Recently I have been reading Quantum Field Theory Of Many-body Systems: From The Origin Of Sound To An Origin Of Light And Electrons
The place where I first learned about this sort of thing was some of the work I did in my thesis where I used the Jordan-Wigner transformation in one dimension (A good read: Michael Nielsen’s notes on the Jordan-Wigner transform.) Suppose you have a one dimensional lattice of fermions, where the fermions only interact between nearest neighbors. Let [tex]$a_i$[/tex] and [tex]$a_i^dagger$[/tex] be the annihilation and creation operators at the site [tex]$i$[/tex]. These being fermions, these operators satisfy [tex]${a_i,a_j^dagger}=delta_{i,j}$[/tex] and [tex]${a_i,a_j}=0$[/tex]. In the Jordan-Wigner transform, we replace each fermion site by a qubit, then we perform the map [tex]$a_irightarrow – prod_{j=1}^{i-1} Z_jfrac{1}{2} (X_i + i Y_i)$[/tex]. One can easily check that this mapping preserves the fermion commutation relations. Under this mapping, we can map our nearest neighbor fermion model to a nearest neighbor qubit model. It is exactly this kind of mapping, for more interesting systems, that Wen is excited about.
An interesting question to ask is how to perform the above mapping for lattices of dimension higher than one. To this end, you will notice that the mapping used above has a linear ordering and hence is not well adapted to such a task. In particular if you try to use the mapping in this manner, you will end up creating qubit Hamiltonians with very nonlocal interactions. In fact, many have tried to create higher dimensional Jordan-Wigner transforms, but in general, there were always limiations with these attempts. To this end, the recent paper cond-mat/0508353 by F. Verstraete and J.I. Cirac is very exciting. These two authors show that it is possible to convert any local fermion model into a local model with qubits (or qudits), i.e they effectively solve the problem of creating a Jordan-Wigner transform on higher dimensional lattices.
One of the points that Wen likes to raise from this work is the question of whether fermions are actually fundamental. From what I understand, while there are examples of fermions arising from these local interacting boson modes, it is not known how to do this with chiral fermions. Strangely I’ve always been more inamored with fermions than with bosons (holy cow am I a geek for writing that sentence.) But perhaps my love of bosons will have to start growing (oh, that’s even worse!)
O brave new quantum world!
Despite being a LaTeX nut myself, I would be wary of recommending that someone should learn the whole thing if they just want to put one or two equations in their presentations.
If you are using Word/Powerpoint then there is an equation editor that comes with office. It’s not installed as standard, so you have to dig out your installation CDs. It’s easy to use, but it takes a long time if you have loads of equations. However, it is fine for just 1 or 2.
Given that you are a Mac user, here are some other options:
– Apple Works 6 has the same equation editor as Office (but Apple Works sucks). On the plus side, it handles putting equations inline with text much better than Powerpoint for some reason.
– It is possible to insert equations into Keynote/Pages via a similar utility to TeXpoint (I forget what it is called). It has the advantage that the equations are pdfs rather than bmps, so they don’t get blocky when you rescale them as in TexPoint.
– If you have the equation in a pdf document somewhere already then you can just cut and paste it into any program on a Mac provided you have the latest version of preview.
– If you are using OpenOffice (or NeoOfficeJ) because you think Microsoft is evil, then there is an equation editor that is slightly better (but much slower) than the MS Office one. I don’t recommend this option unless you have a fast Mac, but on the plus side it can save in Word/Powerpoint format.
– Finally, if you want to take the LaTex plunge (the best option if you have lots of equations), but can’t be bothered to learn the whole thing then there are a few WYSIWYM (What You See Is What You Mean) editors available that do most of the markup for you. LyX is my favourite one, and it can produce pdfs of equations from which you can use your Mac’s cut and paste ability.
I realize that the whole discussion has got a little off topic, so I’d just like to say that I love the Jordan-Wigner transformation too!
A totally off-topic question: what’s a good way to type in mathematical formulas to use for presentations or papers? I guess maybe some people can do this in matlab or mathematica, but are there other ways to do this? I find it incredibly annoying (being a biologist) and not having ready access to typical math programs, but occasionally needing to put up a formula and making do with a lame graphing calculator program on my Mac. (It does not suffice.)
Hey Suz, most people in physics use LaTeX for their papers. For my talks, I use a plugin for powerpoint, called TeXpoint which inserts pictures of LaTeX files into the powerpoint slides.
Biologists are really funny. You often submit with word files right?
okay, thanks for the advice. Actually I have access to LaTex but tried to start using it for my thesis and decided I didn’t have the extra time to learn LaTex. On the other hand, the time spent it could be viewed as an investment for the future, as I hope to move into a more quantitative field.
Today during a biology presentation someone presented bar graphs with no error bars at all and I nearly choked in my seat.