X790 | 28392 | Amit Hagar

In our thinking about the world, especially in physics, infinities play a crucial role; the continuum of the real numbers as a representation of spacetime is the best known example. Throughout the history of science, however, many have felt that the continuum model is an unphysical idealization, and that there are in fact only a finite number of degrees of freedom in any finite volume. This tension between the discrete and the continuous has its roots in the philosophy of mathematics, and goes back to the early days of Zeno. Penetrating modern theoretical physics, it has fuelled the development of our theories of matter. But while ideas about atomism and discreteness have proven instrumental in the inception of statistical mechanics and the quantum theory, the attempts to incorporate them into the space-time arena and to apply them in field theories have met with strong resistance. Trying to overcome theoretical inconsistencies in the context of the unification of quantum theory with the special theory of relativity, that arise from the tension between a point particle and a continuous field, or from the attempts to quantize fields with infinite degrees of freedom, many physicists had entertained the notion of a fundamental length, but had eventually abandoned it, as it became difficult to reconcile the hypothesis of discrete space with local Lorentz invariance and with unitary quantum theory. Remarkably, in recent years theoretical physics has witnessed a revival of the notion of fundamental length, and many candidates to the future theory of quantum gravity incorporate it in their theoretical structure. This course will investigate the inter-relations between mathematics and physics as they are manifest in the development of the ideas surrounding the notion of fundamental length. Possible research questions: Zeno’s argument against the continuum; Non-standard analyses and the alternative to the real line segment; Argument against discrete spacetime; Discreteness in Statistical Mechanics; Non local field theories; Heisenberg and March on fundamental length; Fundamental length in quantum gravity; Violations of Lorenz invariance in HEP All the physics/math necessary for the course will be taught in class; the class will definitely be accessible to humanities students and conceptually challenging to both humanities and science majors. Each participant in the seminar will be required to make at least one presentation to the seminar, on a the research topic to be chosen in consultation with the instructor, that would result in a research paper (minimum 20 pp). Grading will given according to student’s presentation and final term paper. The extra credit reflects the fact this course is a research seminar, which requires substantial research and writing skills.