Dressed states from gauge invariance

The dressed state formalism enables us to define the infrared finite $S$-matrix for QED. In the formalism, asymptotic charged states are dressed by clouds of photons. The dressed asymptotic states are originally obtained by solving the dynamics of the asymptotic Hamiltonian in the far past or future region. However, there was an argument that the obtained dressed states are not gauge invariant. We resolve the problem by imposing a correct gauge invariant condition. We show that the dressed states can be obtained just by requiring the gauge invariance of asymptotic states. In other words, Gauss's law naturally leads to proper asymptotic states for the infrared finite $S$-matrix. We also discuss the relation between the dressed state formalism and the asymptotic symmetry for QED.

1 Introduction and summary 1

.1 Introduction
S-matrix is a fundamental quantity of quantum field theories in Minkowski spacetime. There is a systematic way to compute it perturbatively by the Feynman rules. However, we then often encounter infrared (IR) divergences for theories with massless particles. A famous example is quantum electrodynamics (QED). Virtual photons with small energy cause divergences of loop diagrams. This problem can be avoided by considering the total cross-section of various processes including the emission of real soft photons [1,2]. Another approach was developed, which enables us to treat directly the IR finite S-matrix [3,4,5,6,7,8]. It is called the dressed state formalism, which will be reviewed in the next subsection.
Although the dressed state formalism was proposed many years ago, it has been recently reconsidered in the connection with the asymptotic symmetry (see, e.g., [9,10,11,12,13,14,15]). It has been recognized that QED has an infinite number of symmetries associated with large gauge transformations [16,17]. Thus, the conservation laws should constrain the S-matrix. On the other hand, scattering amplitudes vanish in the conventional approach because the sum of IR divergences at all orders produces the exponential suppression. It was pointed out in [11] that the vanishing of the amplitudes is consistent with the asymptotic symmetry of QED. Initial and final states used in the conventional approach generally belong to different sectors with respect to the asymptotic symmetry. Therefore, the amplitude between them should vanish since otherwise it breaks the conservation law. It was argued that we need dressed states in order to obtain non-vanishing amplitudes [11].
Motivated by these facts, we will investigate the dressed state formalism in this paper. In particular, we will revisit the gauge invariance in the formalism. We will argue that there is a problem on the gauge invariant condition in [8], and will resolve the problem. In our method, dressed states are obtained just from the appropriate gauge invariant condition. We will also discuss the i prescription for the dressed states. In addition, the relation between the dressed state formalism and the asymptotic symmetry will be considered. In order to explain our results more precisely, we first review the dressed state formalism in subsection 1.2. We then present our results and the outline of the paper in subsection 1.3.

Review of a problem of S-matrix in QED and the dressed state formalism
S-matrix elements of scatterings in quantum field theories are defined by inner products of in-states and out-states: where |α out and |β in are eigenstates with energies E α and E β of the Hamiltonian H (which is assumed to be time-independent) such that they can be regarded as eigenstates |α 0 and |β 0 with the same energies E α and E β of a free Hamiltonian H 0 at t → ±∞. More precisely, we should consider wave packets which are superpositions of eigenstates with a smooth function g as follows: dβ g(β) |β in . (1. 2) The condition of in-states and out-states are then given by where we have introduced an arbitrary finite time t s at which the Schrödinger operators are defined. We can formally write the condition as Here, U (t, t ) and U 0 (t, t ) are the full and free time-evolution operators respectively: Using eq.(1.5), the S-matrix element (1.1) can be written as Since |α 0 and |β 0 usually have the Fock state representation, we finally obtain the S-matrix operator on the Fock space H F ock as For computations of the Fock space basis S-matrix (1.9), it is convenient to play in the interaction picture. We divide the Hamiltonian as H = H 0 + V , and define the interaction operator in the interaction picture as (1.10) Then, the operator Ω(t) in (1.6) can be written as where the symbol T represents the time-ordered product. The S-matrix (1.9) can be represented as the Dyson series [18] (1.12) The above is the standard treatment of the S-matrix in QFTs. However, this S-matrix on the Fock space is not well-defined in QED because of the infrared (IR) divergences. If we try to compute the S-matrix elements on the Fock space by the standard perturbation theory, we encounter the IR divergences.
One way to address this problem is giving up the S-matrix as usually adopted in QFT textbooks such as [19,20]. It is argued that in any experiment for particle physics the detector has a minimum energy E d such that photons with energies less than E d cannot be detected, and therefore the measured cross-section is the sum of cross-sections for all events emitting undetectable soft photons [1,2]. This inclusive method actually works and the measured crosssection is IR finite.
Nevertheless, it is better that we have a well-defined S-matrix. Fortunately, there is a way to define an IR finite S matrix. It is called the dressed state formalism [3,4,5,6,7,8]. IR divergences in the conventional approach originate from the assumption that the asymptotic scattering states can be regarded as free particle states at t ∼ ±∞. Since photons are massless particles, i.e., the electromagnetism is a long-range interaction, we should take account of the interaction even in the asymptotic region. It means that we should modify the free time-evolution operator U 0 in (1.6) into another time-evolution operator U as which contains contributions of the long-range interaction in the asymptotic region. In fact, even for scatterings in quantum mechanics (not QFTs), in order to obtain the IR finite S-matrix in the Coulomb potential, we need such a modification [21].
In Faddeev and Kulish's paper [8], it was argued that the asymptotic dynamics of QED can be approximated by the following "interacting" Hamiltonian in the Schrödinger picture 1 : where H s 0 is the usual free Hamiltonian for QED. j µ cl (t, x) is a "classical" current operator given by where the sum in (1.14) runs over all charged particles, and we omit the label for simplicity. b † (and d † ) are creation operators of the charged particles (and antiparticles). 2 This current is "classical" in the sense that it is a diagonal operator on the usual Fock space. Because of the explicit time-dependence of j µ cl , the asymptotic Hamiltonian H s as is time-dependent even in the Schrödinger picture.
The S-matrix is then given by where Ω as (t) is obtained by replacing U 0 in (1.6) into U as with We can proceed further by computing this asymptotic time-evolution operator (1.17). Similar 1 The superscript s represents the Schrödinger picture. 2 The creation and annihilation operators have labels for a spinor basis, if the particle is a fermion.
to the derivation of (1.11), one can find (see [8] for the derivation) that U as (t, t s ) is given by Furthermore, since the commutator [V I as (t 1 ), V I as (t 2 )] commutes with V I as (t) for any t, we obtain 20) and by performing the t-integral, we have In [8], R(t s ) in (1.21) was deleted by a requirement for an initial condition. Permitting this, 3 the S-matrix (1.16) becomes As a result, this S-matrix differs from usual Dyson's one (1.12) only in the dressing factors e R and e iΦ . Thus, if we formally introduce a dressed Hilbert space H F K as the S-matrix on H F K is given by usual one (1.12). 4 Actually, the factors play a similar role as summing the contributions of soft photons, and the S-matrix on H F K is known to be IR finite [3], 5 if we impose the physical state condition. 3 We will see that we do not need to worry about this requirement in our approach. 4 Even on H F K , the notion of particles for charged fields is still valid because j µ cl (t, x) in the dressing factor e R(t) is a diagonal operator on the Fock space. However, the standard interpretation of photons on the Fock space seems to be lost because the dressing factor excites an infinite number of photons. As we will see in subsec. 4.1, the energy of the excited photons by the dressing factor is soft in the limit t → ±∞. Hence, the particle notion for hard photons may be valid. 5 In subsec. 4.1, we will comment on a subtlety of the proof of IR finiteness in [3].

Our method and the differences from Faddeev and Kulish's
Not all of the states in H F K are physical, and thus we have to restrict H F K to the subspace by imposing a gauge invariant condition. However, the treatment for the gauge invariance in [8] seems inappropriate. The free Gupta-Bleuler condition was imposed on H F K as the physical state condition, i.e., physical states, |ψ ∈ H F K , were required to satisfy k µ a µ ( k) |ψ = 0 for any k. (1.26) In [8], to satisfy (1.26), the dressing operator R in (1.22) was modified by introducing a null vector c µ ( k) satisfying k µ c µ = 1. More concretely, the dressing operator was altered by shifting the coefficient p µ p·k in (1.22) to p µ p·k − c µ . We will see that the artificial vector c µ is not needed for an appropriate gauge invariant condition. Our claim is that contributions of long-range interactions should be incorporated into the gauge invariant condition, as dressed states are obtained by taking account of such an interaction. The free Gupta-Bleuler condition (1.26) is not adequate for dressed states. In section 2, we will present the appropriate condition.
Furthermore, we will show that the dressed Hilbert space can be obtained just by requiring the gauge invariant condition. In our approach, it turns out that we do not to need to solve the dynamics of the asymptotic Hamiltonian H as as we reviewed in subsec 1.2. In fact, although the Dyson's S-matrix (1.12) is not a good operator on the usual Fock space H F ock , it is well-defined on the dressed space H F K . 6 The asymptotic Hamiltonian H as is just an approach to derive the dressing factor e R(t) . We think that the gauge invariant condition is a simpler approach to obtain the factor, and the interpretation is clear. The condition essentially just says that if there is a charged particle, there should exist electromagnetic fields around it by Gauss's law. The fields around the charge indeed make up the dress.
In section 3, as a support of this interpretation and also another justification that we do not have to introduce c µ , we will discuss the meaning of the original dressing operator R(t) in eq. (1.22). As shown in [22], the dressing factor for a charged particle with momentum p corresponds to the Liénard-Wiechert potential for the uniformly moving charge with momentum p. We will reconfirm this fact especially taking care of the i prescription.
Besides, our method allows a variety of dresses, and H F K given by (1.25) is just one of them. We will see that in our gauge invariant condition, the physical Hilbert space H as on which Dyson's S-matrix (1.12) acts takes the form where e Ras is a dressing operator, and H f ree is a subspace of the Fock space H F ock such that the free Gupta-Bleuler condition are satisfied (k µ a µ ( k) |ψ = 0, |ψ ∈ H f ree ). The operator R as can be R(t) + iΦ(t, t s ), but not necessary. We will discuss the relation between the ambiguity of dressing and the asymptotic symmetry of QED in subsec 4.2.
We conclude that infrared divergences of the S-matrix in the usual perturbative approach for QED are caused by the usage of the inappropriate asymptotic states. Although the asymptotic states satisfying the free Gupta-Bleuler condition may be used at the tree level, they should not be used at loop levels. If we instead use the correct gauge-invariant states at the same order, we can avoid IR divergences.

Necessity of dresses
We show that states with charged particles must include photons even in the interaction picture to satisfy the gauge invariance. In order to impose the physical state condition in a systematic way, we use the BRST formalism.

Lagrangian and Hamiltonian in covariant gauge
In the BRST formalism with the Feynman gauge, the Lagrangian of QED is given by 7 and L matter is the Lagrangian of charged matter fields. In this paper, matter fields can be any massive complex scalars and fermions without derivative self-interactions, and we do not write down the explicit form of the Lagrangian. They are coupled to the gauge field so that the EoM of the gauge field is given by where j µ is the matter current derived from L matter . For example, if we have a charged scalar φ with charge e such as the matter current is given by We now consider the Hamiltonian. We represent the conjugate momentum fields of A µ , c,c by Π µ , π (c) ,π (c) which are defined from the Lagrangian (2.1) as The Hamiltonian is then given by free part H 0 and the other interacting part V : where and H matter is the free part of the Hamiltonian of matter fields.

Physical states in the Schrödinger picture
We now quantize the system by imposing the canonical (anti)commutation relations. We put the superscript s to represent that an operator is in the Schrödinger picture like A s µ . The equal-time (anti)commutation relations for the gauge fields and the ghost fields are given by 10 The obtained Hilbert space is too large, and the physical Hilbert space H phys is given by the BRST cohomology. In the Schrödinger picture, the BRST operator is expressed as It is convenient to move to the momentum representation because the ghost fields are free. If we write  eq. (2.11) leads to We also introduce the ladder operators of photons as 11 The BRST operator (2.12) is then written as Since the ghost fields are decoupled, we can always restrict the ghost-sector of physical states to the ghost-vacuum annihilated by c( k) andc( k). Therefore, the physical state condition Q s BRST |ψ = 0 becomes k µ a µ ( k) + e iωtsj0s ( k) |ψ = 0 for all k . (2.26) This is the physical state condition in the Schrödinger picture. Note that this condition is different from the usual Gupta-Bleuler condition (1.26). This fact holds in the interaction picture as we will see in next subsection.

Gauge invariant asymptotic states
We now move to the interaction picture such that the S-matrix is given by the usual one (1.12). The asymptotic state |β 0 which is acted on the S-matrix is related to the in-states in the Schrödinger picture, |β in , as (1.5). Since |β in is a physical state in the Schrödinger picture, it satisfies Q s BRST |β in = 0. Thus, from eq. (1.5), |β 0 should satisfy Since Q s BRST commutes with the exact Hamiltonian H s , we have Q s BRST Ω(t) = U (t s , t)Q s BRST U 0 (t, t s ) = Ω(t)Q I BRST (t), (2.28) where Q I BRST (t) is the BRST operator in the interaction picture: By restricting the ghost-sector to the ghost-vacuum, this condition becomes It means that states satisfying the free Gupta-Bleuler condition k µ a µ ( k) |ψ = 0 are generally not the physical asymptotic states. Thus, the charged 1-particle states in the standard Fock space, such as b † ( p) |0 , cannot be the asymptotic physical states.
We will show below that the states satisfying the condition (2.31) are dressed states. In fact, if there is an anti-Hermitian operatorR(t) such that 12 [k µ a µ ( k),R(t)] = −e iωtj0I (t, k), [j 0I (t, k),R(t)] = 0, (2.33) then states in eR (t) H f ree are annihilated by k µ a µ ( k) + e iωtj0I (t, k) where H f ree is a subspace of H F ock satisfying the free Gupta-Bleuler condition. Thus, the Hilbert space satisfying (2.31) is given by Although one may use such a dressing operatorR(t), we can simplify it by recognizing that the current operator j 0I can be approximated in the asymptotic regions (t ∼ ±∞) by the classical current operator j 0 cl given by (1.14). It is convenient to use the hyperbolic coordinates (τ, ρ) to look at the asymptotic behaviors of massive charged fields as follows [17]: (2.35) 12 There are various choices of the dressing operator. One example is wherek µ = (ω, − k).
Then, we can straightforwardly obtain (see appendix C in [23] for details) 13 Therefore, we can rewrite the condition (2.31) as For later convenience, we represent the operator in (2.38) byĜ(t, k) aŝ Noting that the momentum representation of the classical current operator is given bỹ (2.41) Thus, an asymptotic physical Hilbert space satisfying (2.38) is given by Since the phase operator Φ in (1.23) commutes withĜ(t, k) and R(t), Φ is not relevant for the gauge-invariance (2.38). Therefore, the Faddeev-Kulish dressed space H F K in (1.25) is gauge invariant without introducing a vector c µ , if we restrict H F ock to the subspace H f ree .
Besides of the phase operator, there are other choices of the dressing operator R as (t) satisfying 14Ĝ (t, k)e Ras(t) = e Ras(t) k µ a µ ( k). (2.43) Then, we can define another asymptotic physical Hilbert space: which is a solution of the gauge invariant condition (2.38). Although the question that what types of dressing cancel the IR divergences in the S-matrix is beyond the scope of this paper, we will discuss in subsection 4.2 that the existence of many choices is natural from the point of view of asymptotic symmetry. 13 Other components of the current also satisfy similar equations: Here, the subscript f ree means that the current is that of the free theory. 14 One example of R as (t) is obtained by replacingj 0I withj 0 cl in (2.32). This R as (t) is different from the Faddeev-Kulish dressing operator (1.22).

Interpretation of the Faddeev-Kulish dresses
It is shown in [22] that the Faddeev-Kulish dressing factor for a charged particle with momentum p µ corresponds to the classical Liénard-Wiechert potential around the particle. This fact supports our statement that Gauss's law require the dressing factor. In this section, we will reconfirm this fact taking care of the i prescription, and see that we should use different prescriptions for initial and final states, which might be useful for the explicit computation of scattering amplitudes.

Coulomb potential by point charges in the asymptotic region
Here, we will recall the expression of the electromagnetic potential created by a charged point particle with momentum p µ . The classical equation of motion for the gauge field in the Lorenz gauge is given by where y µ (τ ) = pµ m τ = pµ Ep t is the trajectory of the charged particle, which is supposed to pass through the origin at t = 0. The position at t = 0 is not relevant when we consider the asymptotic region. 15 By using the retarded Green's function for the Klein-Gordon equation, the general solutions of (3.1) are given by where A in µ (x) is the incoming free wave, which is specified at t → −∞, and the second term is the Liénard-Wiechert potential created by the particle with momentum p µ and charge e.

Coulomb potential from dressed states with i prescription
Let's consider a dressed state of a single incoming electron 16 with momentum p µ defined by

4)
15 However, the position at t = 0 can contribute to subleading orders, and it was shown in [24] that the position is important for the subleading memory effect. 16 The generalization to multi-particles is trivial.
where R i (t) is an operator dressing the incoming single particle state. The gauge field in the interaction picture can be written as Then, we demand that its expectation value for the above dressed state match the classical gauge field configuration created by a charged point particle with momentum p µ , which is the second term in (3.3), as We can easily check that the following dressing operator satisfies the above condition, This operator matches the dressing operator (1.22) up to the i insertion. How to insert i in the dressing operator is determined by how the initial condition of gauge fields is specified. Thus, the dressed states stand for the states of (anti-)electrons surrounded by relativistic Coulomb fields created by themselves. In other words, in the dressed state, the charged particles are properly dressed by electromagnetic fields in the asymptotic region where the particles have nearly constant velocities. This result is natural since our dressed states are obtained by solving the BRST (gauge invariant) condition without ignoring the interaction in the asymptotic regions. We also would like to comment that this expectation value changes if we modify the dressing operator by introducing a vector c µ as in [8]. This is another reason to think that such a modification is unnatural.
Similarly, we can fix the i prescription for the dressing operator R f (t) for outgoing states. We consider a dressed outgoing state 8) and require that the expectation value of A I µ (x) agree with the advanced potential for the point particle, which is given by The requirement can be satisfied by the following dressing operator (3.11) Thus, the sign of i terms is opposite to that in the initial dressing operator R i given by (3.7). 17

Further discussion
In this paper, we have shown that the Faddeev-Kulish dressed states can be obtained just from the gauge-invariant condition without solving the asymptotic dynamics. While in the original paper [8] it was discussed that the dressing operator R(t) in eq. (1.22) should be modified, we have found that such a modification is not needed. We have also justified the unmodified dressing factor e R(t) with the i prescription by giving the interpretation as the Coulomb fields around charges. In addition, we have shown the possibility of other types of gauge-invariant dressed states.
We close this section with further discussion and comments on future directions.

Softness of dresses
The infrared finiteness of the dressed state formalism is based on Chung's analysis [3]. If we extract soft momentum region k ∼ 0 for the dressing operator (1.22), the operator takes the form Ep t ∼ 1 at k ∼ 0. Roughly, this is the dressing operator used in [3]. In fact, the behavior of the dressing operator at the non-soft momentum region was not specified in [3]. Since only the soft momentum region is relevant for the proof of the IR finiteness, this simplification may be justified.
However, one may worry that the hard momentum contribution in (1.22) affects the physical observables. We can make a rough argument that this is not the case as follows. Actually, owing to the oscillating factor e i p·k Ep t , contributions from nonzero momenta (ω > 0) can be ignored in the limit t → ±∞. 18 This statement can be made more rigorous by using -inserted dressing operator eq. (3.7) or eq. (3.11). We shall use the following identity as a distribution: From the identity, we have Therefore, we can say that only soft photons constitute the dresses in the asymptotic limit t → ±∞.
Nevertheless, it may be dangerous to use the above asymptotic limit directly. The S-matrix on the Fock space is given by Thus, we should first compute the finite time S-matrix element and then take the limits t f → ∞ and t i → −∞. In addition, since eq.(4.5) probably suffers from an infinitely oscillating phase factor, the phase operator such as (1.23) might be needed to make the S-matrix well-defined. As we said in subsec. 2.3, the phase operator cannot be determined from the gauge invariance. We would like to report a computation of the S-matrix in our dressed state formalism including the determination of the phase operator in future.

Asymptotic symmetry
Gauge theories in 4-dimensional Minkowski space have an infinite number of symmetries (see [26] for a recent review). The symmetries are given by "large" gauge transformations such that the gauge parameters can be nonvanishing functions in the asymptotic regions but they preserve the asymptotic behaviors of fields. We now discuss the relation between the asymptotic symmetry in QED and dressed states (see also [9,10,11,12,13,14,15] for related discussions).
The Lagrangian (2.1) is invariant under a class of gauge transformations which keep ∂ µ A µ intact. Neother's charge for the transformations in the Schrödinger picture is given by where the gauge parameter (x) satisfies 2 = 0. This charge Q s as [ ] is BRST exact up to the boundary term: Therefore, if the gauge parameter (x) vanishes in the asymptotic regions, this charge does not play any role on the physical Hilbert space. However, this is not the case if takes nonvanishing values in the asymptotic regions. Such nontrivial charges are called asymptotic charges. As discussed in [23], the asymptotic charges are physical charges, i.e., the asymptotic symmetry generated by the charges is not a redundancy of the Hilbert space but the physical symmetry. For example, if is a constant, the corresponding charge represents the total electric charge. There is no reason to restrict the Hilbert space to the subspace with zero total electric charge. Similarly, we should not restrict the Hilbert space to the subspace annihilated by Q s as [ ]. Therefore, asymptotic charges Q s as [ ] can act nontrivially on the physical Hilbert space. 19 The finiteness condition of Q s as [ ] requires that should approach functions of angular coordinates around the null infinities. It means that we have an infinite number of asymptotic charges corresponding to the number of functions on two-sphere [16]. All of the asymptotic charges commute with the BRST charge: . (4.9) Therefore, the spectrum of the physical Hilbert space is infinitely degenerated.
This fact naturally leads us to classify the asymptotic states by Q I as in the interaction picture. In fact, as discussed in [11], eigenstates of the asymptotic charges resembles dressed states. On the other hand, the Faddeev-Kulish dressed states are generally not eigenstates of Q I as . This is due to the fact that the standard Fock vacuum is not the eigenstate of Q I as for general large gauge parameters . 21 Roughly speaking, eigenstates of Q I as consist of clouds of soft photons, regardless of whether charged particles exist or not. However, the Faddeev-Kulish dressing factor R(t) makes a cloud only when there are charged particles. Thus, we need a dress other than Faddeev-Kulish's in order to prepare an eigenstate of Q I as . As we saw in subsection 2.3 that dressing factors are not uniquely fixed from the gauge invariant condition. We think that this variety implies the above degeneracy, and leave it for a future work to classify gauge invariant dressed states in terms of the asymptotic charges.

Other future directions
We would like to comment on other future directions.
Mandelstam developed a manifestly gauge-independent formalism of gauge theories [27,28]. In the formalism, the dynamical variables of QED are the field strength F µν and path-dependent charged fields such as φ(x; Γ) ≡ e −ie x Γ dξ µ Aµ(ξ) φ(x). (4.10) Such fields attached with Wilson lines are also considered in the context of the bulk reconstruction in the AdS/CFT correspondence (see e.g. [29,30,31]). A similarity between Mandelstam's formalism and the dressed state formalism was discussed in [32]. However, the dressing operator constructed in [32] has additional terms depending on the choice of the path Γ. Thus, the dressing operator seems not to be related directly to Faddeev-Kulish's one (1.22). Furthermore, we should also investigate the relation to the asymptotic symmetry. As explained in [33] for gravitational theories in AdS, operators like (4.10) are transformed under the asymptotic symmetry, and the behavior of the path Γ near the asymptotic boundary is important in determining the transformation law of the symmetry. In [27,28], the behavior of Γ near the asymptotic region was not specified. Thus, it is interesting to understand more precisely the relations among Mandelstam's formalism, the dressed state formalism and the asymptotic symmetry [34]. 20 This is the reason why we adopted the Hamiltonian (2.9). As mentioned in footnote 9, the canonical Hamiltonian H s can has extra boundary terms: H s can = H s − d 3 x ∂ i (Π s 0 A is + Π is A 0s ). The boundary terms affect the commutator (4.9) as [Q s as , d 3 x ∂ i (Π s 0 A is + Π is A 0s )] = i d 3 x ∂ i (Π 0s ∂ i + Π is ∂ 0 ) = {Q s BRST , i d 3 x ∂ i (c s ∂ i )} + i d 3 x ∂ i (Π is ∂ 0 ). Since ∂ 0 = O(r −1 ) at r → ∞, we can neglect the effect of boundary terms if the radial component of the electric field operator,x i Π i , decays as O(r −2 ). This condition is probably satisfied for physical scattering states in a reasonable setup. 21 The asymptotic symmetries for general gauge parameters are spontaneously broken in the standard Fock vacuum [16].
It is important to extend our analysis to other theories. Although the cancellation of IR divergences in the inclusive method [1,2] was extended to more general theories [35,36], we do not know how to define the IR finite S-matrices directly. We think that the dressed state formalism is surely useful for this problem. The dressed state formalism for the perturbative gravity was developed in [37] (see also [12,13]). However, a tensor c µν , which is an analog of a vector c µ in [8], was introduced by imposing a free "gauge invariant" condition which is a gravitational counterpart of the free Gupta-Bleuler condition (1.26). As in QED, we should impose an appropriate physical condition, and we expect that the tensor c µν is unnecessary. Asymptotic symmetries for scalar theories are also studied in [38,39,40]. It was found recently that the asymptotic symmetry of a massless scalar is related to the gauge symmetry of the two-form field dual to the scalar [41,42,43,44,45]. It might be possible to construct dressed states in a massless scalar theory from the gauge-invariant condition for the dual two-form field.