Department of Mathematics
Math 351 - Fall 1998
For example the rotational symmetry of a system about the origin of the force is responsible for the conservation of angular momentum (equal areas in equal time). The first applications will be to classification of crystals and to small molecular vibrations (methane molecule). The only background required is calculus through math 242, linear algebra, and a course in elementary physics.
In four dimensional space time, distance is measured by the form x2 + y2 + z2 - t2. The Poincare group (an infinite group) consists of transformations of space-time that preserve this distance. A central goal of the course is to tell the story of Wigner's classification of the irreducible representations of the Poincare group (these are the building blocks for all the representations). The remarkable result is that the physical properties of mass (continuous) and spin (discrete with half integer steps) arise as parameters of these representations. Wigner's result has "provided a framework for the physical search for elementary particles" (S. Sternberg).
Prerequisites: Math 242 and Math 315; junior standing in mathematics or physics or instructor's permission.
Grading: I will assign problems that develop the student's understanding, but the grade will be (primarily) determined by a 5-10 page paper and a take home final exam.
Recommended Text: Group Theory and Physics by S. Sternberg (paperback edition); published by Cambridge University Press.