Introduction to String Theory and D–Branes - School of Natural ...

where we have used the commutator (58) for the oscillators. From this we see that the time–like ζ’s will
produce a state with negative norm. Such states cannot be made sense of in a unitary theory, and are often
called 3 “ghosts”.
Let us study the first constraint:
The next constraint gives:
(L0 − a)|ζ; k>= 0 ⇒ α ′ k 2 + 1 = a, M 2 =
(L1)|ζ; k>=
α ′
1 − a
α ′ . (66)
2 k · α1ζ · α−1|0; k >= 0 ⇒, k · ζ = 0 . (67)
Actually, at level 1, we can also make a special state of interest: |ψ >≡ L−1|0; k >. This state has the
special property that it is orthogonal to any physical state, since < φ|ψ >=< ψ|φ > ∗ =< 0; k|L1|φ >= 0. It
also has L1|ψ>= 2L0|0; k>= α ′ k 2 |0; k > . This state is called a “spurious” state. So we note that there are
three interesting cases for the level 1 physical state we have been considering:
1. a < 1 ⇒ M 2 > 0 :
• momentum k is timelike.
• We can choose a frame where it is (k, 0, 0, . . .)
• Spurious state is not physical, since k 2 = 0.
• k · ζ = 0 removes the timelike polarization. D − 1 states left
2. a > 1 ⇒ M 2 < 0 :
• momentum k is spacelike.
• We can choose a frame where it is (0, k1, k2, . . .)
• Spurious state is not physical, since k 2 = 0
• k · ζ = 0 removes a spacelike polarisation. D − 1 tachyonic states left, one which is including
ghosts.
3. a = 1 ⇒ M 2 = 0 :
• momentum k is null.
• We can choose a frame where it is (k, k, 0, . . .)
• Spurious state is physical and null, since k 2 = 0
• k · ζ = 0 and k 2 = 0 remove two polarizations; D − 2 states left
So if we choose case (3), we end up with the special situation that we have a massless vector in the
D dimensional target spacetime. It even has an associated gauge invariance: since the spurious state is
physical and null, and therefore we can add it to our physical state with no physical consequences, defining
an equivalence relation:
|φ >∼ |φ > +λ|ψ > ⇒ ζ µ ∼ ζ µ + λk µ . (68)
Case (1), while interesting, corresponds to a massive vector, where the extra state plays the role of a
longitudinal component. Case (2) seems bad. We shall choose case (3), where a = 1.
It is interesting to proceed to level two to construct physical and spurious states, although we shall not
do it here. The physical states are massive string states. If we insert our level one choice a = 1 and see what
3 These are not to be confused with the ghosts of the friendly variety —Faddeev–Popov ghosts. These negative norm states
are problematic and need to be removed.
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