John Klauder

Lectures on Functional Integration

Given at the University of Florida, Spring Semester 2004.

Functional integration is a tool useful to study general diffusion processes, quantum mechanics, and quantum field theory, among other applications. The mathematics of such integrals can be studied largely independently of specific applications, and this approach minimizes the prerequisites needed to follow the material. Although some knowledge of quantum mechanics would prove useful, the material is presented so that most of what is needed is included. The course is designed for a physics audience, with an occasional excursion into material and style of presentation that is more mathematical than a typical physics course might present.

Broadly speaking, topics covered include integration, random variables, stochastic processes, Wiener measure, Feynman path integrals in various forms, a brief survey of scalar quantum field theory, and concludes with the presentation of a soluble mathematical model of a nonrenormalizable quantum field theory that illustrate efforts that currently represent the frontier of research.

Thanks to Khandker Muttalib whose lecture notes appear here.

Audio & image files copyright John Klauder and the University of Florida.

Comments welcome at klauder at phys dot ufl dot edu

Some Recommended Books

G. Roepstorff, "Path Integral Approach to Quantum Physics", Springer-Verlag, Berlin, 1996

R. Feynman and A. Hibbs, "Quantum Mechanics and Path Integrals", McGraw-Hill, New York, 1965

A.V. Skorokhod, "Studies in the Theory of Random Processes", Addison-Wesley Publishing, Reading, Massachusetts, 1965

B. Simon, "Functional Integration and Quantum Physics", Academic Poress, New York, 1979

L. Schulman, "Techniques and Applications of Path Integration'', John Wiley & Sons, New York, 1981

J. Klauder and B-S. Skagerstam, "Coherent States", World Scientific, Singapore, 1985

C. Grosche and F. Steiner, "Handbook of Feynman Path Integrals", Springer-Verlag, Berlin, 1998

J. Klauder, "Beyond Conventional Quantization", Cambridge University Press, Cambridge, 2000

H. Kleinert, "Path Integrals in Quantum Mehcanics, Statistics, and Polymer Physics", 3rd Edition, World Scientific, Singapore, 2003

Lecture Notes & Audio

Audio files are wma format - each lecture is around 50 mins and 12 MB.

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Lecture 1
integration; basic facts and useful integrals

Lecture 2
absolute and conditional integrals; expressions that imitate integrals; probability, random variables, various measures

Lecture 3
characteristic functions

Lecture 4
Gaussian variables; infinitely divisible distributions

Lecture 5
central limit theorem; infinite dimensional integrals

Lecture 6
finite dimensions need not limit to infinite dimensions; support properties

Lecture 7
Bochner-Minlos theorem; nuclear spaces

Lecture 8
from sequences to functions; functional derivatives; functional Fourier transformations

Lecture 9
change of variables; stochastic variable theory; general remarks

Lecture 10
Gaussian stochastic processes; Wiener processes

Lecture 11
continuity of paths; stochastic equivalence; joint probability densities

Lecture 12
Ito calculus; Ito and Stratonovich integrals; Wiener measure

Lecture 13
Brownian bridge; Feynman-Kac formula

Lecture 14
Ornstein-Uhlenbeck processes; realization of general Gaussian processes; generalized stochastic processes; stochastic differential equations

Lecture 15
Poisson stochastic processes; Poisson limits to Wiener; quantum mechanical path integrals; configuration space path integrals

Lecture 16
quadratic actions

Lecture 17
path integral for the harmonic oscillator; Kato-Trotter product formula

Lecture 18
alternative path integral regularizations; phase space path integrals; lattice formulation

Lecture 19
examples: random potential, relativistic particle; discourse on integration

Lecture 20
choice of canonical variables; coordinate free classical mechanics

Lecture 21
coherent state path integrals; coherent states

Lecture 22
change of variables; coherent state propagator

Lecture 23
alternative coherent state path integral; shadow metric

Lecture 24
Wiener measure regularization

Lecture 25
technical conditions; physical analog system

Lecture 26
canonical coordinate transformations; coordinate free coherent state path integral

Lecture 27
imposition of constraints; classical review and examples

Lecture 28
quantization of constrained systems; Faddeev and Dirac procedures

Lecture 29
projection operator method

Lecture 30
projection operator method (continued)

Lecture 31
projection operator method (continued)

Lecture 32
fermion path integrals; Grassmann variables; another approach

Lecture 33
applications to quantum field theory; overview (scalar fields); basic quantum formulation; generating functional; free fields

Lecture 34
Euclidean free fields; Green functions

Lecture 35
interacting theories; perturbation theory (sketch); counterterms

Lecture 36
self-interacting scalar theories; lattice formulation

Lecture 37
renormalization group; rigorous results

Lecture 38
Monte Carlo methods; nonrenormalizability and hard core interactions; quantum mechanical example

Lecture 39
quantum field theory; Solobev type inequality; soluble nonrenormalizable model; free theory; perturbation theory; rigorous solution

Lecture 40
rigorous solution (continued); realization as a lattice limit

Problem Set 1
includes solutions

Problem Set 2
includes solutions