How do you integrate the Hopf term in 2+1 dimensions?

In summary, the conversation is about the integration of the Hopf term in a Lagrangian with a Chern-Simons term in 2+1 dimensional spacetime. This defines the fractional statistics of particles in the system. Despite following the instructions given in a book and reference, the integration seems impossible and the result is zero. The conversation ends with a request for help and a question if anyone has succeeded in finding a closed form solution.
  • #1
Ian Lovejoy
7
0
Hi,

I'm reading Quantum Field Theory in a Nutshell by A. Zee, which is excellent but it is occasionally difficult to fill in the blanks on one's own.

Once such place is the integration of the Hopf term, which results from integrating out the gauge field in a Lagrangian with a Chern-Simons term in 2+1 dimensional spacetime.

The Hopf term is:

[tex]L_{Hopf}=\frac{1}{4\gamma}j_\mu(\frac{\epsilon^{\mu\nu\lambda}\partial_\nu}{\partial^2})j_\lambda[/tex]

From here we are supposed to be able to define a current j representing one particle at rest at the origin while another particle goes halfway around it. Integrating the above with this current we are supposed to obtain [tex]1/4\gamma[/tex]. This defines the fractional statistics of the particles in the system.

Except for a factor of 2 the same treatment appears in section II of this document:

http://arxiv.org/PS_cache/cond-mat/pdf/9501/9501022v2.pdf

It all seems perfectly straightforward but I'm unable to do the integration. For [tex]1/{\partial^2}[/tex] I am using:

[tex]\int{\frac{d^3k}{(2\pi)^3}\frac{-e^{ik(x - y)}}{k^2 + i\epsilon}}[/tex]

No matter what order I do the integration in, I seem to get either an integral that is impossible to do, or an result that is apparently zero. The book and the above reference seem to imply that the result is easily obtained by plugging in the current into the Lagrangian.

Can anyone give me a hint? It would be much appreciated.

Thanks,
Ian
 
Physics news on Phys.org
  • #2
First point to be noted: You cannot interchange the two currents. If you could, by antisymmetry of the epsilon tensor, the expression is identically zero.

The way the above makes sense is if the derivatives act on the right hand current.

Now, for a static charge the only nonzero component of the current is the zeroth component.
For one that is moving, something else, say the x or the y component is also nonzero.

Moving halfway around: say it moves on two consecutive sides of a square - first along the x and then along the y direction...

does this help?
 
  • #3
I'm also stuck on this question, simply can't integrate out a closed form. Anyone succeeded doing it?
 

1. What is the Hopf term in integration?

The Hopf term in integration refers to a type of mathematical term used in the theory of differential geometry. It was first introduced by mathematician Heinz Hopf in the 1930s and is used to describe the topological properties of a manifold.

2. How is the Hopf term integrated into mathematical equations?

The Hopf term is integrated into mathematical equations through the use of differential forms. It is typically integrated over a closed surface or manifold, and the resulting value is used to calculate the Euler characteristic of the manifold.

3. What are the applications of the Hopf term in science?

The Hopf term has various applications in science, particularly in the fields of physics and mathematics. It is used in the study of fluid dynamics, knot theory, and topological quantum field theory, among others.

4. Can the Hopf term be visualized?

Yes, the Hopf term can be visualized in 3-dimensional space using a geometric construction known as the Hopf fibration. This involves mapping points on a 3-sphere to points on a 2-sphere, allowing for a visual representation of the Hopf term.

5. How does the Hopf term relate to other mathematical concepts?

The Hopf term has connections to various other mathematical concepts, such as differential geometry, algebraic topology, and Lie groups. It is also related to the concept of the Hopf algebra, which is used in the study of quantum mechanics and field theory.

Similar threads

Replies
5
Views
287
Replies
1
Views
816
  • Quantum Physics
Replies
3
Views
298
  • Quantum Physics
Replies
1
Views
1K
Replies
2
Views
410
Replies
1
Views
608
Replies
24
Views
2K
  • Quantum Physics
Replies
1
Views
583
  • Quantum Physics
Replies
14
Views
2K
  • Quantum Physics
Replies
11
Views
1K
Back
Top