The quantum mechanics of the free bosonic string is described in the light-cone
formalism [6.5] in terms of the independent coordinates Xi(σ) and x−, and their
canonical conjugates P 0i(σ) and p+, with “time” (x+ = 2α′p+τ ) dependence given
by the hamiltonian (see (6.3.4b))
1
dσ
H =
1 (P 0 2 + X ′2) .
(7.1.1)
α′
2π 2
i
i
Functionally,
1
δ
δ
P 0
,
, X
α′
i(σ) = i δX
j (σ2)
= δij2πδ(σ2 − σ1) .
(7.1.2)
i(σ)
δXi(σ1)
(Note our unconventional normalization for the functional derivative.)
For the open string, it’s convenient to extend σ from [0, π] to [−π, π] as in (6.1.7) by defining
1
δ
1
X(−σ) = X(σ) , P (σ) = √
α′i
+ X′ = √ P ±(±σ) for ± σ > 0 ,
2α′
δX
α′
(7.1.3a)
so that P is periodic (and ∼ P + or P −, as in (6.2.5), for σ > or < 0), or to express the open string as a closed string with modes which propagate only clockwise (or only counterclockwise) in terms of ˆ
X of sect. 6.2:
1
P (σ) = √
ˆ
X′(σ) ,
(7.1.3b)
2α′
7.1. Bosonic
129
which results in the same P (same boundary conditions and commutation relations).
The latter interpretation will prove useful for graph calculations. (However, the form of the interactions will still look different from a true closed string.) Either way, the new boundary conditions (periodicity) allow (Fourier) expansion of all operators in terms of exponentials instead of cosines or sines. Furthermore, P contains all of X
and δ/δX (except x, which is conjugate to p ∼
dσP ; i.e., all the translationally
invariant part). In particular,
π dσ
H =
1 P 2 .
(7.1.4)
2
i
−π 2π
The commutation relations are
[P i(σ1), P j(σ2)] = 2πiδ′(σ2 − σ1)δij .
(7.1.5)
For the closed string, we define 2 P ’s by
1
δ
1
P (±)(σ) = √
α′i
± X′(±σ) = √
ˆ
X(±)′(σ) .
(7.1.6)
2α′
δX(±σ)
2α′
Then operators which are expressed in terms of integrals over σ also include sums
over ±: e.g., H = H(+) + H(−), with H(±) given in terms of P (±) by (7.1.4).
In order to compare with particles, we’ll need to expand all operators in Fourier
modes. In practical calculations, functional techniques are easier, and mode expan-
sions should be used only as the final step: external line factors in graphs, or expansion of the effective action. (Similar remarks apply in general field theories to expansion of superfields in components and expansion of fields about vacuum expectation values.) For P for the open string (suppressing Lorentz indices),
∞
P (σ) =
αne−inσ ,
n=−∞
√
√
α
†
0 =
2α′p ,
αn = (α−n)† = −i nan
,
[p, x] = i ,
[α
†
m, αn] = nδm+n,0
,
[am, an ] = δmn .
(7.1.7a)
We also have
√
∞
ˆ
1
X(σ) = ˆ
X†(σ) = ˆ
X*(−σ) = (x + 2α′pσ) + 2α′
√ (a †e−inσ + a
n n
neinσ )
,
1
δ
X(σ) = 1 ˆ
X(σ) + ˆ
X(
= 1 ˆ
X′(σ) + ˆ
X′(
2
−σ)
,
P 0(σ) = α′i
−σ)
.
δX(σ)
2
(7.1.7b)
130
7. LIGHT-CONE QUANTUM MECHANICS
(The † and * have the usual matrix interpretation if the operators are considered
as matrices acting on the Hilbert space: The † is the usual operatorial hermitian
conjugate, whereas the * is the usual complex conjugate as for functions, which
changes the sign of momenta. Combined they give the operatorial transpose, which
corresponds to integration by parts, and thus also changes the sign of momenta, which are derivatives.) H is now defined with normal ordering:
∞
H = 1 :
α
2
n · α−n : + constant
−∞
∞
= α′p 2
†
2
i +
nan · an + constant = α′pi + N + constant . (7.1.8a)
1
In analogy to ordinary field theory, i∂/∂τ + H = α′p2 + N + constant.
The constant in H has been introduced as a finite renormalization after the infi-
nite renormalization done by the normal ordering: As in ordinary field theory, wherever infinite renormalization is necessary to remove infinities, finite renormalization should also be considered to allow for the ambiguities in renormalization prescrip-tions. However, also as in ordinary field theories, the renormalization is required to respect all symmetries of the classical theory possible (otherwise the symmetry is anomalous, i.e., not a symmetry of the quantum theory). In the case of any light-cone theory, the one symmetry which is never manifest (i.e., an automatic consequence of the notation) is Lorentz invariance. (This sacrifice was made in order that unitarity would be manifest by choosing a ghost-free gauge with only physical, propagating
degrees of freedom.) Thus, in order to prove Lorentz invariance isn’t violated, the commutators of the Lorentz generators J
dσ
ab = 1
X
α′
2π
[aP 0b] must be checked.
All
are trivial except [J−i, J−j] = 0 because X− and P 0− are quadratic in Xi and P 0i by (6.3.5). The proof is left as an (important) exercise for the reader that the desired result is obtained only if D = 26 and the renormalization constant in H and P 0−
(H = − 1
dσ P 0
α′
2π
−) is given by
H = α′p 2
2
i + N − 1 ≡ α′(pi + M 2)
.
(7.1.8b)
The constant in H also follows from noting that the first excited level (a †
1i |0 ) is
just a transverse vector, which must be massless. The mass spectrum of the open
string, given by the operator M2, is then a harmonic oscillator spectrum: The possible values of the mass squared are −1, 0, 1, 2, ... times 1/α′, with spins at each mass level running as high as N = α′M2 + 1. (The highest-spin state is created by the
symmetric traceless part of N a †
1 ’s.)
7.1. Bosonic
131
Group theory indices are associated with the ends of the open string, so the
string acts like a matrix in that space. If the group is orthogonal or symplectic,
the usual (anti)symmetry of the adjoint representation (as a matrix acting on the
vector representation) is imposed not just by switching the indices, but by flipping the whole string (switching the indices and σ ↔ π − σ). As a result, N odd gives
matrices with the symmetry of the adjoint representation (by including an extra “−”
sign in the condition to put the massless vector in that representation), while N even gives matrices with the opposite symmetry. (A particular curiosity is the “SO(1)”
open string, which has only N even, and no massless particles.) Such strings, because of their symmetry, are thus “nonoriented”. On the other hand, if the group is just
unitary, there is no symmetry condition, and the string is “oriented”. (The ends can be labeled with arrows pointing in opposite directions, since the hermitian conjugate state can be thought of as the corresponding “antistring”.)
The closed string is treated analogously. The mode expansions of the coordinates
and operators are given in terms of 2 sets of hatted operators ˆ
X(±) or P (±) expanded
over α(±)n:
X(σ) = 1 ˆ
X(+)(σ) + ˆ
X(−)(
ˆ
X(+)′(σ) + ˆ
X(−)′(
2
−σ)
,
P 0(σ) = 12
−σ)
;
ˆ
X(±)(σ) = ˆ
X(±)†(σ) = ˆ
X(±)*(−σ)
→
α(±)
†
n = α(±)−n
= −α(±)n* .
(7.1.9)
However, because of the periodicity condition (6.2.4c), and since the zero-modes (from (7.1.7)) appear only as their sum, they are not independent:
x(±) = x ,
p(±) = 1p .
(7.1.10)
2
Operators which have been integrated over σ (as in (7.1.4)) can be expressed as sums over two sets (±) of open-string operators: e.g.,
H = H(+) + H(−) = 1α′p 2 + N
2
i
− 2 , N = N(+) + N(−) .
(7.1.11)
Remembering the constraint under global σ translations
d
dσ
δ
i
≡
X′ · i
= ∆N ≡ N(+) − N(−) = 0 ,
(7.1.12)
dσ
2π
δX
the spectrum is then given by the direct product of 2 open strings, but with the states constrained so that the 2 factors (from the 2 open strings) give equal contributions to the mass. The masses squared are thus −4, 0, 4, 8, ... times 1/α′, and the highest spin at any mass level is N = 1α′M2 + 2 (with the corresponding state given by the
2
132
7. LIGHT-CONE QUANTUM MECHANICS
symmetric traceless product of N a †
1 ’s, half of which are from one open-string set
and half from the other). Compared with α′M2 + 1 for the open string, this means
that the “leading Regge trajectory” N(M2) (see sec. 9.1) has half the slope and twice the intercept for the closed string as for the open string. If we apply the additional constraint of symmetry of the state under interchange of the 2 sets of string operators, the state is symmetric under interchange of σ ↔ −σ, and is therefore “nonoriented”; otherwise, the string is “oriented”, the clockwise and counterclockwise modes being distinguishable (so the string can carry an arrow to distinguish it from a string that’s flipped over).
From now on we choose units
α′ = 1
.
(7.1.13)
2
In the case of the open bosonic string, the free light-cone Poincaré generators can be obtained from the covariant expressions (the obvious generalization of the particle expressions, because of (7.1.2), since Xa and P 0a are defined to Lorentz transform as vectors and X to translate by a constant),
π dσ
dσ
Jab = −i
X[a(σ)P 0b](σ) ,
pa =
P 0a(σ) ,
(7.1.14a)
−π 2π
2π
by substituting the gauge condition (6.3.2) and free field equations (6.1.7) (using (7.1.7))
2
ˆ
P
− 2
X
i
+(σ) = p+σ
,
P 2 − 2 = 0
→
P − = − 2p+
→
Jij = −ix[ipj] +
a†n[ianj] ,
J+i = ixip+ ,
J−+ = −ix−p+ ,
J−i = −i(x−pi − xip−) +
a†n[−ani] ,
1
1
p
2
i
− = −
p + 2
na† a
(pia
2p
i
n
ni − 1
,
an− = −
ni + ∆n)
,
+
p+
1
n−1
∞
∆
1
i
i
n = i √
m(n − m)a a
m(n + m)a† a
n
2
m
n−m,i −
m
n+m,i
.
m=1
m=1
(7.1.14b)
(We have added the normal-ordering constant of (7.1.8) to the constraint (6.1.7).)
As for arbitrary light-cone representations, the Poincaré generators can be expressed completely in terms of the independent coordinates xi and x−, the corresponding
momenta pi and p+, the mass operator M, and the generators of a spin SO(D−1),
7.1. Bosonic
133
Mij and Mim. For the open bosonic string, we have the usual oscillator representation of the SO(D−2) spin generators
′
M
îj = −i
XiP j =
a†n[ianj] ,
(7.1.15a)
n
where ′ means the zero-modes are dropped. The mass operator M and the remaining
SO(D−1) operators Mim are given by
′
′
M2 = −2p
2
i
+
P − =
(P i − 2) = 2
na†n ani − 1 = 2(N − 1) ,
′
M
îmM = ip+
XiP − =
a†ni∆n − ∆†nani
.
(7.1.15b)
The usual light-cone formalism for the closed string is not a true light-cone for-
malism, in the sense that not all constraints have been solved explicitly by eliminating variables: The one constraint that remains is that the contribution to the “energy”
p− from the clockwise modes is equal to that from the counterclockwise ones. As a
result, the naive Poincaré algebra does not close [4.10]: Using the expressions
J−i = −i(x−pi − xip−) +
a†(±)n[−a(±)ni] ,
n,±
1
p
2
− = −
p + 4 N(+) + N(−) − 2
,
(7.1.16a)
2p
i
+
we find
4
[J−i, J−j] = −
∆N∆J
p 2
ij
,
(7.1.16b)
+
where ∆Jij is the difference between the (+) and (−) parts of Jij.
We instead define 2 sets of open-string light-cone Poincaré generators J(±)ab and
p(±)a, built out of independent zero- and nonzero-modes. The closed-string Poincaré generators are then [4.10]
Jab = J(+)ab + J(−)ab ,
pa = p(+)a + p(−)a .
(7.1.17a)
Since the operators
∆pa = p(+)a − p(−)a
(7.1.17b)
commute with themselves and transform as a vector under the Lorentz algebra, we
can construct a Poincaré algebra from just Jab and ∆pa. This is the Poincaré algebra whose extension is relevant for string field theory; it closes off shell. (This holds in either light-cone or covariant quantization.) However, as described above, this
134
7. LIGHT-CONE QUANTUM MECHANICS
results in an unphysical doubling of zero-modes. This can be fixed by applying the
constraints (see (7.1.10))
∆pa = 0 .
(7.1.18)
In the light-cone formalism,
∆p+ = ∆pi = 0
(7.1.19a)
eliminates independent zero-modes, while
1
2
0 = ∆p− = −
(M2(+) − M2(−)) = −
∆N
(7.1.19b)
p+
p+
is then the usual remaining light-cone constraint equating the numbers of left-handed and right-handed modes.
The generators of the Lorentz subgroup again take the form (2.3.5), as for the
open string, and the operators appearing in Jab are expressed in terms of the open-
string ones appearing in (7.1.15) as
Mij =
M(±)ij ,
M2 = 2
M2(±) = 4(N − 2) ,
N =
N(±)
,
∆N = N(+) − N(−) ,
MimM = 2
(MimM)(±) .
(7.1.20)
(Since Jab and pa are expressed as sums, so are Mab and M. This causes objects
quadratic in these operators to be expressed as twice the sums in the presence of the constraint ∆pa = 0 → p(±)a = 1p
2 a.)
These Poincaré algebras will be used to derive the OSp(1,1|2) algebras used in
finding gauge-invariant actions in sects. 8.2 and 11.2.
7.2. Spinning
In this section we’ll describe a string model with fermions obtained by introducing a 2D supersymmetry into the world sheet [7.1,5.1], in analogy to sect. 5.3, and derive the corresponding Poincaré algebra. The description of the superstring obtained by this method isn’t manifestly spacetime supersymmetric, so we’ll only give a brief discussion of this formalism before giving the present status of the manifestly supersymmetric formulation. In both cases, (free) quantum consistency requires D = 10.
For simplicity, we go directly to the (first-)quantized formalism, since as far as
the field theory is concerned the only relevant information from the free theory is how
7.2. Spinning
135
to construct the covariant derivatives from the coordinates, and then the equations of motion from the covariant derivatives. After quantization, when (in the Schrödinger picture) the coordinates depend only on σ, there still remains a 1D supersymmetry
in σ-space [7.2]. The covariant derivatives are simply the 1D supersymmetrization of the P operators of sect. 7.1 (or those of sect. 6.1 at the classical level, using Poisson brackets):
{ ˆ
Da(σ1, θ1), ˆ
Db(σ2, θ2)} = ηabd22πδ(σ2 − σ1)δ(θ2 − θ1) ,
(7.2.1a)
ˆ
D(σ, θ) = ˆ
Ψ(σ) + θP (σ)
→ [P , P ] = as before , { ˆ
Ψa(σ1), ˆ
Ψb(σ2)} = ηab2πδ(σ2 − σ1) .
(7.2.1b)
The 2D superconformal generators (the supersymmetrization of the generators (6.1.7b) of 2D general coordinate transformations, or of just the residual conformal transformations after the lagrange multipliers have been gauged away) are then
1 ˆ
Dd ˆ
D = ( 1 ˆ
Ψ
P 2 + 1 i ˆ
Ψ′
2
2
· P ) + θ(12
2
· ˆ
Ψ) .
(7.2.2)
There are 2 choices of boundary conditions:
ˆ
D(σ, θ) = ± ˆ
D(σ + 2π, ±θ) → P (σ) = P (σ + 2π) ,
ˆ
Ψ(σ) = ± ˆ
Ψ(σ + 2π) . (7.2.3)
The + choice gives fermions (the Ramond model), while the − gives bosons (Neveu-
Schwarz).
Expanding in modes, we now have, in addition to (7.1.7),
∞
ˆ
Ψ(σ) =
γ
†
ne−inσ
→
{γam, γbn} = ηabδm+n,0 , γ−n = γn
,
(7.2.4)
−∞
where m, n are integral indices for the fermion case and half-(odd)integral for the bosonic. The assignment of statistics follows from the fact that, while the γn’s are
√
creation operators γ
†
n = dn
for n > 0, they are γ matrices γ0 = γ/ 2 for n = 0
(as in the particle case, but in relation to the usual γ matrices now have Klein
transformation factors for both dn’s and, in the BRST case, the ghost C, or else the dn’s and C are related to the usual by factors of γ11). However, as in the particle case, a functional analysis shows tha