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Mathematical Introduction to Quantum Electrodynamics

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Mathematical Introduction to Quantum Electrodynamics MATHEMATICAL INTRODUCTION TO QUANTUM ELECTRODYNAMICS JAN DEREZINSKI´ Slides of the minicourse at IHP 21-25.06.2010 Contents 3 0 Introduction 5 1 General scattering theory 29 2 Axioms of quantum field theory 65 3 Neutral scalar bosons 89 4 Massive photons 137 5 Massless photons 175 6 Linearly perturbed quadratic Hamiltonians 209 7 Charged scalar bosons 221 8 Dirac fermions 267 Chapter 0 Introduction 5 Quantum Electrodynamics (QED) is an extremely successful theory. It depends on very few adjustable parameters { the fine structure constant and the masses of charged particles { and it can be used to compute with great precision many physical quantities, explaining a wide range of phenomena. It has a rich and complex formal structure, even though it can be developed from a very small number of postulates without any arbitrariness, The agreement of the QED computations with the experimental data is astounding. Its most spectacular successes are the computation of the anomalous magnetic moment of the electron and of the energy levels of light atoms (the Lamb shift). Its range of applicability is however much wider: one can argue that a large part of modern physics can be derived from QED. QED has serious problems. It has been noticed already by F. Dyson that the perturbation expansion computed in QED is probably divergent. Even worse: as noted by L. D. Landau, if we try to sum up this series we encounter a singularity, the so-called Landau pole. The common wisdom says that QED does not exist as a consistent satisfactory nonperturbative theory. QED is an example of a quantum field theory (QFT). It is a relatively sophisticated example of a QFT because of the gauge invariance. It is natural to ask whether QED, or more generally, other QFT's, fit into the framework of usual quantum mechanics and of some common-sense postulates: whether they can be described on a Hilbert space, with a bounded from below Hamiltonian, satisfy the Poincar´e covariance, and the Einstein causality. In the literature two basic sets of such postulates can be found: the Wightman axioms and the Haag-Kastler axioms. They are viewed by many physicists as an irrelevant pedantry of narrow-minded mathematicians. In my opinion, this is a wrong assessment { in reality, most of these postulates express basic intuitions coming from quantum mechanics. One can view these axioms not as an unshakable foundation, but as a set guiding principles. Unfortunately, so far the only known nontrivial models of Wightman or Haag-Kastler axioms in 1+3 dimensions are free theories. Interacting quantum field theories, such as eg. the λ:φ4: theory, satisfy these axioms at best in the sense of formal power series. There is probably no hope that a model of interacting QFT will be rigorously and non-perturbatively constructed in a near future. In particular, it is commonly believed that it is impossible to construct the λ:φ4: theory satisfying the Wightman axioms (because of the Landau pole). The situation with QED is, to my understanding, even worse, since it does not even satisfy Wightman axioms on the formal level, because of the gauge invariance. According to the Wightman axioms, the full informations about a QFT can be encoded in the so-called Wightman functions. Actually, in usual physics textbooks, QFT is studied following a somewhat different philosophy. According to them, the objective of QFT is computation of various quantities related to the scattering operator (of course, as formal, probably divergent power series). These quantities are easier to describe in massive theories. In theories containing massless particles, the situation is messed-up by the so-called the infra-red problem. Basic quantities computed in QFT massive QFT QFT possessing massless particles N-point time-ordered N-point time-ordered Green's functions Green's functions (with a physical normalization); (without a physical normalization); scattering amplitudes; ||||- scattering cross-sections; inclusive scattering cross-sections; (complex) energy levels; (complex) energy levels. In standard modern textbooks, N-point time-ordered Green's functions are treated as the ba- sic objects of a given QFT. Other quatities, such as the energy levels and scattering amplitudes and cross-sections can be in principle derived from Green's functions. Scattering amplitudes (that is matrix elements of the scattering operator) do not, however, exist in the massless case (even as formal power series). Physically measurable are inclusive cross-sections and energy levels. In QED (both massive and massless), there is a problem with Green's functions, because they depend on the gauge (both in the massless and massive case). Neither the (inclusive) scattering amplitudes nor energy levels depend on the gauge. One can name 4 major issues in QED that need a special attention: 1. the ultra-violet problem; 2. the infra-red problem; 3. the gauge-invariance; 4. bound states of many particle systems. The ultra-violet problem is common to all kinds of interacting relativistic quantum field theories. It is the main reason why we seem to be limited to a perturbative treatment and why the renormalization is necessary. Historically, it posed the biggest obstacle in the development of QED, and was mastered only in the late 40's, with major contributions of Schwinger, Feynman, Tomonaga and Dyson. The infra-red problem is common to all QFT's possessing massless particles. In QED it is due to the presence of massless photons and is absent in the massive QED. It appears already in non-relativistic quantum mechanics { in scattering theory with Coulomb forces. These forces are long-range, which makes the usual definition of scattering operators impossible. Its another manifestation is the appearance of inequivalent representations of canonical commutation re- lations when we consider scattering of photons against a classical current that has a different direction in the past and in the future. Thus, even in these toy non-relativistic situations the usual scattering operator is ill-defined. Therefore, it is not surprising that (much bigger) problems are present in the full relativistic case. One can cope with the infra-red problem by approximating massless QED with the massive one and restricting computations only to inclusive cross-sections justified by an imperfect resolution of the measuring device. The expression gauge invariance has in the context of QED several meanings. 1. The most common, discussed in classical electrodynamics, is the fact that if we add to a 4-potential solving the Maxwell equation a total derivative, then it still solves the Maxwell equations. Of course, this no longer holds for the Proca equations { the massive gener- alization of the Maxwell equations. Therefore, it is often stressed that gauge-invariance implies that the photons are massless. 2. There exists another meaning of the gauge-invariance: we can multiply charged fields by a space-time dependent phase factor and compensate it by changing the external potentials. 1. and 2. go together in the full Lagrangian of QED, which is invariant with respect to these two gauge transformations applied simultaneously. 3. In QED one often uses the term \gauge invariance" in yet another meaning: we have the freedom of the choice of the (free) photon propagator. This meaning applies both to the massive and massless QED. Some of these propagators are distinguished. In particular, the Feynman propagator plays a special role in the renormalization in the ultraviolet regime. There are several distinct approaches to treat the problem of the gauge invariance QED, all of them equivalent what concerns the physical quantities. These approaches are quite different in the massive and massless case. Nevertheless, all physical quantities depend continuously on the mass, down to zero. Bound states energies are defined as the positions of singularities of Green's functions. They usually have a nonzero imaginary part. The understanding of bound states seems to be one of the major problems of QFT. It is in fact dificult to study bound states by purely perturbative methods, since the basic quantities that we compute involve elementary particles as asymptotic states. The same problem would appear if we wanted to compute the scattering matrix of an N-body Schr¨odingeroperator in a purely perturbative way. QED is a quantized theory that describes charged particles interacting with photons. Charged particles are either fermions described by the Dirac equation or bosons described by the Klein- Gordon equation. From the point of view of applications, the case of charged fermions is much more important, nevertheless, from the theoretical point of view, charged bosons can be considered on the same footing as charged fermions. Photons are massless vector bosons, described by the Maxwell equation. The zero mass of photons leads to the infra-red divergencies, which are difficult from the theoretical point of view. The usual way to cope with them is to assume that photons are massive, described by the Proca equation. Then we can treat the (physical) massless QED as the limit of the (non- physical) massive one. Therefore, it is useful to discuss in parallel the massless and massive QED. In full QED both photons and charged particles are quantized. It is however useful to consider the formalism, where only one kind of particles is quantized. In fact, the following three theories can be viewed as the main pillars of the full QED: 1. charged scalar bosons interacting with an external potential; 2. Dirac fermions interacting with an external potential; 3. (massless or massive) photons interacting with an external current. All the above three theories are (or at least can be) well understood in the non-perturbative sense. They have a well-dynamics on a Hilbert space, or at least in an appropriate C∗-algebra. They can be used to compute quite a number of physically relevant quantities and to explain tricky theoretical questions.
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