String Theory Demystified

78 String Theory Demystifi ed
Hence, α µ acts like a lowering operator. A similar exercise shows that the
m
negative frequency mode α µ
, ≥ 1acts like a raising operator:
− m m
( )= + ( )
N α i ( i m) α i
m −m m m −m
m
The lowering operator α µ
m destroys the vacuum or ground state, which is the state
with im = 0:
α µ
m
0 = 0
(4.15)
We can construct higher-energy states using the raising operators which are the
negative modes, α µ
− m ≥ m
µ µ µ µ µ
, 1, that is, α−1 0 , α−1α−1 0 , α−1α−2 0 , and so on. A
string state also carries momentum, so we can label a state by im , k . Considering
the ground state, supposing that the string carries momentum k µ , the momentum
operator acts as
µ µ
p 0, k = k 0,
k
(4.16)
Earlier we remarked that since the Minkowski metric η µν appears in the
commutation relations, negative norm states can exist. We can demonstrate this
explicitly as follows. Consider the fi rst excited state with momentum k µ , that is,
0 0 † 0
0, k . Using ( α ) = α we fi nd the norm of this state to be
α −1
− 1
1
0
0 0
α 0, k = 0, k α α 0, k =−1
(4.17)
−11−1 We can rid the theory of the negative norm states by applying the Virasoro
constraints. The classical expressions for the Virasoro constraints are
1
1
Lm = ∑αm−n⋅ αn Lm
= ∑αm−n⋅α
n
(4.18)
2
2
n
In the quantum theory, the Virasoro constraints are promoted to Virasoro
operators. However, since the modes must satisfy the given commutation relations,
some care must be applied when writing the Virasoro operators as derived from the
classical expressions. The technique of normal ordering is used. This will ensure
that the eigenvalues of the Virasoro operators will be fi nite. The prescription of
normal ordering is simple:
• Move all lowering operators (positive frequency modes) to the right.
• Move all raising operators (negative frequency modes) to the left.
n