Lecture 19
8.251 Spring 2007
Lecture 19 - Topics
• Critical Dimension
• Constructing the State Space
• Tachyons
In the middle of quantizing the open string:
αn− = √
p− =
1
2α� p1+ L⊥
n
1
(L⊥ + a)
2p+ α� 0
What is a? a is arbitrary for now.
H = 2p+ p− α� = L⊥
0 +a
� I I
⊥
L⊥
0 =α p p +N
N⊥ =
∞
�
k=1
k
aI+ aI
� k�� k�
(Number operator)
Implicit sum over I
[N ⊥ , aJη + ] = naJη +
1 ⊥
1
(L) + a) − pI pI = � (a + N ⊥ )
�
α
α
D
−
2
⊥
⊥
[L⊥
m(m2 − 1)δm+n,0
m , Ln ] = (m − n)Lm+n +
12
M 2 = −p2 = 2p+ p− − pI pI =
With a lie algebra:
[A1 [B1 C]] + [B1 [C1 A]] + [C1 [A1 B]] = 0
I
τ
I
σ
�I
[L+
m , X (τ, σ)] = ξm Ẋ + ζm X
τ
ζm
(τ, σ) = −ie(imτ ) cos(mσ)
σ
ζm
(τ, σ) = e(imτ ) sin(nσ)
1
Lecture 19
8.251 Spring 2007
ζ0σ = 0, ζ0τ = −i
I
I
[L⊥
0 , X (τ, σ)] = −iẊ
i∂τ ζ = [ζ, H]
Reparameterization of worldsheet:
τ
σ
τ
σ
X I (τ + ζm
, σ + ζm
) = X I (τ, σ) + ζm
Ẋ I + ζm
X �I
�
��
�
I
[L⊥
m ,X ]
Critical 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:
M
µν
�
π
dσ(Xµ Pµτ − Xµ Pµτ )
� π
1
=
dσ(Xµ Ẋν − Xν Ẋµ )
2πα� 0
∞
�
1 µ µ
ν
= X0µ pν − X0ν pµ − i
(α−n αn − α−n
αnµ )
n
n=1
=
0
Calculation sketched in book.
[M −I , M −I ] = 0
M −I = M µν |µ=−,ν=I = X0− pI −
� �� �
Hermitian
X0I p−
� ��
�
Not Hermitian
∞
�
1 − I
I
(α−n αn − α−n
αn− )
n
n=1
�
��
�
−i
Hermitian
Rearrange so everything Hermitian:
2
Lecture 19
M −I = X0− pI −
8.251 Spring 2007
1
4p+ α
⊥
I
(X0I (L⊥
0 +a)+(L0 +a)X0 )− √
�
[M −I , M −I ] = α� p+2
∞
1 1 �1 ⊥ I
I
(L−n αn −α−n
L⊥
n)
2α� p+ n=1 n
∞
�
1
1 1
I
J
J
I
(α−m
αm
− α−m
αm
) · (m(1 − (D − 2))) + ( (D − 2) + a)
24
m
�
��
� � 24 ��
�
m=1
“A”
mA +
1
B=0
m
“B”
m = 1, 2, 3, 4, 5 . . .
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 = −1
24
H = 2p+ p− α� = L⊥
0 −1
M2 =
1
(−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
+
I
I+
Operators: xI0 , pI , x−
0 , p , a n , an
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Lecture 19
8.251 Spring 2007
� + �
�p , p�τ (Ground states ∀ values of momenta)
By definition, annihilated by aIn :
�
�
aIn �p+ , p�τ = 0
n = 1, 2, 3, . . .
�
�
�
1 �
1
M 2 �p+ , p�τ = � �p+ , p�τ Scalar field of M 2 =
α
−α�
� + �
�p , p�τ ↔ Scalar Field
⎛
⎜
� + � ⎜
�p , pτ : ⎜
⎜
⎝
(2)+
(3)+
a1
(2)+
a2
..
.
(2)+
an
(25)+
a1
(3)+
a2
..
.
···
···
..
.
a1
(25)+
a2
..
.
(3)+
···
a1
a1
(25)+
⎞
⎟
⎟
⎟
⎟
⎠
General basis state of the state space:
|λ� =
∞ �
25
�
�
�
λn,I � +
(aI+
p , p�τ
n )
n=1 I=2
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.
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Lecture 19
8.251 Spring 2007
String Hilbert space = infinite-dimension vector space (spanned by infinite set
of linearly-independet basis |λ�’s)
String theory describes an infinite number of different particles.
N ⊥ = 1:
� + � D−2states I+
�
aI+
�τ
→
ap+ ,pτ |Ω�
⊥ p ,p
M2 =
1
(−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:
�
�
�
�
J+ � +
�p+ , pτ
aI+
p , pτ , aI+
1 , a1
2
�
��
�
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)
V (φ) =
1 2 2
M φ + 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 approx­
imations.
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