• 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.


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


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,075 questions , 2,226 unanswered
5,348 answers , 22,744 comments
1,470 users with positive rep
818 active unimported users
More ...

  Conserved currents in higher-spin theories

+ 5 like - 0 dislike

After the proposal of Maldacena (AdS/CFT), there have been numerous attempts to find out gravity duals of various kinds of CFT. Klebanov and Polyakov gave one such correspondence here. The claim is this: The singlet sector of the critical $O(N)$ model with the $(\phi^{a}\phi^{a})^{2}$ interaction is dual to the minimal bosonic theory in $AdS_4$. Of course, we have to take the large $N$ limit. Simplest such $O(N)$ invariant theory (free, with no interactions) is: \begin{equation*} S= \frac{1}{2}\int d^3 x \sum_{a=1}^{N}(\partial_{\mu}\phi^{a})^{2} \end{equation*} How do I derive the conserved currents in this theory? $\phi^{a}$ are $N$-component vector fields and NOT $N \times N$ matrix fields.

This post imported from StackExchange Physics at 2014-07-28 11:12 (UCT), posted by SE-user Debangshu
asked May 26, 2013 in Theoretical Physics by DebangshuMukherjee (165 points) [ no revision ]
O.k. I figured it out myself. It seems this formalism was developed by Fronsdal long time back. However, another question has popped up. Spin-$s$ massless fields are given by totally symmetric tensor $\phi_{n_1 n_2..n_s}$. There is a condition though on these fields. They have to be "double traceless" i.e I contract them with $g_{\mu_1 \mu_2}g_{\mu_3 \mu_4}$. Why do I need to impose this condition?

This post imported from StackExchange Physics at 2014-07-28 11:12 (UCT), posted by SE-user Debangshu
Can you post your results? I don't know anything about this Fronsdal stuff, but the c.c.'s are not so difficult to write down, namely $J_\mu = \phi^a \partial_\mu \phi^a,$ $J_{\mu \nu} = \phi^a \partial_\mu \partial_\nu \phi^a$ etc. The derivatives need to act on both fields, so they are conserved (but don't know the TeX to do this). They are traceless because $\partial^2 \phi^a = 0.$ If $\phi^a$ is real the odd spins vanish identically.

This post imported from StackExchange Physics at 2014-07-28 11:12 (UCT), posted by SE-user Vibert
Vibert, one can write the currents as follows, $j_{\mu_1...\mu_s}=\bar{\phi}(x) (\overleftarrow{\partial_\mu}-\overrightarrow{\partial_\mu})^s\phi(x)$. These are not traceless, the trace is given by superpotential. One can easily make them traceless, but the formulas are no so nice

This post imported from StackExchange Physics at 2014-07-28 11:12 (UCT), posted by SE-user John

1 Answer

+ 2 like - 0 dislike

That looks weird but you have to impose the double trace constraint in order to have the right number of degrees of freedom. This is easy to see by going into light-cone and analyzing the equations of motion.

This post imported from StackExchange Physics at 2014-07-28 11:12 (UCT), posted by SE-user John
answered Jul 10, 2013 by John (25 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:
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