Seminar on Operator Theory and Mathematical Physics
 
              Wednesday, November 24, 2004, 13:00 -15:00
 
              University of Haifa, Education - Science Building, Room 570
 
1.   D. Keren (Department of Computer Science):
              Basic Notions of Quantum Mechanics (3rd and last talk)
              See: http://www.cs.haifa.ac.il/~dkeren/qm-01.pdf
 
2.   J. Feinberg (Math-Physics, Oranim):
              A brief Introduction to Relativistic Quantum Field Theory
              and the Renormalization Group (first talk in a sequence of 3 talks).
 
A Brief Introduction to Relativistic Quantum Field Theory and the
Renormalization Group. 
 
In this series of lectures I will explain the rudiments of
relativistic quantum field theory. I will focus primarily on the
scalar field - i.e., a field which belongs to the trivial
representation of the Lorentz groups SO(3,1). I will almost
exclusively apply the path integral formalism. I will start with
the free field, whose equation of motion is linear: the
Klein-Gordon wave equation. This simplest theory explains why
two scalar sources which exchange scalar quanta attract each
other (this is Yukawa's old explanation of nuclear forces which
predicted the existence of pi mesons). Following similar lines
of explanation, one cas show that two like electrical charges
(which exchange virtual photons) should repel, while two masses
(which exchange virtual gravitons) should attract.
 
I will then move on to the nonlinear, or self-interacting scalar
field theory. When the interaction is weak, the nonlinearity
could be treated perturbatively. I will derive the perturbation
series in the nonlinearity of various physical quantities. The
terms in the perturbation series can be represented pictorially
in terms of Feynman diagrams. Feynman diagrams are graphs, which
may contain closed loops. The Feynman rules assign to each such
diagram a complex number. The Feynman rules formally assign to
any Feynman diagram, which contains one loop or more, an
infinite number. Thus, these objects require regularization to
render them finite. The perturbation series (in terms of the
regularized diagrams) is only asymptotic - it has a zero radius
of convergence. Its large order behavior is related to the
existence of classical solutions of the equations of motion.
 
Regularization of Feynman diagrams requires the introduction of
an arbitrarily large energy or momentum scale, which has to do
with the definition of the quantum field theory at very short
distances. Quantum field theories in which this short distance
definition does not affect physical quantities measured at a
coarser, longer distance scale, are called renormalizable
quantum field theories. This brings in the issue of the
perturbative renormalization group. I will thus discuss
elementary aspects of perturbative renormalization in scalar
quantum field theory.
 
Finally, I will touch upon the so-called exact, or Wilsonian
renormalization group, which extends renormalization theory
beyond perturbation theory.
 
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Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel <techm@math.technion.ac.il>
Announcement from: Jonathan Arazy <jarazy@math.haifa.ac.il>