1 / 6

Noether’s Theorem

Noether’s Theorem. If there exists r infinitesimal transformations of the following form:. and. Then:. Are r conserved currents -- . Conservation of energy and momentum:. Let each e r correspond to a displacement in the r direction. Gauge Invariance of the First Kind.

khuyen
Download Presentation

Noether’s Theorem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Noether’s Theorem If there exists r infinitesimal transformations of the following form: and Then: Are r conserved currents -- Conservation of energy and momentum: Let each er correspond to a displacement in the r direction

  2. Gauge Invariance of the First Kind Global Gauge Invariance Complex scalar Lagrangian: Transformation: Preserves Lagrangain

  3. Gauge Invariance of Second Kind Local Gauge Invariance Now allow gauge transformation to be position dependent: To get gauge invariant Lagrangian, must add “gauge” field to cancel second term: Gauge invariant Lagrangian:

  4. Interpretation Free Field term Kinetic energy of EM field EM field – current interaction New type term – four point function coming from gauge invariance

  5. SU(n) Gauge Invariance E&M Single Phase U(1) GroupN Now let an N-vector of states Require invariance under local SU(n) transformations Need an extra gauge interaction to cancel out additional term:

  6. SU(n) Field Strength Tensor We want the field strength tensor to be invariant under gauge transformation After a long proof we find that: Is gauge invariant

More Related