From Faddeev-Kulish to LSZ. Towards a non-perturbative description of colliding electrons. (English) Zbl 1375.81239
Summary: In a low energy approximation of the massless Yukawa theory (Nelson model) we derive a Faddeev-Kulish type formula for the scattering matrix of \(N\) electrons and reformulate it in LSZ terms. To this end, we perform a decomposition of the infrared finite Dollard modifier into clouds of real and virtual photons, whose infrared divergencies mutually cancel. We point out that in the original work of Faddeev and Kulish the clouds of real photons are omitted, and consequently their wave-operators are ill-defined on the Fock space of free electrons. To support our observations, we compare our final LSZ expression for \(N=1\) with a rigorous non-perturbative construction due to Pizzo. While our discussion contains some heuristic steps, they can be formulated as clear-cut mathematical conjectures.
MSC:
| 81V10 | Electromagnetic interaction; quantum electrodynamics |
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
| 81U20 | \(S\)-matrix theory, etc. in quantum theory |
| 81U10 | \(n\)-body potential quantum scattering theory |
| 30H20 | Bergman spaces and Fock spaces |
References:
| [1] | Abdesselam, A.; Hasler, D., Analyticity of the ground state energy for massless Nelson models, Commun. Math. Phys., 310, 511-536 (2012) · Zbl 1256.81076 |
| [2] | Alazzawi, S.; Dybalski, W., Compton scattering in the Buchholz-Roberts framework of relativistic QED, Lett. Math. Phys., 107, 81-106 (2017) · Zbl 1365.81078 |
| [3] | Araki, H., Mathematical Theory of Quantum Fields (1999), Oxford University Press · Zbl 0998.81501 |
| [4] | Bagan, E.; Lavelle, M.; McMullan, D., Charges from dressed matter: construction, Ann. Phys., 282, 471-502 (2000) · Zbl 0990.81144 |
| [5] | Buchholz, D., Gauss’ law and the infraparticle problem, Phys. Lett. B, 174, 331-334 (1986) |
| [6] | Buchholz, D.; Roberts, J. E., New light on infrared problems: sectors, statistics, symmetries and spectrum, Commun. Math. Phys., 330, 935-972 (2014) · Zbl 1296.81044 |
| [7] | Chen, T.; Fröhlich, J.; Pizzo, A., Infraparticle scattering states in non-relativistic QED: I. The Bloch-Nordsieck paradigm, Commun. Math. Phys., 294, 761-825 (2010) · Zbl 1208.81211 |
| [8] | Chen, T.; Fröhlich, J.; Pizzo, A., Infraparticle scattering states in non-relativistic QED: II. Mass shell properties, J. Math. Phys., 50, Article 012103 pp. (2009) · Zbl 1189.81240 |
| [9] | Dereziński, J.; Gérard, C., Scattering Theory of Classical and Quantum N-Particle Systems (1997), Springer · Zbl 0899.47007 |
| [10] | Dollard, J. D., Asymptotic convergence and the Coulomb interaction, J. Math. Phys., 5, 729-738 (1964) |
| [11] | Dybalski, W.; Pizzo, A., Coulomb scattering in the massless Nelson model I. Foundations of two-electron scattering, J. Stat. Phys., 154, 543-587 (2014) · Zbl 1301.81316 |
| [12] | Dybalski, W.; Pizzo, A., Coulomb scattering in the massless Nelson model II. Regularity of ground states, Preprint · Zbl 1301.81316 |
| [13] | Dybalski, W.; Pizzo, A., Coulomb scattering in the massless Nelson model III. Ground state wave-functions and non-commutative recurrence relations, Preprint · Zbl 1391.81208 |
| [14] | Faddeev, L. D.; Kulish, P. P., Asymptotic conditions and infrared divergencies in quantum electrodynamics, Theor. Math. Phys., 5, 153-166 (1970) · Zbl 0197.26201 |
| [15] | Ferrari, R.; Picasso, L. E.; Strocchi, F., Some remarks on local operators in Quantum Electrodynamics, Commun. Math. Phys., 35, 25-38 (1974) |
| [16] | Fröhlich, J., On the infrared problem in a model of scalar electrons and massless, scalar bosons, Ann. Inst. Henry Poincaré A (N.S.), 19, 1-103 (1973) · Zbl 1216.81151 |
| [17] | Fröhlich, J., Existence of dressed one electron states in a class of persistent models, Fortschr. Phys., 22, 158-198 (1974) |
| [18] | Fröhlich, J.; Morchio, G.; Strocchi, F., Charged sectors and scattering states in quantum electrodynamics, Ann. Phys., 119, 241-284 (1979) |
| [19] | Fröhlich, J.; Pizzo, A., Renormalized electron mass in non-relativistic QED, Commun. Math. Phys., 294, 439-470 (2010) · Zbl 1208.81212 |
| [20] | Gabai, B.; Sever, A., Large gauge symmetries and asymptotic states in QED, J. High Energy Phys., 12, Article 095 pp. (2016) · Zbl 1390.83141 |
| [21] | Gomez, C.; Letschka, R., Memory of the infrared, Preprint · Zbl 1383.83056 |
| [22] | Gomez, C.; Panchenko, M., Asymptotic dynamics, large gauge transformations and infrared symmetries, Preprint |
| [23] | Haag, R., Quantum field theories with composite particles and asymptotic conditions, Phys. Rev., 112, 669-675 (1958) · Zbl 0085.43602 |
| [24] | Hasler, D.; Herbst, I., Absence of ground states for a class of translation invariant models of non-relativistic QED, Commun. Math. Phys., 279, 769-787 (2008) · Zbl 1143.81022 |
| [25] | Kapec, D.; Perry, M.; Raclariu, A.-M.; Strominger, A., Infrared divergencies in QED, revisited, Preprint |
| [26] | Kibble, T. W.B., Coherent soft-photon states and infrared divergences. IV. The scattering operator, Phys. Rev., 175, 1624-1640 (1968) |
| [27] | Lehmann, H.; Symanzik, K.; Zimmermann, W., Zur Formulierung quantisierter Feldtheorien, Nuovo Cimento, 1, 205-225 (1955) · Zbl 0066.44006 |
| [28] | Mirbabayi, M.; Porrati, M., Shaving off black hole soft hair, Preprint |
| [29] | Morchio, G.; Strocchi, S., The infrared problem in QED: a lesson from a model with Coulomb interaction and realistic photon emission, Ann. Henri Poincaré, 17, 2699-2739 (2016) · Zbl 1355.81163 |
| [30] | Panchenko, M., The infrared triangle in the context of IR safe S matrices, Preprint |
| [31] | Pizzo, A., Scattering of an infraparticle: the one particle sector in Nelson’s massless models, Ann. Henri Poincaré, 4, 553-606 (2005) · Zbl 1072.81057 |
| [32] | Pizzo, A., One-particle (improper) states in Nelson’s massless model, Ann. Henri Poincaré, 4, 439-486 (2003) · Zbl 1057.81024 |
| [33] | Polley, L., Modelle zum Infrarotproblem in der Quantenelektrodynamik (1980), University of Würzburg, PhD thesis |
| [34] | Ruelle, D., On the asymptotic condition in quantum field theory, Helv. Phys. Acta, 35, 147-163 (1962) · Zbl 0158.45702 |
| [35] | Strominger, A., Lectures on the infrared structure of gravity and gauge theory, Preprint · Zbl 1408.83003 |
| [36] | Schroer, B., Infrateilchen in der Quantenfeldtheorie, Fortschr. Phys., 11, 1-31 (1963) |
| [37] | Steinmann, O., Perturbative Quantum Electrodynamics and Axiomatic Field Theory (2000), Springer · Zbl 0946.81079 |
| [38] | Yennie, D. R.; Frautschi, S. C.; Suura, H., The infrared divergence phenomena and high-energy processes, Ann. Phys., 13, 379-452 (1961) |
| [39] | Zwanziger, D., Scattering theory for quantum electrodynamics. II. Reduction and cross-section formulas, Phys. Rev. D, 11, 3504-3530 (1975) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
