• 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
5,054 questions , 2,207 unanswered
5,347 answers , 22,720 comments
1,470 users with positive rep
818 active unimported users
More ...

  Why is the projective symmetry group (PSG) called projective?

+ 3 like - 0 dislike

As discussed by Prof.Wen in the context of the quantum orders of spin liquids, PSG is defined as all the transformations that leave the mean-field ansatz invariant, IGG is the so-called invariant gauge group formed by all the gauge-transformations that leave the mean-field ansatz invariant, and SG denotes the usual symmetry group (e.g., lattice space symmetry, time-reversal symmetry, etc), and these groups are related as follows SG=PSG/IGG, where SG can be viewed as the quotient group.

However, in math, the name of projective group is usually referred to the quotient group, like the so-called projective special unitary group $PSU(2)=SU(2)/Z_2$, and here $PSU(2)$ is in fact the group $SO(3)$.

So physically why we call the PSG projective rather than the SG? Thank you very much.

This post imported from StackExchange Physics at 2016-02-26 10:00 (UTC), posted by SE-user Kai Li
asked Jun 1, 2014 in Mathematics by Kai Li (980 points) [ no revision ]

1 Answer

+ 2 like - 0 dislike

It depends on what group you consider the starting point, and that depends on context.

One context is the mathematical one (forget all you know about spins etc) where we start with the vector space $\mathbb C^2$ and the natural action of $SU(2)$ on it. If we then look at the projective vector space $\mathbb C^2/\mathbb C^* = \mathbb CP^1$, then the action of $SU(2)$ is given by $PSU(2) = SU(2)/ \mathbb Z_2$. Since this group now acts on the projective vector space, we can call it a projective group.

In the physical context--in the case of rotation symmetry--our starting group is not $SU(2)$ but rather $SO(3)$. The way this acts on the Hilbert space of a spin $1/2$ particle is by a nice linear representation $\rho: SO(3) \to \mathbb CP^1$ (this is in fact the identity map, since as you point out $\mathbb CP^1 \cong SO(3)$!). However physicists don't really like to think about projective Hilbert spaces, and so we prefer to think of our symmetry as acting on the linear Hilbert space: $\tilde \rho : SO(3) \to \mathbb C^2$. However, it turns out that the best you can do is that $\rho$ is a projective representation (which means the group structure is only respected up to a complex scalar). Hence you can say we traded in the projective Hilbert space for a projective group action. Again, physicist don't like thinking about projective representations, and so instead we use the linear representation of the covering group. Indeed, if we first extend $SO(3)$ to $SU(2)$ then it can act linearly on the linear space $\mathbb C^2$.

I am sure you know that that is why we use $SU(2)$ instead of $SO(3)$, but I wanted to go through the reasoning explicitly, to demonstrate that our original symmetry group is $SO(3)$, but since using that induces ''projectiveness'' down the road (either on the space or in the way it acts), we instead use the extended symmetry group $SU(2)$, which we can call the projective symmetry group of our system since it encodes all the projective realizations of our original symmetry group.

In conclusion: it's not that one name is better than the other, and you are correct in noticing that they are not talking about the same thing, it's just that it depends on the context when you tag the label ''projective''. In the former case we call it the projective group since it is the way in which the original group acts on the projective vector space. In the latter case we might call the extended symmetry group projective because its (linear) representations correspond to all projective representationsn of our original symmetry group.

This post imported from StackExchange Physics at 2016-02-26 10:00 (UTC), posted by SE-user Ruben Verresen
answered Dec 1, 2015 by Ruben Verresen (205 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