String Theory Demystified

CHAPTER 4 String Quantization
expect for a vibrating string, except a system that resembles a harmonic oscillator.
µ ν µν
Let’s continue forward. Since [ αm, αn] = η mδ
, we can write
m+ n,
0
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µ ν µ ν µν
⎡⎣ αm, α ⎤ n⎦ = ⎡⎣ αm, α ⎤ − m⎦ = mη
(4.13)
At this point, it is important to take a step back and recognize a key fact. The
commutation relation not only bears some resemblance to the harmonic oscillator
of quantum mechanics, but there is a very important difference. Notice that the
presence of the metric η µν means that we can have negative commutators. This is
=−, 1 it follows that
the case for the time components. That is, since η 00
0 0
⎡⎣ α , α−
⎤ ⎦ =−
m m m
This is going to turn out to be important because it can lead to negative norm
states.
Now, in analogy with the harmonic oscillator from ordinary quantum mechanics,
we defi ne a number operator. These are given in terms of the modes as
Nm = α−m⋅αm where we take m ≥ 1. The eigenstates of the number operator satisfy
Nm im = im im
The total number operator is defi ned by summing over all possible Nm = α−m⋅α m:
∞
∞
∑ ∑
N = Nm= α−m⋅α (4.14)
m
m=
1 m=
1
Following a procedure used in elementary quantum mechanics, we can use
µ ν µ ν
µ µ
Nm = α−m⋅α mtogether
with [ αm, αn] = [ αm, α− m] = mη
to show that the α α
µν
− m and m
are raising and lowering operators, respectively. This follows from
( )= ⋅ ( )
N α i α α α i
m m m − m m m m
= ( αm⋅α−m − m) ( αm
im
)
= αm( α−α
) i − mα i
m m m m m
= αmimim− mαmim= ( im− m) ( αmim)