Monday, December 05, 2005

The Nielsen-Ninomiya theorem

In recent posts in this series, we have been looking at naive, Wilson and staggered fermions. One of the things we have seen is how difficult it is to get rid of the doubler fermions; staggering did a good job at this, but still retained some of the doublers with all the problems they bring, while the Wilson term got rid of the doublers, but only at the expense of spoiling chiral symmetry, which brought on even worse problems. Why should the discretisation of fermions be so hard?

The answer lies in a theorem about lattice fermions, the celebrated Nielsen-Ninomiya no-go theorem, which states that it is impossible to have a chirally invariant, doubler-free, local, translation invariant, real bilinear fermion action on the lattice. The theorem comes from topological arguments: A real bilinear fermion action can be written as
$$S = \sum_{x,y} \bar{\psi}(x)M(x,y)\psi(y)$$
with hermitean $$M$$. Translation invariance means that
$$M(x,y)=D(x-y)$$
and locality requires that the Fourier transform $$\tilde{D}(p)$$ of $$D(z)$$ be a regular function of p throughout the Brillouin zone. Chiral symmetry
$$\{\tilde{D}(p),\gamma^5\}=0$$
requires that
$$\tilde{D}(p)=\sum_\mu \gamma_\mu d_\mu(p)$$
Since the Brillouin zone has the topology of a 4-torus, we thus have a vector field $$d_\mu$$ on the torus. Now it is possible to assign an "index" of +1 or -1 to every zero of this vector field, and the Hopf-Poincare index theorem states that the sum over the indices of the zeros of a vector field on a manifold is equal to the Euler characteristic of the manifold. The Euler characteristic of any n-torus is zero, and therefore the zeros of $$d_\mu$$ must come in pairs of opposite index, which is precisely the origin of the doublers.

OK, so what does all this mathematics mean? Well, prima facie it seems to leave us with the choice between chiral symmetry and freedom from doublers (since locality, translation invariance and hermiticity are too important to abandon). There is, however, a clever way around this, which will be the topic of our next post.

4 comments:

Georg said...

In this post I have for the first time been using images to display equations. Since this means extra work uploading the images, I will continue this only if people really want it; so tell me if you strongly prefer this to TeX'ed equations. Otherwise I will probably revert to TeX-style equations.

Jacques Distler said...

Though I'm perfectly fluent in TeX, I find reading TeX source, rather than the rendered equations to be a drain on my attention.

Still, it must be a lot of work to render the equations, and then upload the image files.

Maybe I should offer you guys a hosted, TeX-enabled blog on Golem?

Send me an email, if you're interested, and we can discuss it.

Anonymous said...

Hi Georg,

Thank you for taking the time to generate the pictures!

I actually played around with setting up a small script yesterday to parse your page and generate the pictures for the latex equations.

for example,
http://www.elventails.com/lol_script/daily/

Its cronned to parse a copy daily. I could update it to inline with the text, etc..which i'll probably do at some point.

Ideally, we can get the blogger.com folks to look into allowing mathml.

Cheers,

andrewhix said...

I like the images! please keep using images.