### 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