to note that this ground state is degenerate in the sense that
ground states for fixed J ¼ J
1;2
þ J
3;4
all have the same
energy E
J
. We will be parameterizing the rotating string in
terms of the two radii R
1;2
, R
3;4
in the two planes. In terms
of these, the classical target space energy and angular
momentum are given by
¯
E ¼ 2πγR;
¯
J
1;2
¼ πγR
2
1;2
;
¯
J
3;4
¼ πγR
2
3;4
; ð10Þ
with
R ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R
2
1;2
þ R
2
3;4
q
: ð11Þ
Here the bar stands for the values corresponding to the
classical (background) configuration
¯
X, cf. (2). In our
parametrization, the relation between the world sheet
Hamiltonian H, the quantum correction E
q
to the target
space energy E, and the quantum corrections J
q
1;2
, J
q
3;4
to
the angular momenta J
1;2
, J
3;4
is
E
q
¼
1
R
ðH þ J
q
1;2
þ J
q
3;4
Þ; ð12Þ
yielding
E
2
¼ð
¯
E þ E
q
Þ
2
¼ 4π
2
γ
2
R
2
þ 4πγðH þ J
q
1;2
þ J
q
3;4
ÞþOðR
−2
Þ
¼ 4πγðJ
1;2
þ J
3;4
þ H
0
ÞþOðR
−1
Þ: ð13Þ
By comparison with (9), one can directly read off the
intercept a from the expectation value of H
0
, i.e.,
a ¼ −
1
2
hH
0
i: ð14Þ
As the case of elliptic strings is notationally and
computationally much more involved, we chose to first
present our calculation for the case of circular strings, i.e.,
for R
1;2
¼ R
3;4
, and to discuss the modifications necessary
for the treatment of elliptic strings in a separate section. The
article is thus structured as follows: In the next section, we
discuss the perturbations of classical rotating circular
solutions, and in Sec. III the calculation of the correspond-
ing intercept. Section IV deals with the generalization to the
elliptic case. These calculations are for simplicity per-
formed for D ¼ 6. The straightforward generalization to
general D is discussed in Sec. V. Finally, in Sec. VI,we
compare the spectrum of physical excitations in the
linearized semi-classical theory and the covariant quanti-
zation scheme. We conclude in Sec. VII. An Appendix
contains some intermediate results of our treatment of the
elliptic case.
II. PERTURBATIONS OF CLASSICAL ROTATING
CIRCULAR STRINGS
For an embedding X∶Σ → ðR
D
; ηÞ the Nambu-Goto
action (3) yields the equations of motion
□
g
X ¼ 0: ð15Þ
We also recall the target space momentum and angular
momentum derived from the action (3):
P
i
¼
Z
2π
0
δS
δ∂
0
X
i
dσ ¼ −γ
Z
2π
0
ffiffiffi
g
p
g
0ν
∂
ν
X
i
dσ; ð16aÞ
L
ij
¼
Z
2π
0
δS
δ∂
0
X
j
X
i
− i ↔ j
dσ
¼ γ
Z
2π
0
ffiffiffi
g
p
g
0ν
ðX
j
∂
ν
X
i
− X
i
∂
ν
X
j
Þdσ: ð16bÞ
Here we assumed Σ to be parametrized by ðτ; σÞ ∈
R × ½0; 2πÞ. The target space energy is given by E ¼ P
0
.
We parametrize the rotating circular solution as
¯
Xðτ; σÞ¼
R
ffiffiffi
2
p
0
B
B
B
B
B
B
B
B
@
ffiffiffi
2
p
τ
sin τ cos σ
− cos τ cos σ
cos τ sin σ
sin τ sin σ
0
1
C
C
C
C
C
C
C
C
A
; ð17Þ
where σ ∈ ½0 ; 2πÞ and R is given by (11). Note that in our
parametrization, the world sheet coordinates τ, σ are
dimensionless. For simplicity, we here assumed that the
target space-time is six dimensional. As discussed in
Sec. V, it is straightforward to add further dimensions
(or to remove the sixth). The induced metric on the world
sheet, in the coordinates introduced above, is a multiple of
the Minkowski metric,
¯
g
μν
¼
R
2
2
η
μν
: ð18Þ
In particular, the equation of motion (15) can be easily
checked. Energy and angular momentum of the above
solution were given in (10), with R
1;2
¼ R
3;4
¼
R
ffiffi
2
p
.
Our goal is to perform a (canonical) quantization of the
fluctuations φ around the classical background
¯
X, cf. (2).
As discussed in the introduction, we may, at the level of the
free theory, restrict to normal fluctuations. We parametrize
these as
φ ¼ f
s
v
s
þ f
a
v
a
þ f
b
v
b
þ f
c
v
c
; ð19Þ
SEMICLASSICAL ENERGY OF CLOSED NAMBU-GOTO STRINGS PHYS. REV. D 100, 106005 (2019)
106005-3
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