E-writing Action on a different slice of space-time, Quantum Hall Effect

In summary, the conversation discusses equations 5.46 to 5.48 from the QHE notes by D.Tong and their derivation. It focuses on how equation 5.46 is expanded to equation 5.48, which involves the use of the Levi-Civita tensor and the trace of a matrix. The conversation also explores the vanishing of spatial derivatives and the presence of a time derivative in the first term of equation 5.48. The conversation concludes with a detailed explanation of the derivation process, including the use of integration by parts and the definition of ##f_{\mu \nu}##.
  • #1
binbagsss
1,259
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Hi, I'm looking at QHE notes D.Tong and wondering how he gets from equation 5.46 to 5.48 ( http://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf )

##S_{CS}=\frac{k}{4\pi}\int d^3 x \epsilon^{\mu \nu \rho} tr(a_{\mu}\partial_{\nu}a_{\rho} -\frac{2i}{3}a_{\mu}a_{\nu}a_{\rho})##.
manifold ## \bf{R} \times \Sigma ## where ##\bf{R}## is time and ##\Sigma## is a spatial compact manifold.

##S_{CS}=\frac{k}{4\pi}\int dt \int\limits_{\sigma} d^2x tr(\epsilon^{ij} a_i \frac{\partial}{\partial t} a_j + a_0 f_{12)}##

I'm very stuck , I'm not sure where to begin. Any hint or explanation very much appreciated.

For example how have we gone from levi - civita tensor in '3-d to 2-d', how have we gone to only a time derivative in the first term- all spatial derivatives vanish? why would this be?

also very confused about the last term. I can't see no reason why spatial derivatives would vanish so I guess instead it's made use of compactness combined with the levi-civita symbol causing something to vanish? I guess such vanishing might also be the reason we able to write in terms of the '2-d' levi-civita symbol instead, but I'm pretty clueless.

Many thanks in advance.
 
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  • #2
binbagsss said:
how have we gone to only a time derivative in the first term- all spatial derivatives vanish?
Perhaps you have forgotten the definition of ##f_{\mu \nu}##? It is
$$
f_{\mu \nu} = \partial_{\mu} a_{\nu} - \partial_{\nu} a_{\mu} - i [a_{\mu},a_{\nu}],
$$
so the ##f_{12}## term contains spatial derivatives.

Let's just write everything out. Eq (5.46) is
$$
S_{CS} = \frac{k}{4 \pi} \int d^3 x \, \epsilon^{\mu \nu \rho} \mathrm{tr} \left( a_{\mu} \partial_{\nu} a_{\rho} - \frac{2 i}{3} a_{\mu} a_{\nu} a_{\rho} \right).
$$
First of all,
$$
\epsilon^{\mu \nu \rho} a_{\mu} a_{\nu} a_{\rho} = a_0 a_1 a_2 + a_1 a_2 a_0 + a_2 a_0 a_1 - a_0 a_2 a_1 - a_2 a_1 a_0 - a_1 a_0 a_2.
$$
Now, if this expression is inside of a trace, we can use the cyclicity of the trace to combine terms which are the same up to a cyclic permutation. So
$$
\epsilon^{\mu \nu \rho} \mathrm{tr} \left( a_{\mu} a_{\nu} a_{\rho} \right) = 3 \mathrm{tr} \left( a_0 [a_1,a_2] \right).
$$

A similar computation on the other term finds
$$
\epsilon^{\mu \nu \rho} a_{\mu} \partial_{\nu} a_{\rho} = a_0 \partial_1 a_2 - a_0 \partial_2 a_1 + a_1 \partial_2 a_0 - a_1 \partial_t a_2 + a_2 \partial_t a_1 - a_2 \partial_1 a_1.
$$
At this point, we use the fact that everything sits inside of a spatial integral over a compact manifold ##\Sigma## (see the text prior to (5.48)), so we can freely integrate by parts on each term without any "boundary term" appearing. So clearly
$$
\epsilon^{\mu \nu \rho} \int dt \int_{\Sigma} d^2 x \, a_{\mu} \partial_{\nu} a_{\rho} = \int dt \int_{\Sigma} d^2 x \, \left( 2 a_0 \left( \partial_1 a_2 - \partial_2 a_1 \right) + a_2 \partial_t a_1 - a_2 \partial_1 a_0 \right).
$$
Combining the terms, we then find
$$
S_{CS} = \frac{k}{4 \pi} \int dt \int_{\Sigma} d^2 x \, \mathrm{tr}\left( 2 a_0 f_{12} - \epsilon^{ij} a_i \frac{\partial}{\partial t} a_j \right)
$$
So I actually get a different expression than Tong, unless perhaps there is some notational different which is important (e.g. the Levi-Civita symbol has implicit minus signs in Minkowski signature?). I believe that my expression is equivalent to Equations (21) and (64) in this review article.
 
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Related to E-writing Action on a different slice of space-time, Quantum Hall Effect

1. What is "E-writing Action" and how does it relate to the Quantum Hall Effect?

"E-writing Action" refers to the electronic writing or manipulation of data on a computer or other electronic device. This term is often used in the context of studying the Quantum Hall Effect, which is a phenomenon observed in two-dimensional electron systems where the electrons behave as if they have fractional charges and exhibit quantized conductance.

2. How does the Quantum Hall Effect work?

The Quantum Hall Effect occurs when a two-dimensional electron system is subjected to a strong magnetic field. The electrons in this system are confined to move along the surface, and the magnetic field causes them to follow circular orbits. As the strength of the magnetic field increases, the orbits become quantized, meaning that the electrons can only occupy certain energy levels. This leads to the observation of fractional charges and quantized conductance.

3. What is a "slice of space-time" and why is it important in the context of the Quantum Hall Effect?

A "slice of space-time" refers to a specific point or moment in the four-dimensional space-time continuum. In the context of the Quantum Hall Effect, it is important because the phenomenon is observed in two-dimensional systems, which can be thought of as a slice of the three-dimensional space we live in. Additionally, the quantum behavior of the electrons is influenced by the dimensionality of the system and the strength of the magnetic field, making the "slice of space-time" a crucial factor in understanding the effect.

4. How does studying the Quantum Hall Effect on a different slice of space-time contribute to our understanding of the phenomenon?

Studying the Quantum Hall Effect on a different slice of space-time allows us to explore the behavior of the electrons in different conditions, such as varying magnetic field strengths or different dimensions. This can provide insights into the underlying physics and mechanisms behind the effect, as well as potential applications in fields such as quantum computing.

5. What are some potential applications of understanding the Quantum Hall Effect on a different slice of space-time?

Understanding the Quantum Hall Effect on a different slice of space-time has potential applications in fields such as quantum computing, where the phenomenon can be harnessed to create more efficient and powerful devices. It can also help us better understand and manipulate electron behavior in other systems, leading to advancements in electronics and materials science.

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