String Theory- Tachyons

� � � � Lecture 19 8.251 Spring 2007 Lecture 19 -Topics Critical Dimension • • Constructing the State Space • Tachyons In the middle of quantizing the open string: 1 αn−= 1√2α� p+ L⊥n 1 p− =2p+α� (L⊥0 + a) What is a? a is arbitrary for now. H =2p + p−α� = L⊥0 + a L⊥0 = α�p I p I + N⊥ ∞N⊥ = kakI+ akI (Number operator) k=1 Implicit sum over I [N⊥,a Jη +]= na Jη + 11 M22 =2p + I p I =(L⊥I p I =(a + N⊥)= −pp− − pα� ) + a) − pα� D − 2 2[L⊥,L⊥]=(m − n)L⊥+ 12 m(m − 1)δm+n,0mn m+n With a lie algebra: [A1[B1C]] + [B1[C1A]] + [C1[A1B]] = 0 [L+ ,XI (τ,σ)] = ξτ X˙I + ζσ X�I mmmζτ (τ,σ)= −ie(imτ ) cos(mσ)mζσ (τ,σ)= e(imτ) sin(nσ)m 1 � � � � � Lecture 19 8.251 Spring 2007 ζ0 σ =0,ζ0 τ = −i [L⊥0 ,XI (τ,σ)] = −iX˙I i∂τ ζ =[ζ,H] Reparameterization of worldsheet: XI (τ + ζτ ,σ + ζσ )= XI (τ,σ)+ ζτ X˙I + ζσ X�I mmmm[L⊥,XI ]mCritical Dimension For a long time, wrote equations assuming D = 4. But came across problems in the quantum theory. Lovelace found some problems went away when D = 26 (considered a bad joke then) Lorentz Charge: � πMµν = dσ(XµPµτ − XµPτ )µ 0 � π=1 dσ(XµX˙ν − Xν X˙µ)2πα� 0 = X0 µp ν − X0 ν pµ ∞1(αµ αµn − αν αµn)− in −n−nn=1 Calculation sketched in book. [M−I ,M−I ]=0 ∞1 M−I = Mµν µ=−,ν=I = X0−p I X0 I p− (α−αIn − αI αn−)|� � � − � � � − in −n−nn=1 Hermitian Not Hermitian � � � Hermitian Rearrange so everything Hermitian: 2 � � Lecture 19 8.251 Spring 2007 1 11 ∞1 M−I = X0−p I (X0 I (L⊥0 +a)+(L⊥0 +a)X0 I )− (L⊥αnI −αI L⊥n )− 4p+α� √2α� p+ n −n−nn=1 ∞1 11[M−I ,M−I ]= α�p +2 (αI αJ αI )(m(1 − (D − 2)))+ ( (D − 2) + a)m − αJ m=1 −m−mm· � 24 � ��m 24 � � “A” “B” 1 mA + B =0 m =1, 2, 3, 4, 5 ... m m =1: A + B =0 1 m =2:2A + B =0 2 A =0,B = 0: D − 2 =1 D = 26 24 ⇒ (26 − 2) + a =0 a = −124 ⇒ H =2p + p−α� = L⊥0 − 1 1 M2 =(−1+ N)⊥ α� Einstein’s theory makes sense in any number of dimensions. String theory fixes the dimension. No one really knows whether this 26-dimensional theory has anything to do with the 10-or 11-dimensional theory. What is a Kaluza-Klein Tower of States? Constructing the State Space Operators: x0I , pI , x−0 , p+ , anI , anI+ 3 ��� ��� ��� ��� ��� ��� � ��� Lecture 19 8.251 Spring 2007 p +,�pτ (Ground states ∀ values of momenta) I By definition, annihilated by a : n I +, �pτ = 0 n = 1, 2, 3, . . . a p n M2 1 p +, �pτ Scalar field of M2 1 −α� +,�pτ = =p α� p +,�pτ Scalar Field ↔ ⎞⎛ ⎜⎜⎜⎜⎝ (2)+ (3)+ (25)+aaa11 1··· (2)+ (3)+ (25)+aaa22 ··· 2 . ... . ... . ... (2)+ (3)+ (25)+aaan 11··· ⎟⎟⎟⎟⎠ p + ,pτ : General basis state of the state space: |λ� = 25∞� n=1 I=2 (anI+)λn,I p + ,p�τ λn,I integers ≥ 0 ∀n ≥ 1,I =2,..., 25. To make mathematicians happy, restrict to case where states have only finite number of creation operations acting on the ground states. λ�∃ only finite number of λn,I = 0. ∀|� 4 � � � � � � � Lecture 19 8.251 Spring 2007 String Hilbert space = infinite-dimension vector space (spanned by infinite setof linearly-independet basis |λ�’s)String theory describes an infinite number of different particles.N⊥ = 1:I+ I+ a ��p +,�pτ � D−2states a p+ ,pτ Ω�⊥ →|1 M2 =(−1+ N⊥)=0 α� �� =1 One-photon massless states! (Only massless bosons are photons). Started with classical strings, quantized them, and out popped photons! Will do the same with closed strings to get gravitons. N⊥ = 2: a1 I+ ,a 1 J+ �p + ,pτ ,a 2 I+ �p + ,pτ 324 states Symmetric, traceless tensor in D − 1 dimensions. Tachyons: The more mass (up to the y axis), the less relevant it is to observed particle physics. 5 Lecture 19 8.251 Spring 2007 So might turn out that tachyons are the most relevant particles. D-branes are made of tachyons! (Maybe) 1 V (φ)= M2φ2 + θ(φ3)2 Instead of saying tachyons are massles, go faster than speed of light, etc.Think of tachyons as an instability.Instability in ... what? The D-branes (1999).Too difficult to compute beyond T25.Tachyon Conjecture: Zwiebach et al used computer simulations to get very high probability approximattions Dec 2005: Analytic solution found. Energy exactly right. Tachyon conjecture verified: Tachyon instability is the instability of the D-brane and if it “rolls down” it destroys the D-brane. D-brane collsions related to inflation? Maybe. 6