# Experimental test on lower dimensional QFT's$.$

In principle, it is possible to formulate a quantum theory in an arbitrary number of spacetime dimensions. This is useful because it allows us to abstract general features of the theory from those that are specific to $d=4$. It is also useful because lower dimensional systems are usually easier to handle and they provide with nice and tractable toy models.

But reality is four-dimensional, so it is not trivial to test the predictions of lower (or higher) dimensional models. In the non-relativistic regime, we may use semi-conductors to localise the dynamics in a plane, line or even a point, so as to study two-dimensional, one-dimensional, and zero-dimensional quantum mechanics. The experiments nicely match the theoretical predictions, as one would expect.

In the relativistic regime, though, I don't know how to test the predictions of the theory in lower dimensions. Say, we study Compton scattering in $d=2+1$ dimensions, or the electron anomalous magnetic moment in $d=1+1$. Is it possible to set up lower dimensional experiments to test these predictions? Or are numerical simulations the only way to proceed?

Note: The question is how to test relativistic predictions, so the experiments should be sensible to relativistic corrections, and in such a way that the paradigm has to be quantum field theory. Semi-classical/effective treatments, such as including a fine-structure term $\hat{\boldsymbol P}^4/8m^3c^2$ in the hydrogen atom in the Schrödinger equation, are not what I'm looking for here. Furthermore, I am concerned with "fundamental" relativistic quantum mechanics, as opposed to an emergent Lorentz symmetry in, say, graphene.

• How about systems like heavy atoms or nuclei that exhibit spherical symmetry yet also show relativistic effects requiring use of the radial Dirac equation for theoretical study? – Lewis Miller Feb 5 '18 at 16:05
• in your view, what is the dimensionality of an experimental setup where collinear high-energy (relativistic) particles collide? – AlQuemist Feb 9 '18 at 9:22
• @AlQuemist good question. I'd say that an idealised one-to-one collision could be considered lower dimensiona; but in practice we collide bunches of particles, so the actual experiment really is $3+1$ dimensional. – AccidentalFourierTransform Feb 9 '18 at 14:50
• I think with the stringent provisions that OP sets (eg., effective low-dim. field theories are not accepted), one cannot achieve better experiments than the (approximately) collimated high-energy beams scattering from each other. Remember that at such high-energies scales, confining particles to lower dimensions is extremely difficult, or practically impossible. – AlQuemist Feb 9 '18 at 18:09
• @AlQuemist effective field theories are accepted. What is not accepted is effective point-particle mechanics, such as the Schrödinger equation or the Dirac equation. I want a QFT answer, not a QM one. Cheers! – AccidentalFourierTransform Feb 9 '18 at 18:11

Quantum simulations with ultracold atoms

Maybe not a direct experimental test, but there are some ideas to use quantum simulations not only to test many-body physics (see e.g. review by Gross & Bloch 2017), but also to simulate gauge theories (see e.g. Zohar,Cirac,Reznik 2015, or this master thesis by Cirac's student).

The idea behind this is to trap ultracold atoms on a lattice and design interactions between them that effectively give you a gauge theory. This may not include all the features of a fully relativistic QFT, but may allow to study some of their properties (e.g. pair production in QED, Kasper 2016).

The effective interaction Hamiltonian can be designed by choosing a variety of parameters:

• trapping potential of the optical lattice
• what atoms to use
• magnetic field (tunes direct interactions via a Feshbach resonance)

In this case it is indeed easier to have low dimensional systems.

Just to show that these are not just theoretical ideas, but some examples have already been implemented:

Disclaimer: I am by not an expert on the topic, the papers above are just examples and by no means an adequate representation of the literature.