Hamiltonian anomalies of bound states in QED
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ELEMENTARY PARTICLES AND FIELDS Theory
Hamiltonian Anomalies of Bound States in QED∗ V. I. Shilin1), 2) and V. N. Pervushin1) Received May 29, 2012
Abstract—The Bound State in QED is described in systematic way by means of nonlocal irreducible representations of the nonhomogeneous Poincare group and Dirac’s method of quantization. As an example of application of this method we calculate triangle diagram Para-Positronium → γγ. We show that the Hamiltonian approach to Bound State in QED leads to anomaly-type contribution to creation of pair of parapositronium by two photon. DOI: 10.1134/S1063778813090172
1. INTRODUCTION The bound states in gauge theories are usually considered in the framework of representations of the homogeneous Lorentz group in one of the Lorentzinvariant gauges [1]. In this paper, we suggest a systematic scheme of the bound state generalization of S-matrix elements, which is based on irreducible representations of the nonhomogeneous Poincare´ group in concordance with the first Quantum Electrodynamic (QED) quantization [2, 3] and first QED description of bound states [4, 5]. We obtain bound states by means of excluding time component of fourpotential [2, 3, 6] and Hubbard–Stratonovich transformation [7, 8], that is in agreement with general principles [9]. The time component is chosen in correspondence with the Markov–Yukawa constraint of irreducibility [10, 11]. The aim of the article is to research the experimental consequences of such Poincare´ group irreducible representations of QED (see also review [12]). As a test of this scheme we calculate process P → γγ, where P is parapositronium, that describes triangle diagram, with one real and the other virtual photons. The structure of the article is as follows. In Section 2 we exclude time component of four-potential from QED action. For derived action in Section 3 we make the Hubbard–Stratonovich transformation and positronium field appears. Then we make semiclassical quantization of the resulting system in Section 4. In Section 5 we calculate the triangle anomaly. And in Section 6 we calculate the contribution in process γγ → P P inspired by triangle anomaly. ∗
The text was submitted by the authors in English. Joint Institute for Nuclear Research, Dubna, Russia. 2) Moscow Institute of Physics and Technology (State University), Dolgoprudny, Russia. 1)
2. EXCLUDING TIME COMPONENT OF Aμ IN QED We start with usual QED Lagrangian: 1 LQED = − Fμν F μν 4 − Aμ j μ − ψ(iγ μ ∂μ − m)ψ.
(1)
Below we will work in frame of reference in which the bound state as a whole is at rest. The Lagrangian (1) contains nonphysical degree of freedom. This degree can be excluded by substitution of the solution of classical equation of motion in action. Variation action providing by (1) over A0 leads to one component of Maxwell equation: ΔA0 − ∂0 ∂k Ak = −j0 , where Δ = ∂1 ∂1 + ∂2 ∂2 + ∂3 ∂3 . If we take the gauge: (2) ∂k Ak = 0, then the previous equation simplifies: ΔA0 = −j0 . The solution of this equation is: 1 A0 = − j0 , Δ where 1 1 j (x, y, z) = − Δ 4π
(3)
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