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- Type of Document: M.Sc. Thesis
- Language: Farsi
- Document No: 43277 (04)
- University: Sharif University of Technology
- Department: Physics
- Advisor(s): Rouhani, Shahin
- Abstract:
- This dissertation on the Schrodinger-Virasoro algebra will begin by a brief historical review on the development of the symmetry arguments and its final overwhelming success in the 20th century physics. Testifying many successes of symmetry arguments and group theoretic methods in physics such as the theory of relativity, many physicists began to ask whether covariance under a larger group of symmetry transformation would explain a broader era of physical phenomena? As will be reviewed in the beginning of the second chapter, invariance under conformal group - which includes Poincaré group as a subgroup and thus underlies Lorentzian geometry as its natural background geometry – has been implemented successfully in understanding the scale invariance and universal behavior of statistical systems at equilibrium whether they undergo a phase transition in 2 dimensions. Turning to time dependent phenomena, replacing isotropic by anisotropic scale transformations changes fundamentally the geometry; Lorentzian geometry has to be replaced by Newton-Cartan geometry, which accepts several groups of symmetries. As will be explained in chapter 2, some of these groups of symmetries have been observed in physical phenomena, leading to the study of the so called non-relativistic conformal field theory. Chapter 3 will continue by a deeper look at Newton-Cartan geometry and derivation of some of its natural groups of symmetry, among which, Schrodinger-Virasoro group - which is an infinite dimensional group - has been regarded as the right candidate for replacing conformal group in a theory for local scale invariance in physical phenomena. In chapter 4, we will study its group theoretic structure and its relations with some other algebras. While the former point of view has not met any further success, mainly because of the lack of physical systems, which accept Schrodinger-Virasoro symmetry, new lines have been raised in looking at the Schrodinger-Virasoro group. In the final chapter, which is dedicated to the study of the applications of the Schrodinger-Virasoro group, we will show that the free Schrodinger equation is invariant under a representation of this group. In the remainder of this chapter, we will review some other applications of Schrodinger-Virasoro algebra, which goes beyond the line of researches in the framework of non-relativistic conformal field theories.
- Keywords:
- Schrodinger Symmetry ; Symmetry ; Newton-Cartan Gravity ; Galilean Conformal Symmetry (GCS) ; Non-Relativistic Conformal Symmetry ; Schrodinger-Virasoro Algebra
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