Equation of motion and quantization of the system defined by interpreting Hopf invariant as an action functional

Question

Consider a field theory defined on the circle $S^1$, with one field taking values also in $S^1$. Let the action equal to the winding number of the field. This can be thought of special case of the Hopf invariant for higher dimensional sigma models, for instance, you can analogously take a purely topological Hopf-invariant action for a field theory on $S^3$ where the field values lie in $S^2$.

• How does one calculate the equation of motion of \(\theta\) from this action functional?
• How does the above action get quantized?
• Is this a quantum field theory at all?

Comments

What is S1 supposed to be? This makes no sense if I take it to be the circle, the Hopf invariant is zero (Later: sorry, misunderstood definition--- it's the linking number of a collection of displaced points on a circle, which makes sense and is a function of the winding number).

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I believe the Hopf invariant for $S^1 \to S^1$ is just the winding number.

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@Ryan Thomgren: Yes, you're right, it is the winding number, but the OP wrote the invariant wrong--- it's just the integral of $d\theta$, the $\theta$ makes no sense (or, rather, if theta is a one form, as you interpret it then $d\theta$ makes no sense, it would just be zero). The wedge product in the original formula made no sense (1d!). The OP miscopied the formula--- n=1, so this is a degenerate case for the Hopf invariant formula, where the two wedge space are $\Lambda^0$ and $\Lambda^1$, and the formula is simply $\int d\theta$, or counting the winding number as you go around the base space.

I edited the question, removing the mistaken identification of the spaces, removing the high-falutin' language, and asking whether the winding number action makes sense. If OP wants original wording back, just revert, it's just a suggestion!

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"I edited this infernal question to be clear, removing the mistaken identification of the spaces, removing the high-falutin' language, and asking whether the winding number action makes sense. Also downvoted it, because it's a bad question, and probably insincere, designed to test whether we can detect gibberish."

Hi Ron, I'm not sure if either insulting the OP or accusing him of trying to 'trick' the community is a nice way to treat people on this site, let alone attract new users, even when you don't think the question makes a lot of sense. 

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I think a good idea idea to deal with questions that confuse many things and to show that we can recognise cofused things, is to just edit and/or clarify as needed in comments and answers, basically as you did ;-). If it is very bad gibberish we can also make use of the Request for close vote thread. I personally dont think this OP meant any harm ...

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@Danu one of the main reasons for setting up this site was to do away with "niceness", where people can be direct, without getting banned by politically motivated moderators.. All the admins Dimension10, Ron, Dilaton have been banned on Physics stack exchange for "not being nice" at some point lol.

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@physicsnewbie could it be that you are Physics Lover on Physics SE ... ;-)? Anyway, the main reason to revive TP.SE with a slightly broadend scope outside the SE network, is that the SE company rather tries to maximize mass visibility (the content has to be useful to people who find it by google etc), which is not in line with setting up a high-level academic community that produces high-level content which is naturally useful only to a limited number of people. But lets not discuss these things here, you could set up an extra meta thread for continuing this discussion if you like (as we do not yet have a good chat feature).

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@dilaton Yep I'm physicslover on PSE ;)

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@Danu Well, I don't really consider Ron's comments to be particularly offensive, and I don't think that we should really have rules preventing rudeness.  

Of course, I disagree with him about the poster's intentions.  

@physicsnewbie I really don't understand the purpose of your "lol" here...   

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yes, calling a question "bad" or "infernal" is completely justified if it is really bad, but we probably shouldn't put our speculations about others' intention or sincerity here(of course unless it's an obvious troll), so -1 to Ron's comment. 

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@dimension10 the "lol" is to lighten the atmosphere a bit, and not give the impression that I see it as serious you and the other admins being banned elsewhere in the past for not "being nice".

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@Danu: I am sorry. I retract my comments. Anonymous is totally welcome, and if anonymous would like the original wording back, that's ok too. The "Infernal" was just a jokey way of expressing my own frustration, not a slight on the OP.

Answer 1

The action

$\int d\theta$

has a gauge symmetry $d\theta \mapsto d\theta + df$. The gauge equivalence classes are discrete (labelled by the winding number), so the equations of motion don't say anything (and the gauge field is automatically flat for dimensional reasons).

The action is integer-valued, so you get a different "TQFT" for every parameter $\alpha \in U(1)$ by

$S_\alpha = \alpha \int d\theta$.