String Theory Demystified

76 String Theory Demystifi ed
µ ν
modes αm (and αn
for the closed string), what we have done is promote them to
operators. By extension, since the X µ ( σ, τ)are
defi ned in terms of α µ
m, the X µ ( σ, τ)
are now to be thought of as operators as well. Therefore the next task in our program
is to determine the state space of the system, that is the states upon which the α µ
m and
by extension the X µ ( σ, τ)
act. This procedure is really pretty similar to what you’re
used to from your previous studies of quantum theory.
The fi rst item to notice is that the commutation relations have some similarity to
the harmonic oscillator you learned about in fi rst semester quantum mechanics.
Temporarily dispensing with our convention of setting = 1, recall that we can
defi ne the creation and annihilation operators for the harmonic oscillator as
† m i
m i
aˆ
xˆ
pˆ aˆ
xˆ
ˆ
m m p
ω ⎛
= −
⎞
ω ⎛
⎝
⎜
⎠
⎟ = +
⎞
⎝
⎜
⎠
⎟
2 ω
2
ω
These operators satisfy the commutation relation:
The hamiltonian of the system is given by
†
[ aˆ, a ˆ ] = 1 (4.11)
ˆ ⎛ † 1
H = ω aˆ aˆ+
⎞
⎝ 2⎠
We introduce the number operator ˆ †
N = aˆ aˆ
which has eigenstates n :
ˆN n = nn
(4.12)
where n = 012 , , ,.... A system consisting of an infi nite collection of harmonic
oscillators is called a fock space.
Continuing, the number operator and its eigenstates allow us to write down the
quantized energy levels of the harmonic oscillator, which are given by
ˆ ⎛
Hn Nˆ 1⎞
⎛ 1⎞
= ω + n n n Enn ⎝
⎜
⎠
⎟ = ω
+
⎝
⎜
⎠
⎟ =
2
2
You should also recall that the system has a ground state, which is the lowest
possible energy state. This is denoted by 0 .
A comparison of Eqs. (4.9) and (4.11) indicates that we have a similar system in
the case of the string. This should not be surprising, since what else would you