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Wigner Rotation in Dirac Fields Hor Wei Hann View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ScholarBank@NUS WIGNER ROTATION IN DIRAC FIELDS HOR WEI HANN (B. Sc. (Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2006 Acknowledgements This project has been going on for several years and I have had the fortune to work on it with the help of following people. First of all, I am grateful to my supervisors Dr. Kuldip Singh and Prof. Oh Choo Hiap for o®ering this project. Dr. Kuldip has given a lot of advice and ¯rm directions on this project, and I would not have been able to complete it without his guidance. It was my pleasure to be able to work with someone like him who is extremely knowledgeable but yet friendly with his students. Secondly, I wish to thank some of my friends Mr. Lim Kim Yong, Mr. Liang Yeong Cherng and Mr. Tey Meng Khoon for their assistance towards the completion of this thesis. I learned a great deal about MATHEMATICATM, WinEdtTM and several other useful programs through these people. I feel indebted to my family for sticking through with me and providing the necessary advice and guidance all the time. I have always found the strength to accomplish whichever goals through their encouragements. Last but not least, I would like to mention a very special friend of mine, Ms. Emma Ooi Chi Jin, who has been a gem. I am appreciative of her companionship and the endless supply of warm-hearted yet cheeky comments about this project. The satisfaction that I derived from the completion of this project is as much hers as mine. i Contents Acknowledgements i Summary iv List of Symbols vii 1 Basic Quantum Field Theory 1 1.1 Introduction .................................... 1 1.1.1 Lorentz and Poincar¶eGroups ....................... 2 1.1.2 Multiple-Particle SchroÄedingerEquation ................. 5 1.2 Classical Field Theory ............................... 6 1.2.1 Conservation Laws in Classical Fields .................. 8 1.2.2 Field Quantization for Dirac Field .................... 12 2 Representations of Poincar¶eGroup 22 2.1 Conventional Approach .............................. 22 2.1.1 Evaluating the Wigner Matrix ....................... 26 2.2 The Action of the Wigner Transformation on the Dirac Spinors ......... 31 ii 2.2.1 Comparing Wigner Transformation and its E®ect on Dirac Spinors ... 35 3 Spin Measurements in Dirac Fields 39 3.1 Spin Projection for Single Particle with Arbitrary Momentum .......... 39 3.1.1 The E®ect of Lorentz Transformation on the Dirac Spin Operator .... 45 3.2 Single Particle State with Fixed Momentum ................... 47 3.2.1 Bell Correlations for Spin Singlet States ................. 51 4 Spin in Poincar¶eGroup 54 4.1 De¯ning Spin Observables in Relativistic Settings ................ 54 4.1.1 Spin Projection for Dirac Spinors ..................... 54 4.1.2 Lie Algebra of the Poincar¶eGroup .................... 58 4.1.3 Dirac Spin Operator and Pauli-Lubanski Vector ............. 61 4.2 Unitary Transformations on the Dirac Spinors .................. 62 5 Conclusion 68 iii Summary Spin as de¯ned in non-relativistic quantum mechanics is the two-dimensional irreducible representation of angular momentum of a quantum state. The Einstein, Podolsky and Rosen (EPR) correlations [1] and Bell-CHSH inequality [2, 3] are important in the ¯elds of quantum information and entanglement, in which the spins or helicities of the quantum systems are manipulated to produce the necessary entangled states. However, a Minkowskian space-time picture is required to describe a free relativistic particle with spin. E. Wigner in his seminal 1939 paper [4] explored the irreducible unitary representations of the Poincar¶egroup, which would describe all possible physical transformations of a quantum state. Thus, the spin generator can be realized as a linear combination of the Poincar¶egenerators, as discussed in Bogolubov [5] and Terno [6], and it is named as the Pauli-Lubanski vector. Other researchers like Czachor [7, 8], Doyeol [9], Terashima [10] have utilized di®erent forms of the Pauli-Lubanski vector. However, the spin observable as de¯ned by its generator, has to satisfy certain commutation relations in which the above-mentioned papers have not addressed. It is important to note that a judicial choice for the spin observable, constructed from the Poincar¶egenerators, is to apply an appropriate Lorentz boost to the Pauli-Lubanski vector as explained by Bogolubov [5] and Terno [6]. One can thus de¯ne the spin states (or the helicities) of any quantum system with respect to this well-de¯ned spin generator. Another approach to addressing relativistic states is to consider quantum ¯elds associated with underlying particles. For instance describes particles with the Klein-Gordon ¯eld for spin 1 0 particles or the Dirac ¯eld theory that describes particles of spin 2 . The operators attributed to the physical observables, for example, energy, momentum and angular momentum, can be constructed accordingly which would yield the conventional observables as in a non-relativistic scenario. The angular momentum operator in a ¯eld-theoretic formalization, as shown in this thesis, produces the same exact measurements as compared to the spin generator de¯ned from iv the Poincar¶egenerators. The motivation for this thesis is to show the logical consistency be- tween the ¯eld-theoretic approach and the group description of relativistic quantum mechanics. In this thesis, the group of transformations that leaves the particle momentum invariant is examined. The elements of this group is named as the Wigner elements, and it can be de¯ned in terms of the Poincar¶egenerators as shown by Soo and Lin [11]. Chapter one introduces the basics of quantum ¯eld theory which covers the ¯eld-theoretic realization of some physical quantities like energy and angular momentum. It is shown by Noether's theorem that one can derive these observables for physical symmetries based on the concept of conserved currents. For chapter two, the Wigner elements for the Poincar¶egroup is introduced by way of induced representations. The equivalence between the induced representation of the Wigner elements for Dirac particles and the Lorentz transformation on Dirac spinors is then explored. In chapter three, the ¯eld-theoretic realization of spin for Dirac particles is elucidated. The spin operator can then be used in the context of spin measurements and also entanglement, especially during the discussions on Bell correlations and Clauser-Horne-Shimony-Holt (CHSH) inequality. It is demonstrated that this spin operator exhibits similar measurement results that agree with the works of Terno [6], Czachor [7, 8], Doyeol [9] and Terashima [10] whose spin operator is based on the Pauli-Lubanski vector. In the fourth chapter, the Dirac spin operator derived from the ¯eld-theoretic approach is shown to be equivalent to the induced representation of the Pauli-Lubanski vector of the Poincar¶egroup, under appropriate coordinate representation. Finally, it is also shown that the Dirac spin operator is equivalent to the Wigner spin operator up to a proportional factor. These two spin operators agree exactly when the measurement axis is taken as along the direction of the particle momentum. v List of Figures 3.1 Rotation of Bloch vector ............................... 50 3.2 Bell Observable for Dirac Spin Operator ...................... 52 3.3 Bell Observable for Dirac Spin Operator with momentum on xy-plane ...... 53 4.1 Geometrical Picture of the Unitary Rotation on Dirac Spinors .......... 64 4.2 Three-Dimensional Plot of the Cosine of Rotation Angle '1 ............ 65 4.3 Two-Dimensional Plot of the Cosine of Rotation Angle '1 ............ 66 vi List of Symbols The Greek indices (¹; º; ®; ¯) etc. used in this report are normally reserved for space-time coor- dinate labels 0; 1; 2; 3, with x0 the time coordinate. They are also applicable to the labeling 1; 2; 3; 4 that denotes the spinorial components in the Dirac equations. The Latin indices (i; j; k) are used for three-dimensional spatial coordinates 1; 2; 3 that represent the x; y and z coordinates respectively. The letters in bold denote the three-dimensional vector of that quantity. For special cases like the three-dimensional vector of an operator, an arrow is placed over it. An example is σ~ = (σ1; σ2; σ3) where σi are the standard Pauli matrices. A hat over any vector indicates the corresponding unit vector. However, a widehat over any symbol denotes its status as a group generator. The space-time metric ´¹º is diagonal and its metric signature is given by f1; ¡1; ¡1; ¡1g. Repetition of the indices denotes a summation over the index , unless otherwise stated. A dot over any quantity denotes the time-derivative of that quantity. Dirac matrices γ¹ are de¯ned so that γ¹γº + γºγ¹ = 2´¹º. The complex conjugate, transpose and Hermitian adjoint of a matrix or vector M are written as M ¤, M T and M y respectively. The element in any matrix can be labeled by specifying its row-column position, for example the i i-th row, j-th column of matrix M is given by M j. The symbol I is reserved for a square identity matrix. For any non-singular square matrix M, the notation kMk denotes its eigenvalues. In general, the Planck's constant ~ and speed of light c are taken as one unless otherwise stated. vii Chapter 1 Basic Quantum Field Theory Quantum ¯eld theory has been visualized as a marriage between the theory of special relativity and quantum mechanics. By studying the proper Lorentz group and the Poincar¶e group, it is possible to incorporate the Minkowskian space-time structure in quantum ¯eld theory by way of induced group representations. Their group elements represent the Lorentz transformations and space-time translations associated with the particle states.
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