What happened to the position basis?

[Bobhawke]

In QFT states in Hilbert space are labelled by particle type and number, momentum, spin and internal quantum numbers. But surely it is equally possible to replace the momentum label by position - why does one never hear of the position basis being used in QFT, and why is there a momentum operator but not a position operator? Why are there only creation and annihalation operators for particles of momentum p, but not for particles of position x?

 

[Haelfix]

Its a controversial subject and you should do a search on this board and the quantum one for past threads about this (they are quite lengthy).

Their is actually such a purported objected and its called the Newton-Wigner position operator, but it has some technical issues and is still argued about in the literature to this day. Further calculating things in this basis is quite unlovely compared to the momentum basis.

Anyway, the really basic, intuitive and naive problem with a position operator in qft (say as taught in the Coleman lectures) is that you can imagine placing a single particle in a rigid box and let it vibrate and move around. Then you squeeze the box. Pretty soon, you are looking at smaller and smaller wavelengths, which means particle creation. So you start to wonder, exactly what position am I measuring in the first place.

 

[hamster143]

I think it's mainly because 99% of QFT research in the last fifty years was on perturbation theory, and you can only do perturbation theory in momentum basis. In a free theory, eigenstates of momentum evolve with time in a very simple manner (a particle with momentum p remains a particle with momentum p, evolve it and you just get a multiplicative phase), whereas eigenstates of position are messy (you start with a particle localized in point x, you evolve the system, and you get a complex superposition of particles localized in all points inside x's future light cone, and some particles outside it). If you perturb about that free theory, you get small probabilities for momentum eigenstates to interact and become different momentum eigenstates, which are, to the first order in coupling constant, just Feynman tree diagrams. You can't simplify the theory like that in position basis.

You can easily convert between one and the other (it's just a linear transformation), but position operators just happen to be much less useful for everyday computations.

Now, of course, once you get into strongly coupled scenarios such as lattice QCD, tables are turned and position basis suddenly becomes interesting. And if you want to quantize gravity, you can't (or shouldn't) do that in momentum basis either, because real physics is local and momentum basis is inherently not - it just happens to work because we assume in regular QFT that spacetime is Minkowski. If you want to do quantum gravity (seriously - not just an IR approximation), Minkowskianness, Fourier transforms, etc. all go out the window and you're forced to do quantum field theory in a way that's counter to all your training.

There are some interesting aspects of QFT in position basis, such as Feynman checkerboard, polymer model of fermions, ... But overall I don't think that area is all that heavily researched.

 

[genneth]

Errr. What? A lot textbooks start things in the position basis, deriving all perturbation theory, including Feynman diagrams, before moving into the momentum basis. See, ooo, I don't know, Peskin & Schroeder. The interaction vertex is, after all, just a delta in the position basis.

 

[Bobhawke]

Ive searched but I really can't find any of these threads you speak of. Do you know if there are any good summary papers about this kind of thing?

 

[Haelfix]

I mentioned Colemans lectures (found on the Harvard website), also a literature search on Newton-Wigner.

When one works in the position basis in QFT, everything is quite formal (there are delta functions all over the place that are problematic in the interacting context), but it is utilized for instance as Hamster indicates in quantum gravity. Matrix theory is almost entirely done in the position basis and that's fine. It is, after all, a Fourier transform of something that is (relatively speaking) well defined.

The actual operator generating position is where it gets a little dirty, even in the free case. One of the problems is existence, and thas where it gets controversial and gets tied into the sticky problem of localization (and hence philosophy of science -- what 'is' a particle). The Newton Wigner operator has issues with massless particles and helicity for instance.

Historically the perceived failure of localization was a motivation for introducing quantum field theory in the first place!