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Lecture 2 - Topics Energyandmomentum Compactdimensions,orbifolds Quantummechanicsandthesquarewell

Reading: Zwiebach,Sections: 2.4- 2.9x =1

2(x0x1)

x+ l.c. time

Leavex2 andx3 untouched.

ds2 =

(dx0)2 + (dx1)2 + (dx2)2 + (dx3)2

=vdxdxvu, v= 0,1,2,3

2dx+dx = (dx0 +dx1)(dx0dx1)= (dx0)2(dx1)2

ds2 =2dx+dx + (dx2)2 + (dx3)2=vdxdxv

u, v= +,,2,3

=0 1 0 0

1 0 0 00

0 1 00 0 0 1

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++ = =+I = =II= 2,3

+ =+ =122 =33 = 1

Givenvectora,transformto:a :=1

2(a0a1)

Einsteins equations in 3 space-time dimensions are great. But 2 dimensionalspaceisnotenoughforlife. Luckily,itworksalsoin4dimensions(d5,d6, ...).Whydontwelivewith4spacedimensions?Ifwelivedwith4spacedimesnions,planetaryorbitswouldntbestable(whichwouldbeaproblem!)Maybetheresanextradimensionwherewecanunifygravityand...Maybeifso,thentheextradimensionswouldhavetobeverysmalltoosmalltosee.String theory has extra dimensions and makes theory work. Though caution:thisis aprettybigleap.Trees in a BoxLookattrees inabox

Movealittleandseeanotherbehindit

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Infact,seerowthatareallidentical! Leavesfallidenticallyandeverything.

Dot Product3

a b=ab + aibii=1

=a+bab+ +a2b2 +a3b3=ab

a =aa+ =+a =+a =a

a+ =aa =a+

dxvlc =

dx+Lightraysabitlike inGalileanphysics- gofrom0to.

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Lecture2 8.251Spring2007

Energy and Momentum

Event1atxEvent

2

at

x

+dx

(after

some

positive

time

change)

dx isaLorentzvectorThe dimension along the room, row is actually a circle with one tree, so notactually infinity.See lightraywsthatgoesaroundcirclemultipletimestoseemultipletrees.Crazywaytodefineacircle

Thiscircle isatopologicalcircle- nocenter,noradiusIdentifytwopoints,P1 andP2. Saythesame(P1 P2) ifandonly ifx(P1) =x(P2)+(2R)n(nZ)Writeas:

xx+(2R)nDefine: FundamentalDomain=aregionsit.1. Notwopoints initareidentified2. Every point in the full space is either in the fundamental domain or has arepresentationinthefundamentaldomain.Soonourxline,wewouldhave:

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Lecture2 8.251Spring2007

ds2 =c2dt2 + (dx)2=c2dt2 +v2(dt)2=c2(12)(dt)2

ds2

isapositivevaluesocantakesquareroot:ds= 12dt

Intoco-movingLorentzframe,dosamecomputationandfind:ds2 =c2(dtp)2 + (dx)2 =c2(dtp)2

dtp: Propertimemovingwithparticle. Alsogreaterthan0.

ds=cdtpdx

=LorentzVectordsDefinevelocityu-vector:

cdcxu =

dxDefinitemomentumu-vector:

m dx dxp =mu = =m

12 dt dt1

=12

Ruleto

get

the

space

were

trying

to

construct:

Takethef d,include itsboundary,andapplytheidentification

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Note: Easytogetmixedupifrulenotfollowedcarefully.Consider2 with2 identifications:

(x, y)(x+L1, y)(x, y)(x, y+L2)

Blue: Fundamentaldomainforfirst identificationRed: Fundamentaldomainforsecond identification

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dx0 dxp =m ,

dt dt= (mc , mv)

E= , p

c2

E: relativisticenergy= c12

p: relativisticmomentumScalar:

p p = (p0)2 + (p)2E2

=c2 +p2

2 2 2 2m c m v=

12 + 122 2 12c=m

122 2c=m

Everyobserveragreesonthisvalue.

Light Lone Energyx0 =time, E

c =p0+x =time, E

clc =p+? Nope!hJustifyusingQM:(t, x) =ei(Etp0x)

CanthinkoftheIDsastransformations- pointsmove. Heressomethingthatmovessomepointsbutnotall.Orbfolds1.

ID:x xFD:

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ThinkofIDastransformationx xThisFDnotanormal1Dmanifoldsinceoriginisfixed. Callthishalftime/Zzthequotient.2.

ID:xxrotatedaboutoriginby2/nInpolarcoordinates:

z=x+iy2i zn

zeFundamentaldomaincanbechosentobe:

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Cone!Wefocusonthesetwosincequitesolvable instringtheory.

p=h/iSE:

Eih =

x0 cih

=Ec t

Soforourx+,wantihx+ =ElccE

Etp x= ct+p x c=px=(p+x+px +. . .)

Nowhave isolateddependenceonx+,socantakederivative:+ih =e (p+x+ +. . .)

ih =p+

x+So:

Elc p+ =p=

Supposehave linesegmentof lengtha. Particleconstrainedtothis:

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ComparetophysicsofworldwithparticleconstrainedtothincylinderofradiusRand lengtha(2D)

Canbedefinedas:

withID(x, y)(x, y+ 2R)So:

SE=h2 2 + 2 =E2m x2 y 2

1.kx

k =sin a 2

h2 kEk =

2m a2.

k,l =sin kx cos lya Rk,l =sin

kxsin ly

a RIfstateswith l=0thengetsamestatesascase1, but if l=0getdifferentE 2 valuefrom

Rl contribution. Onlynoticeableatveryhightemperatures.

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