Let us now construct the Hilbert space of our quantum theory. That is, we seek the irreducible representations of the operators
• xµ= (xµ)†, pµ= (pµ)† as well as
• αµ
m= (αµm)†, α˜µm= ( ˜αµm)†, which form an infinite family of harmonic oscillators.
Let us recall some basics facts about the harmonic oscillator:
• The harmonic oscillator is defined in terms of the operators a, a†with commutation relations
[a†, a] = −1.
• The number operator N = a†a is hermitian and diagonalisable with eigenstates such that
N |n i = n |n i and commutation relations [N, a] = −a [N, a†] = a†
a : lowering operator
a† : raising operator (3.18)
• The Fock space is constructed form the vacuum |0 i with the property a |0 i = 0 as the space of states of the form a†. . . a†|0 i.
Returning to the bosonic string, the key insight is that for each mode number m and for each dimension µ we are facing a separate harmonic oscillator defined by the commutation relations
[αµm, ανn] = m δm+n,0ηµν ηµν= diag(−1, 1, . . . , 1), (3.19)
and similarly for ˜αµmin the closed string. For spacelike µ = i we have [αim, αi−m] = +m.
Thus, the operators ai |m|= 1 √ |m|α i |m|and (a i |m|) † satisfy [ai |m|, (a i |m|)
†] = 1, which identifies them
as annihilation and creation operators. We prefer to work with the unnormalised αi
m, though,
i.e. with
αi−|m|: creation operators, αi|m|: annihilation operators.
In addition, we need to furnish a representation of the Heisenberg algebra formed by xµ, pµ.
Combining everything we define a ground state |0; pµi with the following properties
• ˆΠµ|0; p i = pµ|0; pµi,
• αµ
m|0; pµi = 0 ∀m > 0.
That is the state |0; pµi has momentum pµ. The Fock space is then generated by action of the
independent creation operators αµ−m, m > 0, on this state. It is spanned by the set of states
{Y
m
(αµ−m)nm,µ|0; p i}. (3.20)
Note that a priori there is a ground state for each value of pµ, and pµ and the oscillators and independent. We will soon see that this is not true in the physical Hilbert space any longer. To characterise a particular state one introduces its polarisation tensor, e.g. ξµα
µ
−1|0; p i or
ζµνα µ
−1αν−1|0; p i etc.
This poses an immediate problem. Take e.g. |ψ i = ξµα µ
−k|0; p i with k > 0 and ξµ= (1, 0, . . . , 0)
and compute the norm hψ|ψi = h 0; p| a0 ka 0 −k|0; p i = h 0; p| [a0k, a 0 −k] |0; p i = −kh0; p|0; pi < 0.
The appearance of such negative norm states or ”ghosts” is unacceptable in a quantum theory as they spoil unitarity.
To appreciate the problem and find its solution we recall that exactly the same issue arises in the Gupta-Bleuler quantisation of QED.
• To define a canonical formulation of U (1) gauge theory it is necessary to start not from the gauge invariance Lagrangian, but rather from
L = −1 4F µνF µν− 1 2(∂ · A) 2.
This corresponds to (partially) fixing the U (1) gauge symmetry by imposing the gauge fixing constraint ∂ · A = 0.
• While the naive Fock space suffers from ghosts, these are absent from the set of physical states defined by imposing the gauge fixing constraint at the quantum level.
In string theory we have likewise (partially) fixed the underlying diffeomorphism and Weyl sym- metry by imposing the gauge fixing condition hab= ηab. But even at the classical level we still
have to impose the Virasoro Virasoro constraints Tab = 0, i.e. ⇔ Lm = 0 ∀ m ∈ Z (for open
strings and similarly including ˜Lm for he closed string).
• Thus we must impose the constraints Lm = 0 as an operator equation. Our first guess
Lm|ϕ i = 0 ∀ m is inconsistent due to the central term in Virasoro algebra. But this would
be too strict anyways as the analogue of the classical condition would be, in the spirit of Ehrenfests’s theorem, rather to impose the vanishing of the expectation value of the Virasoro generators.
• Therefore it is sufficient to require, for the open string,
Lm|ϕ i = 0 m > 0, (3.21)
(L0− a) |ϕ i = 0. (3.22)
I.e. a state is called |ϕ i physical if and only if it satisfies
(Lm− a δm,0) |ϕ i = 0 ∀ m ≥ 0 . (3.23)
• Indeed physical states satisfy h ϕ| Lm− aδm,0|ϕ i = 0 ∀ m.
• Note that in implimenting the zero level Virasoro constraint we allow for a yet-to-be de- termined normal ordering constant a, following the logic discussed around (3.12).
• For the closed string we require
(Lm− a δm,0) |ϕ i = 0, ∀ m ≥ 0, (3.24)
( ˜Lm− ˜a δm,0) |ϕ i = 0, ∀ m ≥ 0. (3.25)
• A priori, a and ˜a might be different. However, we insist that invariance under σ → σ + ∆ continues to hold in the quantum theory. Otherwise the theory would suffer from a gravi- tational anomaly, i.e. an anomaly of spatial diffeomorphism invariance of the worldsheet.1
Then
(L0− ˜L0) |ϕ i = 0 (3.26)
requires
a = ˜a. (3.27)
Mass shell condition
As in the classical theory the quantum mass shell condition arises as the level-zero Virasoro constraint involving L0. Thus the normal ordering constant a effects the mass of the string
states.
1Note that in canonical quantisation the time coordinate τ is singled out from spatial coordinates - here
just σ. Unlike the gravitational anomaly affecting the latter, an anomaly in τ -reparametrisation invariance by (L0+ ˜L0− 2a) |ϕ i = 0 is accepatable,
i) For the open string we consider (NN) boundary conditions in directions µ, (DD) boundary conditions in directions i and (ND) or (DN) boundary conditions in directions a and compute
L0= X n>0 α−n· αn+ 1 2α 2 0= ∞ X n=1 [αµ−n(αn)µ+ αi−n(αn)i] + X r∈N0+12 αa−r(αr)a + α0pµpµ+ α0(T ∆xi)2, i.e. L0:= N + α0p2+ α0(T ∆x)2
with N the number operator and pµp
µ = −M2 the invariant mass. Then solving the
level-zero Virasoro constraint (L0− a) |ϕ i = 0 for p2= −M2yields
α0M2|ϕ i = (N + α0(T ∆x)2− a) |ϕ i, (3.28) with N =P
n>0Nn+
P
r∈N0+12Nr. . . the number of excitations. The contribution α
0(T ∆x)2
takes into account the energy from the tension of the string stretched in the (DD) directions. ii) For the closed string we obtain
L0= X n>0 α−n· αn+ α0 4p 2= N + α0 4p 2, ˜ L0= X n>0 α−n· αn+ α0 4p 2= ˜N +α0 4p 2. Level matching (3.26), (N − ˜N ) |ϕ i = 0, (3.29) and the Virasoro constraints (L0− a) |ϕ i = 0 = ( ˜L0− a) |ϕ i yield
α0M2= 4(N − a). (3.30)