User Tools

Site Tools


advanced_tools:jet_bundles

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
advanced_tools:jet_bundles [2017/11/08 17:35]
jakobadmin [Why is it interesting?]
advanced_tools:jet_bundles [2018/05/03 11:45] (current)
paul
Line 1: Line 1:
 ====== ​ Jet Bundles ====== ====== ​ Jet Bundles ======
  
-<tabbox Why is it interesting?> ​ 
-<​blockquote>​ 
  
- +<​tabbox ​Intuitive
- +
-Given a [[advanced_tools:​fiber_bundles|field bundle]] as above, we know what type of quantities the corresponding fields assign to a given spacetime point. Among all consistent such field configurations,​ some are to qualify as those that “may occur in reality” if we think of the field theory as a means to describe parts of the observable universe. Moreover, if the reality to be described does not exhibit “action at a distance” then admissibility of its field configurations should be determined over arbitrary small spacetime regions, in fact over the infinitesimal neighbourhood of any point. This means equivalently that the realized field configurations should be those that satisfy a [[:​equations|specific differential equation]], hence an equation between the value of its derivatives at any spacetime point. +
- +
-In order to formalize this, it is useful to first collect all the possible derivatives that a field may have at any given point into one big space of “field derivatives at spacetime points”. This collection is called the jet bundle of the field bundle.<​cite>​https://​ncatlab.org/​nlab/​show/​field+%28physics%29#​AFirstIdeaOfQuantumFields</​cite></​blockquote>​ +
-<​tabbox ​Layman+
  
 <note tip> <note tip>
Line 15: Line 8:
 </​note>​ </​note>​
   ​   ​
-<​tabbox ​Student+<​tabbox ​Concrete
  
  
Line 37: Line 30:
 encodes a (non-linear) differential operator $D_f : \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(F)$ by encodes a (non-linear) differential operator $D_f : \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(F)$ by
 sending any section $\phi$ of $E$ to the section ​ $f \circ j^\infty(\phi)$ of $F$.<​cite>​https://​arxiv.org/​abs/​1601.05956</​cite></​blockquote>​ sending any section $\phi$ of $E$ to the section ​ $f \circ j^\infty(\phi)$ of $F$.<​cite>​https://​arxiv.org/​abs/​1601.05956</​cite></​blockquote>​
-<tabbox Researcher> ​ 
- 
-<note tip> 
-The motto in this section is: //the higher the level of abstraction,​ the better//. 
-</​note>​ 
- 
-  ​ 
-<tabbox Examples> ​ 
  
 --> Example1# --> Example1#
Line 56: Line 41:
 <-- <--
  
-<tabbox FAQ> ​+<tabbox Abstract>​  
 + 
 +<note tip> 
 +The motto in this section is: //the higher the level of abstraction,​ the better//. 
 +</​note>​ 
 + 
 +<tabbox Why is it interesting?>​  
 +<​blockquote>​ 
 + 
 + 
 + 
 +Given a [[advanced_tools:​fiber_bundles|field bundle]] [...], we know what type of quantities the corresponding fields assign to a given spacetime point. Among all consistent such field configurations,​ some are to qualify as those that “may occur in reality” if we think of the field theory as a means to describe parts of the observable universe. Moreover, if the reality to be described does not exhibit “action at a distance” then admissibility of its field configurations should be determined over arbitrary small spacetime regions, in fact over the infinitesimal neighbourhood of any point. This means equivalently that the realized field configurations should be those that satisfy a [[:​equations|specific differential equation]], hence an equation between the value of its derivatives at any spacetime point. 
 + 
 +In order to formalize this, it is useful to first collect all the possible derivatives that a field may have at any given point into one big space of “field derivatives at spacetime points”. This collection is called the jet bundle of the field bundle.<​cite>​https://​ncatlab.org/​nlab/​show/​field+%28physics%29#​AFirstIdeaOfQuantumFields</​cite></​blockquote>​ 
 + 
 +/* <tabbox FAQ> ​*/
   ​   ​
-<tabbox History> ​+/* <tabbox History> ​*/
  
-</​tabbox>​+</​tabbox> ​
  
  
advanced_tools/jet_bundles.1510158919.txt.gz · Last modified: 2017/12/04 08:01 (external edit)