NFYK12003U Quantum Geometry – A Statistical Field Theory Approach

Volume 2014/2015

The purpose of this course is to show that the quantum mechanics of the relativistic particle can be understood entirely geometrically as a sum over random paths, and that the quantum mechanics of strings similarly can be understood entirely geometrically as a sum over random surfaces. In particular this will lead to an understanding of two-dimensional quantum gravity coupled to conformal field theories, the only theory we presently know which couples matter and geometry in a fully consistent way.

Definition of the path integral in quantum mechanics. Application to relativistic particles and to strings. This covers selected parts of the book ”Quantum Geometry”.

Learning Outcome

The purpose of this course is that the student obtains a basic understanding of quantum field theory and string theory from a statistical mechanics point of view, i.e. quantum field theory represented as a theory of random walks and string theory as a theory of random surfaces.
In particular, this means that when the course is finished it is expected that the student is able to:

  • understand how to quantize the bosonic and fermionic particles using random walk reprentations.
  • understand how to analyse more general random ensembles, like branched polymers and relate them to particle propagation
  • understand the concept of random surfaces and how it relates to string theory
  • have a basic understanding of the fact that bosonic string theory cannot exist in space-time dimensions larger than two
  • understand the essentials of non-critical string theory
  • understand how non-critical string theory is related to two-dimensional quantum gravity coupled to matter with central charge less than one
  • understand how to define the concept of Hausdorff dimension and the concept of fractal dimensions of an ensemble of geometric objects
  • be able to calculate the fractal dimension for random works and for random surfaces.

The student is expected to be able to derive and explain the fundamental representation of particles and strings in terms of random geometry as well as the universality of these results.

This course will provide the students with a competent background for further studies within this research field, e.g. an M.Sc. project

Jan Ambjorn, B. Durhuus and T. Jonsson,Quantum: Geometry – A Statistical Field Theory Approach, Cambridge Monographs on Mathematical Physics, 1998

The course only requires basic knowledge of differentiation, integration and complex numbers and some basic knowledge of statistical mechanics
  • Category
  • Hours
  • Exam
  • 0,5
  • Lectures
  • 48
  • Preparation
  • 157,5
  • Total
  • 206,0
7,5 ECTS
Type of assessment
Oral examination, 30 minutes
Without preparation time
Without aids
Marking scale
7-point grading scale
Censorship form
No external censorship
More internal examiners
Criteria for exam assesment

The highest mark (12) is given for excellent exam performance that demonstrates full mastering of the above mentioned teaching goals with no or only small irrelevant gaps.
The grade 2 is given to a student who has achieved only minimally the course goals.