Pre-requisites for path integral formulation?

[hawkingfan]

Pre-requisites for path integral formulation?

Does anybody have any idea of the pre-requisites to learn Feynmann's path integral formulation? (properly) Right about now, I'm still learning about Lagrangian and Hamiltonian mechanics which focuses on the principle of least action. Right now, the only knowledge that I have about quantum mechanics are the basic qualitative principles and the simple quantitative description of them. (which uses calculations that do not exceed multivariable calculus) As for my math proficiency, I'm just starting on real analysis, functions of a complex variable and I am halfway through ordinary differential equations. Are there any more pre-requisites that do not include what I do not already know or what I'm currently studying? I'd appreciate your feedback. Thanks.

 

[Civilized]

First you should learn the Schroedinger-Heisenberg formulation of QM, and to understand what the path-integral is used to calculate you need to know what a green's function / propagator / kernel is, which is a general concept in physics and partial differential equations.

Real Analysis is good, and so is Lagrangian/Hamiltonian mechanics. I would say to concentrate on learning PDE and Shroedinger-Heiseberg QM in dirac notation.

 

[hawkingfan]

Thanks. I appreciate it.

 

[RUTA]

After you've familiarized yourself with QM via Shrodinger and Heisenberg, to include Dirac notation, read Chap I.2 (only 6 pp) in Quantum Field Theory in a Nutshell by A. Zee, Princeton UP, 2003, ISBN 0-691-01019-6. You may also want to read the first 6-7 pp of Chap 8 in Principles of Quantum Mechanics (either edition) by R. Shankar, Plenum, 1994 (2nd ed), ISBN 0-306-44790-8.

 

[peteratcam]

The Feynman path integral formulation of QM is often the first time people meet functional integrals, and it can be confusing when lots of new physics and maths all come at the same time.

Personally I'd recommend learning a bit about functional integrals completely away from the context of QM as well, for example, by learning a bit about statistical field theory. As long as you're familiar with the Boltzmann distribution, it will be easy to guide you to writing down a path integral without even noticing, by considering a harmonic chain (masses connected by strings). Let me know if you want me to walk you through it.

Absolutely invaluable is being fluent with gaussian integrals, that is you should make sure you can derive the following:
$$\int_{-\infty}^{\infty}dv\,e^{-\frac{a}{2}v^2+bv} = \sqrt{\frac{2\pi}{a}}e^{\frac{b^2}{2a}}$$
and then internalise it forever.

Once you have that, generalise it to an n-component vector v and an n x n positive definite real symmetric matrix A (learn enough linear algebra that you know what one is):
$$\int d{\mathbf v}\,e^{-\frac{1}{2}{\mathbf v}^{T}{\mathbf A}{\mathbf v}+{\mathbf b}^T{\mathbf v}} = \sqrt{\frac{(2\pi)^n}{{\rm det} {\mathbf A}}}e^{\frac{1}{2}{\mathbf b}^T{\mathbf A}^{-1}{\mathbf b}}$$

You will struggle to understand path integrals if you can't yet derive the above bits of maths. It is a good exercise to prove both, and plenty of people on this board should be able to give you hints if you get stuck.