Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Discrete version of Feynman path integrals

+ 4 like - 0 dislike
1381 views

I've decided to put a very limited amount of my time into understanding the path integral formulation of quantum mechanics. I'm interested in the mathematical formalism more than the physics, so I'd like to understand how the following abstract and fairly general scenario for a discrete quantum system can be translated into the path integral formalism, assuming it can.

A system starts has a time-dependent state that can be represented by an $n\times n$ density matrix $\rho(t)$. It begins in state $\rho_0$ and evolves over time according to $$ i\hbar \frac{\partial \rho}{\partial t} = H\rho - \rho H,$$ where the Hamiltonian $H$ is some constant Hermitian matrix, yielding $$ \rho(t) = e^{-iHt/\hbar}\rho_0 e^{iHt/\hbar}. $$ At time $t'$ I make a complete set of orthogonal measurements, which amounts to (optionally) doing a basis change on $\rho(t')$ and then interpreting its diagonal elements as probabilities.

My question is, can someone show me how to reinterpret the above in path integral terms? For example, what is a "path" in a discrete system like this, and, given an arbitrary Hamiltonian, how does one assign each path an action?

Alternatively, can anyone recommend a treatment that summarises the path integral formalism with a minimum of physics (by which I mean details about how specific Hamiltonians behave and where they come from, etc.)?

This post imported from StackExchange Physics at 2014-04-08 05:14 (UCT), posted by SE-user Nathaniel
asked May 16, 2012 in Theoretical Physics by Nathaniel (495 points) [ no revision ]
The discrete form of the Feynman path integral is just time-dependent perturbation theory. This is not worked out so great, I'll try to write an answer when I get a chance, but the paths are labelled by the time of discrete state transitions, and the integral over intermediate time labels introduces the phase factors of the time-dependent perturbation theory. Something like this is covered in an obscure early 1950s article by Feynman, where he introduces a time-ordered product and mathematical tricks to quickly derive time-independent perturbation theory from PI like formalism

This post imported from StackExchange Physics at 2014-04-08 05:14 (UCT), posted by SE-user Ron Maimon
@RonMaimon that sounds like the kind of explanation I'm looking for - I look forward to it.

This post imported from StackExchange Physics at 2014-04-08 05:14 (UCT), posted by SE-user Nathaniel

2 Answers

+ 1 like - 0 dislike

I guess you might be interested in this kind of an introduction,

http://www.math.sunysb.edu/~tony/whatsnew/column/feynman-1101/feynman1.html

This post imported from StackExchange Physics at 2014-04-08 05:14 (UCT), posted by SE-user user6818
answered May 16, 2012 by UnknownToSE (505 points) [ no revision ]
Thanks, that's very helpful - skimming it's given me some idea of what kind of mathematics is involved - but it takes my request for "a minimum of physics" a bit too far, in that it never mentions Hamiltonians at all, leaving me unsure for the moment of how to connect these ideas to the scenario I described in my question...

This post imported from StackExchange Physics at 2014-04-08 05:14 (UCT), posted by SE-user Nathaniel
+ 0 like - 0 dislike

Check out the original paper by Feynman, google "The Space-Time Formulation of Nonrelativistic Quantum Mechanics".

This post imported from StackExchange Physics at 2014-04-08 05:14 (UCT), posted by SE-user mtrencseni
answered May 16, 2012 by mtrencseni (10 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsO$\varnothing$erflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...