Infrared problem in perturbative quantum field theory
Pawe lDuch Max-Planck Institute for Mathematics in the Sciences Inselstr. 22, 04103 Leipzig, Germany Institut f¨urTheoretische Physik, Universit¨atLeipzig Br¨uderstr.16, 04103 Leipzig, Germany [email protected] July 21, 2021
Abstract We propose a mathematically rigorous construction of the scattering matrix and the interacting fields in models of relativistic perturbative quantum field theory with massless fields and long-range interactions. We consider quantum electrodynamics and a certain model of interacting scalar fields in which the standard definition of the scattering matrix is not applicable because of the infrared problem. We modify the Bogoliubov construction using the ideas of Dollard, Kulish and Faddeev. Our modified scattering matrix and modified interacting fields are constructed with the use of the adiabatic limit which is expected to exist in arbitrary order of perturbation theory. In the paper we prove this assertion in the case of the first- and the second-order corrections to the modified scattering matrix and the first-order corrections to the modified interacting fields. We study the physical properties of our construction. We conclude that the electrons and positrons are always surrounded by irremovable clouds of photons. Moreover, the physical energy-momentum operators do not coincide with the standard ones and their joint spectrum does not contain the mass hyperboloid. arXiv:1906.00940v3 [math-ph] 20 Jul 2021
1 Contents
1 Introduction3 2 Outline of Epstein-Glaser approach8 2.1 Wick products ...... 8 2.2 Time ordered products ...... 9 2.3 S-matrix and interacting fields with adiabatic cutoff ...... 11 2.4 Adiabatic limit ...... 12 3 Models with infrared problem 13 3.1 Scalar model ...... 14 3.2 Quantum electrodynamics ...... 15 4 Infrared problem in scattering theory 17 4.1 Classical and quantum mechanics ...... 18 4.2 Quantum field coupled to classical current ...... 19 4.3 Relativistic perturbative QFT ...... 20 5 Construction of S-matrix and interacting fields 24 5.1 Modified scattering theory ...... 24 5.2 Asymptotic currents ...... 28 5.3 Coulomb phase ...... 31 5.4 Domains in Fock space ...... 34 5.5 Definition of modified S-matrix and modified interacting fields . . . . . 35 6 First- and second-order corrections to S-matrix 46 6.1 First-order corrections ...... 48 6.2 Second-order corrections ...... 51 7 First-order corrections to interacting fields 62 7.1 Scalar model ...... 63 7.2 QED ...... 65 8 Energy-momentum operators 67 8.1 Scalar model ...... 68 8.2 QED ...... 70 9 Physical interpretation of construction 71 9.1 Space of asymptotic states ...... 71 9.2 Cross sections ...... 74 9.3 Inclusive cross sections ...... 75 10 Summary and outlook 78 A Value of a distribution at a point 78 B BCH formula and Magnus expansion 79 C Long-range tail 80 D Incoming and outgoing LSZ fields 80 E Propagators 80
2 1 Introduction
Quantum electrodynamics (QED) is one of the best tested theory known in physics. The success of QED is to large extent based on very accurate theoretical predictions of the anomalous magnetic moments of leptons and transition frequencies between various energy levels in light hydrogenlike atoms. In contrast to other sectors of the standard model the scattering experiments do not constitute the most stringent tests of QED and, as a matter of fact, the state of the art theoretical description of the scattering processes in QED is still far from satisfactory. One of the long-standing fundamental open problems in QED is the definition of the scattering operator. Because of the long- range character of interactions mediated by massless photons the standard method of constructing this operator is plagued by infrared (IR) divergences [10]. In view of these problems, one usually abandons the definition of the scattering matrix and computes only the so-called inclusive cross sections [102, 98]. Although, this pragmatic approach is sufficient for most applications, it is not fully satisfactory. The physical interpreta- tion of the inclusive cross sections is obscure since their construction relies on the use of charged states with sharp momenta which do not exist in the theory without the IR regulator. Moreover, the inclusive cross sections provide only partial information about the scattering process. There are processes that cannot be described without using the scattering matrix. An example of such a process is a shift of a trajectory of a charged particle moving in a weak low-frequency electromagnetic field [91, 59]. It would be also challenging to study a quantum analogs of the memory effects predicted in classical electrodynamics [6, 50, 96] without having a full scattering theory in QED at hand. The inclusive cross sections are completely insensitive to the IR degrees of freedom. Yet such degrees of freedom can be of physical importance. Indeed, Hawk- ing, Perry and Strominger [56] have argued recently that it is possible to solve the black hole information paradox using the IR degrees of freedom of the gravitational field. The significance of the soft degrees of freedom has been also recently observed by a number of authors who studied the relation between the Weinberg soft photon theorem [98] and the so-called large gauge transformations [49, 19] (a more complete list of references can be found in the lecture notes [96]). The recent revival of interest in the IR degrees of freedom calls for a construction of the scattering matrix in QED in which these degrees of freedom are properly taken into account. A strategy for the construction of a IR-finite scattering matrix in QED was sug- gested long time ago by Kulish and Faddeev [70]. Their construction is based on the Dollard method [28] which was originally formulated for the Coulomb scatter- ing in quantum mechanics. The basic idea is to define a modified scattering matrix by comparing the full dynamics of the system with some simple but nontrivial refer- ence dynamics, called the Dollard dynamics. The Kulish-Faddeev construction was reformulated and further developed e.g. in [80, 66] and more recently in [53, 67, 61]. However, the method of Kulish and Faddeev has never been put on firm mathematical grounds and tested at least in low orders of perturbation theory. In fact, up to our knowledge, a rigorous construction of the scattering matrix in relativistic QED which does not suffer from the ultraviolet (UV) or IR problem has never been given. In the present paper, we attempt to formulate such a construction using the ideas of Kulish, Faddeev, Dollard and insights obtained more recently by Morchio and Strocchi [78, 77] in their analysis of a toy model of QED. Our modified scattering matrix is constructed with the use of the adiabatic limit as in the method developed by Bogoliubov [11].
3 Since QED most likely does not make sense outside perturbation theory [71] we for- mulate our proposal in the perturbative setting. We stress that QED is a well-defined model of quantum field theory (QFT) whose perturbative construction is under full control. What is not understood is the large-time behavior of its dynamics. The main model of our interest is QED. However, we consider also a certain model of interact- ing scalar fields which we call the scalar model. In principle our method should be also applicable to other models with long-range interactions in the four-dimensional Minkowski space which, like QED and the scalar model, have the interaction vertices containing exactly one massless and two massive fields. In the paper we propose a method of constructing the following objects in QED and the scalar model: ˆ • the Hilbert spaces of asymptotic outgoing and incoming states Hout/in(%) param- eterized by sector measures % (a sector measure is a signed measure on the mass 2 2 0 hyperboloid Hm consisting of four-momenta p such that p = m and p > 0),
ˆ iPˆ a • a strongly-continuous representation of the translation group U(a) = e · acting ˆ ˆµ in Hin/out(%), with generators P satisfying the relativistic spectrum condition, ˆ ˆ ˆ • the translationally-invariant scattering matrix S : Hin(%) → Hout(%) (the inclu- sive cross section constructed with the use of Sˆ formally coincides with the one obtained using the standard procedure), ˆ • the translationally-covariant retarded and advanced interacting fields Cret/adv(x): ˆ ˆ Hin/out(%) → Hin/out(%), where C is a (gauge invariant) polynomial in the ba- sic fields and their derivatives (the interacting fields are local, satisfy the field equations and their Wightman and Green functions coincide with the standard ones).
We investigate various properties of the above objects. Below we summarize the results in the case of QED and comment on the case of the scalar model. The measure % is related to the long range tail of the electromagnetic field (the flux of the electric field) Z lim r2 Fˆµν (x + rn) = −e d%(p) f µν(p/m; n), r ret/adv →∞ Hm where e is the elementary charge, n is a normalized spatial four-vector and f µν(v; ·) is the Coulomb field of a unit point-charge moving with constant velocity v. Since ˆ the long range tail is a classical observable [17] the spaces Hin/out(%) with different % correspond to different superselection sectors. By the Gauss law %(Hm) is equal to the 1 total electric charge . Thus, we demand that %(Hm) is an integer (there is no such ˆ constraint in the scalar model). The spaces Hin/out(0) are called the vacuum sector. They contain a unique vector Ωˆ such that PˆµΩˆ = 0, called the vacuum. The photon creation and annihilation operators are denoted by2
# ˆ ˆ aˆ (k): Hin/out(%) → Hin/out(%).
1In the paper, by the electric charge we always mean the difference between the number of electrons and positrons. 2A# stands for the operator A or its hermitian conjugate.a ˆ#(k) are (unbounded) operators when smeared with Schwartz test functions.
4 They satisfy the standard commutation relations, have the energy-momentum trans- fer on the lightcone and are obtained as the incoming/outgoing LSZ limits of the retarded/advanced interacting electromagnetic field (the massless scalar field in the case of the scalar model). They add or subtract one photon with four-momentum k. We also define creators and annihilators of the electrons ˆ# ˆ ˆ b (η, %1; p): Hin/out(%) → Hin/out(% ± %1) satisfying the standard (anti-)commutation relations. They depend on the choice a 4 R 4 real-valued profile η ∈ S(R ), d x η(x) = 1, and some measure %1 on Hm, %1(Hm) = 1 (in the case of the scalar model all particles are chargeless and we can set %1 = 0). ˆ# The operators b (η, %1; p) do not have the energy-momentum transfer on the mass hyperboloid and do not commute witha ˆ#(k). They add or subtract one massive particle together with a coherent cloud of photons surrounding it and depending on ˆ ˆ ˆ ˆ η and %1. For example, for Ω ∈ Hin/out(0) the state b∗(η, %1; p)Ω ∈ Hin/out(%1) is an (improper) eigenvector of the photon annihilation operatora ˆ(k) with eigenvalue Z p p0 e η˜(k) ε(k) · − d%1(p0) , p · k Hm p0 · k whereη ˜ is the Fourier transform of η and εµ(k) is the polarization vector of the photon annihilated bya ˆ(k). There is no mass hyperboloid in the joint spectrum of the energy- momentum operators Pˆµ regardless of the choice of the sector. It is convenient to define the following Hilbert spaces containing sectors with all possible eigenvalues of the electric charge operator Q ˆ M ˆ Hout/in(ˆ%) := Hout/in(%0 + q%1), q Z ∈ where% ˆ = %0 +Q%1 and %0 and %1 are fixed measures on Hm such that %0(Hm) = 0 and %1(Hm) = 1 (in the scalar model Q = 0, %0 need not satisfy the condition %0(Hm) = 0 ˆ ˆ and Hout/in(ˆ%) = Hout/in(%0)). Since it is difficult to discriminate experimentally sectors with the same electric charge but different long-range tail of the electromagnetic field ˆ for most practical application one can restrict attention to the spaces Hout/in(ˆ%) with one fixed% ˆ. Our standard choice in the case of the scalar model is% ˆ = 0, which corresponds to the vacuum sector. In the case of QED the standard choice is% ˆ = Q%mv, where v is some four-velocity and %p is the Dirac measure at p ∈ Hm. The corresponding spaces Hout/in(ˆ%) contain the super-selection sectors in which the flux of the electric field is independent of the spatial direction in the reference frame of the observer moving with four-velocity v. ˆ The asymptotic spaces Hin/out(ˆ%) are isomorphic to the standard Fock space H. However, the isomorphism is non-canonical and depends on the choice of a profile η. In particular, the physical photon creators and annihilators are not identified with the standard creation and annihilation operators of photons acting in the Fock space. We exploit this isomorphism in our construction. We first define a family of (modified) scattering operators Smod(η) and interacting fields Cret/adv(η; x) in the standard Fock space H. The operators Smod(η) and Cret/adv(η; x) depend on η but satisfy certain compatibility condition. Using this compatibility condition we prove that the corre- ˆ ˆ ˆ sponding operators S and Cret/adv(x) acting in Hin/out(ˆ%) are independent of choice a ˆ profile η used to identify H and Hin/out(ˆ%) (we keep the measure% ˆ determining the superselection sector fixed).
5 Our modified scattering matrix corresponds roughly to the following expression
Z t2 Z 0 (formally) i(t2 t1)H Smod = lim Texp i dt HD(t) e− − Texp i dt HD(t) , (1.1) t1 0 t1 t2→−∞+ → ∞ where H is the full Hamiltonian of the system and HD(t) is an appropriately chosen time-dependent Dollard Hamiltonian. The above formula makes sense in quantum mechanics where it can be used to define the scattering matrix for systems of parti- cles influenced by long-range potentials such as the Coulomb potential [28, 26]. In order to give meaning to expression (1.1) in perturbative QFT we use the Bogoliubov method [11]. We first construct the modified scattering matrix with the adiabatic cutoff as as 1 Smod(η, g) = R(η, g)Sout(η, g)S(g)Sin (η, g)R(η, g)− . (1.2) It is defined in the standard Fock space (in the case of QED we use the BRST frame- work) and depends on the switching function g ∈ S(R4) and the profile η. The switching function plays the role of the IR regulator and is also called the adiabatic cutoff. The Bogoliubov operator S(g) is formally defined by Z S(g) := Texp ie d4x g(x)L(x) , where Texp is the time-ordered exponential and the RHS of the above equation is interpreted as a formal power series in the coupling parameter e. The interaction vertex L is expressed in terms of the free local fields defined in the Fock space. These fields play an auxiliary role in the construction and do not coincide with the asymptotic LSZ fields of the interacting retarded/advanced fields that we construct. The Bogoliubov operator is the scattering operator in an unphysical theory in which the coupling constant e is replaced with the function e g(x). In order to solve the familiar UV problem in the construction of this operator we use the Epstein-Glaser method [42, 14, 63] (the use of the Epstein-Glaser method is not essential in our construction). as The operators Sout/in(g) are called the Dollard modifiers and up to finite self-energy renormalization are given by Z as 4 Sout/in(g) = Texp −ie d x g(x)Lout/in(x) , (1.3)
where Lout(x) and Lin(x) are the asymptotic outgoing and incoming vertices and Texp is the anti-time-ordered exponential. The asymptotic vertices describe the emission or absorption of a photon by an electron or positron whose momentum is unchanged in this process. They are given by some non-local functionals of fields which can be expressed in a simple way in terms of the creation and annihilation operators. Finally, 1 the operators R(η, g) and R(η, g)− , which depend implicitly on the sector measure% ˆ, are used to implement a coherent transformation to sectors with desired long-range tail of the electromagnetic field (the massless scalar field in the case of the scalar model). as There is no UV problem in the construction of the operators Sout/in(g) and R(η, g). The physical scattering matrix Smod(η) is defined by the following adiabatic limit
(Ψ|Smod(η)Ψ0) := lim (Ψ|Smod(η, g)Ψ0), (1.4) 0 &
6 where the one-parameter family of the switching functions g, ∈ (0, 1), is defined by 4 setting g(x) = g(x) with g ∈ S(R ) such that g(0) = 1. The states Ψ and Ψ0 have wave functions that vanish when momenta of the electrons or positrons are sufficiently close to one another. The modified interacting fields Cret/adv(η) can be constructed in a similar way. It is expected that the adiabatic limit (1.4) exists in arbitrary order of perturbative expansion in the coupling parameter e in both QED and the scalar model. In the paper we prove this claim up to the second order of perturbation theory. It is plausible that one could show the existence of the adiabatic limit (1.4) in all orders of perturbation theory by adapting the method of [32, 43], which was used to construct the scattering matrix in purely massive models. Note that on the formal level the existence of the limit (1.4) is equivalent to the factorization of the IR divergences. Such a factorization was established at the physics level of rigor in [102, 98] and is usually taken for granted in the physics literature [99, 81]. Thus, the existence of the adiabatic limit our modified scattering matrix is well-motivated on physical grounds. Apart from a lack of the rigorous proof of this fact our analysis of the IR structure of QED and the scalar model is complete and complies with the results obtained previously in the axiomatic framework or simplified models. The plan of the paper is as follows. In Sec.2 we outline to the Epstein and Glaser approach to perturbative QFT. In Sec.3 we introduce the models which are investigated in the paper. In Sec.4 we discuss the IR problem in the perturbative construction of the scattering matrix in QED and the scalar model. In Sec.5 we define the modified scattering matrix and the modified interacting fields in the scalar model and QED. Sec.6 contains a construction of the first and second order corrections to the modified scattering matrix. The first order corrections to the modified interacting fields are presented in Sec.7. In Sec.8 we investigate the translational covariance of our construction. In Sec.9 we define the spaces of asymptotic states, give their physical interpretation and present a method of constructing the inclusive cross section using our scattering matrix. The paper closes with a summary. AppendixA contains a useful theorem about the value of a distribution at a point defined with the use of the adiabatic limit. In AppendixB we recall the BCH formula and the Magnus expansion. In AppendixC we define the long-range tail of a field. In AppendixD we discuss the LSZ limit of fields. In AppendixE we give explicit expressions for various propagators used in the paper. Notation:
• The Minkowski spacetime is identified with R4. It is equipped with the in- µ ν 0 0 1 1 2 2 3 3 ner product given by x · y = gµνx x = x y − x y − x y − x y . If x = (x0, x1, x2, x3) ∈ R4 is a four-vector, then ~x = (x1, x2, x3) ∈ R3. • The future and past light cones in the Minkowski spacetime are denoted by 4 2 0 V ± = {p ∈ R : p > 0, ±p > 0}. Their closures are denoted by V ±. 4 2 2 0 • The invariant measure on the mass shell Hm := {p ∈ R : p = m , p ≥ 0} is 1 4 0 2 2 + denoted by dµm(p) := (2π)3 d p θ(p )δ(p − m ), m ≥ 0. We have H0 = ∂V . We p 2 2 4 set Em(~p) = m + |~p| . We say that v ∈ R is a four-velocity iff v ∈ H1. • The Heaviside theta function is denoted by θ. • The space of test functions with compact support and Schwartz functions on RN N N are denoted by Cc∞(R ) and S(R ), respectively.
7 N N • Let t ∈ S0(R ) be a Schwartz distribution and g ∈ S(R ). We use the notation R dN x t(x)g(x) for the pairing between distributions and test functions. N • The Fourier transform of the Schwartz distribution t ∈ S0(R ) is denoted N R N by t˜. For any g ∈ S(R ) it holdsg ˜(q) := d x exp(iq · x)g(x) and g(x) = R dN q (2π)N exp(−iq · x)˜g(q). • The positive-definite inner product in the (Krein-)Fock space is denoted by h·|·i and the corresponding norm by k·k. The covariant non-degenerate inner-product, which need not be positive definite, is denoted by (·|·). In the case of models without vector fields the above inner products coincide. The hermitian conjugate of an operator A with respect to the positive-definite/covariant inner product is # denoted by A† and A∗, respectively. Moreover, A stands for A or A∗.
• Let Y1 and Y2 be complex-linear spaces. The space of linear operators Y1 → Y2 is denoted by L(Y1, Y2). We write L(Y, Y) = L(Y). The space of complex-valued anti-linear functionals on Y is denoted by Y∗. The space of sesquilinear forms on Y can be identified with L(Y, Y∗). • The symbols s-lim, w-lim denote the limits in the strong and weak operator topologies. • The ring of formal power series in the coupling constant e with coefficients in the ring R will be denoted by R e . The coefficient of order en of a formal power series a ∈ R e is denoted by a[Jn].K J K • We denote the electric charge by −eQ, where e > 0 is the elementary charge. The free BRST charge is denoted by QBRST. The free ghost charge is denoted by Qgh. The operator ρ(p) is defined by Eqs. (3.2) and (3.5) in the scalar model and QED, respectively. We set dρ(p) := ρ(p) dµm(p).
2 Outline of Epstein-Glaser approach
In this section we outline a construction of interacting models of perturbative quantum field theory in the Epstein-Glaser framework. More detailed exposition can be found in [37, 42, 86, 87, 33]. In Sec. 2.3 we recall the Bogoliubov method of constructing the scattering operator and the interacting fields. The precise definition of the adiabatic limit is given in Sec. 2.4.
2.1 Wick products In perturbative QFT the interacting models are built with the use of the free fields. It is useful to consider first the algebra F of symbolic fields which is by definition a commutative algebra generated by symbols corresponding to the components Ai of α fields used in the definition of a given model and their derivatives ∂ Ai, where α is a multi-index. To every symbolic field B ∈ F we associate the Wick polynomial :B(x): which is an operator-valued Schwartz distribution [100, 94] on a suitable domain D0 in the Fock Hilbert space H with the positive-definite inner product h·|·i. Given a domain D ⊂ H let L(D) be the space of linear maps D → D. By an operator-valued Schwartz distribution on D we mean a map T : S(RN ) → L(D), such that for any
8 Ψ, Ψ0 ∈ D the map Z N N S(R ) 3 g 7→ hΨ| d x g(x) T (x)Ψ0i ∈ C (2.1) is a Schwartz distribution. The space of operator-valued Schwartz distributions on N N N 0 D is denoted by S0(R ,L(D)). If T (x) ∈ S0(R ,L(D)) and T 0(x0) ∈ S0(R ,L(D)), N+N 0 then T (x)T 0(x0) ∈ S0(R ,L(D)), by the nuclear theorem. The vacuum state in the Fock space H is denoted by Ω. We assume that the Fock space is equipped with a non-degenerate sesquilinear inner product (·|·) which in general need not be positive definite. The hermitian conjugation and the notion of unitarity are defined with respect to this product. The unitary representation of the inhomogeneous SL(2, C) group (the covering group of the Poincar´egroup), denoted by U(a, Λ), where a ∈ R4 and Λ ∈ SL(2, C) is a Lorentz transformation, is defined in the Fock space in the standard way. In the case of pure translations we write U(a) ≡ U(a, 1). Consider the following dense, Poincar´e-invariant domain in the Fock space Z 4 4 D0 := spanC d x1 ... d xn f(x1, . . . , xn) Ai1 (x1) ...Ain (xn)Ω : 4n n ∈ N0, i1, . . . , in ∈ {1,..., p}, f ∈ S(R ) . (2.2)
As shown in [100] Wick polynomials are are well defined as operator-valued Schwartz distributions on D0. In fact, the following theorem holds. 4n Theorem 2.1. [42] Let t ∈ S0(R ) be translationally invariant, i.e.
t(x1, . . . , xn) = t(x1 + a, . . . , xn + a) 4 for all a ∈ R . Then for any B1,...,Bn ∈ F 4n t(x1, . . . , xn):B1(x1) ...Bn(xn): ∈ S0(R ,L(D0)). (2.3)
2.2 Time ordered products In perturbative QFT the scattering operator and the interacting fields are constructed with the use of the time-ordered products. The time-ordered products are multi-linear maps
n 4n T: F 3 (B1,...,Bn) 7→ T(B1(x1),...,Bn(xn)) ∈ S0(R ,L(D0)), where n ∈ N0, satisfying the following axioms: A.1 T(∅) = 1, T(B(x)) = :B(x):,
T(B1(x1),...,Bn(xn), 1(xn+1)) = T(B1(x1),...,Bn(xn)), where 1 on the LHS of the above equality is the unity in F. A.2 Poincar´ecovariance:
1 U(a, Λ) T(B1(x1),...,Bn(xn))U(a, Λ)−
= T((ρ(Λ)B1)(Λx1 + a),..., (ρ(Λ)Bn)(Λxn + a)), (2.4)
where B1,...,Bn ∈ F, ρ is the representation of SL(2, C) acting in F and U(a, Λ) is the unitary representation of the Poincar´egroup in D0.
9 A.3 Symmetry:
T(B1(x1),...,Bn(xn)) = T(Bπ(1)(xπ(1)),...,Bπ(n)(xπ(n))), where π is any permutation of the set {1, . . . , n}.
A.4 Causality: If none of the points x1, . . . , xm is in the causal past of any of the points xm+1, . . . , xn then
T(B1(x1),...,Bn(xn))
= T(B1(x1),...,Bm(xm)) T(Bm+1(xm+1),...,Bn(xn)).
A.5 Wick expansion: T(B1(x1),...,Bn(xn)) is uniquely determined by the vacuum expectations of the time-ordered products of the sub-polynomials of B1,...,Bn
T(B1(x1),...,Bn(xn)) s1 sn X (s1) (sn) :A (x1) ...A (xn): = (Ω| T(B1 (x1),...,Bn (xn))Ω) . s1! . . . sn! s1,...,sn The sub-polynomials of the interaction vertex of QED are listed in (2.6). A.6 Bound on the Steinmann scaling degree that controls the UV behavior of the time-ordered products: n+1 X sd( (Ω| T(B1(x1),...,Bn(xn),Bn+1(0))Ω) ) ≤ dim(Bj), (2.5) j=1 4 where sd(t) is the Steinmann scaling degree [92, 14] of t ∈ S0(R ) and dim(B) is the dimension of a polynomial B (the dimension dim(B) may be strictly larger than the canonical dimension; see Sec. 3.1). A.7 Unitarity:
T(B1(x1),...,Bn(xn))∗ = T(Bn∗(xn),...,B1∗(x1)), where T is the anti-time-ordered product (cf. e.g. [42]). A.8 Covariance with respect to the discrete group of CPT transformations (the charge conjugation, the spatial-inversion and the time-reversal). A.9 Field equations: If one of the arguments of a time-ordered product is a basic field, then this time-ordered product is related to some time-ordered product with one argument less (for details see e.g. the normalization condition N.FE introduced in Sec. 4.3 of [31]).
A.10 Ward identities in QED [38]: Let B1,...,Bn be sub-polynomials of the QED µ interaction vertex L = JµA = ψAψ/ , i.e. µ B1,...,Bk ∈ {1,Aµ, ψa, ψa,J , (Aψ/ )a, (ψA/)a, L}. (2.6) It holds k x µ X ∂µ T(J (x),B1(x1),...,Bk(xk)) = q(Bj) δ(x − xj) T(B1(x1),...,Bk(xk)), j=1 (2.7) where q(B) is the additive electric charge number defined such that q(Aµ) = 0, q(ψa) = −q(ψa) = −1.
10 The above conditions were stated for bosonic fields and have to be slightly modified in the case of fields with Fermi statistics (see e.g. [31]). We refer the reader to [42, 86, 14, 63] for the construction of the time-ordered products satisfying the above axioms. Let us mention that the time-ordered products are not defined uniquely. Assume that all the time-ordered products with at most n arguments are fixed. Then two possible definitions of the vacuum expectation value
4(n+1) (Ω| T(B1(x1),...,Bn+1(xn+1))Ω) ∈ S0(R ) (2.8) differ by a distribution of the form
X γ cγ∂ δ(x1 − xn+1) . . . δ(xn − xn+1) γ γ ω | |≤ for some constants cγ ∈ C indexed by multi-indices γ, |γ| ≤ ω, where
n+1 X ω := 4 − (4 − dim(Bj)) (2.9) j=1 is the order of singularity. The constants cγ have to be chosen in such a way that the new set of time-ordered products satisfies all the axioms. Note that if ω < 0, then (2.8) is determined uniquely by the time-ordered products with at most n argu- ments. Complete characterization of the freedom in defining the time-ordered products can be found in [62, 82, 34].
2.3 S-matrix and interacting fields with adiabatic cutoff The scattering matrix of a model of perturbative QFT with the interaction vertex L ∈ F is formally given by the Dyson [41] formula S = Texp ie R d4x L(x), where L ∈ F is the interaction vertex. However, the above expression is ill-defined even in purely massive models because the time-ordered products T(L(x1),..., L(xn)) are operator-valued Schwartz distribution and generically cannot be integrated with con- stant functions. In order to obtain a well-defined expression for the scattering matrix in QFT we follow the idea of Bogoliubov [11] and multiply the interaction vertex L by the switching function g which is a real-valued Schwartz function defined on the spacetime. The above prescription regularizes the theory in the IR regime. We call this type of the IR regularization the adiabatic cutoff. Note that in the theory with adiabatic cutoff there are only short-range interactions. The scattering matrix with adiabatic cutoff, which is also called the Bogoliubov S-matrix, is defined by [11]
Z S(g) = Texp ie d4x g(x)L(x)
X∞ inen Z := d4x ... d4x g(x ) . . . g(x ) T(L(x ),..., L(x )) ∈ L(D ) e . (2.10) n! 1 n 1 n 1 n 0 n=0 J K The above expression is well-defined as a formal power series of operators mapping the domain D0 into D0. The physical scattering operator is obtained by taking the limit g(x) → 1 of the S(g) in an appropriate sense to be discussed in Sec. 2.4.
11 Bogoliubov proposed also an elegant method of constructing the interacting fields. For any polynomial in the basic fields and their derivatives C ∈ F the corresponding retarded and advanced interacting fields are given by [11]
δ 1 Cret(g; x) := (−i) S(g)− S(g; h) ∈ L(D0) e , δh(x) h=0 J K (2.11) δ 1 Cadv(g; x) := (−i) S(g; h)S(g)− ∈ L(D0) e , δh(x) h=0 J K where Z Z S(g; h) := Texp ie d4x g(x)L(x) + i d4x h(x)C(x) (2.12) is the so-called extended scattering matrix. If the switching function g has a compact support then the retarded and advanced fields coincide with the free field :C(x): out- side the past and future of supp g, respectively. If C ∈ F is a basic field, then the corresponding retarded/advanced field Cret/adv(g; x) satisfies the interacting equation of motion with the coupling constant e replaced with eg(x). For example, in the case of the massless ϕ4 model we have e ϕ (g; x) + g(x)(ϕ3) (g; x) = 0. x ret/adv 3! ret/adv Moreover, the interacting fields satisfy the local commutativity condition, i.e. for any C,C0 ∈ F it holds [42, 38]
[Cret(g; x),Cret0 (g; x0)] = 0, [Cadv(g; x),Cadv0 (g; x0)] = 0 if x and x0 are spatially separated. Let us mentioned that the retarded and advanced 4 fields Cret/adv(g; h) with adiabatic cutoff g ∈ S(R ) can be used to construct [39, 46] the net of abstract algebras F(O) of interacting fields localized in bounded spacetime regions O ⊂ R4. The net does not depend on the switching function and satisfies the Haag-Kastler axioms [55] in the sense of formal power series.
2.4 Adiabatic limit In order to define the physical scattering operator or the vacuum representation of the net F(O) one uses the so-called strong adiabatic limit. To this end, one introduces a one-parameter family of the switching functions g, ∈ (0, 1), defined by g(x) = g(x), where g ∈ S(R4) is an arbitrary real-valued Schwartz function such that g(0) = 1. We say that the adiabatic limit of S(g) or Cret/adv(g; h) exists if the limits
lim S(g), lim Cret/adv(g; h) 0 0 & & exist in appropriate sense to be specified (see e.g. Eqs. (2.13) or Eqs. (5.27)) and do not depend on the choice of g used in the definition of g. The existence of the adiabatic limit of the scattering operator in massive models was proved in [43]. In what follows, we summarize the main results of [32], where the above-mentioned proof is simplified and generalized.
12 Consider an arbitrary model of perturbative QFT which contains only massive fields. Let us define the following domain in the Fock space
Z DH := spanC dµm(p1) ... dµm(pn) h(p1, . . . , pn) n × a∗(p1) . . . a∗(pn)Ω : n ∈ N0, h ∈ SH(Hm× ) ,
n where SH(Hm× ) consists of H¨older-continuous functions which decay rapidly at infinity. 4 For any Ψ ∈ DH, C ∈ F and h ∈ S(R ) the following limits:
SΨ := lim S(g)Ψ ∈ DH e ,Cret/adv(h)Ψ := lim Cret/adv(g; h)Ψ ∈ DH e (2.13) 0 0 & J K & J K exist in each order of perturbation theory and define the physical scattering operator
S ∈ L(DH) e and the physical interacting field operators Cret/adv(h) ∈ L(DH) e . The above objectsJ K are covariant with respect to the Poincar´etransformations, i.e.J K
1 U(a, Λ)SU(a, Λ)− =S, 1 1 U(a, Λ)Cret/adv(h)U(a, Λ)− =(ρ(Λ− )C)ret/adv(ha,Λ),
1 where ha,Λ(x) = h(Λ− (x − a)), ρ is the representation of the Lorentz group acting in F and U(a, Λ) is the standard representation of the Poincar´egroup acting in the Fock space. Moreover, the scattering operator satisfies the following conditions
SΩ = Ω,S|p) = |p) expressing the stability of the vacuum and one-particle states. The interacting field operators fulfill the interacting field equations and the local commutativity condition. Their LSZ limits coincide with the free fields. Thus, the physical particle interpretation of the states in the Fock space coincides with the standard one. The above method provides a direct and complete perturbative construction of mas- sive models of QFT in the vacuum representation. However, because of the IR problem it is not applicable to most models with massless fields. In Sec.5 we present a modifi- cation of the above method which can be used to construct the scattering matrix and interacting fields in QED and the scalar model introduced in Sec.3. The construction allows us to draw a number of interesting conclusions about the IR structure of these models. The properties of the scattering matrix and interacting fields in QED and the scalar model turn out to be quite different than the one listed above. In particular, the particle interpretation of states is more involved and the energy-momentum operators do not coincide with the standard ones and have different spectrum. Moreover, there is an abundance of the superselection sectors.
3 Models with infrared problem
In this section we recall basic facts about the perturbative construction of QED in the BRST framework and give the definition of the scalar model.
13 3.1 Scalar model The scalar model is a model of QFT which we use to study the IR problem in per- turbation theory. As we will see its IR structure is quite similar to that of QED. Its classical action is given by
Z 1 1 S[ϕ, ψ] = d4x (∂ ψ(x))(∂µψ(x)) − m2ψ2(x) 2 µ 2 1 e + (∂ ϕ(x))(∂µϕ(x)) + ψ2(x)ϕ(x) , 2 µ 2 where the fields ϕ(x) and ψ(x) are real and scalar and e is the coupling constant. The field ϕ(x) is massless and the field ψ(x) has mass m. The Euler-Lagrange equations have the following form e ( + m2)ψ(x) = eψ(x)ϕ(x), ϕ(x) = ψ2(x). 2 The perturbative quantization of the model is straightforward. The Hilbert space is the tensor product of the Fock spaces describing the massless particles and the particles of mass m, H = Γs(h0) ⊗ Γs(hm), where Γs/a(h) denotes the symmetric/antisymmetric Fock space built over the one- particle Hilbert space h. The Hilbert spaces h0 and hm are, respectively, the com- pletions of Cc∞(H0) and Cc∞(Hm) with respect to the topologies given by the scalar products Z Z (f1|f2) = dµ0(k) f1(k)f2(k), (f1|f2) = dµm(p) f1(p)f2(p).
By analogy to QED, the massless particles are called the photons and the massive particles – the electrons. The creation and annihilation operators of the photons are denoted by a∗(k), a(k), whereas the creation and annihilation operators of the electrons are denoted by b∗(p), b(p). The free quantized fields are given by Z Z ik x ip x ϕ(x) = dµ0(k) a∗(k)e · + h.c. , ψ(x) = dµm(p) b∗(p)e · + h.c. . (3.1)
1 2 The interaction vertex of the model coincides with L = 2 ψ ϕ = Jϕ, where, by analogy 1 2 to QED, the scalar quantity J = 2 ψ is called the current. Note that, as in QED, the interaction vertex contains two massive fields and one massless. For future reference, let us also define the operator 1 ρ(p) := b∗(p)b(p). (3.2) 2m Because the canonical dimension of the interaction vertex is equal to three the scalar model is super-renormalizable according to the standard classification. However, as explained in [33, 31], the normalization condition (2.5) is to restrictive to construct the Wightman and Green functions in the scalar model if the dimension dim is the canonical dimension, i.e. dim(ϕ) = dim(ψ) = 1. For this reason, in the case of the scalar model we assign dimensions to the fields in a different way. In this paper we
14 assume that dim(ϕ) = 1, dim(ψ) = 3/2 and each derivative increases the dimension by one. With this choice of of the dimensions the scalar model is a renormalizable (not super-renormalizable) theory. Moreover, as showed in [33, 31], it is possible to construct the physical Wightman and Green functions in this model using the weak adiabatic limit. The standard unitary representation of the Poincar´egroup acting in H is denoted by U(a, Λ). The generators of the space-time translations U(a) are given by Z Z µ µ µ P = dµ0(k) k a∗(k)a(k) + dµm(p) p b∗(p)b(p).
3.2 Quantum electrodynamics The standard classical action of QED has the following form Z 1 S[A , ψ , ψ ] = d4x ψ(x)(iD/ − m)ψ(x) − F (x)F µν(x) , (3.3) µ a a 4 µν where Aµ(x) is the electromagnetic potential, Fµν(x) = ∂µAν(x) − ∂νAµ(x) is the electromagnetic field tensor, Dµ = ∂µ − ieAµ is the covariant derivative and ψa(x), ψa(x) are Dirac spinor fields. We assign to the fields their canonical dimensions, i.e dim(Aµ) = 1 and dim(ψa) = dim(ψa) = 3/2. The coupling constant, denoted by e, coincides with the elementary charge which is equal to minus the charge of electron. Because of the gauge freedom the Euler-Lagrange equations for the above action do not admit a well-posed initial value formulation. In order to deal with the this problem one uses the BRST method [4, 97,5] which is a generalization of the approach developed by Gupta and Bleuler [54,9]. To this end, one considers a modified theory with the action Z 4 S[Aµ, ψa, ψa, c, c¯] = d x ψ(x)(iD/ − m)ψ(x) 1 − (∂ A (x))(∂µAν(x)) + i(∂ c(x))(∂µc¯(x)) , (3.4) 2 µ ν µ where c andc ¯ are the ghost and and anti-ghost fields. The Euler-Lagrange equations for the above action are of the form
µ µ A (x) = −eψ(x)γ ψ(x), (iD/ − m)ψ(x) = 0, c(x) = 0, c¯(x) = 0. The above system of partial differential equations is normally-hyperbolic and has a well-posed initial value formulation. The quantization of the above model is described in detail e.g. in [86, 87]. The free fields are defined in the Krein-Hilbert-Fock space which is a tensor product of the photon, electron/positron and ghost/anti-ghost Fock spaces H = Γs(hph) ⊗ Γa(hel) ⊗ Γa(hgh), where hph, hel and hgh are the one-photon, one-electron/positron and one-ghost/anti- ghost Hilbert spaces, respectively. The positive-definite scalar products on hph and hgh are given by 3 Z Z X µ µ X hf|giph := dµ0(k) f (k)g (k), hf|gigh := dµ0(k) fa(k)ga(k), µ=0 a=1,2
15 respectively. The corresponding covariant Krein scalar products are defined as follows Z Z µ ν (f|g)ph := − dµ0(k) gµνf (k)g (k), (f|g)gh := dµ0(k) abfa(k)gb(k), where gµν is the Minkowski metric with the signature (+, −, −, −) and ab is the antisymmetric symbol with 12 = 1. The Fock space of photons contains all four po- larizations. The Fock space of electrons is completely standard. In particular, the positive-definite scalar product defined in this space is covariant. The hermitian con- jugation of an operator A defined using the positive-definite scalar product is denoted by A†. On the other hand, the Krein conjugation defined using the covariant Krein scalar product is denoted by A∗. The creation and annihilation operators of photons are denoted by aµ† (k), aµ(k), µ = 0, 1, 2, 3. It holds aµ† (k) = aµ∗ (k) for µ = 1, 2, 3 and a0†(k) = −a0∗(k). The creation and annihilation operators of ghost are denoted by ca(k), ca† (k), a = 1, 2. It holds ca† (k) = abcb∗(k). The above creation and annihilation operators satisfy the following commutation relations
µ ν [aµ(f ), aν∗(g )] = (f|g)ph, [ca(fa), cb∗(gb)]+ = (f|g)gh, where [·, ·]+ denotes the anti-commutator. The creation and annihilation operators of electrons/positrons are denoted by b∗(σ, p) = b†(σ, p), b(σ, p), d∗(σ, p) = d†(σ, p), d(σ, p), where σ = 1, 2 corresponds to the two spin states of electron/positron. The spin vectors are denoted by ua(σ, p), va(σ, p). The free fields are the following operator- valued Schwartz distributions Z ik x ik x Aµ(x) = dµ0(k) aµ∗ (k)e · + aµ(k)e− · ,
Z X ip x ip x ψa(x) = dµm(p) b∗(σ, p)ua(σ, p)e · + d(σ, p)va(σ, p)e− · , σ=1,2 Z X ip x ip x ψa(x) = dµm(p) d∗(σ, p)va(σ, p)e · + b(σ, p)ua(σ, p)e− · , σ=1,2 Z ik x ik x c(x) = dµ0(k) c1∗(k)e · + c1(k)e− · , Z ik x ik x c¯(x) = dµ0(k) c2∗(k)e · + c2(k)e− · . For future reference, let us also introduce the operator X ρ(p) := (b∗(σ, p)b(σ, p) − d∗(σ, p)d(σ, p)). (3.5) σ=1,2 R The electric charge operator Q = dµm(p) ρ(p) coincides with the difference between the number of electrons and positrons. Thus, the physical electric charge equals −eQ. µ µ µ µ The interaction vertex of QED is given by L = ψγ ψAµ = J Aµ, where J = ψγ ψ is the spinor current. The definition of the scattering matrix and the interacting fields with adiabatic cutoff in the model described by the action (3.4) do not pose any diffi- culties. However, it is not obvious how these objects are related to the corresponding
16 objects in QED which is supposed to be defined by the action (3.3). In order to address this problem one introduces the free BRST charge Z µ QBRST = dµ0(k) k (aµ∗ (k)c1(k) + aµ(k)c1∗(k)) which is well defined as an element of L(D0). The covariant inner product is semi- positive-definite on KerQBRST i.e. if Ψ ∈ KerQBRST, then (Ψ|Ψ) ≥ 0. Moreover, if Ψ ∈ KerQBRST ∩ KerQgh, where Qgh is the ghost number operator, then the condition (Ψ|Ψ) = 0 implies that Ψ ∈ RanQBRST. We conclude that the covariant inner product phys induces a positive-definite inner product on D0 , where
phys KerQBRST ∩ KerQgh D0 = . RanQBRST ∩ KerQgh
phys phys Elements of D0 are equivalence classes denoted by [Ψ] with Ψ ∈ D0. Let H be phys the Hilbert space obtained by the completion of the pre-Hilbert space D0 with the inner product induced from the covariant inner product (·|·). If an operator B ∈ L(D0) commutes with the free BRST and ghost charge, then it induces a unique operator phys [B] ∈ L(D0 ) defined by the equality [B][Ψ] := [BΨ].
Let εµ(1, k) and εµ(2, k) be real vectors of two arbitrarily chosen physical polarizations of photons. More specifically, we demand that εµ(1, k), εµ(2, k) and kµ are linearly independent and µ µ kµε (s, k) = 0, εµ(s, k)ε (s0, k) = −δss0 . The operators # µ # a (s, k) := ε (s, k)[aµ (k)], s = 1, 2, are the creation and annihilation operators of the physical free photons and satisfy the standard commutation relations. Abusing the notation we identify b∗(σ, p) ≡ [b∗(σ, p)] and d∗(σ, p) ≡ [d∗(σ, p)]. The polynomials in the operators a∗(s, k), b∗(σ, p), d∗(σ, p) smeared with Schwartz functions generate a dense subspace in the physical Hilbert space when acting on the vacuum [Ω]. The standard representation of the Poincar´egroup acting in the Krein-Hilbert-Fock space H is denoted by U(a, Λ). The transformations U(a, Λ) ∈ L(D0) commute with the free BRST charge. Since the representation U(a, Λ) is Krein-unitary it induces a unitary representation [U(a, Λ)] acting in the physical Hilbert space. The generators of the subgroup of space-time translations [U(a)] have the following form Z µ X µ [P ] = dµ0(k) k a∗(s, k)a(s, k) s=1,2 Z X µ + dµm(p) p (b∗(σ, p)b(σ, p) + d∗(σ, p)d(σ, p)). σ=1,2
4 Infrared problem in scattering theory
In this section we review the origin of the IR problem in the scattering of particles in models with long-range interactions.
17 4.1 Classical and quantum mechanics The IR problem in QED can be traced back to classical mechanics. Consider a classical particle of mass m moving in the repulsive Coulomb potential.3 Let ~x,~p ∈ R3 be its position and momentum. The Hamiltonian of the system under consideration is given by the following expression
|~p|2 e2 1 H := H + V (~x),H := ,V (~x) = . (4.1) fr fr 2m 4π |~x| We call the position of the particle as a function of time,
R 3 t 7→ ~x(t) ∈ R3, its trajectory. If the particle is moving in the short range potential V (~x) such that 1 δ |V (~x)| < const |~x|− − for some δ > 0, then its trajectory is asymptotic in the future and in the past to the unique trajectories of the free particles whose evolution is 3 governed by Hfr, i.e. for some ~xout/in,~vout/in ∈ R it holds
lim |~x(t) − ~xout/in − ~vout/int| = 0. t →±∞ One easily verifies that the velocity of the particle moving in the Coulomb potential acquires specific values at the future and past infinity. However, the asymptotic values are approached so slowly that the distance between the particle influenced by the Coulomb potential and a freely moving particle always diverges irrespective of the choice of the initial position and velocity of the free particle – the trajectory of the particle moving in the Coulomb potential cannot be asymptotic to a trajectory of any free particle. In fact, one shows that
lim ~x(t) − ~xout/in(t) = 0, t →±∞ where the asymptotic trajectories are of the form
2 e ~v / ( ) := + − out in log(| | ) (4.2) ~xout/in t ~xout/in ~vout/int 3 t /ζ 4πm |~vout/in|
3 for some ~xout/in,~vout/in ∈ R and ζ ∈ R+. A similar problem arises for the Coulomb scattering in quantum mechanics. The position and momentum of the particle, ~x and ~p, are now interpreted as operators defined in the Hilbert space H = L2(R3). The free and full evolution operators are by denoted by Ufr(t) := exp(−itHfr), U(t) := exp(−itH), respectively. The standard Møller and scattering operators in quantum mechanics, which are well-defined in the case of short-range potentials [85], are constructed in the following way
Ωout/in := s-lim U(−t)Ufr(t),S := Ω∗ Ωin. t out →±∞ In the case of quantum-mechanical particle moving in the Coulomb potential the IR problem manifests itself by the nonexistence of the above limits. The problem is related to the fact that the wave-function satisfying the Schr¨odingerequation with
3We consider only the case of the repulsive Coulomb potential in order to avoid minor complications caused by the presence of the bound states.
18 the Coulomb potential approaches at large distances the free-particle wave-function only up to a logarithmically divergent phase factor [88] called the Coulomb phase. The above-mentioned divergent phase factor depends only on the momentum of the particle and does not contribute to the the differential cross section for the Coulomb scattering which is finite.
4.2 Quantum field coupled to classical current In the case of QED the infinite Coulomb phase is not the only manifestation of the IR problem. Another aspect of the IR problem, which we describe in this section, is the infinite photon emission. In theories without massless particles the principle of conservation of energy guarantees that the number of particles emitted during scat- tering is always finite. In QED or the scalar model nothing prohibits a production of infinitely many low-energetic photons. In fact, one proves that the probability of an emission of a low-energetic photon in a non-trivial scattering of charged particles is proportional to the inverse of the energy of the emitted photon. Consequently, the expected number of emitted photons is generically infinite and the probability of the emission of any finite number of photons is equal to zero [10]. As a result, in the case of QED or the scalar model both the scattering matrix and the differential cross section are plagued by IR divergences. In order to illustrate the above-mentioned problem we consider a simple model describing the second-quantized electromagnetic field F µν(x) coupled to a classical conserved current J µ(x) of spatially compact support. The field F µν(x) satisfies the Maxwell equations
µν ν [µ αβ] ∂µF (x) = −eJ (x), ∂ F (x) = 0. (4.3)
µ We assume that the current J (x) has future and past asymptotes ρin/out ∈ C(Hm) defined by 2 3 µ m µ lim λ J (±λv) = v ρout/in(mv), (4.4) λ 2(2 )3 →∞ π where v ∈ H1 and the prefactor was introduced to comply with the notation used in the rest of the paper (note that the parameter m does not appear in the model considered in this section). The above assumption with non-trivial ρout/in is typically satisfied by physically relevant currents [44, 57] (see also Sec. 5.2.3). The unique solution of (4.3) satisfying no-incoming radiation condition is given by Z µν µν 4 ret [µ ν] Fret (x) = Ffr (x) − 2e d y D0 (x − y) ∂ J (y), (4.5)
µν where Ffr (x) is the standard free quantum field defined in the Fock space. Using µν the formulas from AppendixD we verify that the past LSZ limit of the field Fret (x) µν µν coincides with the free quantum field Fin (x) = Ffr (x) whereas the future limit gives Z µν µν 4 [µ ν] Fout(x) = Ffr (x) − 2e d y D0(x − y) ∂ J (y).
µν µν One shows that the asymptotic LZS fields Fin (x) and Fout(x) are unitarily related if and only if ρin ≡ ρout. Since the above condition is generically violated we conclude that the scattering operator does not exist. If the initial state of the electromagnetic
19 radiation is described by a vector in the standard Fock representation, then the final state is typically a vector is some non-Fock coherent representation which depends on both the incoming and outgoing asymptotes of the current. µν In the case of the electromagnetic field Fret the long-range tail is informally defined by the following limit with n being a unit spatial four-vector (for the precise definition of the long-range tail see AppendixC) Z 2 µν µν lim r Fret (x + rn) = −e dρin(p) f (p/m; n), (4.6) r →∞ Hm where xµvν − vµxν f µν(v; x) := (4.7) ((x · v)2 − x2)3/2 is the Coulomb field of a unit point-charge moving with velocity v and dρout/in(p) := ρout/in(p) dµm(p). The above field commutes with all local fields, and as a result is a classical observable determining the super-selection sector. The above statement remains true in QED [17] and its consequences are the superselection of the velocity of the electron and the fact that the charged states cannot be eigenstates of the mass operator (the so-called infraparticle problem).
4.3 Relativistic perturbative QFT In the case of perturbative QED or the scalar model already the first order correction to the standard scattering operator is ill-defined. In the second order even its matrix elements between regular states are divergent.
4.3.1 Problems in first-order of perturbation theory Let us investigate the first-order correction to the Bogoliubov S-matrix in the scalar model. All possible processes contributing in this order are listed in Sec. 6.1. In what follows we consider only the decay of an electron into a photon and an electron. Let p and k be the momenta of the outgoing electron and photon, respectively, and p0 be the momentum of the incoming electron. All of the momenta are assumed to be on-shell. The energy-momentum conservation p+k = p0 implies that p·k = 0. Thus, p = p0 and k = 0 which suggest that the amplitude of the decay should vanish. However, one has to bear in mind that the energy and momentum are conserved only in the adiabatic limit, if this limit exists. The amplitude for the decay is clearly non-vanishing if > 0. The wave function of the outgoing particles is given explicitly by Z Z [1] F(p, k) = i dµm(p0)(p, k|S (g)|p0) f(p0) = i dµm(p0)g ˜(p + k − p0) f(p0), where f ∈ S(R4) is the wave function of the incoming electron. The expression for the wave function of the outgoing state can be rewritten in the form
Z d4q F (p, k) = i g(q) θ(p0 + k0 − q0)δ(2p · (k − q) + (k − q)2) f(p + k − q). (2π)3
20 After performing the integral over q0 with the use of the Dirac delta we arrive at
i Z d3~q 1 ( ) = F p, k 3 (2π) 2Em(~p + ~k − ~q) −E (~p) − |~k| + E (~p + ~k − ~q) × g˜ m m , ~q f(p + k + q),
p 2 2 where Em(~p) = m + |~p| . Using the above equation we prove that the following limits exist Z Z 1 2 lim dµm(p)dµ0(k) h(p, k)F(p, k), lim dµm(p)dµ0(k) |F(p, k)| > 0, 0 0 & & where h ∈ S(Hm × H0) is arbitrary. Since S(Hm × H0) is dense in the Hilbert space 2 L (Hm × H0, dµm × dµ0) we obtain
w-lim F = 0, and lim F does not exist, 0 0 & & 2 where w-lim and lim stand for the weak and strong limits in L (Hm × H0, dµm × dµ0), respectively. The same result holds in the case of QED. The non-existence of the of the above adiabatic limits is a manifestation of the infinite photon emission described in Sec. 4.2. The adiabatic limit of the first order correction to the standard scattering matrix does not exist also in the massless ϕk theory with k ≤ 4. Consider the wave function of the unphysical process of the decay of the vacuum into m massless particles
[1] F(k1, . . . , kk) = (k1, . . . , kk|S (g)Ω) =g ˜(k1 + ... + kk). (4.8)
One easily checks that the L2 norm of the above wave-function does not converge to zero in the limit & 0 if k ≤ 4. Since F converges weakly to zero the adiabatic limit 2 4 4 F does not exist in L (H0× , µ0× ). Note that in the case of the scalar model in higher orders of perturbation theory there appear effective vertices of the above type with any k. However, for k ≤ 4 one can use the renormalization freedom to make these contributions IR-finite (e.g. the case k = 2 is considered in Sec. 6.2.2).
4.3.2 Problems in higher orders of perturbation theory To illustrate the IR problem at the second order of perturbation theory let us consider the scattering of two electrons. By analogy to QED we call this process the Møller scattering. We will show that even the matrix elements of this correction between regular states are ill-defined in the adiabatic limit. The relevant Feynman diagrams are depicted in Figure8: (A), (B) in Sec. 6.2.6. For simplicity let us assume that the wave function of the two incoming electrons f ∈ S(Hm × Hm) is real-valued and supported outside the set of coinciding momenta. Then the wave function of the two outgoing electrons is given by Z [2] F(p1, p2) := dµm(p10 )dµm(p20 )(p1, p2|S (g)|p10 , p20 ) f(p10 , p20 ) Z d4k 1 = −i dµ (p0 )dµ (p0 ) g˜ (p − k − p0 )˜g (p + k − p0 ) f(p0 , p0 ). m 1 m 2 (2π)4 2 2 1 1 k2 + i0 1 2
21 2 2 2 Using the above expression one shows that ReF converges in L (Hm× , dµm× ) both strongly and weakly but the limit depends on the choice of g. This implies that the adiabatic limit of ReF does not exist (there is no distinguished switching function g). 2 Moreover, the L (Hm × Hm) norm of ImF diverges like log ||. Note thatg ˜(q) con- 4 verges to (2π)δ(q) in S0(R ). However, after replacingg ˜ by its limit in the expression for ImF we obtain an ill-defined integral 1 Z d4k 1 δ((p + p ) · k)δ((p − p ) · k − k2) f(p + k, p − k). 8 (2π)2 k2 1 2 1 2 1 2 The above expression corresponds formally to the infinite Coulomb phase. The second- order correction to the M¨ollerscattering in QED has a very similar IR divergence. In both models there are other divergent contributions at the second order. The above results are unchanged if one uses a different IR regulator. In general, the matrix elements of the standard scattering operator between regular states are generically well-defined only if there are no contributing Feynman diagrams with in- ternal photon propagators joining the electron lines with momenta close to the mass shell p2 = m2 [102, 98, 99]. The modified scattering matrix introduced in Sec.5 is free from all of the above-mentioned IR problems.
4.3.3 Overview of various approaches Because of the infinite Coulomb phase and the infinite photon emission the standard scattering matrix and the standard differential cross sections are not well-defined in QED or the scalar model. In this section we give a brief overview of different ap- proaches that have been developed to solve or circumvent this problem. For a more detailed exposition, we refer the reader to [76, 95, 60, 96]. A thorough analysis of the IR problem in classical electrodynamics can be found in [57, 44]. To avoid the IR problem one usually restricts attention to the so-called infrared safe observables such as the inclusive cross section. The inclusive differential cross section with the threshold E > 0 is usually computed with the use of the following formula [99] (or various modifications thereof formally equivalent up to terms of order E)
∞ Z incl X 1 dσp ,p (E; p1, . . . , pn) := lim dσp ,p (k1, . . . , km, p1, . . . , pn), (4.9) 1 2 0 m! k0+...+k0 E 1 2 & m=0 1 m≤ where p , p and p , . . . , p are the four-momenta of the incoming and observed out- 1 2 1 n going particles (their energies are supposed to be large compared to the threshold E), k1, . . . , km are the four-momenta of the unobserved photons and dσ is the standard differential cross section in a theory with some IR regulator such as: a mass of the photon [102], an artificial lower bound on the photon energies [98, 99], d − 4 in the dimensional regularization [51, 90] or the adiabatic cutoff [36, 35]. The limit → 0 is supposed to exist if the threshold E is fixed and positive. The construction of the inclusive cross sections requires several idealizations. As we explain in Sec. 9.3, their finiteness is a very delicate issue. It relies heavily on the exact energy-momentum conservation and the assumption that one can prepare an incoming state with sharp energy-momentum content. Nevertheless, it is expected that the inclusive cross sec- tions are well defined in arbitrary order of perturbation theory. A reasoning indicating that this indeed may be the case was presented by Yiennie, Fratschi and Suura [102]
22 and later simplified by Weinberg [98, 99]. So far no rigorous proof of this statement was given. Since particle detectors always have a finite sensitivity soft photons with energy below some threshold may always escape undetected. Consequently, states with differ- ent content of soft photons are difficult to discriminate experimentally and have to be all taken into account as a possible final states. Thus, the inclusive differential cross sections correspond to quantities that are usually measured experimentally. Neverthe- less, they do not provide a fully satisfactory description of the scattering processes. First of all, the physical interpretation of the inclusive cross section is not clear since its definition as a sum of the standard cross sections is meaningless in a theory without the IR regularization. In Sec. 9.3 we propose a different construction which gives the same numerical predictions as the standard construction. It involves the scattering matrix constructed in the paper and makes sense in an unregulated theory. Let us also stress that the inclusive cross sections do not provide a complete information about the scattering. One can imagine [59, 91] a scattering process whose only outcome is a measurable shift of the trajectories of the charged particles. Such a shift corresponds to the change of the phase of the wave function and does not give any contribution to the cross section. Let us briefly describe some more satisfactory attempts to solve the IR problem in perturbative QED. In the case of models with only massive particles the matrix elements of the scattering operator can be easily expressed in terms of the Green functions with the use of the LSZ reduction formulas [72]. Note that the IR problem in the perturbative construction of the Wightman and Green functions is completely under control in most models with massless fields including QED or non-abelian Yang- Mills theories [7, 74, 12, 13, 93, 31]. Moreover, as shown in [16] the standard LSZ limit of the photon field is well-defined. However, this is not the case for the Dirac field which has a non-standard asymptotic behavior. In fact, the Fourier transforms of the perturbative corrections to the interacting Feynman propagator of the electron are logarithmically divergent on the mass shell. Furthermore, it is expected that the full Feynman propagator of the electron defined in a non-perturbative way (assuming that QED can be formulated non-perturbatively) would be less singular on the mass shell than the free Feynman propagator and, in particular, it would not have a pole there [69]. For the above reasons, the application of the standard LSZ formula leads to divergent expressions in perturbation theory whereas non-perturbatively it is expected to produce vanishing scattering matrix elements. The latter statement is consistent with the fact that the scattering of charged particles is always accompanied by the infinite photon emission. The way out is to modify the LSZ reduction formulas by taking into account the non-standard asymptotic behavior of the Dirac field. One can find in the literature a number of interesting proposals for the construction of the asymptotic Dirac field which creates and annihilates the physical electrons [103, 104, 1, 58,2,3, 78, 24]. However, the constructions usually involve non-local functionals of interacting fields and it is not clear how to give them a proper mathematical meaning in perturbative QFT. Another strategy is based on the observation that the electromagnetic radiation emitted by scattered charged particles can be always accommodated in the Hilbert space of some coherent representation of the electromagnetic field. As argued by Chung and Kibble [23, 68], one can define a scattering matrix elements between appropriately chosen coherent states depending on the asymptotic velocities of charged particles. The
23 drawback of this approach is an involved definition of the space of physical states and the fact that the infinite Coulomb phase, which enters the expression for the scattering matrix elements, if there are two or more charged particles in the incoming or outgoing states, has to be dropped by hand. In this paper, we follow the approach put forward by Kulish and Faddeev [70] and later investigated e.g. in [66, 80, 53, 67, 61] (see also [79]). The method proposed by Kulish and Faddeev is based on the Dollard strategy [28]. The basic object is the so- called modified scattering matrix which is constructed by comparing the full dynamics of the system with some non-trivial asymptotic dynamics. One of the advantages of this approach is the fact that it takes care of both the infinite Coulomb phase and the infinite photon emission. We describe this approach in more detail in Sec. 5.1 and give its mathematically rigorous reformulation in the case of perturbative QFT in Sec. 5.5. We have discussed above different strategies of solving the IR problem in perturba- tive QED. Many important results about the IR structure of QED were derived in the framework of axiomatic QFT. These results rely on the assumption that there exists a non-perturbative version of QED which satisfies a number of physically motivated conditions. Let us list the most important results obtained in this framework. As shown by Buchholz [17] states with non-zero electric charge cannot be eingenstates of the mass operator. Consequently, charged particles such as electrons and positrons are not elementary particles in the sense of the Wigner definition [101] – they cannot be identified with vectors in the Hilbert space of some irreducible unitary represen- tation of the universal cover of the Poincar´egroup. For the above reasons charged particles are usually called in the literature infraparticles [89]. Another known fact is the abundance of superselection sectors. Indeed, using the assumption of locality one proves that the tail of the electromagnetic field which decays in spatial directions like the inverse distance squared is a classical observable characterizing different possible superselection sectors of the theory. Consequently, the sectors in QED are not fully characterize by their total charge. In fact, for each given physically attainable value of the total charge there are uncountably many sectors. The characterization of the space of physical states in QED was given in [15, 18, 58]. A related result obtained in the axiomatic framework is the proof that the Lorentz transformation cannot be unitarily implemented in sectors with non-zero total charge [47, 48].
5 Construction of S-matrix and interacting fields
In this section we propose a mathematically rigorous construction of the modified scattering matrix in the scalar model and QED. We use the modified scattering the- ory [28, 70] and combine it with the method of the adiabatic switching of the interaction proposed by Bogoliubov [11].
5.1 Modified scattering theory The standard scattering theory is not applicable to most models with long range interactions. The source of the problem is the fact that even long before or after the scattering event the actual evolution of the system is not well approximated by the free evolution. The standard solution, which was originally proposed by Dollard [28], is to compare the evolution of the system with some non-trivial but simple asymptotic evolution.
24 In this section we describe the Dollard method [28] for the Coulomb scattering of a quantum-mechanical particle. We use the notation introduced in Sec. 4.1. The asymp- totic evolution for the model at hand is generated by the following time dependent Dollard Hamiltonian e2 1 H (t) := H + . D fr 4π |t~p/m| Note that the Dollard Hamiltonian is obtained by replacing the position ~x of the particle in the expression (4.1) for full Hamiltonian with t~v, where ~v = ~p/m is the velocity of the particle. The corresponding unitary evolution operator is given by
Z t2 UD(t2, t1) := exp −i dt HD(t) t1 since the Dollard Hamiltonians at different times commute. The modified Møller op- erators Ωout/in/mod and the modified scattering matrix Smod for the Coulomb scattering are obtained by comparing the full dynamics of the system U(t) := exp(−itH) with the above asymptotic dynamics
Ωout/in,mod(ζ) := s-lim U(−t)UD(t, ±ζ),Smod(ζ) := Ωout,mod(ζ)∗Ωin,mod(ζ), t →±∞ where ζ ∈ R+ is some fixed parameter which has to be positive in order to avoid a non-integrable singularity in time at zero of the Dollard Hamiltonian. The physical interpretation of the parameter ζ can be inferred from Eq. (4.2) describing the long time behavior of the classical particle moving in the Coulomb potential. Let us mention that the modified scattering matrix Smod(ζ) can be also defined by the weak limit of the expression in Eq. (1.1). The modified Møller operators and scattering matrix have the following properties
Ωout/in,mod(ζ)Ufr(t) = U(t)Ωout/in,mod(ζ),Smod(ζ)Ufr(t) = Ufr(t)Smod(ζ) and for ζ, ζ0 ∈ R+
Ωout/in,mod(ζ0) = Ωout/in,mod(ζ)Vout/in(ζ, ζ0),S(ζ) = Vout(ζ, ζ0)S(ζ0)Vin(ζ0, ζ), (5.1) where the unitary intertwining operators Vout/in(ζ, ζ0) are defined by
2 e log(ζ0/ζ) Vout/in(ζ, ζ0) := s-lim UD(ζ; 0, t)UD(ζ0; t, 0) = exp ∓i . t 4 | m| →±∞ π ~p/ The ζ dependence of the modified scattering matrix may seem slightly unsatisfac- tory. Note that different choices of this parameter correspond to different identifica- tions of the asymptotic states as vectors in the Hilbert space H = L2(R3). In order to make the ζ dependence of this identification explicit and construct the scattering ma- ˆ trix that is ζ independent we define the following spaces Hin/out of asymptotic outgoing and incoming states,
ˆ ˆ ˆ 2 3 ˆ ˆ Ψ ∈ Hin/out iff Ψ: R+ → L (R ) and Ψ(ζ) = Vout/in(ζ, ζ0)Ψ(ζ0). ˆ The scalar product in Hin/out is given by
(Ψˆ |Ψˆ 0) := (Ψ(ˆ ζ)|Ψˆ 0(ζ))
25 and is independent of the choice of ζ ∈ R+ by the unitarity of Vout/in(ζ, ζ0). In par- ˆ 2 3 ticular, the Hilbert spaces Hin/out are non-canonically isomorphic to L (R ). Using property (5.1) we define the unique Møller and scattering operators
ˆ ˆ 2 3 ˆ ˆ ˆ Ωout/in : Hin/out → L (R ), Ωout/inΨ := Ωmod,out/in(ζ)Ψ(ζ), ˆ ˆ ˆ ˆ ˆ ˆ S : Hin → Hout, (SΨ)(ζ) := Smod(ζ)Ψ(ζ).
ˆ 2 3 Note that all states Ψ(ζ) ∈ L (R ), ζ ∈ R+, correspond to the same interacting state ˆ ˆ 2 3 Ωout/inΨ ∈ L (R ). In Sec. 9.1 we present a similar reformulation of the construction of the modified scattering matrix in the case of the scalar model and QED. The Dollard method of constructing the modified Møller and scattering operators can be generalized (see e.g. [26]) to systems consisting of an arbitrary number of non-relativistic particles interacting via pair potentials which can be of the long-range type. The modified scattering theory for a single Dirac particle moving in the Coulomb potential was developed in [29]. The generalization to other long-range potentials was considered in [52, 25]. The main aim of the present paper is to formulate a mathematically rigorous generalization of the Dollard method which is applicable to models of perturbative QFT such as QED. The Dollard strategy has been applied to QED for the first time by Kulish and Faddeev [70]. Their approach has been revisited by many authors but has not been put on firm mathematical ground so far. One of the difficulties is the fact that in interacting models of relativistic perturbative QFT it is not possible to define the interacting part of the Hamiltonian of the system as operator on a dense domain in the Fock space4. Ignoring this problem Kulish and Faddeev suggested to construct the scattering matrix in QED with the use of the formula
Z t2 (formally) I Smod lim Texp ie dt H (t) = t D,int 1 0 t2→−∞+ → ∞ Z t2 Z 0 I I × Texp −ie dt Hint(t) Texp ie dt HD,int(t) , (5.2) t1 t1 where I I Hint(t) = Ufr(−t)HintUfr(t),HD,int(t) = Ufr(−t)HD,int(t)Ufr(t) are the interaction part of the total Hamiltonian in the interaction picture and the in- teraction part of the Dollard Hamiltonian in the interaction picture. The formula (5.2) is formally equivalent to (1.1). Kulish and Faddeev assumed that Z Z I (formally) 3 3 µ Hint(t) = d ~x L(t, ~x) = d ~x ψ(t, ~x)γ ψ(t, ~x)Aµ(t, ~x)
4This is essentially a consequence of the Haag theorem (see e.g. [94]), which implies the non- existence of the interaction picture in QFT with no IR cutoff. Note that in spacetimes of dimension two one can define the Hamiltonian with a spatial IR cutoff. If the dimension is greater than two, then even the IR regularized Hamiltonian is ill defined. Indeed, as a consequence of worse UV properties of the free field in dimensions greater than two the Wick polynomials have to be smeared in both space and time [65]. Recall that the well-defined Bogoliubov S-matrix with adiabatic cutoff given by Eq. (2.10) involves spacetime smearing of the time-ordered products and does not rely on the existence of the interaction Hamiltonian.
26 and used the following Dollard Hamiltonian
Z µ 0 ~p·~k I (formally) p i k t H (t) dµ (p)dµ (k) ρ(p) a∗ (k) e − p0 + h.c. , D,int = m 0 p0 µ where ρ(p) is defined by (3.5). In order to motivate the form of the Dollard Hamiltonian I they investigated the asymptotic behavior in time of Hint(t) using heuristic reasoning based on stationary phase arguments [70, 64]. One can make this reasoning precise I I but only when both Hint(t) and HD,int(t) are treated as forms on some domain in the Fock space, say D0 and not as densely defined operators. This is not sufficient to make sense of the formula (5.2). Other inconsistencies in the original Kulish and Faddeev proposal have been recently pointed out by Dybalski [40]. Our construction of the modified scattering operator in QED and the scalar model presented in Sec. 5.5 is based to some extent on the ideas of Kulish and Faddeev [70] and Morchio and Strocchi [77, 78]. The main challenge is the implementation of the Dollard method in a relativistic model with no UV cutoff. Another difficulty is related to the perturbative nature of QED. Our modified scattering matrix is given by the as as adiabatic limit of the expression (1.2). The factors Sout(η, g), S(g) and Sin (η, g) in that expression play the role of the respective factors in the expression under the limit 1 in Eq. (5.2). The factors R(η, g) and R(η, g)− , which have no counterpart in Eq. (5.2), are used to implement a coherent transformation to sectors with non-trivial long-range tail of the interacting electromagnetic field Aµ in the case of QED or the interacting massless field ϕ in the case of the scalar model. In this way we are able to treat on equal footing a large class of superselection sectors that arise naturally for example in the simple model introduced in Sec. 4.2. Moreover, in the case of QED the factors 1 R(η, g) and R(η, g)− in Eq. (5.2) are essential to ensure the BRST invariance of the as constructed scattering matrix. The factors Sout/in(η, g) are formally given by Eq. (1.3) µ µ with Lout/in(x) := Aµ(x)Jout/in(η; x), where the asymptotic currents Jout/in(η; x) are regularized versions5 of Z Z µ µ µ µ Jas(x) = dµm(p) jas(p; x) ρ(p), jas(mv; x) := v dτ δ (x − τv) . R µ Note that jas(mv; x) is a current of a point-particle moving with constant velocity v. as In order to see a relation between our Dollard modifiers Sout/in(g) and the first and the third factor under the limit in Eq. (5.2) note that formally Z (formally) 3 µ HD,int(t) = d ~xAµ(t, ~x) Jas(x).
Let us mention that the ideas of Kulish and Faddeev have been tested in several models of QED that can be defined non-perturbatively. We would like to point the construction of the one-electron states in the Nelson model [83] and non-relativistic QED [22, 21] and the recent proposal for the LSZ-type formula which in principle can be used to construct many-electron states [40]. Another interesting result has been recently obtained by Morchio and Strocchi who developed the modified scattering theory for a model of QED [78] using the strategy formulated in [77]. The model describes charged particles interacting with each other via a pair potential of the same long range behavior as the Coulomb potential and coupled to the quantized
5The regularized currents are smooth and depend on a profile η (see Def. 5.1)
27 electromagnetic field. The analysis of this model is simplified by the lack of the recoil of charged particles in the emission or absorption of photons. Despite this feature the model retains the key IR properties of QED. Let us stress that all of the above-mentioned models have some fixed UV-cutoff and are not covariant under Lorentz transformations. In contrast, the technique developed in this paper works for relativistic models and allows to solve both the IR and UV problem in these models.
5.2 Asymptotic currents µ In this section we define the currents Jout/in(η; x), which are used in the definition of as µ the Dollard modifiers Sout/in(η, g), and the current Jsector(η, x), which is used in the definition of the operator R(η, g) implementing a transformation to a sector with some non-trivial long range-tail of the massless field. In order to motivate our definitions of the asymptotic currents let us consider for a moment the model describing the quantized electromagnetic field coupled to an external classical current introduced in Sec. 4.2 (in this paragraph we use the notation introduced in that section). As follows from Eq. (4.6) the long-range tail µν of the solution Fret of the equations of motion of this model (4.3) with no-incoming radiation condition is correlated to the velocity of the incoming electrons (the velocity superselection problem). In order to avoid this correlation in our construction of the modified scattering matrix, we drop the no-incoming radiation condition. Instead, we choose a solution of the equations of motion (4.3) of the form µν µν free radiation field of Fret,mod = Fret − µ , Jin depending on ρin where ρout/in, defined by Eq. (4.4), are the future/past asymptotes of the external current J µ. Assuming that the total charge of the current J µ vanishes we choose µ the asymptotic current Jin (depending only on the past asymptote ρin of the external µ µν current J ) in such a way that the long-range tail of Fret,mod vanishes. The past µν µν µν and future LSZ limits of Fret,mod, denoted by Fin and Fout, are equivalent to free fields in some coherent, generically non-Fock, representations which depend only on the asymptotes ρin and ρout, respectively (note that the representation class of the µν outgoing LSZ limit of the original field Fret given by Eq. (4.5) depends on both ρin and ρout). The modified S-matrix intertwines the fields µν free radiation field of µν free radiation field of Fin + µ and Fout − µ . Jin depending on ρin Jout depending on ρout The generalization to sectors with non-trivial long-range tail of the electromagnetic field involves the use of the sector current J depending on the sector measure % in- troduced in Sec. 5.5. In the quantum theory the asymptotes ρin/out are replaced by µ the operator ρ defined by (3.5). We consider families of asymptotic currents Jout/in parameterized by profiles.
Definition 5.1. A profile is a real-valued Schwartz function η ∈ S(R4) which satisfies the normalization condition R d4x η(x) ≡ η˜(0) = 1. There is no natural distinguished choice of a profile. A profile plays in our con- struction of the modified scattering matrix in perturbative QFT a similar role to the
28 parameter ζ ∈ R+ that appears in the definition of the modified Møller operators in Sec. 5.1.
5.2.1 QED We first introduce the outgoing, incoming and asymptotic numerical currents Z µ µ jout/in(η, mv; x) := v dτ θ(±τ)η (x − τv) , R Z µ µ jas(η, mv; x) := jout(η, mv; x) + jin(η, mv; x) = v dτ η (x − τv) , R which depend on a profile η and a four-velocity v and are smooth. The Fourier trans- forms of the above currents have the following form
i vµ η˜(q) ˜jµ (η, mv; q) = ± , ˜jµ (η, mv; q) = vµ (2π)δ(v · q)˜η(q). out/in v · q ± i0 as The outgoing, incoming and asymptotic operator-valued currents are defined by Z µ µ Jout/in/as(η; x) = dρ(p) jout/in/as(η, p; x), Hm where dρ(p) := ρ(p) dµm(p) with ρ(p) defined by (3.5). Note that ∂ ∂ J µ (η; x) = 0, J µ (η; x) = ±η(x)Q. ∂xµ as ∂xµ out/in
5.2.2 Scalar model The functions and operators we introduce in this section are Lorentz scalars. Never- theless, by analogy to QED we call them currents. Let us first define the outgoing, incoming and asymptotic numerical currents Z jout/in(η, mv; x) := dτ θ(±τ)η (x − τv) , (5.3) R jas(η, mv; x) := jout(η, mv; x) + jin(η, mv; x).
The Fourier transforms of the above currents have the following form
iη ˜(q) ˜j (η, mv; q) = ± , ˜j (η, mv; q) = (2π)δ(v · q)˜η(q). out/in v · q ± i0 as The outgoing, incoming and asymptotic operator-valued currents are given by Z Jout/in/as(η; x) := dρ(p) jout/in/as(η, p; x), Hm where dρ(p) := ρ(p) dµm(p) with ρ(p) defined by (3.2).
29 5.2.3 Asymptotic behavior of currents In this section we show that the matrix elements of the free currents and the asymptotic currents have the same timelike asymptotic behavior. We consider in detail the case of the scalar model. The obtained results generalize to QED in a straightforward manner. Let us recall that by the analogy to QED we call the operator :J(x): = 1/2 :ψ2(x): the current in the scalar model. It plays a similar role to the spinor current :J µ(x): = :ψ(x)γµψ(x): in QED. Let Z |f) = dµm(p) f(p) b∗(p)Ω, f ∈ Cc∞(Hm), be a state with one electron and let v be a four-velocity. We first determine the asymptote of :J(x):
m4 Z 3 3 iλm(v1 v2) v lim λ (f| :J(±λv): |f) = lim λ dµ1(v1)dµ1(v2) f(mv1)f(mv2)e − · λ λ 2 →∞ →∞ m = |f(mv)|2. 4(2π)3 The last equality follows from the standard result about the asymptotic behavior of a solution of the Klein-Gordon equation (see e.g. Appendix 1 to Section XI.3 in [85]). Next, we find the asymptotes of the asymptotic currents. To this end, we shall use the following expression for the asymptotic numerical current Z jas(η, mu; λv) = dτ η (λ(v − (v · u)u) − τu) R Z = dτ η −τ(1 + |~u|2)1/2, −λ(1 + |~u|2)1/2~u − τ~u , R which is valid for any four-velocities v, u such that v = (1,~0). We observe that
m Z λ3 dµ (u) |f(mu)|2j (mu, λv) 2 1 as Z 3 m dτd ~u / 2 = f(m(1 + |~u|2/λ2)1 2, m~u/λ) 4(2π)3 (1 + |~u|2/λ2)1/2 × η −τ(1 + |~u|2/λ2)1/2, −(1 + |~u|2/λ2)1/2~u − τ~u/λ .
Hence, by the Lebesgue dominated convergence theorem it holds
3 3 lim λ (f|Jout/in(η; ±λv)|f) = lim λ (f|Jas(η; ±λv)|f) λ λ →∞ Z→∞ 3 m 2 = lim λ dµ1(u) |f(mu)| jas(η, mu, λv) λ 2 →∞ m Z m = |f(mv)|2 dτd3~uη(τ, ~u) = |f(mv)|2. 4(2π)3 4(2π)3 The last equality is a consequence of the normalization of the profile R d4x η(x) = 1. Concluding, for Ψ, Ψ0 ∈ D0 it holds
2 3 3 m lim λ (Ψ| :J(±λv): Ψ0) = lim λ (Ψ|Jout/in(η; ±λv)Ψ0) = (Ψ|ρ(mv)Ψ0). λ λ 2(2 )3 →∞ →∞ π 30 In the case of QED it holds
2 3 µ 3 µ m µ lim λ (Ψ| :J (±λv): Ψ0) = lim λ (Ψ|J (η; ±λv)Ψ0) = v (Ψ|ρ(mv)Ψ0). λ λ out/in 2(2 )3 →∞ →∞ π We comment on the asymptotic behavior of the interacting current in Sec.7.
5.3 Coulomb phase In this section we investigate the properties of the relativistic Coulomb phase which will be used in our definition of the modified scattering matrix in the scalar model and QED. In order to motivate its definition, stated below, let us consider a system of non-relativistic quantum-mechanical particles interacting via the Coulomb potential. As showed in [30], one can construct the modified scattering operator for this systems using the method of the adiabatic switching of the interaction6. For simplicity, let us assume that there are only two particles of the same mass m and denote by ~x1, ~x2 and ~p1, ~p2 their position and momenta operators. The modified scattering matrix can be obtained by the following adiabatic limit