Lecture 2 8 251 Spring 2007 Lecture 2 Topics Energy and momentum Compact dimensions orbifolds Quantum mechanics and the square well Reading Zwiebach Sections 2 4 2 9 1 x x0 x1 2 x l c time Leave x2 and x3 untouched ds2 dx0 2 dx1 2 dx2 2 dx3 2 v dx dxv u v 0 1 2 3 2dx dx dx0 dx1 dx0 dx1 dx0 2 dx1 2 ds2 2dx dx dx2 2 dx3 2 v dx dxv u v 2 3 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 Lecture 2 8 251 Spring 2007 I I I 2 3 1 22 33 1 Given vector a transform to 1 a a0 a1 2 Einstein s equations in 3 space time dimensions are great But 2 dimensional space is not enough for life Luckily it works also in 4 dimensions d5 d6 Why don t we live with 4 space dimensions If we lived with 4 space dimesnions planetary orbits wouldn t be stable which would be a problem Maybe there s an extra dimension where we can unify gravity and Maybe if so then the extra dimensions would have to be very small too small to see String theory has extra dimensions and makes theory work Though caution this is a pretty big leap Trees in a Box Look at trees in a box Move a little and see another behind it 2 Lecture 2 8 251 Spring 2007 In fact see row that are all identical Leaves fall identically and everything Dot Product a b a b 3 a i bi i 1 a b a b a2 b2 a3 b3 a b a a a a a a a a a a dx dx Light rays a bit like in Galilean physics go from 0 to vlc 3 Lecture 2 8 251 Spring 2007 Energy and Momentum Event 1 at x Event 2 at x dx after some positive time change dx is a Lorentz vector The dimension along the room row is actually a circle with one tree so not actually in nity See light rayws that goes around circle multiple times to see multiple trees Crazy way to de ne a circle This circle is a topological circle no center no radius Identify two points P1 and P2 Say the same P1 P2 if and only if x P1 x P2 2 R n n Z Write as x x 2 R n De ne Fundamental Domain a region sit 1 No two points in it are identi ed 2 Every point in the full space is either in the fundamental domain or has a representation in the fundamental domain So on our x line we would have 4 Lecture 2 8 251 Spring 2007 ds2 c2 dt2 d x 2 c2 dt2 v 2 dt 2 c2 1 2 dt 2 ds2 is a positive value so can take square root ds 1 2 dt In to co moving Lorentz frame do same computation and nd ds2 c2 dtp 2 d x 2 c2 dtp 2 dtp Proper time moving with particle Also greater than 0 ds cdtp dx Lorentz Vector ds De ne velocity u vector u cdcx dx De nite momentum u vector m dx dx p mu m dt 1 2 dt 1 1 2 Rule to get the space we re trying to construct Take the f d include its boundary and apply the identi cation 5 Lecture 2 8 251 Spring 2007 Note Easy to get mixed up if rule not followed carefully Consider 2 with 2 identi cations x y x L1 y x y x y L2 Blue Fundamental domain for rst identi cation Red Fundamental domain for second identi cation 6 Lecture 2 8 251 Spring 2007 dx0 d x p m dt dt mc m v E p c E relativistic energy c 2 1 2 relativistic momentum p Scalar p p p0 2 p 2 E2 p 2 c2 m2 c2 m2 v 2 1 2 1 2 1 2 m2 c2 1 2 m2 c2 Every observer agrees on this value Light Lone Energy x0 time Ec p0 x time Eclc p Nope Justify using QM t x e i 0 x h Et p Can think of the IDs as transformations points move Here s something that moves some points but not all Orbfolds 1 ID x x FD 7 Lecture 2 8 251 Spring 2007 Think of ID as transformation x x This FD not a normal 1D manifold since origin is xed Call this half time Zz the quotient 2 ID x x rotated about origin by 2 n In polar coordinates z x iy 2 i n z e Fundamental domain can be chosen to be 8 z Lecture 2 8 251 Spring 2007 Cone We focus on these two since quite solvable in string theory p h i SE ih E x0 c ih E c t So for our x want ih x Elc c E Et p x ct p x c p x p x p x Now have isolated dependence on x so can take derivative e ih i h p x p x So Elc p p Suppose have line segment of length a Particle constrained to this 9 Lecture 2 8 251 Spring 2007 Compare to physics of world with particle constrained to thin cylinder of radius R and length a 2D Can be de ned as with ID x y x y 2 R So h2 SE 2m 2 2 2 E x2 y 1 k x k sin a 2 h2 k Ek 2m a 2 k x ly k l sin cos a R k x ly k l sin sin a R If states with l 0 then get same states as case 1 but if l 0 get di erent E 2 value from Rl contribution Only noticeable at very high temperatures 10