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: ^ • 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), ^ iP^ a • a strongly-continuous representation of the translation group U(a) = e · acting ^ ^µ in Hin=out(%), with generators P satisfying the relativistic spectrum condition, ^ ^ ^ • the translationally-invariant scattering matrix S : Hin(%) ! Hout(%) (the inclu- sive cross section constructed with the use of S^ formally coincides with the one obtained using the standard procedure), ^ • the translationally-covariant retarded and advanced interacting fields Cret=adv(x): ^ ^ 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 F^µν (x + rn) = −e d%(p) f µν(p=m; n); r ret=adv !1 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 ^ 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 ^ constraint in the scalar model). The spaces Hin=out(0) are called the vacuum sector. They contain a unique vector Ω^ such that P^µΩ^ = 0, called the vacuum. The photon creation and annihilation operators are denoted by2 # ^ ^ a^ (k): Hin=out(%) ! Hin=out(%): 1In the paper, by the electric charge we always mean the difference between the number of electrons and positrons.