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 advanced_tools:jet_bundles [2017/11/08 17:58]jakobadmin [Why is it interesting?] advanced_tools:jet_bundles [2018/05/03 11:45]paul Both sides previous revision Previous revision 2018/05/03 11:45 paul 2017/12/04 08:01 external edit2017/11/08 17:58 jakobadmin [Why is it interesting?] 2017/11/08 17:35 jakobadmin [Why is it interesting?] 2017/11/08 17:35 jakobadmin [Why is it interesting?] 2017/11/08 17:32 jakobadmin [Why is it interesting?] 2017/11/08 17:28 jakobadmin [Why is it interesting?] 2017/11/08 16:59 jakobadmin [Student] 2017/11/08 16:59 jakobadmin [Student] 2017/11/08 16:55 jakobadmin ↷ Page name changed from advanced_tools:jet_bundels to advanced_tools:jet_bundles2017/11/08 16:55 jakobadmin created Next revision Previous revision 2018/05/03 11:45 paul 2017/12/04 08:01 external edit2017/11/08 17:58 jakobadmin [Why is it interesting?] 2017/11/08 17:35 jakobadmin [Why is it interesting?] 2017/11/08 17:35 jakobadmin [Why is it interesting?] 2017/11/08 17:32 jakobadmin [Why is it interesting?] 2017/11/08 17:28 jakobadmin [Why is it interesting?] 2017/11/08 16:59 jakobadmin [Student] 2017/11/08 16:59 jakobadmin [Student] 2017/11/08 16:55 jakobadmin ↷ Page name changed from advanced_tools:jet_bundels to advanced_tools:jet_bundles2017/11/08 16:55 jakobadmin created Line 1: Line 1: ====== ​ Jet Bundles ====== ====== ​ Jet Bundles ====== -  ​ - <​blockquote>​ - + <​tabbox ​Intuitive> - + - 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​ + - <​tabbox ​Layman> + Line 15: Line 8: ​ ​ ​ - <​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​ sending any section $\phi$ of $E$ to the section ​ $f \circ j^\infty(\phi)$ of $F$.<​cite>​https://​arxiv.org/​abs/​1601.05956​ -  ​ - - - The motto in this section is: //the higher the level of abstraction,​ the better//. - ​ - - ​ -  ​ --> Example1# --> Example1# Line 56: Line 41: <-- <-- -  ​ + ​ + + + The motto in this section is: //the higher the level of abstraction,​ the better//. + ​ + + ​ + <​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​ + + /*  ​*/ ​ ​ -  ​ + /*  ​*/ - ​ +  ​