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Some Applications of Coherent States in Quantum Electrodynamics Dennis Leland Nordstrom Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1970 Some applications of coherent states in quantum electrodynamics Dennis Leland Nordstrom Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Elementary Particles and Fields and String Theory Commons Recommended Citation Nordstrom, Dennis Leland, "Some applications of coherent states in quantum electrodynamics " (1970). Retrospective Theses and Dissertations. 4188. https://lib.dr.iastate.edu/rtd/4188 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. 70-18,897 NORDSTROM, Dennis Leland, 1941^ SOME APPLICATIONS OF COHERENT STATES IN QUANTUM ELECTRODYNAMICS. Iowa State University, Ph.D., 1970 Physics, elementary particles University Microfilms, A XEROX Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED SOME APPLICATIONS OF COHERENT STATES IN QUANTUM ELECTRODYNAMICS by Dennis Lei and Nordstrom A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requi rements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: High Energy Physics Approved: Signature was redacted for privacy. In Charge of Major Woi Signature was redacted for privacy. lepartment Signature was redacted for privacy. Dè^ ofGratiufffe Ç^lege Iowa State University Of Science and Technology Ames, Iowa 1970 i i TABLE OF CONTENTS Page 1. INTRODUCTION 1 II. CONVENTIONAL TREATMENT OF INFRARED DIVERGENCE PROBLEM 8 III. COHERENT STATES 20 IV. COHERENT STATE TREATMENT OF INFRARED DIVERGENCE PROBLEM 34 V. WAVE PACKET TREATMENT OF INFRARED DIVERGENCE PROBLEM 48 VI. SUMMARY 65 VII. BIBLIOGRAPHY 72 VIII. ACKNOWLEDGMENTS 76 IX. APPENDIX 77 1 I. INTRODUCTION Quantum electrodynamics has proven to be a highly successful theory of interacting fields. Even though the theory has been beset with diver­ gence difficulties, predictions from the theory have been found to be in excellent agreement with experiment. These divergences have been satis­ factorily resolved by various means. One such means is to remove the infinite zero-point quantities, e.g., infinite zero-point energy, by slightly modifying the formal theory by requiring that all operators re­ presenting physical observables be normal-ordered. Another type of diver­ gence, the ultraviolet divergence, is handled by a renormalization pro­ cedure which consists of isolating the divergence from finite physical quantities and redefining physical constants in the theory, for example, the electron mass and charge. A third type of divergence, the infrared divergence, has also been shown to cause no serious difficulties; yet its removal lacks a completely satisfactory physical justification and is therefore the basis for the discussion presented here. In the following treatment it will be assumed that all physical quantities have been properly renormalized so that the usual Feynman-Dyson perturbation techniques may be employed. As an introduction, first consider the infrared divergence for real photons from a classical viewpoint. The semi-classical predictions of the infrared divergence phenomena may be understood by considering the de­ flection of a moving electron due to its interaction with a potential. The field of the electron is altered by the collision and the resulting change appears as emitted electromagnetic radiation. For wavelengths sufficiently 2 long în comparison to the dimensions of the scattering region and for negligible collision time the radiation will depend only on the initial and final momenta of the electron and the direction in which the radiation is observed. A well-known result (1, Sec. 69) is that in this limit the energy emitted per unit frequency is independent of frequency. In terms of photons the number of photons emitted per unit frequency range is inversely proportional to the frequency so that the photon spectrum is of the form dk which diverges as k -» Oi k Semi-classical arguments also predict the angular distribution of the radiation. In the high energy limit the electron's field becomes contracted along the direction of motion. This leads to the direction of emission being confined to a narrow cone along either the initial or final direction of motion. This behavior is evident in the classical amplitude for emission of radiation which is proportional to P'' e _ P • g e (1) p' • k p • k where p and p' are the initial and final momenta, respectively, e is the polarization vector for the emitted radiation whose wave number is k and e is the charge of the electron. The covariant notation used throughout this investigation is discussed in the appendix. The probability for emission of a photon of momentum k in the scattering process is then pro­ portional to « 2 P • G e p • k k (2) 3 This expression exhibits the infrared behavior and the strong tendency for emission of radiation parallel to either the initial or final momentum of the electron. The same type of infrared behavior occurs in a complete quantum mechanical treatment of the emission of a low frequency real photon due to the scattering of an electron from a potential. A calculation of the matrix element for the emission process (2, p. Î23) yields a result pro­ portional to expression 1 times the basic matrix element for the scattering process with no emission. Thi-s result then displays the same characteris­ tics as are found in the classical treatment of the emission of radiation, in particular, that the infrared divergence is associated with emission of soft photons from external charge lines only. A similar infrared behavior exists for the emission and reabsorption of soft virtual photons. In addition to the external line dependence the emission and absorption of low frequency photons does not cause a significant disturbance in the electron's motion. Therefore, the soft photons should be emitted and absorbed independently thus possessing a Poisson distribution. It is this similarity that is used to eliminate the infrared divergence problem when lowest order radiative corrections to a given process are made in the perturbative framework. Because of the zero mass of the photon and the limitations of the experimental apparatus there are always photons of sufficiently low energy emitted in a scattering process which escape de­ tection. The calculated cross section for emission of an undetected pho­ ton then contains an infrared divergence which exactly cancels the infra­ red divergence due to the soft virtual photon corrections. 4 This then îs the basic idea behind the resolution of the Infrared di­ vergence problem In the conventional treatment originally given by Bloch and Nordsleck (3)» Since emitted photons of arbitrarily small energy may escape detection, the final state in any scattering experiment involving charged particles can never be completely known. By making several approxi­ mations, such as the neglect of pair effects and electron recoil effects, Bloch and Nordsleck showed that the cross section for emission of a finite number of photons in a scattering process is zero. This, of course, is due to the soft virtual photons. However, they were able to show that the probability to emit any number of soft photons suns to a non-zero result. They further showed that this result was very nearly equal to that obtained by Ignoring all radiative corrections. This is due to the cancellation of the real and virtual infrared divergences as mentioned in the preceding paragraph. After Bloch and Nordsleck solved the infrared divergence problem using several semi-classical approximations, a number of attempts were made to solve the problem employing a fully quantum field-theoretic treat­ ment. Although several examples of the cancellation of real and virtual photons to lowest order may be found In the literature, two excellent works, among others, by Jauch and Rohrllch (4 and 5, Ch. 16) and Yennle, FrautschI, and Suura (6) present general treatments of the infrared di­ vergence problem. These general treatments show how an additional soft real or virtual photon contribution may be isolated as a factor multi­ plying the basic matrix element for an arbitrary process. They then demonstrate how the cancellations are made In the cross sections. 5 The conventional treatment of the Infrared divergence problem which consists of summing the cross sections over all possible final states prompted Chung (7) to conjecture that the divergence is quite probably due to the use of incorrect initial and final states because of the undetermined number of soft phptons emitted. He then proposed letting these states be coherent states parametrized in such a way as to eliminate the infrared di­ vergence in the matrix element for some arbitrary process. The coherent states are well-known from quantum optics and are defined as the eigenstates of the photon annihilation operator. The coherent states were found to be quite appropriate since they allow for the presence of an unlimited number of photons. Storrow (8) also used the coherent states to demonstrate the possibility of formulating an S-matrix theory free of infrared divergences so that processes involving mixing of strong and electromagnetic inter­ actions could be treated. The
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