Nonrelativistic conformal symmetry in 2+1 dimesional field theory

This thesis is devoted to the study of conformal invariance and its breaking in nonrelativistic field theories. It is a well known feature of relativistic field theory that theories which are conformally invariant at the classical level can acquire a conformal anomaly upon cluantization and renormalization. The anomaly appears through the introduction of an arbitrary, but dimensionful, renormalization scale. One does not usually associate the concepts of renormalization and anomaly with nonrelativistic quantum mechanics, but there are a few examples where these concepts are useful. ''he most well known case is the two-dimensional 3-function potential. In two dimensions the S-function scales like the kinetic term of the Hamiltonian, and therefore the problem is classically conformally invariant. Another example of classical conformal invariance is the famous Aharonov-Bohm (AB) problem. In that case each partial wave sees a 1/r 2 potential. We use the second quantized formulation of these problems, namely the nonrelativistic field theories, to compute Green's functions and derive the conformal anomaly. In the case of the AB problem we also solve an old puzzle, namely how to reproduce the result of Aharonov and Bohm in perturbation theory. The thesis is organized in the following manner. Chapter 1 is an introduction to nonrelativistic field theory, nonrelativistic conformal invariance, contact interactions and the AB problem. In Chapter 2 we discuss nonrelativistic scalar field theory, and how its quantization produces the anomaly. Chapter 3 is devoted to the AB problem, and the resolution of the perturbation puzzle. In Chapter 4 we generalize the discussion of Chapter 3 to particles carrying nonabelian charges. The structure of the nonabelian theory is much richer, and deserves a separate discussion. We also comment on the issues of forward scattering and single-valuedness of wavefunctions, which are important for Chapter 3 as well. Thesis Supervisor: Roman Jackiw Title: Professor Acknowledgments I wish to express my deep gratitude to Roman Jackiw who has guided my work for three years, and whose technical expertise and critical reading of my manuscripts I shall surely miss. I would also like to thank Barton Zwiebach for his guidance and collaboration on my final project here at MIT, and for his patience in helping me enter a new field. I thank Dr. Gustavo Lozano and Dongsu Bak whose collaboration on the papers leading to chapters 3 and 4, respectively, I found extremely enjoyable. In addition I[ would like to thank my fellow graduate students at the Center for Theoretical Physics, especially Gilad Lifschytz, with whom I've had several discussions (every day! ) on many topics in and out of physics, Krishnan Ranganathan, Esko Keski-Vakkuri, Edi Halyo, Dan Kabat and Ruben Levi. I would like to thank the Department of Physics and the Center for Theoretical Physics for their financial support through the Karl Taylor Compton fellowship and a Research Assistantship during my five year stay at MIT. Finally, I would like to thank my parents Amalia and David Bergman for introducing me to physics. my wife Ruth, who gave me the best four years of my life. To