Spin-3/2 Field Theories and The Gauge Invariance
A Thesis by
Haryanto Mangaratua Siahaan 20206014
In Partial Fulfillment of the Requirements for the Degree Magister Sains
Physics Study Program Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung February 2008
Spin-3/2 Field Theories and The Gauge Invariance
Approved by:
Triyanta, Ph.D., Adviser
Date Approved
To Putri, David, and Victor.
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”Djoudjou ma tu Ahu, asa Hu alusi ho, asa Hupabotohon tu ho hata angka na bolon dohot na songkal, angka na so binotomi”. (Berserulah kepada-Ku, maka Aku akan menjawab engkau dan akan memberitahukan kepadamu hal-hal yang besar dan yang tidak terpahami, yakni hal-hal yang tidak kau ketahui) Panurirang Djeremia (Yeremia) 32 : 3 ”We all have dreams. In order to make dreams come into reality, it takes an awful lot of determination, dedication, self-discipline, and effort”. James Cleveland Owens (1913-1980) ”... Ai tung so boi pe au inang da, marmido marjam tangan Tarsongon dongan-dongan hi da, marsedan marberlian Asal ma sahat gelleng hi da, sai sahat tu tujuan Anakhon hi... do hamoraon di au ...” Nahum Situmorang
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ABSTRACT
Spin-3/2 particle theory is still an unclear subject since its first formulation by W. Rarita and J. Schwinger in 1941. The problems are about extra degree of freedom for describing spin-3/2 particles and the lack of relativistic property of wave solutions when interaction with external (electromagnetic) field is introduced (known as VeloZwanziger problem). In this thesis we consider Rarita-Schwinger (RS) formalism as a standard description of spin-3/2 particle which widely used for the delta (1232) baryon resonance. Also, we discuss a new approach by Napsuciale-Kirchbach (NK) based on the squared Pauli-Lubanski (PL) eigen equation to describe the spin-3/2 field. Then we do a test of gauge invariance of the theories by performing gauge transformation of polarization vector in the corresponding Compton scattering amplitude. We find that both theories are gauge invariant in their full propagators and vertices. Since NK formalism is free from Velo-Zwanziger problem, it seems that this theory could be a better description of spin-3/2 particle after some experimental verifications. Keyword : Spin-3/2 theory, Rarita-Schwinger formalism, Napsuciale-Kirchbach formalism, Compton scattering amplitude, gauge invariance.
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ABSTRAK
Teori untuk partikel berspin-3/2 merupakan sebuah subjek yang belum begitu jelas semenjak formulasi awalnya oleh W. Rarita dan J. Schwinger pada tahun 1941. Permasalahan-permasalahan yang terkait yaitu mengenai derajat kebebasan ekstra dalam penggambaran partikel-partikel berspin-3/2 serta pelanggaran terhadap sifat relativistik dari solusi gelombang ketika suku interaksi dengan medan (elektromagnetik) eksternal dimasukkan (dikenal sebagai masalah Velo-Zwanziger (VZ)). Dalam thesis ini kita menggunakan formalisme Rarita-Schwinger (RS) sebagai sebuah penjelasan standar dari partikel berspin-3/2 yang telah digunakan secara luas untuk resonansi baryon delta (1232). Juga, kita membahas sebuah pendekatan baru oleh Napsuciale dan Kirchbach (NK) berdasarkan persamaan eigen dari kuadrat operator Pauli-Lubanski (PL) untuk menggambarkan pertikel berspin-3/2. Kemudian kita melakukan sebuah tes keinvarianan gauge dari teori-teori ini melalui transformasi vektor polarisasi foton di dalam amplitudo hamburan Compton yang terkait. Kita menemukan bahwa kedua teori ini invarian gauge dalam propagator dan verteks penuhnya. Karena formalsime NK bebas dari masalah VZ, maka sepertinya teori ini akan dapat menjadi pejelasan yang lebih baik dari partikel berspin-3/2 setelah beberapa verifikasi secara eksperimen. Kata kunci : Teori spin-3/2, formalisme Rarita-Schwinger, formalisme NapsucialeKirchbach, amplitudo hamburan Compton, keinvarianan gauge.
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ACKNOWLEDGEMENTS
I am very grateful to my advisor, Triyanta, Ph.D. , for his supervision and teaching me a great deal of physics (especially from his lectures on Classical Mechanics, Quantum Mechanics, and Quantum Field Theory). For answering a lot of questions, while giving me a freedom to explore my own ideas. It has been a pleasure and an honor to be his student. I am also grateful to F. P. Zen, D.Sc. and Dr. rer. nat. Bobby E. Gunara for sharing me their knowledge in theoretical physics. I thank to Jusak S. Kosasih, Ph.D. and Alexander A. Iskandar, Ph.D. for their willingness to referee this thesis, and also for their comments. I am also grateful to Terry Pilling, Ph.D. (North Dakota State University) and Prof. M. Napsuciale (Univ. de Guanajuato) for very interesting discussions about spin-3/2 field theory. I would like to thank my friends Algemen, Sigit, Rinto Sinurat, Cin Pau, Benz P.J., Sony Pro, Arma (for providing me some papers from PROLA), and others. Also to my colleagues (Mr. Guntur, Mr. Suwandi, Mr. Sony, Mr. J. Silalahi, Mr. Gunarto, Ms. Tyas, and others) and students (I am sorry for my merely fast speaking in front of the class) at Batununggal St. Aloysius Junior High School for creating a very good working environment. Thanks to our librarian, Ms. Silvi, for her assistance and moral support. Also to Mr. Daryat for his helpful administration assistances. I also owe all the people in the GII Hok Im Tong night prayer group special thanks. I am thankful to my family (especially my parents), whose constant support was invaluable. Above all, I thank God for making all the things possible.
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TABLE OF CONTENTS DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ABSTRAK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . .
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I
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
II
RARITA-SCHWINGER FORMALISM . . . . . . . . . . . . . . . .
3
2.1
Free Rarita-Schwinger Field . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Lagrangian and Propagator . . . . . . . . . . . . . . . . . . . . . . .
7
2.3
Interaction with External Field . . . . . . . . . . . . . . . . . . . . .
8
2.4
Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.5
Velo-Zwanziger Problem . . . . . . . . . . . . . . . . . . . . . . . . .
12
III NAPSUCIALE-KIRCHBACH FORMALISM . . . . . . . . . . . .
15
3.1
Incompatibility of Rarita-Schwinger Formalism with Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.2
Poincare Symmetry and Pauli Lubanski Operator . . . . . . . . . .
20
3.3
Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.4
Free Lagrangian, Noether Current, and Propagator . . . . . . . . . .
28
3.5
Interacting spin-3/2 field . . . . . . . . . . . . . . . . . . . . . . . .
29
3.6
Propagation and The Gyromagnetic Factor . . . . . . . . . . . . . .
34
3.7
Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
IV TEST OF THE GAUGE INVARIANCE . . . . . . . . . . . . . . .
39
4.1
The Importance of Gauge Invariance . . . . . . . . . . . . . . . . . .
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4.2
Compton Scattering Amplitude and Gauge Invariance in The R-S Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
Compton Scattering Amplitude and Gauge Invariance in The N-K Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.3
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V
CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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APPENDIX A — COURANT-HILBERT CRITERION FOR HYPERBOLICITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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