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

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

126 submissions , 106 unreviewed
3,683 questions , 1,271 unanswered
4,655 answers , 19,737 comments
1,470 users with positive rep
440 active unimported users
More ...

Master Constraint in LQG: How to derive a path integral?

+ 2 like - 0 dislike
40 views

If I have the Hamiltonian operator of Loop Quantum gravity $H(x)$ with $x \in \mathbb{R}^3$ then the Master constraint is expressed by the time evolution operator $\int \mathcal{D}t exp(t \int d^3x H^\dagger(x) H(x))$ can be turned into a path integral. First I do a gauge fixing and I let $t = const.$ such that integration over this variable can be omitted.

Now if I know the eigenstates and the corresponding eigenvalues of the operator $I = \int d^3x H^\dagger(x) H(x)$ I can derive a path integral. Some spin network states are eigenvectors of $I$. The inner product between spin network states $e_i = |\Gamma_i>$ are defined by

$<\Gamma_i|\Gamma_j> = \delta_{ij} (*)$

up to diffeomorphism. The spectral theorem tells us that

$I = \sum_{k=1}^\infty \lambda_k e_k \otimes e_k^\dagger$

with eigenvalues $\lambda_k$. But if I use the orthogonality condition (*) then I will get a very simple statement that spin network states will not be changed under the action of $I$. Should I use a Jordan Block matrix for the spectral theorem where off-diagonal contributions occur on multiple eigenvalues in the diagonalized matrix? Or is the error in the condition (*)? Spin networks must change if time elapses. How a path integral can be defined here where I sum over all spin network states and not over Ashtekar variables and tedrads?

asked Mar 16 in Mathematics by PatrickLinker (20 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$\hbar\varnothing$sicsOverflow
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).
To avoid this verification in future, please log in or register.




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

Your rights
...