String Theory- Closed Strings II

Lecture 21 8.251 Spring 2007 Lecture 21 -Topics Wrap-up of Closed Strings Wrapup of Closed Strings String Coupling Constant Dimensionless number that sets the strength of the interaction. Example of coupling const ants: 1. Fine structure constant a = & o &. Derives from interactions between charged particles, magnetic fields. Could imagine particles with e = 0. Action S = S dpx(-F~) -mc S dS + e A,(,)dxp. Doesn't talk about in-teraction between charged particle and field (both are free). 2. Binding energy of an electron in a H atom. Ebinding 0( e4 since V 0( r o $ in Bohr atom. < Similar effect in string theory unless gravitons interacting with matter, no grav-ity! D-dim Newton's Constant: [G(D)] = L~-~ (natural units). D = 10: [G(lO)] = L~ + G('O) = g2(a')4 where g is the string coupling constant. Planck length 1, related to string length 1,: 1: = g21: + 1, o gil, Particle View: Lecture 21 8.251 Spring 2007 String View: Close string splits into 2 strings. Same coupling constant g'd string interactions. really is g for closed strings. +j for open strings (too complicated for this class) How do you fix g? $(x) =dilation * a:'~i:+ 10) massless state. g = edX)). Field sets value of coupling constant. So g changes with 4(x). Not a constant, but a dynamical Lecture 21 8.251 Spring 2007 value. But usually work with constant field. Superstrings Everyone uses superstrings more than superstrings. Takes a long time to de-velop all background, so will present intuitively-reasonable results from QFT. Pauli exclusion principle: multiple fermions cannot occupy the same state. X"(r, a): Classical variables. Commute X1(r, a)~ (r', a') = x J(r', ar)X1(r, a). The X's behave as boson fields in (7, a) space. In quantum theory, things don't quite commute. For operators A, B commuta-tor [A, B] = A B -Be A. A and B commute if and only if [A, B] = 0. Define: $: (r, a), $; (7, a) . Classical, anticommuting variables $g (7, a). Let B1, B2 be classical anticommuting variables. Then B1 B2 = -B2 B1, Bl Bl = -B1 B1 + Bl Bl = 0. Same for all indices. Set of anticomm. variables Bi: B.B. --B.B. 23-3 2 BiBi (not summed) = 0 Example with matrices: Lecture 21 8.251 Spring 2007 7172 = -7,271 YlYl # 0 # 722 Quantum operators f f2 {fl, f2) = flf2 + f2fl Operators anticommute if and only if {fl, f2) = 0. Quantized a scalar field. Got particle state of nk particles (up1) + nl + n2)...(aPk)+)"klfi) (up2) Electron Dirac Field p: momentum, s: spin. Creation operators have nonzero anticomm. operators with annihilation oper-ators. All f +'s anticommute. This relates to the Pauli exclusion principle and Fermi statistics. Action: S = S~osonic + S~ermionic Lecture 21 8.251 Spring 2007 Action for LC coordinates. Tells you pretty much everything about dynamics of LC variables. What Dirac would have written for a fermion in 2D on the worldsheet, but we can a fermion in spacetime. Dirac equation usually written as $iyd$ Usually, Ad,A = ;&A2 but here the A's ($Ls) are anticommuting so A2 = 0. Instead, Ad, A = d, (AA) -(8, A) A. Ad, A + (8, A) A = 8, ( AA) . Varying a$:, 6$,' as, 1 dTdo(ar:(a,+aa)$:+aa:(a-aa)$2)+-1 a=r = -J J I ) 7i-27i-2 a=O Lecture 21 8.251 Spring 2007 Satisfies BC without too much violence. 1 w: (77 a*) = +w: (7' a*) I Choose sign to be positive since action doesn't care, but then can't change sign of field. We have two choices: Continuous field over [-T, T] since $: (r, 0) = $: (r, 0) 9 fermion field is periodic if choose positive sign, antiperiodic if choose negative sign. Suppose choose periodic (Rammond sector) Suppose choose antiperiodic (Neveu-Schawrz sector)