also often called the [[frameworks:​hamiltonian_formalism|Hamiltonian]]. ​  +Here in this context the total energy $T+V \equiv H$ is also often called the [[formalisms:​hamiltonian_formalism|Hamiltonian]]. ​  
  
 We can understand the connection between the Lagrangian and the Hamiltonian better by defining that ${\partial L \over \partial \dot{q}} \equiv p$ is We can understand the connection between the Lagrangian and the Hamiltonian better by defining that ${\partial L \over \partial \dot{q}} \equiv p$ is
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   * http://​snu.edu/​Websites/​snuokc/​images/​Physics/​Noethers-Theorem.pdf   * http://​snu.edu/​Websites/​snuokc/​images/​Physics/​Noethers-Theorem.pdf
   * A great book on Noether'​s theorems is "Emmy Noether'​s Wonderful Theorem"​ von Dwight E. Neuenschwander.   * A great book on Noether'​s theorems is "Emmy Noether'​s Wonderful Theorem"​ von Dwight E. Neuenschwander.
 +
 +
 +----
 +
 +**Noether'​s Theorem in the Hamiltonian formalism**
 +
 +Noether'​s theorem is especially transparent in the [[formalisms:​hamiltonian_formalism|Hamiltonian formalism]]. ​
 +
 +A symmetry is a transformation that leaves the Hamiltonian unchanged
 +
 +$$ \delta H =0. $$
 +
 +In general, the variation of $H$ is
 +
 +$$ \delta H = \frac{\partial H}{\partial q_i} \delta q_i + \frac{\partial H}{\partial p_i} \delta p_i . $$
 +
 +We can rewrite this using the [[equations:​hamiltons_equations|generalized Hamiltons equations]] as
 +
 +\begin{align} \delta H &= \frac{\partial H}{\partial q_i} \delta q_i + \frac{\partial H}{\partial p_i} \delta p_i \notag \\
 +& \alpha \frac{\partial H}{\partial q_i} \frac{\partial G}{\partial p_i} - \alpha \frac{\partial H}{\partial p_i} \frac{\partial G}{\partial q_i} + \mathcal{O}(\alpha^2)\notag \\
 +& \equiv \alpha \{ H,G\},
 +\end{align}
 +where $ \{ H,G\}$ denotes the [[advanced_notions:​poisson_bracket|Poisson bracket]] and $G$ is a generating function.
 +
 +Now, we can see that $G$ generates a __symmetry__ if  $ \{ H,G\} =0$. 
 +
 +In addition, we know that the equation of motion for $G$ is 
 +
 +$$\dot G = \{ G,H\} $$
 +which simply means that the Hamiltonian generates time translations. ​
 +
 +Since the Poisson bracket is antisymmetric we can conclude that if $G$ is a symmetry: $ \{ H,G\} =0$ we automatically also have $ \{ G,H\} =0$ and thus
 +$$\dot G = 0. $$
 +
 +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}}
  
  
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 </​blockquote>​ </​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) to Noether’s second theorem)
  
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 Therefore, $A$ is a constant of motion (= does not change as time passes on) only if $\{A, H\} =0$. Therefore, $A$ is a constant of motion (= does not change as time passes on) only if $\{A, H\} =0$.
  
-However, we can equally consider the reverse Poisson bracket \{H, A\}. Here $A$ appears in the second slot and therefore this Poisson bracket represents the rate of change of $H$ along the flow generated by $A$. So if $A$ generates a symmetry of a system we expect that $\{H, A\}$  is true.  This comes about because if the Hamiltonian remains unchanged by the flow generated by $A$ then also the [[equations:​hamiltons_equations|equations of motion]]: $\dot q = \frac{\partial H}{\partial p}, \quad \dot p = - \frac{\partial H}{\partial q}$ remain unchanged under the same flow. +However, we can equally consider the reverse Poisson bracket ​$\{H, A\}$. Here $A$ appears in the second slot and therefore this Poisson bracket represents the rate of change of $H$ along the flow generated by $A$. So if $A$ generates a symmetry of a system we expect that $\{H, A\}$  is true.  This comes about because if the Hamiltonian remains unchanged by the flow generated by $A$ then also the [[equations:​hamiltons_equations|equations of motion]]: $\dot q = \frac{\partial H}{\partial p}, \quad \dot p = - \frac{\partial H}{\partial q}$ remain unchanged under the same flow. 
  
  
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   ​   ​
 <tabbox Why is it interesting?> ​ <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. ​ Noether'​s most famous first theorem connects each symmetry of a system with a conserved quantity. ​
  
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 </​blockquote>​ </​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> ​ <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 by Yvette Kosmann-Schwarzbach
 </​tabbox>​ </​tabbox>​
theorems/noethers_theorems.1523186517.txt.gz · Last modified: 2018/04/08 11:21 (external edit)