Lectures on String Theory

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we get
Since a general state in the NS sector has the form
|Φ〉 = α i 1
−n1 · · · α i N
−nN b j 1
−r1 · · · b j M
−rM |0〉
G|Φ〉 = (−1) M−1 |Φ〉.
Thus, all states with M even are projected out, in particular, the tachyon. This
removes the tachyon and all the states with half-integer α ′ M 2 . Indeed,
α ′ M 2 =
r M
∑
i=1
r i − 1 2 }{{}
a
and for M even the sum of half-integers is always an integer α ′ M 2 is half-integer
number.
In the Ramond sector the operator G is defined as follows
G = (−1) F = b 1 0 · · · b 8 0(−1) P ∞
n=1 bi −n bi n .
The transversal zero modes b i 0 form the Clifford algebra {b i 0, b j 0} = δ ij . Thus, these
operators can be represented by SO(8) γ-matrices Γ i which have size 16 by 16. These
matrices act on the 16-dimensional Majorana (i.e. real) spinor whose components
can be thought to combine two Weyl projections, which are precisely |a〉 and |ȧ〉:
( ) |a〉8s
|ψ〉 = .
|ȧ〉 8c
In the Majorana representation these matrices Γ i can be taken in the block-diagonal
form as
( ) 0 γ
Γ i i
=
(γ i ) t .
0
The fact that Γ i obey the standard Glifford algebra {Γ i , Γ j } = 2δ ij implies that 8×8
real matrices γ i satisfy the following algebra
γ i (γ j ) t + γ j (γ i ) t = 2δ ij .
If we introduce the standard Pauli matrices
σ 1 =
0 1
1 0
, σ 2 =
0 −i
1 0
, σ
i 0
3 =
0 −1
then the matrices γ i can be defined as
γ 1 = −iσ 2 ⊗ σ 2 ⊗ σ 2 , γ 2 = iI ⊗ σ 1 ⊗ σ 2 ,
γ 3 = iI ⊗ σ 3 ⊗ σ 2 , γ 4 = iσ 1 ⊗ σ 2 ⊗ I ,
γ 5 = iσ 3 ⊗ σ 2 ⊗ I , γ 6 = iσ 2 ⊗ I ⊗ σ 1 ,
γ 7 = iσ 2 ⊗ I ⊗ σ 3 , γ 8 = I ⊗ I ⊗ I .