Fulltext
Available from: Makoto Natsuume, May 02, 2013arXiv:hepth/0509188v1 26 Sep 2005
UTKomaba/059
KEKTH1038
September 2005
Physical state representations and gauge ﬁxing
in string theory
Masako Asano, Mitsuhiro Kato
†
and Makoto Natsuume
‡
Faculty of Liberal Arts and Sciences
Osaka Prefecture University
Sakai, Osaka 5998531, Japan
†
Institute of Physics
University of Tokyo, Komaba
Meguroku, Tokyo 1538902, Japan
‡
Theory Division, Institute of Particle and Nuclear Studies
KEK, High Energy Accelerator Research Organization
Tsukuba, Ibaraki 3050801, Japan
ABSTRACT
We reexamine physical state representations in the covariant quantization of
bosonic string. We especially consider one parameter f amily of gauge ﬁxing
conditions for the residual gauge symmetry due to null states (or BRST exact
states), and obtain explicit representations of observable Hilbert space which
include those of the DDF states. This analysis is aimed at giving a necessary
ingredient for the complete gauge ﬁxing procedures of covariant string ﬁeld theory
such as temporal or lightcone gauge.
Page 1
1 Introduction
It is well known that null states appearing in the physical Hilbert space of the string theory
correspond to the gauge degrees of freedom. For instance, level one null state L
−1
pi in
the open bosonic string gives the gauge transformation for the massless vector mode on
the same level when p
2
= 0. Here L
n
(n=integer) is the Virasoro operator and pi is the
oscillator vacuum with the momentum eig envalue p
µ
(µ = 0, 1, ···, 25). The null states
are certainly members of physical states in the sense that they satisfy the physical state
condition L
n
physi = 0 for positive integer n and the onshell condition (L
0
− 1)physi = 0,
while they do not contribute to the physical amplitude. In this sense the onshell physical
state is said to have an ambiguity in its representation.
Fixing the gauge degrees of freedom associated with the null state is not hing but ta king
a representative for the above mentioned ambiguity. One of the wellknown representative
of physical state is the socalled DDF st ate [1] which is generated by applying the transverse
DDF operator s to the tachyon state. As will be seen in the subsequent sections, the DDF
states are characterized by supplementary linear condition other than the physical state
condition.
From the point of view of the string ﬁeld t heory (SFT), that extra condition as well
as physical sta t e condition plays a role of complete gauge ﬁxing condition of the inﬁnite
dimensional gauge symmetry. For the sake of concreteness, let us take the simplest action
for the string ﬁeld Ψ (See e.g., ref.[2])
S =
1
2
Ψ(L
0
− 1)Ψ, (1)
with the condition
L
n
Ψ = 0 (n = 1, 2, ···). (2)
This is the (partially) gaugeﬁxed covariant action. Actually these equations lead to the
following action and gauge condition for the massless vector ﬁeld A
µ
contained as a mode
in Ψ
S =
Z
d
26
x
1
2
A
µ
A
µ
, (3)
∂
µ
A
µ
= 0. (4)
As is known, the action still has a residual gauge invariance
A
µ
→ A
µ
+ ∂
µ
λ with
λ = 0. (5)
1
Page 2
This is also tr ue for the SFT level; eqs.(1) and (2) has residual gauge invariance
Ψ → Ψ + L
−1
Ξ
1
+ (L
−2
+
3
2
L
−1
2
)Ξ
2
, (6)
provided
(
L
n
Ξ
1
= 0
L
0
Ξ
1
= 0
and
(
L
n
Ξ
2
= 0
(L
0
+ 1)Ξ
2
= 0
for positive integer n. The degrees of freedom of Ξ’s are not hing but the null states (or exact
states in the BRST quantization [3]) mentioned at the beginning. Thus putting some extra
conditions, like the DDF representation, corresponds to the complete gauge ﬁxing of SFT.
In the present paper, we will investigate a certain class of complete ga uge ﬁxing condi
tions. In particular, one parameter family o f linear ga ug e condition is analyzed, which is
essentially temporal gauge (or chronological gauge) in the sense that the string excitation
of a timelike direction is restricted to only zero mode. This family includes the DDF rep
resentation as a limit so that the relation between temporal gauge and the lightlike gauge
will also be clariﬁed.
One of the motivations for studying the tempo r al gauge and its cousin is that better
understanding of the gauge may provide a clue towards the resolution of the longstanding
problem on the canonical quantization of SFT [4]. Since timelike excitation is restricted to
the zero mode, it can be t aken a s a time parameter of canonical quantization procedure and
also the interaction becomes lo cal with respect to the time parameter.
For those who are not familiar with the problem may wonder whether there are anything
wrong with the SFT because it reproduced the correct quantum amplitudes in perturbative
sense. In deriving such amplitudes, however, one assumes that the Feynmann rules can be
read oﬀ from the a ction as has been done in the usual local ﬁeld theories. (See for example
ref.[5].) There is generally no justiﬁcation for such an assumption in nonlocal theories.
The existence of light cone SFT may support the validity of the assumption if the exact
relationship from the covariant SFT to the lightcone SFT through the gauge ﬁxing in the
SFT level, because in the latter for mulation lightlike variable x
+
is the time parameter of
the quantization procedure and locality of the interaction with respect to x
+
is satisﬁed.
In o r der to try these scenario, a s a ﬁrst step, we will clarify the structure and the rep
resentation of the physical states in the temporal gauge in keeping the relation to the DDF
states clear, as the latter representation can be regarded as lightlike gauge in the SFT level.
This paper is organized as follows. After discussing some generality of gauge ﬁxing
and identifying the concrete condition for the DDF states in the next section, we prove in
2
Page 3
section 3 that a certain class of gauge ﬁxing conditions are complete in the sense that the
state space speciﬁed by each gauge condition is equivalent to the observable positive deﬁnite
Hilbert space. It will be also shown there that the representation of observable space g iven
by the DDF states can be obtained by a certain limiting procedure from more general
representations, which may cast new light on the relationship between lightlike gauge and
temporal gauge in the SFT. Section 4 is devoted to the summary and discussions.
2 Physical states in covariant gauge
The tota l state space H(p) for the old covariant quantization (OCQ) of perturbative bosonic
string theory (D = 26) is given by the Fock space Fock(α
µ
−n
; p
µ
) spanned by the states of
the form
φ
N
; p
µ
i =
25
Y
µ=0
∞
Y
n=1
(α
µ
−n
)
N
µ
n
0; p
µ
i. (7)
Here, N
µ
n
is a nonnegative integer and 0; p
µ
i is the ground state annihilated by all α
µ
n
(n > 0) with momentum p
µ
. We often divide H(p) into the space with level N =
P
n,µ
nN
µ
n
as H(p) = ⊕
N≥0
H
(N)
(p). Among H( p) , positive deﬁnite Hilbert space H
obs
(p) is deﬁned by
the quotient H
obs
(p) = H
phys
(p)/H
null
(p), which we sometimes call observable Hilbert space.
Here H
phys
(p) is the set of states satisfying the physical state condition
L
n
φi
phys
= 0 (n > 0) (8)
and the onshell condition
(L
0
− 1)φi
phys
= 0 (9)
which restricts the level N of the states as α
′
p
2
+ N −1 = 0. The space H
null
(p)[⊂ H
phys
(p)]
is the set of null states that are identiﬁed as physical states of the form
χi
null
= L
−1
ξ
1
i+ (L
−2
+
3
2
L
2
−1
)ξ
2
i (10)
where L
n
ξ
1
i = (L
n
+ δ
n,0
)ξ
2
i = 0 (n ≥ 0). A null state has zero inner product with any
state in H
phys
(p) (
null
hχφi
phys
= 0). This is seen from (8) and (10) with the deﬁnition of
inner product in H(p): L
†
−n
= L
n
(α
†
−n
= α
n
) and h0; p0; pi = 1.
Due to the existence of null states, we have an ambiguity φi
phys
∼ φi
phys
+ χi
null
in
choosing explicit representations of observable Hilbert space H
obs
(p). As we have seen in
the introduction in terms of SFT, appearance of null states in our OCQ scheme (o r exact
3
Page 4
states for BRST quantization) indicates the existence o f residual gauge symmetry which is
left unﬁxed at the classical level. Thus, choosing explicit representation of H
obs
(p) exactly
corresponds to ﬁxing this residual gauge symmetry. In fa ct, in addition to the physical state
condition, we need supplementary ‘gauge condition’ which exactly ﬁxes whole gauge degrees
of freedom and nothing more nor less:
H
phys
(p) ∩ {‘gauge condition’} ∼ H
obs
(p). (11)
We would like to ﬁnd a class of such conditions and corresponding representations of H
obs
(p)
in a systematic manner.
For example, we know that the set of socalled DDF states can be taken as an explicit
representation of H
obs
(p). They are given by multiplying DDF operators
A
ˆ
i
−n
=
1
2π
I
dz∂X
ˆ
i
(z)e
−ink·X(z)
(12)
on the tachyon ground state  0, ¯pi as
φ; p = ¯p − Nki = A
ˆ
i
1
−n
1
A
ˆ
i
2
−n
2
···A
ˆ
i
l
−n
l
0, ¯pi. (13)
Here,
X
µ
= x
µ
− ip
µ
ln z + i
X
n6=0
1
n
α
µ
n
z
−n
, (14)
ˆ
i = 1, ···, 24, N = n
1
+ ··· + n
l
, k
2
= 0 (with k
ˆ
i
= 0), ¯p
2
= 2 and ¯p · k = 1. Hereafter, we
set α
′
= 1/2 (α
µ
0
= p
µ
). These DDF states satisfy the physical state condition and form a
basis of H
obs
(p) fo r p = ¯p − Nk. D DF states up to level N = 2 are as f ollows:
N = 0
: 0, ¯pi, (15)
N = 1 : A
ˆ
i
−1
0, ¯pi = α
ˆ
i
−1
0, ¯p − ki, (16)
N = 2 : A
ˆ
i
−2
0, ¯pi =
α
ˆ
i
−2
− 2(k · α
−1
)α
ˆ
i
−1
0, ¯p − 2ki, (17)
A
ˆ
i
−1
A
ˆ
j
−1
0, ¯pi =
α
ˆ
i
−1
α
ˆ
j
−1
+
1
2
δ
ˆ
i
ˆ
j
h
(k ·α
−1
)
2
− (k · α
−2
)
i
0, ¯p − 2ki. (18)
In fact, the set of DDF states can be extracted by imposing an additional condition
k · α
n
φ; ¯p − Nki = 0 (n > 0) (19)
on the space o f physical states H
phys
(p) when p = ¯p − Nk [6]. This is an example of
supplementary gaug e condition that completely ﬁxes the ambiguity of null states as discussed
before.
4
Page 5
3 Representations of observable Hilbert space
Next we consider a class of supplementary conditions which are linear in oscillator variables.
They are simple generalizations of (1 9) in the previous section, but still nontrivial in the
sense that the proof of (11 ) does not go in the same way as for the DDF states since one
cannot deﬁne DDFlike o perators in general cases.
Concretely, we consider the following condition
˜ǫ · α
n
φ; p
µ
i = 0 (n > 0) (20)
with a constant timelike or lightlike vector ˜ǫ (i.e., ˜ǫ
2
≤ 0). For ˜ǫ
µ
∝ k
µ
, this condition
reduces to (19) and restricts states not to include α
−
−n
= (α
0
−n
− α
25
−n
)/
√
2 or α
+
−n
= (α
0
−n
+
α
25
−n
)/
√
2 respectively for k
µ
∝ δ
µ
0
+ δ
µ
25
or k
µ
∝ δ
µ
0
−δ
µ
25
. Also, for ˜ǫ
µ
= δ
µ
0
, the condition
(20) restricts states not to include any timelike oscillators (α
0
−n
). We do not consider the
case ˜ǫ
2
> 0 since the condition for such a case is not practical as a gauge condition, though
the condition itself works well to satisfy (11) with some appropriate assumptions.
3.1 Main theorem
The main claim of the present paper is the following theorem:
Theorem 1 Let H
˜ǫ
(p) denotes the subspace of H(p) spanned by the states satisfying both
(L
n
− δ
n,0
)φi
phys
= 0 (n ≥ 0)
and
˜ǫ · α
n
φ; p
µ
i = 0 (n > 0)
for ˜ǫ
2
≤ 0. Then, H
˜ǫ
(p) ∼ H
obs
(p) provided ˜ǫ · p 6= 0.
We divide H
˜ǫ
(p) by level N as H
˜ǫ
(p) = ⊕
N≥0
H
(N)
˜ǫ
(p) and prove the theorem for each N.
Before going into general proof, let us ﬁrst see the simple cases N = 0 and N = 1. For
N = 0, we only have ground state 0, pi in H
(0)
(p) (with p
2
= 2) and it satisﬁes (8) and (20)
trivially: H
(0)
˜ǫ
(p) = {0, pi}(= H
(0)
obs
(p)). For N = 1, general onshell states satisfying (8) are
represented as
φ; pi = ξ · α
−1
0, pi (21)
with p
2
= 0 and ξ · p = 0. Among these states, there is a null state p · α
−1
0, pi and the
space H
(1)
obs
(p) is identiﬁed up to the ambiguity ξ
µ
∼ ξ
µ
+ p
µ
. The condition (20) gives the
5
Page 6
constraint on ξ
µ
as ˜ǫ ·ξ = 0, which ﬁxes the ambiguity completely since ˜ǫ ·p 6= 0 is assumed.
Thus, H
(1)
˜ǫ
(p) ∼ H
(1)
obs
(p). Explicitly, H
(1)
˜ǫ
(p) = {ξ · α
−1
0, pi  p · ξ = ˜ǫ · ξ = 0} for onshell p
(p
2
= 0) with ˜ǫ · p 6= 0. We have proven H
˜ǫ
(p) ∼ H
obs
(p) for N = 0, 1. No t e that here we
have not used t he condition ˜ǫ
2
≤ 0. For general N, we ﬁrst give a proof for ˜ǫ
2
< 0 and then
extend it to ˜ǫ
2
= 0 since the latter can be considered as a limit of the former. For N ≥ 2,
the condition ˜ǫ · p 6= 0 is always satisﬁed for onshell states if ˜ǫ
2
≤ 0.
3.2 Proof for ˜ǫ
2
< 0
First, we will make some deﬁnitions for preparation
1
. We will ﬁx the timelike vector ˜ǫ
µ
as
˜ǫ
µ
= (cosh β, 0, ···, 0, sinh β) [= ˜ǫ
µ
(β)] (22)
with 0 ≤ β < ∞ without losing generality. Correspondingly, we deﬁne a spacelike vector
ǫ
µ
(β) = (sinh β, 0, ···, 0, cosh β). (23)
We take a particular choice of spacetime coordinates (t
β
, s
β
, x
ˆ
i
) ≡ (˜ǫ(β) · x, ǫ(β) · x, x
ˆ
i
)
which are obtained by boost transformation from the original coordinates x
µ
. Commutation
relations for α
˜µ
n
(˜µ = t
β
, s
β
,
ˆ
i) are given as
[α
˜µ
m
, α
˜ν
n
] = mδ
m+n,0
η
˜µ˜ν
(24)
where
α
t
β
n
= cosh β α
0
n
+ sinh β α
25
n
(= ˜ǫ(β) · α
n
), (25)
α
s
β
n
= sinh β α
0
n
+ cosh β α
25
n
(= ǫ(β) · α
n
). (26)
Thus, in particular,
[˜ǫ(β) ·α
m
, ˜ǫ(β)·α
n
] = −mδ
m+n,0
, [ǫ(β)·α
m
, ǫ(β) ·α
n
] = mδ
m+n,0
, [˜ǫ(β) ·α
m
, ǫ(β) ·α
n
] = 0.
(27)
We divide total state space into ‘timelike’ and ‘spacelike’ part:
H(p) = H
t
β
(p
t
β
) ⊗ H
Σ
β
(p
i
β
) (28)
1
In fact, to prove the theorem 1 for ˜ǫ
2
< 0 , it is suﬃcient to take ˜ǫ
µ
= δ
0
µ
since other cas es can be o bta ined
by boost transformations from this. We however consider every ˜ǫ
2
< 0 explicitly for later convenience.
6
Page 7
where i
β
= (s
β
,
ˆ
i), H
t
β
(p
t
β
) = Fock(α
t
β
−n
; p
t
β
) and H
Σ
β
(p
i
β
) = Fock(α
i
β
−n
; p
i
β
). We also
divide L
n
as L
n
= L
t
β
n
+ L
Σ
β
n
where
L
t
β
n
= −
1
2
∞
X
m=−∞
: α
t
β
n−m
α
t
β
m
: , L
Σ
β
n
=
1
2
∞
X
m=−∞
: α
i
β
n−m
α
i
β
m
: . (29)
We further deﬁne the space F
β
(p) as
F
β
(p) = {f
β
; pi  α
t
β
n
f
β
; pi = L
n
f
β
; pi = 0 (n > 0)}. (30)
The relation between this F
β
(p) and H
˜ǫ(β)
(p) is
H
˜ǫ(β)
(p) = {φi ∈ F
β
(p) (L
0
− 1)φi = 0 }. (31)
The space F
(N)
β
(p) is a subspace of 0, p
t
β
i ⊗ H
(N)
Σ
β
(p
i
β
) since f
β
; pi does not contain any
‘timelike’ oscillator α
t
β
−n
. Thus, F
β
(p) is positive deﬁnite a nd cannot contain null states
(10).
With the above deﬁnitions, we will now begin to prove theorem 1, i.e., H
(N)
˜ǫ(β)
(p) ∼ H
(N)
obs
(p).
First, we will give the following lemma:
Lemma 1 States of the form
L
−n
1
···L
−n
r
L
t
β
−m
1
···L
t
β
−m
q
f
β
; pi, f
β
; pi ∈ F
β
(p) (32)
(n
s
≤ n
s+1
, m
s
≤ m
s+1
) are linearly independent and span a basis of H(p) if p
t
β
6= 0.
The proof is given in Appendix A.
With t he above lemma, we will write every state in H(p) as a sum of states of the form
(32). In particular, we divide any physi ∈ H
phys
(p) written in this form into two classes as
physi = gi + χi (33)
where gi consists of terms without any L
−n
, i.e.,
gi =
X
C
m
1
,···,m
q
L
t
β
−m
1
···L
t
β
−m
q
f
β
; pi (34)
with constants C
m
1
,···,m
q
and the χi part consists o f terms including at least one L
−n
. Both
gi and χi satisfy onshell condition. Also, we see that L
1
gi and (L
2
+
3
2
L
2
1
)gi do not
contain any L
−n
and L
1
χi and (L
2
+
3
2
L
2
1
)χi again consist of terms with at least one L
−n
.
Thus L
n
physi = 0 implies that gi and χi are both physical and the state χi is null since
7
Page 8
all L
n
(n ≥ 1) are generated by L
1
and L
2
. For gi part, 0 = (L
t
β
n
+ L
Σ
β
n
)gi = L
t
β
n
gi for any
n > 0 since L
Σ
β
n
f
β
; pi = 0. This contradicts the nondegeneracy of c = 1 Verma module
2
V(1, h < 0) if there exist any L
t
β
−m
in gi. This means that gi contains no L
t
β
−m
and
gi = f
β
; pi. (35)
Thus we have shown that any physical state can be written as an element of H
˜ǫ(β)
(p) ⊂ F
β
(p)
up to a null state:
physi = f
β
; pi + χi, f
β
; pi ∈ H
˜ǫ(β)
(p). (36)
In other word, we have shown that H
˜ǫ(β)
(p) ∼ H
obs
(p) since we know that there are no null
states in H
˜ǫ(β)
(p). We have proven theorem 1 for ˜ǫ
2
< 0.
Note that in some parts of the above proof we have used the similar argument given in
ref.[6, 7] where essentially the same statement as our theorem 1 f or the set of D DF operators
(which corresponds to our case o f ˜ǫ
2
= 0 and p
µ
= ¯p
µ
− Nk
µ
) has been proved. Comparing
to that case, our proof fo r ˜ǫ
2
< 0 is rather simpler since the positivedeﬁniteness of F
β
(p) is
trivial (and also we know the nondegeneracy of V(1, h < 0)).
3.3 Properties of H
˜ǫ(β)
(p)
In this subsection, we present some properties of the space H
˜ǫ(β)
(p) as a representative of
observable Hilbert space H
obs
(p).
The dimension of H
obs
(p) coincides with that of the transverse Hilbert space H(p
ˆ
i
) =
Fock(α
ˆ
i
−n
; p
ˆ
i
): For each level N, dim H
(N)
obs
(p) = P
24
(N) where P
D
(n) is the coeﬃcient of q
n
in
Q
n≥1
(1 −q
n
)
−D
. We would like to choose a basis of H
˜ǫ(β)
(p) in order to analyze the space
systematically. For this aim, we have the following lemma [8]
Lemma 2 Assume that p
t
β
(= ˜ǫ · p) 6= 0 and p
s
β
(= ǫ · p) 6=
r−s
√
2
where r and s are positive
integers with rs < N. Then, a state f
β
; pi ∈ H
(N)
˜ǫ(β)
(p) has at least one term consisting only
of transverse oscillators, i.e.,
f
β
; pi = 
ˆ
φ; pi + (terms with at least one α
s
β
−n
) (37)
where 
ˆ
φ; pi is a nonzero state in Fock(α
ˆ
i
−n
; p).
2
V(c, h) is a linear space spanned by the states constructed by acting Vira soro op erators (L
−n
, n > 0) of
central charge c on the highest weight state hi.
8
Page 9
With this result, for p
s
β
6=
r−s
√
2
, we can choose a basis of H
(N)
˜ǫ(β)
(p) as follows: We specify each
basis element f
β
; pi
ˆ
λ
N
of H
(N)
˜ǫ(β)
(p) by the term

ˆ
φ; pi
ˆ
λ
N
={(
ˆ
i
1
,n
1
),···,(
ˆ
i
l
,n
l
)}
= α
ˆ
i
1
−n
1
···α
ˆ
i
l
−n
l
0, pi, (n
s
≤ n
s+1
,
l
X
s=1
n
s
= N) (38)
and write
f
β
; pi
ˆ
λ
N
= 
ˆ
φ; pi
ˆ
λ
N
+ (terms with at least one α
s
β
−n
). (39)
With ﬁxed
ˆ
λ
N
, the terms with α
s
β
−n
in f
β
; pi
ˆ
λ
N
are uniquely determined by the condition
L
n
f
β
; pi
ˆ
λ
N
= 0. Note that for p
s
β
 >
N−1
√
2
we can always choose t he above basis since
p
s
β
6=
r−s
√
2
and p
t
β
6= 0 for such a case.
For example, we explicitly represent the space H
(N)
˜ǫ(β)
(p) for N = 1, 2 by the basis given
above. For N = 1 with p
s
β
6= 0,
H
(N=1)
˜ǫ(β)
(p) = {f
β
; pi
ˆ
λ
1
=(
ˆ
i,1)
} (40)
where
f
β
; pi
(
ˆ
i,1)
=
"
α
ˆ
i
−1
− p
ˆ
i
α
s
β
−1
p
s
β
#
0, pi. (41)
For N = 2 with p
s
β
6= 0, ±
1
√
2
,
H
(N=2)
˜ǫ(β)
(p) = {f
β
; pi
ˆ
λ
2
={(
ˆ
i,2)}
, f
β
; pi
ˆ
λ
2
={(
ˆ
i,1),(
ˆ
j,1)}
} (42)
where
f
β
; pi
(
ˆ
i,2)
=
α
ˆ
i
−2
−
2
p
s
β
α
ˆ
i
−1
α
s
β
−1
+
4p
ˆ
i
2p
2
s
β
− 1
α
s
β
−1
α
s
β
−1
−
p
ˆ
i
(2p
2
s
β
+ 1)
p
s
β
(2p
2
s
β
− 1)
α
s
β
−2
0, pi (43)
and
f
β
; pi
{(
ˆ
i,1),(
ˆ
j,1)}
=
α
ˆ
i
−1
α
ˆ
j
−1
−
2
p
s
β
p
{
ˆ
i
α
ˆ
j}
−1
α
s
β
−1
+
δ
ˆ
i
ˆ
j
+ 2p
ˆ
i
p
ˆ
j
2p
2
s
β
− 1
α
s
β
−1
α
s
β
−1
−
p
2
s
β
δ
ˆ
i
ˆ
j
+ p
ˆ
i
p
ˆ
j
p
s
β
(2p
2
s
β
− 1)
α
s
β
−2
0, pi.
(44)
3.4 Proof for ˜ǫ
2
= 0
Now we prove the theorem for the remaining case: ˜ǫ
2
= 0. In this case, we may say that we
already have a proof in ref.[6, 7]. We will however give a proof based on the new picture where
the representation of physical states H
˜ǫ
(p) for ˜ǫ
2
= 0 can be understood as a limit of that
for ˜ǫ
2
< 0. In other word, we will identify the space H
˜ǫ(β=∞)
(p) as a limit ‘lim
β→∞
H
˜ǫ(β)
(p).’
In order to deﬁne such a limit consistently, we choose a set of particular states as a basis of
space H
˜ǫ(β)
(p) and take the limit
3
for each basis element of the space for ﬁxed momentum p
µ
.
3
The limit we c onsider is diﬀerent from boost transformation since we keep the momentum p
µ
ﬁxed.
9
Page 10
Now, we will explain how to deﬁne the limit explicitly. We consider the space H
(N)
˜ǫ(β)
(p)
with ﬁxed o nshell momentum p
µ
for each N. Here the momentum fr ame has to be chosen
in order to satisfy ˜ǫ(β) · p 6= 0 for arbitrary β (≤ ∞ ), i.e., we take p
0
+ p
25
6= 0. Then we
take β large enough (β > β
N
0
) to satisfy ǫ(β) · p >
N−1
√
2
for each p
µ
and N. We can always
take such β
N
0
since lim
β→∞
ǫ(β) · p = ∞ for any p with p
0
+ p
25
6= 0. From t he discussion
of the previous subsection, we can take the set of states {f
β
; pi
ˆ
λ
N
} as a basis of H
(N)
˜ǫ(β)
(p) for
β > β
N
0
. Each state f
β
; pi
ˆ
λ
N
contains β through the parts of ǫ(β) · α
−n
(n ≥ 0) and thus
the state can be expanded with respect to e
β
. We can prove from the property of physical
state condition that the terms with positive powers of e
β
cannot appear in the expansion of
f
β
; pi
ˆ
λ
N
and
lim
β(>β
N
0
)→∞
f
β
; pi
ˆ
λ
N
< ∞. (45)
Also, the terms with odd powers of e
β
do not appear in the expansion and thus the expansion
takes the form
f
β
; pi
ˆ
λ
N
= f
(0)
; pi
ˆ
λ
N
+ e
−2β
f
(1)
; pi
ˆ
λ
N
+ e
−4β
f
(2)
; pi
ˆ
λ
N
+ ···
=
∞
X
k=0
e
−2k β
f
(k)
; pi
ˆ
λ
N
. (46)
The leading term f
(0)
; pi
ˆ
λ
N
is given by the limit (45) and contains the term 
ˆ
φ; pi
ˆ
λ
N
of (38).
By deﬁnition, each t erm f
(k)
; pi
ˆ
λ
N
does not contain β and satisﬁes physical state condition
L
n
f
(k)
; pi
ˆ
λ
N
= 0. (4 7)
Furthermore, from the condition ˜ǫ(β) ·α
n
f
β
; pi
ˆ
λ
N
= 0, we have
(α
0
n
+ α
25
n
)f
(k)
; pi
ˆ
λ
N
+ (α
0
n
− α
25
n
)f
(k−1)
; pi
ˆ
λ
N
= 0. (48)
In particular, the leading term f
(0)
; pi
ˆ
λ
N
(= lim
β→∞
f
β
; pi
ˆ
λ
N
) satisﬁes
(α
0
n
+ α
25
n
)f
(0)
; pi
ˆ
λ
N
(∝ ˜ǫ(β → ∞) · α
n
f
(0)
; pi) = 0. (49)
The limit of the inner product of two states f
β
; pi
ˆ
λ
N
and f
β
; pi
ˆ
λ
′
N
can be explicitly
calculated as
lim
β→∞
ˆ
λ
N
hf
β
; p f
β
; pi
ˆ
λ
′
N
(=
ˆ
λ
N
hf
(0)
; p f
(0)
; pi
ˆ
λ
′
N
)
=
ˆ
λ
N
h
ˆ
φ; p 
ˆ
φ; pi
ˆ
λ
′
N
= f
ˆ
λ
N
δ
ˆ
λ
N
,
ˆ
λ
′
N
(50)
10
Page 11
where f
ˆ
λ
N
is a positive integer. This means that the space spanned by the states lim
β→∞
f
β
; pi
ˆ
λ
N
with all
ˆ
λ
N
has dimension P
24
(N) and is nondegenerate. Thus,
{ lim
β→∞
f
β
; pi
ˆ
λ
N
} = H
(N)
˜ǫ(β=∞)
(p) ∼ H
(N)
obs
(p) (51)
from (47) and (49). We have proven theorem 1 for ˜ǫ
2
= 0.
The characteristic point o f our proof comparing to the one in the literature [6, 7] is that
the nondegeneracy of the space H
˜ǫ(β=∞)
(p) is easily seen from that of H
˜ǫ(β<∞)
(p) and each
state in H
˜ǫ(β=∞)
(p) is represented as a limit of the corresponding state in H
˜ǫ(β<∞)
(p). In
fact, the space H
˜ǫ(β=∞)
(p) coincides with a set of DDF states if p
ˆ
i
= 0. Explicitly,
f
(0)
; pi
ˆ
λ
N
={(
ˆ
i
1
,n
1
),···,(
ˆ
i
l
,n
l
)}
= A
ˆ
i
1
−n
1
···A
ˆ
i
l
−n
l
0, p + Nki (52)
where k is a lightlike vector deﬁned by k ∝ lim
β→∞
˜ǫ(β) (i.e., k
µ
∝ (1, 0, ···, 0, 1)) and
k · p = 1.
For example, for N = 1, we explicitly take the β → ∞ limit of (41 ) : By using
lim
β→∞
α
s
β
−n
p
s
β
= lim
β→∞
ǫ(β) · α
−n
ǫ(β) · p
= k · α
−n
, (53)
we obtain
lim
β→∞
f
β
; pi
(
ˆ
i,1)
=
α
ˆ
i
−1
− p
ˆ
i
(k · α
−1
)
0, pi (54)
and this coincides with DDF state A
ˆ
i
−1
0, p + ki if we take p
ˆ
i
= 0. For N = 2, we can
similarly take the limit of (43) and (44) and the result for p
ˆ
i
= 0 is
lim
β→∞
f
β
; pi
(
ˆ
i,2)
=
α
ˆ
i
−2
− 2(k · α
−1
)α
ˆ
i
−1
0, pi (55)
and
lim
β→∞
f
β
; pi
{(
ˆ
i,1),(
ˆ
j,1)}
=
α
ˆ
i
−1
α
ˆ
j
−1
+
1
2
δ
ˆ
i
ˆ
j
h
(k · α
−1
)
2
− (k · α
−2
)
i
0, pi, (56)
which coincide with DDF states (17) and (18).
4 Summary and Discussions
In the present paper, we have investigated the o ld covariant quantization of bosonic string
theory and identiﬁed a class of additional conditions which precisely ﬁx the residual gauge
symmetry corresponding to the ambiguity of null states. By imposing such an additional
11
Page 12
condition on the space of physical states, we obtain a space which can be taken a s an
explicit representation of observable Hilbert space H
obs
(p). Explicitly, we have proven that
the condition ˜ǫ ·α
n
φ; p
µ
i = 0 for a constant timelike or lightlike ˜ǫ exactly plays the role of
the additional gauge condition which precisely ﬁx the ambiguity of null states if ˜ǫ is chosen
as ˜ǫ·p 6= 0. As a result, fo r each ˜ǫ, we have identiﬁed the space H
˜ǫ
(p) which gives a complete
set of physical states as a particular representation of H
obs
(p).
For timelike ˜ǫ = ˜ǫ(β <∞), the additional condition is related to the temporal gauge in the
sense that the corresponding representation of observable Hilb ert space H
˜ǫ(β<∞)
(p) does not
include timelike oscillators α
t
β
−n
. O n the other hand, the condition for lightlike ˜ǫ = ˜ǫ(β =∞)
is related to the lightcone gauge and in this case the space H
˜ǫ(β=∞)
(p) consists of physical
states without α
−
−n
. For each case, we have also identiﬁed a par ticular basis of H
˜ǫ
(p), which
would be useful for analyzing the theory (especially SFT) in the corresponding gauge. In
particular, the space H
˜ǫ(β=∞)
(p) for p
ˆ
i
= 0 is equivalent to the set of DDF states. As for the
other cases, our result means that we have systematically obtained a class of complete sets
of physical states other than the DDF states. We have also seen that the bases we used f or
H
˜ǫ(β<∞)
(p) and for H
˜ǫ(β=∞)
(p) are in oneto one correspondence, i.e., we have shown that
each state in H
˜ǫ(β=∞)
(p) (for p
+
6= 0) is obtained as a certain limit of the corresponding state
in H
˜ǫ(β<∞)
(p) except for a part icular value of momentum vector. This means that there is a
close relation between those two types of representations of physical states and it might be
possible that there is a substantial structure for the states in H
˜ǫ(β<∞)
(p) as well as for DDF
states. Further discussion on this direction will be reported [8].
To apply our discussion to the quantization of SFT, it may be convenient to lift our
problem to the framework of BRST quantization where the physical state condition is written
in a form of one equation Qφi = 0 and the residual gauge symmetry is represented by
exact states Qχi as φi ∼ φi + Qχi. Even in the case of BRST quantization, we can
naturally prove the corresponding statement as our theorem 1 itself and obtain the same
result H
˜ǫ
∼ H
obs
, though in t his case we have t o impose appropriate conditions in the total
state space including g host states as a dditional gauge conditions. Actually, in ref.[9, 10],
BRST quantization of string theory on curved background represented by the CFT of the
form (c
0
= 1, h
0
< 0) ⊗ (c
K
= 25, h
K
> 0) was considered and the claim that there were no
negativenorm states in the observable Hilbert space was made. The logic used there was
that the states with ghosts (b
−n
, c
−n
) or timelike states (α
0
−n
) can decouple from observable
Hilbert space. Our present work for β = 0 corresponds to giving explicit representation of
12
Page 13
the corresponding o bservable Hilbert space (without b
−n
, c
−n
and α
0
−n
) that had not been
explicitly speciﬁed in [9, 10 ]. Furthermore, to proceed our discussion, we would like to ﬁnd
out whether the possible additional gauge conditions are expressed in simpler forms in terms
of BRST quantization.
As stated in the introduction, our analysis is a ﬁrst step towar d a way of canonically
quantizing SFT in the temporal gauge where the diﬃculty associated with the timelike
nonlocality may be avoided. We may, however, learn from the analysis in the main section
about the lightlike gauge ﬁxing of the covariant SFT as well. As is shown, the DDF states are
the representation o f physical states with the lightlike gauge ﬁxing condition. This means
that the modes of the string ﬁeld in this gauge will be expanded by the DDF states, so that
the ﬁeld in each mode has only transverse polarization. As far as the author’s knowledge is
concerned, there is no literature which derives the light cone SFT by appropriately ﬁxing the
gauge in the covariant SFT. The detailed analysis of these issues will be reported elsewhere.
Acknowledgements
The work is supported in pa rt by the Grantsin Aid for Scientiﬁc Research (1774014 2 [M.A.],
13135205 and 163 40067 [M.K.], 13135224 [M.N.]) from the Ministry of Education, Culture,
Sports, Science and Technology (MEXT) and from the Japan Society fo r the Promotion of
Science (JSPS).
Appendix A Proof of lemma 1
First, note that for each f
β
; pi ∈ F
β
(p), a set of states
L
Σ
β
−n
1
···L
Σ
β
−n
r
L
t
β
−m
1
···L
t
β
−m
q
f
β
; pi (A.1)
is equivalent to the set of states (32) as a linear space since L
−n
= L
t
β
−n
+ L
Σ
β
−n
. Thus, it is
suﬃcient to prove that the states (A.1) for all f
β
; pi ∈ F
β
(p) are linearly indep endent and
span a basis of H(p) if p
t
β
6= 0.
Recall that the total state space is divided into timelike c = 1 and spacelike c = 25
part: H(p) = H
t
β
(p
t
β
) ⊗ H
Σ
β
(p
i
β
).
For c = 1 part, H
t
β
(p
t
β
) can be represented by Verma mo dule V(c = 1, h
0
) with highest
weight h
0
= −
1
2
(p
t
β
)
2
since we know that V(c = 1, h
0
) is nondegenerate for h
0
< 0 from
13
Page 14
Kac’s determinant formula, i.e.,
H
t
β
(p
t
β
) = {L
t
β
−m
1
···L
t
β
−m
q
0, p
t
β
i}. (A.2)
For c = 25 part, we would like to show that the space H
Σ
β
(p
i
β
) is spanned by the set of
states
L
Σ
β
−n
1
···L
Σ
β
−n
r
f
β
, p
i
β
i(≡ λ
Σn
i
= {n
1
, ···, n
r
}, f
β
i) (A.3)
with all f
β
, p
i
β
i ∈ F
β
(p
i
β
). Note that the set of above states (A.3) forms the Verma module
V(c=25, h) with h = M +
1
2
(p
i
β
)
2
for each f
β
, p
i
β
i. Dividing with each level N, the equation
we would like to show is
H
(N)
Σ
β
(p
i
β
) =
N
M
M=0
n
λ
N−M
, f
(M)
β
i
f
(M)
β
, p
i
β
i ∈ F
(M)
β
(p
i
β
)
o
. (A.4)
We use the induction on N to show eq.(A.4). For N = 0, the equation is true trivially
since H
(0)
Σ
β
(p
i
β
) = {0, p
i
β
i} and  0, p
i
β
i ∈ F
(0)
β
(p
i
β
). Then we suppose that the equation
holds for level less than N and consider the states at level N. We represent a state in
H
(N)
Σ
β
(p
i
β
) as
ψ
N
i = g
N
i + o
N
i, g
N
i ∈ G
(N)
, o
N
i ∈ O
(N)
. (A.5)
Here G
(N)
is generated by the states of the form λ
N−M
, f
(M)
β
i with M < N:
G
(N)
=
N−1
M
M=0
n
λ
N−M
, f
(M)
β
i
o
(A.6)
and O
(N)
is the complement of G
(N)
in H
(N)
Σ
β
(p
i
β
). A state g
N
i has nontrivial inner products
only within G
(N)
and G
(N)
is nondegenerate since V(c = 25, h > 0) does. Thus, O
(N)
is
orthogonal to G
(N)
:
H
(N)
Σ
β
(p
i
β
) = G
(N)
⊕ O
(N)
. (A.7)
Consider a state L
Σ
β
−m
ψ
N−m
i ∈ G
(N)
with ψ
N−m
i ∈ H
(N−m)
Σ
β
(p
i
β
) (m ≥ 1). Since O
(N)
is
orthogonal to G
(N)
,
(L
Σ
β
−m
ψ
N−m
i)
†
o
N
i = hψ
N−m
L
Σ
β
m
o
N
i = 0 (A.8)
for any state o
N
i ∈ O
(N)
. From the fact that L
Σ
β
m
o
N
i ∈ H
(N−m)
Σ
β
(p
i
β
) and that H
(N−m)
Σ
β
(p
i
β
)
is nondegenerate, we must conclude that
L
Σ
β
m
o
N
i = 0 (m ≥ 1), (A.9)
14
Page 15
which indicates that o
N
i is nothing but an element of F
(N)
β
. Thus,
H
(N)
Σ
β
(p
i
β
) = G
(N)
⊕ F
(N)
β
(A.10)
and this means that the equation (A.4) holds for N.
Combining with the result for c = 1 part, we have completed the proof of lemma 1.
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[3] M. Kato and K. Ogawa, “Covariant quantization of string based on BRS invariance,”
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