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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.

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.

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