Rotational invariance in d=2+1 dimensions (Cherns-Simons term)

[binbagsss]

Homework Statement

see above

Relevant Equations

see above

Hi, this is probably a stupid question, but, does rotational invariance in $d=2+1$ mean to only rotate the spatial coordinates and not the time.

I mean bascially I want to show that $\int d^3 x \epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}$, yes epsilon the antisymmetric tensor, is invaraint under rotation.

Before I write out all the terms incorrectly I'd like to make sure my $x^u$ coordinate transformations are right. So, the convention is to rotate anti-clockwise right?, so in 2-d I would have , for a 90 degree rotation,

$x \to -y , y \to -x$?

(How do I generalise actually, to all rotations, instead of just showing it specifically for this 90 degrees. )

many thanks.

e.g considering parity i have

$x^0 \to x^0 , x^1 \to -x^1 , x^2 \to x^2 ,$ and therefore $A_0 \to A_0, A_1 \to -A_1. A_2 \to A^2.$, and I can finish it off to show it's not invariant. Just want to make sure my first step for rotation is correct. many thanks.

 

[Antarres]

It depends on the field you're transforming. I would say that rotation invariance is meant to be spatial rotations(rotations that involve time are usually called boosts). As for how to transform your field in general, it depends on the representation of the group. By the notation I would assume that this is vector representation(but I can't be sure), so in that case the field would transform as:
$$A^\mu = \Lambda^\mu_{\hphantom{\mu}\nu}A^\nu$$
Where $\Lambda$ is Lorentz matrix for rotations defined with angle as a parameter. In general the law of transformation is defined within theory, by inspecting the definition of the fields you're working with - not every variable has the same law, there can be different representations of the rotation group.

 

[binbagsss]

It depends on the field you're transforming. I would say that rotation invariance is meant to be spatial rotations(rotations that involve time are usually called boosts). As for how to transform your field in general, it depends on the representation of the group. By the notation I would assume that this is vector representation(but I can't be sure), so in that case the field would transform as:
$$A^\mu = \Lambda^\mu_{\hphantom{\mu}\nu}A^\nu$$
Where $\Lambda$ is Lorentz matrix for rotations defined with angle as a parameter. In general the law of transformation is defined within theory, by inspecting the definition of the fields you're working with - not every variable has the same law, there can be different representations of the rotation group.

right so how would i go about showing explicit invariance using a matrix represtation? in compared to the above say.

the partial derivative would also transform wouldn't it?

 

[king vitamin]

Your intuition seems correct. If you're looking at the Chern-Simons action, then the $A_{\mu}$'s transform as vectors, and $\partial_{\mu}$ transforms inversely to a vector. In application to quantum Hall systems (since I know you're looking at Tong's notes), I'd usually say that "rotational invariance" refers only to spatial coordinates. But in this example, it seems simple enough to prove full (orthochronous) Lorentz invariance, and then rotations will follow trivially. Then you can investigate parity/time-reversal separately.