Dirac equation in one dimension

[wdlang]

i am now studying dirac equation and klein paradox

if we confine to one dimension, we only need one alpha matrix, not three

so in lower dimensions, maybe the dirac spinor is not of four components but fewer?

i am curious about this question because it seems that as for the Klein paradox, we have a dimensional problem but the textbooks still work in a three dimensional fashion.

 

[Dickfore]

Yes, you are right. The Lorentz Group SO(1,1) has only 1 generator corresponding to the only possible boost, since there is no rotation possible in 1D space, thus being isomorphic to the U(1) group, which in turn, is isomorphic to the exponential map $\mathbb{R} \backslash (2 \pi \mathbff{Z}) \rightarrow \mathbb{C}^{\ast}$:

$$\theta \in (-\pi, \pi] \rightarrow \exp(i \theta)$$

 

[Spinnor]

i am now studying dirac equation and klein paradox

if we confine to one dimension, we only need one alpha matrix, not three

so in lower dimensions, maybe the dirac spinor is not of four components but fewer?

i am curious about this question because it seems that as for the Klein paradox, we have a dimensional problem but the textbooks still work in a three dimensional fashion.

I asked a similar question and got quite an interesting reply,

https://www.physicsforums.com/showthread.php?t=403293

 

[xepma]

This is semi-related stuff, but I just feel like "spreading the word" ;)

The Dirac equation in 1+1 and 2+1 dimensions are indeed quite interesting on their own. They even have experimental implementations!

The Dirac equation in 2+1 dimensions for instance pops up as an effective field theory for the electrons in graphene. In this material the electrons are constrained to a two-dimensional surface. You can construct a model in which the electrons are effectively massless and move with a constant velocity (the Fermi velocity). The field theory you need to work with is then the Dirac equation for massless particles in 2+1 dimensions. See for instance this article (with references to the Klein paradox)
http://arxiv.org/abs/0812.1116

The Dirac equation in 1+1 dimensions is realized in for instance the S=1/2 Heisenberg spin chain. This is pretty amazing stuff. The spin chain is critical for certain values of the temperature and coupling constant. The low lying excitations are fermionic (the spin degrees of freedom can be mapped to a fermionic problem). The low-energy regime of this critical state has an emergent symmetry: Lorentz invariance! In effect, the low-lying excitations are described by a 1+1 D field theory: the Dirac equation. Which is again massless.

 

[wdlang]

This is semi-related stuff, but I just feel like "spreading the word" ;)

The Dirac equation in 1+1 and 2+1 dimensions are indeed quite interesting on their own. They even have experimental implementations!

The Dirac equation in 2+1 dimensions for instance pops up as an effective field theory for the electrons in graphene. In this material the electrons are constrained to a two-dimensional surface. You can construct a model in which the electrons are effectively massless and move with a constant velocity (the Fermi velocity). The field theory you need to work with is then the Dirac equation for massless particles in 2+1 dimensions. See for instance this article (with references to the Klein paradox)
http://arxiv.org/abs/0812.1116

The Dirac equation in 1+1 dimensions is realized in for instance the S=1/2 Heisenberg spin chain. This is pretty amazing stuff. The spin chain is critical for certain values of the temperature and coupling constant. The low lying excitations are fermionic (the spin degrees of freedom can be mapped to a fermionic problem). The low-energy regime of this critical state has an emergent symmetry: Lorentz invariance! In effect, the low-lying excitations are described by a 1+1 D field theory: the Dirac equation. Which is again massless.

i am concerned with one problem

it is ok to write an equation of the form of dirac equation

but this does not guarantee the wave function will transform in a way such that the equation is form invariant in a different reference frame

if it is not the case, maybe many good properties of the true dirac equation will be lost?

 

[tom.stoer]

In 1+1 dim. a Dirac spinor has two components.

One representation of gamma matrices is

$$\gamma^0 = \sigma^x$$
$$\gamma^1 = -i \sigma^y$$
$$\gamma^5 = \sigma^z$$

Here the Minkowski metric $$(1, -1)$$ is used.

 

[xepma]

i am concerned with one problem

it is ok to write an equation of the form of dirac equation

but this does not guarantee the wave function will transform in a way such that the equation is form invariant in a different reference frame

if it is not the case, maybe many good properties of the true dirac equation will be lost?

In these examples I mentioned the Dirac equation is an emergent field theory. You are correct in saying that these emergent theories are not invariant under the "true" Lorentz transformations of space-time. However, these examples refer to condensed matter systems. In these systems Lorentz symmetry is already broken by the lattice of atoms and one is not interested in this symmetry anymore because of it. You are correct when you say that a boost in some direction destroys some of the nice features of the system. But since we never consider these transformations to constitute a symmetry of the system it doesn't really matter.

Now, the Dirac equation itself is -- by construction-- invariant with respect to "Lorentz transformations". However, these" Lorentz transformations" are of a different nature then the ones you mention.They are are not the usual boosts to different coordinate systems. For instance, the "speed of light" for these Lorentz transformations is played by the Fermi velocity of the electrons.

The Dirac equation is still invariant with respect to a Lorentz group. That means its solutions also form representations of this group. All nice features you know from high energy physics are carried over to this system.

(sidenote: I shouldn't have said that the electrons become massless in graphene. Instead, the low-lying excitations of the electron fluid can be considered as massless quasiparticles).