# 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; · 2.06 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.

Page 2

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

11

Page 13

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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Page 14

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

13

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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