Lectures on String Theory

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string spectrum and makes it discrete. Simultaneously, states to be identified with
photon emerge in the quantum spectrum because of the downward shift of the M 2
producing thereby the massless states with a proper spin labels.
Consider open strings. The Hamiltonian is
The basis vectors of the Hilbert space are
H = L 0 − 1 = α ′ p 2 + N − 1 .
|ψ〉 =
∞∏ ∏25
n=1 µ=0
(α µ†
n ) λ n,µ
|p〉 ,
are non-negative integers. Generic state |ψ〉 is not physical. A physical state is a
state which obeys the Virasoro constraints (4.9) and is not a descendent.
Let us look for some examples of physical states. The first one is the ground state
|p〉. The only non-trivial constraint is
(L 0 − a)|p〉 = (α ′ p 2 − a)|p〉 = 0 .
Since p 2 = −M 2 we get that the on-shell condition for this state is M 2 = − a α ′ . Later
on studying the light-cone quantization we find that the normal-ordering constant
a must be equal to one. Thus, the mass-squared of the ground state is negative:
M 2 = − 1 α ′ . The corresponding hypothetic particle moving faster than light is called
tachyon.
The next state to consider is ζ µ α µ −1|p〉. We have
(L 0 − 1)ζ µ α µ −1|p〉 = (α ′ p 2 + N − 1)ζ µ α µ −1|p〉 = α ′ p 2 ζ µ α µ −1|p〉 = 0
from which we deduce the on-shell condition p 2 = 0, i.e. the corresponding particle
is massless. Further condition gives
L 1 ζ µ α µ −1|p〉 = (α 0 α 1 + α −1 α 2 + · · · )ζ µ α µ −1|p〉 = √ 2α ′ ζ µ p µ |p〉 = 0 .
Thus, for a physical state the momentum p µ and the polarization vector ζ µ must be
related as ζ µ p µ = 0 which is nothing else as the Lorentz gauge condition. All higher
Virasoro modes L n , n ≥ 2 are automatically annihilate the state. We, however, have
not described the physical state completely. The massless vector particle which is
photon must have d−2 independent polarizations while the Lorentz gauge lives d−1
polarizations only. We should now recall that physical states are defined modulo the
null states.
Consider a state (κ is any constant)
|d〉 =
κ √
2α
′ L −1|p〉 = κp µ α µ −1|p〉 , p 2 = 0 .