Lecture 14 8 251 Spring 2007 Lecture 14 Topics Momentum charges for the string Lorentz charges for the strings Angular momentum of the rotating string Discuss and the string length s General gauges Fixing and natural units Reading Section 8 4 8 6 and 9 1 S d 0 d 1 d p L a a coordinates a elds coord index 0 1 p a eld index a 1 m i index for various symmetries a i hai Leaves L invar to rst order 1 L a L a 0 a a 3 J Ji i Ji Similar to mechanics L a a L q q Claim Given this transformation leaves L invar to rst order then 2 Ji 0 i Conserved Current Check this yourself using 1 and the E L equations of motion Done in book as well Conserved charge too Qi JiO d 1 d 2 d p Answer independent of time 1 Lecture 14 8 251 Spring 2007 dQ 0 d O Nambu Gotta action d 0 d 1 L 0 x 1 x S d a d This means 0 1 x a 0 d spatial dimension Let s look for asymmetry A variation of the eld that leaves the eld invar x constant Constant translations of a worldsheet should by asymmetric Why would NambuGotta action care if rigidally moved worldsheet through time or space So 0 x 0 x 0 1 x 0 So x indeed asymmetric Apply 3 J L x L x L L 0 1 J J P P x x J Conservation law J 0 gives us collection of conservation laws for P P a a P 0 1 P d conserved quantity indexed by spacetime index P conserved momentum for the string not dependent on since conserved J 0 Check P is conserved P d 0 1 P d 0 P 0 1 dP d 1 2 Lecture 14 8 251 Spring 2007 This yields the free BCs This is the hardest part of the course After this it gets easier A momentum is in general a variation of a Lagrangian with respect to a velocity eg x L conserved has units of momentum We will see this is indeed the relative momentum of a piece of string When we had J LQ3 J TQL2 Now we have P P P P L P P T Call P momentum density and P momentum current Okay we have dP 0 d But would like dP 0 dt Conserved for Lorentz observer Is this the case Yes dP Sure could work in static gauge t dt 0 But what about an arbitrary curve on worldsheet Look for a generalization formula clue from divergence theorem A ux of vector eld 3 Lecture 14 8 251 Spring 2007 x y A dy A dx R P d P d R Ax Ay dxdy x y P P d d 0 Given an arbitrary curve claim momentum given by P d P d P 2 1 2 P d P d 0 1 0 P 1 P 2 Usually will use P general formulation 1 0 P d with constant but nice to have this 4 Lecture 14 8 251 Spring 2007 Lorentz Transformation x L x Leaves x x invar Vary x subject to x x x x x 2 x x 2 x x where x x If we want x x 0 we make antisymmetric Claim x x 0 So Nambu Gotta action invar and get new set of symmetries x x J L x P x P x x J 1 x P x P 2 No physical relevance to 12 So de ne m x P x P Conserved currents m 0 1 m0 d M 0 Conserved Charge Mij mij d 0 123 1 xi Pj xj Pi d ijk Lk 0 1 totally antisymmetric eg 5 Lecture 14 8 251 Spring 2007 M12 1 x1 P2 x2 P1 d 12l Lk L3 0 r p L So Mij angular momentum conserved Angular momentum of rotating string 1 x1 P2 x2 P1 d M12 L3 J 0 1 x t cos 1 ct ct cos sin 1 1 Parametrized String T0 x T0 cos sin ct cos ct P c2 t c 1 1 1 2 1 T0 x1 P2 x2 P1 cos2 c 1 J 1 E2 2 T0 c E 1 T0 6