Physical state representations and gauge fixing in string theory
ABSTRACT We re-examine physical state representations in the covariant quantization of bosonic string. We especially consider one parameter family 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 light-cone gauge. Comment: 16 pages
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ABSTRACT: A single-parameter family of covariant gauge fixing conditions in bosonic string field theory is proposed. It is a natural string field counterpart of the covariant gauge in the conventional gauge theory, which includes the Landau gauge as well as the Feynman (Siegel) gauge as special cases. The action in the Landau gauge is largely simplified in such a way that numerous component fields have no derivatives in their kinetic terms and appear in at most quadratic in the vertex.Progress of Theoretical Physics 12/2006; · 2.48 Impact Factor
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arXiv:hep-th/0509188v1 26 Sep 2005
UT-Komaba/05-9
KEK-TH-1038
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 599-8531, Japan
†Institute of Physics
University of Tokyo, Komaba
Meguro-ku, Tokyo 153-8902, Japan
‡Theory Division, Institute of Particle and Nuclear Studies
KEK, High Energy Accelerator Research Organization
Tsukuba, Ibaraki 305-0801, Japan
ABSTRACT
We re-examine physical state representations in the covariant quantization of
bosonic string.We especially consider one parameter family 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 light-cone gauge.
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1Introduction
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|p? in
the open bosonic string gives the gauge transformation for the massless vector mode on
the same level when p2= 0. Here Ln(n=integer) is the Virasoro operator and |p? is the
oscillator vacuum with the momentum eigenvalue pµ(µ = 0,1,···,25). The null states
are certainly members of physical states in the sense that they satisfy the physical state
condition Ln|phys? = 0 for positive integer n and the on-shell condition (L0− 1)|phys? = 0,
while they do not contribute to the physical amplitude. In this sense the on-shell physical
state is said to have an ambiguity in its representation.
Fixing the gauge degrees of freedom associated with the null state is nothing but taking
a representative for the above mentioned ambiguity. One of the well-known representative
of physical state is the so-called DDF state [1] which is generated by applying the transverse
DDF operators 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 theory (SFT), that extra condition as well
as physical state 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
2Ψ(L0− 1)Ψ,(1)
with the condition
LnΨ = 0(n = 1,2,···). (2)
This is the (partially) gauge-fixed covariant action. Actually these equations lead to the
following action and gauge condition for the massless vector field Aµcontained as a mode
in Ψ
S =
?
d26x1
2Aµ
Aµ, (3)
∂µAµ= 0.(4)
As is known, the action still has a residual gauge invariance
Aµ→ Aµ+ ∂µλwith
λ = 0. (5)
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This is also true for the SFT level; eqs.(1) and (2) has residual gauge invariance
Ψ → Ψ + L−1Ξ1+ (L−2+3
2L−12)Ξ2, (6)
provided
?
LnΞ1 = 0
L0Ξ1 = 0
and
?
LnΞ2 = 0
(L0+ 1)Ξ2 = 0
for positive integer n. The degrees of freedom of Ξ’s are nothing 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 gauge fixing condi-
tions. In particular, one parameter family of linear gauge condition is analyzed, which is
essentially temporal gauge (or chronological gauge) in the sense that the string excitation
of a time-like 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 light-like gauge
will also be clarified.
One of the motivations for studying the temporal gauge and its cousin is that better
understanding of the gauge may provide a clue towards the resolution of the long-standing
problem on the canonical quantization of SFT [4]. Since time-like excitation is restricted to
the zero mode, it can be taken as a time parameter of canonical quantization procedure and
also the interaction becomes local 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 action 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 non-local theories.
The existence of light-cone SFT may support the validity of the assumption if the exact
relationship from the covariant SFT to the light-cone SFT through the gauge fixing in the
SFT level, because in the latter formulation light-like variable x+is the time parameter of
the quantization procedure and locality of the interaction with respect to x+is satisfied.
In order to try these scenario, as 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 light-like 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
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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 given
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 light-like gauge and
temporal gauge in the SFT. Section 4 is devoted to the summary and discussions.
2 Physical states in covariant gauge
The total state space H(p) for the old covariant quantization (OCQ) of perturbative bosonic
string theory (D = 26) is given by the Fock space Fock(αµ
the form
25
?
Here, Nµ
−n;pµ) spanned by the states of
|φN; pµ? =
µ=0
∞
?
n=1
(αµ
−n)Nµ
n|0;pµ?. (7)
nis a non-negative integer and |0;pµ? is the ground state annihilated by all αµ
(n > 0) with momentum pµ. We often divide H(p) into the space with level N =?
as H(p) = ⊕N≥0H(N)(p). Among H(p), positive definite Hilbert space Hobs(p) is defined by
the quotient Hobs(p) = Hphys(p)/Hnull(p), which we sometimes call observable Hilbert space.
Here Hphys(p) is the set of states satisfying the physical state condition
n
n,µnNµ
n
Ln|φ?phys= 0(n > 0)(8)
and the on-shell condition
(L0− 1)|φ?phys= 0 (9)
which restricts the level N of the states as α′p2+N −1 = 0. The space Hnull(p)[⊂ Hphys(p)]
is the set of null states that are identified as physical states of the form
|χ?null= L−1|ξ1? + (L−2+3
2L2
−1)|ξ2? (10)
where Ln|ξ1? = (Ln+ δn,0)|ξ2? = 0 (n ≥ 0). A null state has zero inner product with any
state in Hphys(p) (null?χ|φ?phys= 0). This is seen from (8) and (10) with the definition of
inner product in H(p): L†
Due to the existence of null states, we have an ambiguity |φ?phys∼ |φ?phys+ |χ?nullin
choosing explicit representations of observable Hilbert space Hobs(p). As we have seen in
the introduction in terms of SFT, appearance of null states in our OCQ scheme (or exact
−n= Ln(α†
−n= αn) and ?0;p|0;p? = 1.
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states for BRST quantization) indicates the existence of residual gauge symmetry which is
left unfixed at the classical level. Thus, choosing explicit representation of Hobs(p) exactly
corresponds to fixing this residual gauge symmetry. In fact, 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:
Hphys(p) ∩ {‘gauge condition’} ∼ Hobs(p). (11)
We would like to find a class of such conditions and corresponding representations of Hobs(p)
in a systematic manner.
For example, we know that the set of so-called DDF states can be taken as an explicit
representation of Hobs(p). They are given by multiplying DDF operators
Aˆi
−n=
1
2π
?
dz∂Xˆi(z)e−ink·X(z)
(12)
on the tachyon ground state |0, ¯ p? as
|φ;p = ¯ p − Nk? = Aˆi1
−n1Aˆi2
−n2···Aˆil
−nl|0, ¯ p?.(13)
Here,
Xµ= xµ− ipµlnz + i
?
n?=0
1
nαµ
nz−n, (14)
ˆi = 1,···,24, N = n1+ ··· + nl, k2= 0 (with kˆi= 0), ¯ p2= 2 and ¯ p · k = 1. Hereafter, we
set α′= 1/2 (αµ
0= pµ). These DDF states satisfy the physical state condition and form a
basis of Hobs(p) for p = ¯ p − Nk. DDF states up to level N = 2 are as follows:
N = 0 :
|0, ¯ p?,
Aˆi
(15)
N = 1 :
−1|0, ¯ p? = αˆi
Aˆi
−1|0, ¯ p − k?,
αˆi
(16)
N = 2 :
−2|0, ¯ p? =
Aˆi
?
−2− 2(k · α−1)αˆi
?
−1
?
(k · α−1)2− (k · α−2)
|0, ¯ p − 2k?,(17)
−1A
ˆj
−1|0, ¯ p? =αˆi
−1α
ˆj
−1+1
2δˆiˆj? ??
|0, ¯ p − 2k?.(18)
In fact, the set of DDF states can be extracted by imposing an additional condition
k · αn|φ; ¯ p − Nk? = 0(n > 0)(19)
on the space of physical states Hphys(p) when p = ¯ p − Nk [6]. This is an example of
supplementary gauge condition that completely fixes the ambiguity of null states as discussed
before.
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3Representations of observable Hilbert space
Next we consider a class of supplementary conditions which are linear in oscillator variables.
They are simple generalizations of (19) 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 DDF-like operators in general cases.
Concretely, we consider the following condition
˜ ǫ · αn|φ; pµ? = 0(n > 0)(20)
with a constant time-like or light-like vector ˜ ǫ (i.e., ˜ ǫ2≤ 0). For ˜ ǫµ ∝ kµ, this condition
reduces to (19) and restricts states not to include α−
−n)/√2 respectively for kµ∝ δµ0+δµ25or kµ∝ δµ0−δµ25. Also, for ˜ ǫµ= δµ0, the condition
(20) restricts states not to include any time-like oscillators (α0
−n= (α0
−n− α25
−n)/√2 or α+
−n= (α0
−n+
α25
−n). We do not consider the
case ˜ ǫ2> 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.
3.1Main 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
(Ln− δn,0)|φ?phys= 0(n ≥ 0)
and
˜ ǫ · αn|φ; pµ? = 0(n > 0)
for ˜ ǫ2≤ 0. Then, H˜ ǫ(p) ∼ Hobs(p) provided ˜ ǫ · p ?= 0.
We divide H˜ ǫ(p) by level N as H˜ ǫ(p) = ⊕N≥0H(N)
Before going into general proof, let us first see the simple cases N = 0 and N = 1. For
˜ ǫ
(p) and prove the theorem for each N.
N = 0, we only have ground state |0,p? in H(0)(p) (with p2= 2) and it satisfies (8) and (20)
trivially: H(0)
represented as
˜ ǫ(p) = {|0,p?}(= H(0)
obs(p)). For N = 1, general on-shell states satisfying (8) are
|φ; p? = ξ · α−1|0,p? (21)
with p2= 0 and ξ · p = 0. Among these states, there is a null state p · α−1|0,p? and the
space H(1)
obs(p) is identified up to the ambiguity ξµ∼ ξµ+ pµ. The condition (20) gives the
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constraint on ξµas ˜ ǫ·ξ = 0, which fixes the ambiguity completely since ˜ ǫ·p ?= 0 is assumed.
Thus, H(1)
(p2= 0) with ˜ ǫ · p ?= 0. We have proven H˜ ǫ(p) ∼ Hobs(p) for N = 0,1. Note that here we
have not used the condition ˜ ǫ2≤ 0. For general N, we first give a proof for ˜ ǫ2< 0 and then
extend it to ˜ ǫ2= 0 since the latter can be considered as a limit of the former. For N ≥ 2,
the condition ˜ ǫ · p ?= 0 is always satisfied for on-shell states if ˜ ǫ2≤ 0.
˜ ǫ(p) ∼ H(1)
obs(p). Explicitly, H(1)
˜ ǫ(p) = {ξ · α−1|0,p? | p · ξ = ˜ ǫ · ξ = 0} for on-shell p
3.2 Proof for ˜ ǫ2< 0
First, we will make some definitions for preparation1. We will fix the time-like vector ˜ ǫµas
˜ ǫµ= (coshβ,0,···,0,sinhβ) [= ˜ ǫµ(β)](22)
with 0 ≤ β < ∞ without losing generality. Correspondingly, we define a space-like vector
ǫµ(β) = (sinhβ,0,···,0,coshβ).(23)
We take a particular choice of spacetime coordinates (tβ,sβ,xˆi) ≡ (˜ ǫ(β) · x,ǫ(β) · x,xˆi)
which are obtained by boost transformation from the original coordinates xµ. Commutation
relations for α˜ µ
n(˜ µ = tβ,sβ,ˆi) are given as
[α˜ µ
m,α˜ ν
n] = mδm+n,0η˜ µ˜ ν
(24)
where
αtβ
n
= coshβ α0
n+ sinhβ α25
n(= ˜ ǫ(β) · αn),
n(= ǫ(β) · αn).
(25)
αsβ
n
= sinhβ α0
n+ coshβ α25
(26)
Thus, in particular,
[˜ ǫ(β)·αm,˜ ǫ(β)·αn] = −mδm+n,0,[ǫ(β)·αm,ǫ(β)·αn] = mδm+n,0,[˜ ǫ(β)·αm,ǫ(β)·αn] = 0.
(27)
We divide total state space into ‘time-like’ and ‘space-like’ part:
H(p) = Htβ(ptβ) ⊗ HΣβ(piβ)(28)
1In fact, to prove the theorem 1 for ˜ ǫ2< 0, it is sufficient to take ˜ ǫµ= δ0
by boost transformations from this. We however consider every ˜ ǫ2< 0 explicitly for later convenience.
µsince other cases can be obtained
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where iβ = (sβ,ˆi), Htβ(ptβ) = Fock(αtβ
divide Lnas Ln= Ltβ
−n; ptβ) and HΣβ(piβ) = Fock(αiβ
−n; piβ). We also
n + LΣβ
n where
Ltβ
n= −1
2
∞
?
m=−∞
: αtβ
n−mαtβ
m: ,LΣβ
n =1
2
∞
?
m=−∞
: αiβ
n−mαiβ
m: .(29)
We further define the space Fβ(p) as
Fβ(p) = {|fβ; p? | αtβ
n|fβ; p? = Ln|fβ; p? = 0 (n > 0)}.(30)
The relation between this Fβ(p) and H˜ ǫ(β)(p) is
H˜ ǫ(β)(p) = {|φ? ∈ Fβ(p)|(L0− 1)|φ? = 0}.(31)
The space F(N)
‘time-like’ oscillator αtβ
β
(p) is a subspace of |0,ptβ? ⊗ H(N)
−n. Thus, Fβ(p) is positive definite and cannot contain null states
Σβ(piβ) since |fβ; p? does not contain any
(10).
With the above definitions, we will now begin to prove theorem 1, i.e., H(N)
First, we will give the following lemma:
˜ ǫ(β)(p) ∼ H(N)
obs(p).
Lemma 1 States of the form
L−n1···L−nrLtβ
−m1···Ltβ
−mq|fβ; p?,|fβ; p? ∈ Fβ(p)(32)
(ns≤ ns+1,ms≤ ms+1) are linearly independent and span a basis of H(p) if ptβ?= 0.
The proof is given in Appendix A.
With the above lemma, we will write every state in H(p) as a sum of states of the form
(32). In particular, we divide any |phys? ∈ Hphys(p) written in this form into two classes as
|phys? = |g? + |χ?(33)
where |g? consists of terms without any L−n, i.e.,
|g? =
?Cm1,···,mqLtβ
−m1···Ltβ
−mq|fβ; p?(34)
with constants Cm1,···,mqand the |χ? part consists of terms including at least one L−n. Both
|g? and |χ? satisfy on-shell condition. Also, we see that L1|g? and (L2+3
contain any L−nand L1|χ? and (L2+3
Thus Ln|phys? = 0 implies that |g? and |χ? are both physical and the state |χ? is null since
2L2
1)|g? do not
2L2
1)|χ? again consist of terms with at least one L−n.
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all Ln(n ≥ 1) are generated by L1and L2. For |g? part, 0 = (Ltβ
n > 0 since LΣβ
V(1,h < 0) if there exist any Ltβ
n +LΣβ
n )|g? = Ltβ
n |g? for any
n |fβ; p? = 0. This contradicts the non-degeneracy of c = 1 Verma module2
−min |g?. This means that |g? contains no Ltβ
−mand
|g? = |fβ; p?. (35)
Thus we have shown that any physical state can be written as an element of H˜ ǫ(β)(p) ⊂ Fβ(p)
up to a null state:
|phys? = |fβ; p? + |χ?,|fβ; p? ∈ H˜ ǫ(β)(p).(36)
In other word, we have shown that H˜ ǫ(β)(p) ∼ Hobs(p) since we know that there are no null
states in H˜ ǫ(β)(p). We have proven theorem 1 for ˜ ǫ2< 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 for the set of DDF operators
(which corresponds to our case of ˜ ǫ2= 0 and pµ= ¯ pµ− Nkµ) has been proved. Comparing
to that case, our proof for ˜ ǫ2< 0 is rather simpler since the positive-definiteness of Fβ(p) is
trivial (and also we know the non-degeneracy of V(1,h < 0)).
3.3Properties 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 Hobs(p).
The dimension of Hobs(p) coincides with that of the transverse Hilbert space H(pˆi) =
Fock(αˆi
in?
systematically. For this aim, we have the following lemma [8]
−n;pˆi): For each level N, dimH(N)
n≥1(1−qn)−D. We would like to choose a basis of H˜ ǫ(β)(p) in order to analyze the space
obs(p) = P24(N) where PD(n) is the coefficient of qn
Lemma 2 Assume that ptβ(= ˜ ǫ · p) ?= 0 and psβ(= ǫ · p) ?=r−s
integers with rs < N. Then, a state |fβ; p? ∈ H(N)
of transverse oscillators, i.e.,
√2where r and s are positive
˜ ǫ(β)(p) has at least one term consisting only
|fβ; p? = |ˆφ; p? + (terms with at least one αsβ
−n)(37)
where |ˆφ; p? is a non-zero state in Fock(αˆi
2V(c,h) is a linear space spanned by the states constructed by acting Virasoro operators (L−n, n > 0) of
central charge c on the highest weight state |h?.
−n;p).
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With this result, for psβ?=r−s
basis element |fβ; p?ˆλNof H(N)
√2, we can choose a basis of H(N)
˜ ǫ(β)(p) by the term
˜ ǫ(β)(p) as follows: We specify each
|ˆφ; p?ˆλN={(ˆi1,n1),···,(ˆil,nl)}= αˆi1
−n1···αˆil
−nl|0, p?,(ns≤ ns+1,
l?
s=1
ns= N)(38)
and write
|fβ; p?ˆλN= |ˆφ; p?ˆλN+ (terms with at least one αsβ
With fixedˆλN, the terms with αsβ
−n).(39)
−nin |fβ; p?ˆλNare uniquely determined by the condition
N−1
√2
we can always choose the above basis since
√2and ptβ?= 0 for such a case.
For example, we explicitly represent the space H(N)
above. For N = 1 with psβ?= 0,
Ln|fβ; p?ˆλN= 0. Note that for |psβ| >
psβ?=r−s
˜ ǫ(β)(p) for N = 1,2 by the basis given
H(N=1)
˜ ǫ(β)
(p) = {|fβ; p?ˆλ1=(ˆi,1)}(40)
where
|fβ; p?(ˆi,1)=
?
αˆi
−1− pˆiαsβ
−1
psβ
?
|0,p?.(41)
For N = 2 with psβ?= 0,±1
√2,
H(N=2)
˜ ǫ(β)
(p) = {|fβ; p?ˆλ2={(ˆi,2)},|fβ; p?ˆλ2={(ˆi,1),(ˆj,1)}} (42)
where
|fβ; p?(ˆi,2)=
αˆi
−2−
2
psβ
αˆi
−1αsβ
−1+
4pˆi
sβ− 1αsβ
2p2
−1αsβ
−1−
pˆi(2p2
psβ(2p2
sβ+ 1)
sβ− 1)αsβ
−2
|0,p?(43)
and
|fβ; p?{(ˆi,1),(ˆj,1)}=
αˆi
−1α
ˆj
−1−
2
psβ
p{ˆiα
ˆj}
−1αsβ
−1+δˆiˆj+ 2pˆipˆj
2p2
sβ− 1αsβ
−1αsβ
−1−
p2
psβ(2p2
sβδˆiˆj+ pˆipˆj
sβ− 1)αsβ
−2
|0,p?.
(44)
3.4Proof for ˜ ǫ2= 0
Now we prove the theorem for the remaining case: ˜ ǫ2= 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 ˜ ǫ2= 0 can be understood as a limit of that
for ˜ ǫ2< 0. In other word, we will identify the space H˜ ǫ(β=∞)(p) as a limit ‘limβ→∞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 limit3for each basis element of the space for fixed momentum pµ.
3The limit we consider is different from boost transformation since we keep the momentum pµfixed.
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Now, we will explain how to define the limit explicitly. We consider the space H(N)
with fixed on-shell momentum pµfor each N. Here the momentum frame has to be chosen
˜ ǫ(β)(p)
in order to satisfy ˜ ǫ(β) · p ?= 0 for arbitrary β (≤ ∞), i.e., we take p0+ p25?= 0. Then we
take β large enough (β > βN
0) to satisfy |ǫ(β) · p| >N−1
√2for each pµand N. We can always
take such βN
of the previous subsection, we can take the set of states {|fβ; p?ˆλN} as a basis of H(N)
β > βN
0since limβ→∞|ǫ(β) · p| = ∞ for any p with p0+ p25?= 0. From the discussion
˜ ǫ(β)(p) for
0. Each state |fβ; p?ˆλNcontains β through the parts of ǫ(β) · α−n(n ≥ 0) and thus
the state can be expanded with respect to eβ. We can prove from the property of physical
state condition that the terms with positive powers of eβcannot appear in the expansion of
|fβ; p?ˆλNand
lim
β(>βN
0)→∞|fβ; p?ˆλN< ∞.(45)
Also, the terms with odd powers of eβdo not appear in the expansion and thus the expansion
takes the form
|fβ; p?ˆλN
= |f(0); p?ˆλN+ e−2β|f(1); p?ˆλN+ e−4β|f(2); p?ˆλN+ ···
∞
?
The leading term |f(0); p?ˆλNis given by the limit (45) and contains the term |ˆφ; p?ˆλNof (38).
By definition, each term |f(k); p?ˆλNdoes not contain β and satisfies physical state condition
=
k=0
e−2kβ|f(k); p?ˆλN.(46)
Ln|f(k); p?ˆλN= 0. (47)
Furthermore, from the condition ˜ ǫ(β) · αn|fβ; p?ˆλN= 0, we have
(α0
n+ α25
n)|f(k); p?ˆλN+ (α0
n− α25
n)|f(k−1); p?ˆλN= 0.(48)
In particular, the leading term |f(0); p?ˆλN(= limβ→∞|fβ; p?ˆλN) satisfies
(α0
n+ α25
n)|f(0); p?ˆλN(∝ ˜ ǫ(β → ∞) · αn|f(0); p?) = 0.(49)
The limit of the inner product of two states |fβ; p?ˆλNand |fβ; p?ˆλ′
calculated as
Ncan be explicitly
lim
β→∞
ˆλN?fβ; p|fβ; p?ˆλ′
=ˆλN?ˆφ; p|ˆφ; p?ˆλ′
= fˆλNδˆλN,ˆλ′
N(=ˆλN?f(0); p|f(0); p?ˆλ′
N)
N
N
(50)
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where fˆλNis a positive integer. This means that the space spanned by the states limβ→∞|fβ; p?ˆλN
with allˆλNhas dimension P24(N) and is non-degenerate. Thus,
{ lim
β→∞|fβ; p?ˆλN} = H(N)
˜ ǫ(β=∞)(p) ∼ H(N)
obs(p)(51)
from (47) and (49). We have proven theorem 1 for ˜ ǫ2= 0.
The characteristic point of our proof comparing to the one in the literature [6, 7] is that
the non-degeneracy 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 pˆi= 0. Explicitly,
|f(0); p?ˆλN={(ˆi1,n1),···,(ˆil,nl)}= Aˆi1
−n1···Aˆil
−nl|0,p + Nk?(52)
where k is a light-like vector defined by k ∝ limβ→∞˜ ǫ(β) (i.e., kµ ∝ (1,0,···,0,1)) and
k · p = 1.
For example, for N = 1, we explicitly take the β → ∞ limit of (41): By using
αsβ
−n
psβ= lim
β→∞
ǫ(β) · p
we obtain
?
and this coincides with DDF state Aˆi
similarly take the limit of (43) and (44) and the result for pˆi= 0 is
lim
β→∞
ǫ(β) · α−n
= k · α−n, (53)
lim
β→∞|fβ; p?(ˆi,1)=αˆi
−1− pˆi(k · α−1)
?
|0,p?(54)
−1|0,p + k? if we take pˆi= 0. For N = 2, we can
lim
β→∞|fβ; p?(ˆi,2)=
?
αˆi
−2− 2(k · α−1)αˆi
−1
?
|0,p? (55)
and
lim
β→∞|fβ; p?{(ˆi,1),(ˆj,1)}=
?
αˆi
−1α
ˆj
−1+1
2δˆiˆj?
(k · α−1)2− (k · α−2)
??
|0,p?,(56)
which coincide with DDF states (17) and (18).
4Summary and Discussions
In the present paper, we have investigated the old 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
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condition on the space of physical states, we obtain a space which can be taken as an
explicit representation of observable Hilbert space Hobs(p). Explicitly, we have proven that
the condition ˜ ǫ·αn|φ; pµ? = 0 for a constant time-like or light-like ˜ 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 ?= 0. As a result, for each ˜ ǫ, we have identified the space H˜ ǫ(p) which gives a complete
set of physical states as a particular representation of Hobs(p).
For time-like ˜ ǫ = ˜ ǫ(β<∞), the additional condition is related to the temporal gauge in the
sense that the corresponding representation of observable Hilbert space H˜ ǫ(β<∞)(p) does not
include time-like oscillators αtβ
−n. On the other hand, the condition for light-like ˜ ǫ = ˜ ǫ(β=∞)
is related to the light-cone 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 particular 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 pˆi= 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 for
H˜ ǫ(β<∞)(p) and for H˜ ǫ(β=∞)(p) are in one-to-one correspondence, i.e., we have shown that
each state in H˜ ǫ(β=∞)(p) (for p+?= 0) is obtained as a certain limit of the corresponding state
in H˜ ǫ(β<∞)(p) except for a particular 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 Q|φ? = 0 and the residual gauge symmetry is represented by
exact states Q|χ? as |φ? ∼ |φ? + Q|χ?. 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∼ Hobs, though in this case we have to impose appropriate conditions in the total
state space including ghost states as additional gauge conditions. Actually, in ref.[9, 10],
BRST quantization of string theory on curved background represented by the CFT of the
form (c0= 1,h0< 0) ⊗ (cK= 25,hK> 0) was considered and the claim that there were no
negative-norm states in the observable Hilbert space was made. The logic used there was
that the states with ghosts (b−n,c−n) or time-like states (α0
Hilbert space. Our present work for β = 0 corresponds to giving explicit representation of
−n) can decouple from observable
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the corresponding observable Hilbert space (without b−n, c−nand α0
explicitly specified in [9, 10]. Furthermore, to proceed our discussion, we would like to find
−n) that had not been
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 toward a way of canonically
quantizing SFT in the temporal gauge where the difficulty associated with the time-like
nonlocality may be avoided. We may, however, learn from the analysis in the main section
about the light-like gauge fixing of the covariant SFT as well. As is shown, the DDF states are
the representation of physical states with the light-like 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 part by the Grants-in-Aid for Scientific Research (17740142 [M.A.],
13135205 and 16340067 [M.K.], 13135224 [M.N.]) from the Ministry of Education, Culture,
Sports, Science and Technology (MEXT) and from the Japan Society for the Promotion of
Science (JSPS).
Appendix AProof of lemma 1
First, note that for each |fβ; p? ∈ Fβ(p), a set of states
LΣβ
−n1···LΣβ
−nrLtβ
−m1···Ltβ
−mq|fβ; p?(A.1)
is equivalent to the set of states (32) as a linear space since L−n= Ltβ
sufficient to prove that the states (A.1) for all |fβ; p? ∈ Fβ(p) are linearly independent and
span a basis of H(p) if ptβ?= 0.
Recall that the total state space is divided into time-like c = 1 and space-like c = 25
−n+ LΣβ
−n. Thus, it is
part: H(p) = Htβ(ptβ) ⊗ HΣβ(piβ).
For c = 1 part, Htβ(ptβ) can be represented by Verma module V(c = 1,h0) with highest
weight h0= −1
2(ptβ)2since we know that V(c = 1,h0) is non-degenerate for h0< 0 from
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Page 15
Kac’s determinant formula, i.e.,
Htβ(ptβ) = {Ltβ
−m1···Ltβ
−mq|0, ptβ?}. (A.2)
For c = 25 part, we would like to show that the space HΣβ(piβ) is spanned by the set of
states
LΣβ
−n1···LΣβ
−nr|fβ,piβ?(≡ |λΣni= {n1,···,nr},fβ?) (A.3)
with all |fβ,piβ? ∈ Fβ(piβ). Note that the set of above states (A.3) forms the Verma module
V(c=25,h) with h = M +1
we would like to show is
2(piβ)2for each |fβ,piβ?. Dividing with each level N, the equation
H(N)
Σβ(piβ) =
N
?
M=0
?
|λN−M,f(M)
β
?
???|f(M)
β
,piβ? ∈ F(M)
β
(piβ)
?
.(A.4)
We use the induction on N to show eq.(A.4). For N = 0, the equation is true trivially
since H(0)
holds for level less than N and consider the states at level N. We represent a state in
H(N)
|ψN? = |gN? + |oN?,
Here G(N)is generated by the states of the form |λN−M,f(M)
Σβ(piβ) = {|0, piβ?} and |0, piβ? ∈ F(0)
β(piβ). Then we suppose that the equation
Σβ(piβ) as
|gN? ∈ G(N),|oN? ∈ O(N).
? with M < N:
(A.5)
β
G(N)=
N−1
?
M=0
?
|λN−M,f(M)
β
?
?
(A.6)
and O(N)is the complement of G(N)in H(N)
only within G(N)and G(N)is non-degenerate since V(c = 25,h > 0) does. Thus, O(N)is
orthogonal to G(N):
H(N)
Consider a state LΣβ
Σβ(piβ). A state |gN? has non-trivial inner products
Σβ(piβ) = G(N)⊕ O(N).(A.7)
−m|ψN−m? ∈ G(N)with |ψN−m? ∈ H(N−m)
orthogonal to G(N),
(LΣβ
Σβ
(piβ) (m ≥ 1). Since O(N)is
−m|ψN−m?)†|oN? = ?ψN−m|LΣβ
m|oN? = 0 (A.8)
for any state |oN? ∈ O(N). From the fact that LΣβ
is non-degenerate, we must conclude that
m |oN? ∈ H(N−m)
Σβ
(piβ) and that H(N−m)
Σβ
(piβ)
LΣβ
m|oN? = 0(m ≥ 1),(A.9)
14
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