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

• binbagsss
In summary, the conversation discusses the concept of rotational invariance in a 2+1 dimensional space and how it applies to a specific equation involving the antisymmetric tensor. The participants also discuss the transformation laws for fields under rotations and the role of Lorentz matrices in this process. They also touch upon the idea of parity and time-reversal invariance.
binbagsss
Homework Statement
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Relevant Equations
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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.

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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.

Antarres said:
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?

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.

## 1. What is rotational invariance in d=2+1 dimensions?

Rotational invariance in d=2+1 dimensions refers to a property of a physical system that remains unchanged under rotations in a three-dimensional space. In other words, the laws and equations governing the system remain the same regardless of the orientation of the coordinate axes.

## 2. What is the Chern-Simons term?

The Chern-Simons term is a mathematical term used in gauge theories, specifically in three-dimensional space. It describes the interaction between gauge fields and fermions, and is used to explain certain physical phenomena, such as the fractional quantum Hall effect.

## 3. How does the Chern-Simons term relate to rotational invariance?

The Chern-Simons term is a gauge invariant quantity, meaning it remains unchanged under local gauge transformations. This property allows it to preserve rotational invariance in d=2+1 dimensions, making it a crucial component in theories that require this symmetry.

## 4. What are some applications of rotational invariance and the Chern-Simons term?

Rotational invariance and the Chern-Simons term have various applications in different fields of physics. For example, in condensed matter physics, they are used to explain topological phases and fractional statistics. In particle physics, they are used to describe the behavior of quarks and gluons in the strong nuclear force.

## 5. Are there any exceptions to rotational invariance in d=2+1 dimensions?

While rotational invariance is a fundamental property in three-dimensional space, there are some exceptions in certain physical systems. For example, in the presence of external forces or boundary conditions, rotational invariance may not hold. Additionally, in theories that involve higher dimensions or non-commutative spaces, rotational invariance may also be violated.

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