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Chapter 3 Quantum Field Theory Relativistic Quantum Field in Theoretical Physics by fEJSA HU, ETTI !NSTITUTE Trung Van Phan OF TECHNOLOLGY Submitted to the Department of Physics AUG 102015 in partial fulfillment of the requirements for the degree of LIBRAR IES Bachelor of Science in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2015 @ Trung Van Phan, MMXV. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Signature redacted A uthor .......................................... Department of Physics May 8, 2015 C ertified by ...................................... Signature redacted Jeise D. Thaler Assistant Professor Thesis Supervisor Signature redacted Accepted by ............................... Nergis Mavalvala Associate Department Head for Education 2 Relativistic Quantum Field in Theoretical Physics by Trung Van Phan Submitted to the Department of Physics on May 8, 2015, in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics Abstract Quantum field theory is the most well-developed tool in theoretical physics to study about the dynamics at microscopic scales, with particles and quantum behaviors. In this thesis I'll review about the construction of quantum field theory from the S-matrix point of view (building up particles and formulating interactions), then poke at interesting topics that are usually briefly mentioned or even ignored in standard quantum theory of field textbooks. In more details, chapter 1 will be about how a quanta is defined from the analysis of group theory, chapter 2 will be focus on how the language of field to describe the quantum behaviors of quanta is desirable for interactions and arised quite naturally (it should be noted that, string theory can be viewed as a totally different theory in quantum interactions, with conformal symmetric and topological natures, but at low energy scale it can always be reduced back to quantum field theories), and chapter 3 will be about several angles of quantum field theory. Thesis Supervisor: Jesse D. Thaler Title: Assistant Professor 3 Acknowledgments I'd like to express my sincere gratitude to Massachusetts Institute of Technology (MIT) for letting me pursuing my dream about studying theoretical physics here. I would also like to thank Prof. Jesse Thaler and the Physics Department for guiding me with this thesis from start to finish. I'm indebted to Prof. Jesse Thaler, my thesis supervisor, for his understanding, patience, enthusiasm and encouragement during all these years in MIT. To Prof. Leonid Levitov, Prof. Edmund Bertschinger, Prof. lain Stewart, Prof. Hong Liu and Prof. Washington Taylor, I'm extremely grateful for the past discussions and perspectives from you in different topics in theoretical physics, which some of them are mentioned in this thesis. To all my friends for supporting me during hardship, especially Thai Pham, Duy Ha, Dan Doan, Nhat Cao, Tru Dang and Truong Cai. To my family for always having faith and not giving up on me. 4 Contents 1 Quanta 7 1.1 Spacetime Symmetry .................. ....... ... 8 1.1.1 Under Spacetime Transformation ........ ....... ... 8 1.1.2 Poincare Group and Poincare Algebra ..... ........... 10 1.2 Fixed Characteristics ...... ............. ... ....... 11 1.2.1 Casimir operators ..... ............ ... ....... 11 1.2.2 Quanta definition ................. ....... ... 12 1.3 Labeling a Quanta .. .......... ........ ... .... ... 14 1.3.1 4-momentum Label ......... ....... ....... .... 14 1.3.2 Details Dynamical Information of Quanta .. ....... ... 16 1.3.3 More on Spin and Helicity .. ..... ..... ....... ... 18 1.3.4 Quanta in Flat Spacetime and Curved Spacetime ........... 21 2 Field 22 2.1 Construction of Multi-particles Dynamics ........ ....... .... 22 2.1.1 Scattering Setting ................ ....... .... 22 2.1.2 Creation and Annihilation Operator ....... ....... .... 24 2.1.3 Causality Condition ..... ..... ..... ..... ...... 26 2.1.4 The General Form of S-Matrix ........ .. .... ....... 28 2.1.5 Internal Symmetry and Conserved Charge .. ........... 28 2.1.6 Cluster Decomposition ...... ...... .. ........... 30 2.2 Macroscopic Description ......... ........ 31 2.2.1 Creation and Annihilation Fields ... ..... 31 2.2.2 Finite-Dimensional Representations of the Lorentz Group. 32 5 2.2.3 Physical Requirements for the Theory .............. ...... 36 2.2.4 Spacetime Rotation and Particle Self-Rotation .... ........ ... 41 3 Quantum Field Theory 44 3.1 The Free Theory Lagrangian ........... ..... .... ..... 44 3.1.1 Massive Quantum Field ..... .... .. ... ... ... ... 45 3.1.2 Massless Quantum Field ......... ... ... ... ... .. 47 3.2 Collective Behaviors of Quantum ........ ... ... ... ... ... 49 3.2.1 Gauge Freedom .............. ... .... ..... ... 49 3.2.2 Renormalization .............. ... ... 51 3.2.3 Long Distance Physics .......... .......... ... ... .. 57 3.2.4 Maxwell's Electromagnetism and Einstein's General Relativity .. ... ... 62 3.2.5 Classical Field Configuration ....... ... ... 64 3.3 Functional Integration .............. .. .... .... .... .... 6 5 3.3.1 Connection to Canonical Quantization . .... .... ... .... ... 66 3.3.2 Is Functional Integration always Superior? ... .... .... .... ... 70 3.3.3 Mathematical Rigors ........... .. .... ... .... .... .. 74 3.3.4 Effective Field Theory .......... .... .... ... .... ... 75 3.3.5 Quantum Anomaly ............ ... .. ... ... ... ... .. 9 3 3.3.6 W avefunctional ........... ........ .. ... .... ... .. 100 6 Chapter 1 Quanta Unchanged properties under the change of observers and reference frames are of great interests in understanding nature, since they are very useful to formulate an universal theory and also easy to keep track of, experimentally. Since it is expected that the law of physics is the same in every inertial frame, which is a postulation in special relativity (with a depth philosophical reason), hence one can partially define the building blocks of the universe to be objects associated with the set of most the fundamental and fixed characteristics under the transformations of spacetime that relate different inertial frames together (to get the full definition, one has to take into account the interactions). The transformations are described by Lorentz group, which is the result from the other postulation of special relativity (unsurprisingly, based on a constant - the speed of light), and the spacetime translation group, which is seen in flat spacetime. Combining these two gives the Poincare group. From the experiments, it is known that nature exhibits quantum (discretization) behavior, therefore the fundamental building blocks should be one-particle states and multi-particles states. A quanta is the physical realization of an one-particle state. From the quanta point of view, the physical transformation that unchanges the presenting physics is a classical interpretation for symmetry. 7 1.1 Spacetime Symmetry 1.1.1 Under Spacetime Transformation Transformation of Spacetime Coordinates The spacetime symmetry is a global symmetry described by the Poincare transformation, with Lorentz rotation Al and spacetime shift a": T(A, a):x:-+" a' = AI",xv + a , T(A', a')T(A, a) = T(A'A, A'a + a') (1.1) Since Lorentz transformation preserves the 4-vector length, the Lorentz parameter satisfies: -1 0 0 0 0 +1 0 0) (1.2) 0 0 +1 0 0 0 0 +1 To,,AvA , = , --+ A = uir1 ", = AAnan , (Ado)t , = A" = lsoP te hAPt: (1.3) To have the connection with unitary, the determinant det A should be 1. Also note that: n/A "oAvo= qoo -+ -(A o)2 + (A'o) = -I, (A00) 2 > (1.4) 0 There are disjoint regions of the real axis for possible A 0 , and for linking requirement with identity 0 transformation, one needs to have A 0 > 1. From the Lorentz proper orthochronous subgroup, 0 with det A = 1 and A 0 > 1, to go to the other part of the Lorentz group, spatial inversion P and time reversal T are employed: 3 po 1 , Pp= 'P2P> _ 2 7 1 (1.5) From now on, the proper orthochronous Lorentz group is refered as simply Lorentz group, unless mention explicitly. Also, Poincare group is with the the proper orthochronous Lorentz group, not 8 the full original Lorentz group. The infinitesimal transformation of the Lorentz group: At', -+ P' + w", , wA = p, w", = r""w ,p, ; ap -+ 16 (1.6) The Lorentz transformation of the metric indicates that w.,, is anti-symmetric: 0 77t = 77PAP AU = TipcG(P + w"i)(5" + w ,) (1.7) = + wI, + WVA + ... -- w,, (1.8) Transformation in the Hilbert space In quantum terminology, any collection of properties can be represented by a ray of normalized vectors in the physical Hilbert space. The transformation in spacetime coordinates is realized by a unitary linear transformation, with similar composition rule [36]: 1IT) -+ U(A, a) IT) , U(A', a')U(A, a) = U(A'A,A'a+ a') (1.9) Note that, there's a caveat here. The associatity of the symmetry group representation on the physical states is not always exactly equal, because the transformation T (a general symmetry transformation, not just restrict to the spacetime Poincare transformation) take a ray to another ray but don't really put a restriction at the possible phase arise: U(T')U(T) = eOT1,T) U(T'T) (1.10) Because of linearity (also, anti-linearity), the phase should be independence of the state which the transformation acts on. The name for a representation with a general # is the projective representation [36]. The symmetry group cannot
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