Non-relativistic limit of Klein Gordon (and massless limits)

[haushofer]

Hi, I have a conceptual question about taking non-relativistic (NR) limits of the Klein-Gordon equation, inspired by Zee's book on QFT (chapter III.5)

So we have a complex scalar field with the equation of motion

$$(\partial^2 + m^2 ) \phi = 0$$

Then we consider the fact that $\phi \sim e^{-iEt}$. In the NR-limit we write
$E = m + \epsilon$ and note that m is much bigger than epsilon.

So, Zee sees this as a motivation to write

$$\phi(x,t) = e^{-imt}\psi(x,t)$$

because the field $\psi(x,t)$ oscillates much slower in time. Plugging this in the equations of motion gives you

$$[\partial^2_t - 2im\partial_t +\nabla^2 ]\psi = 0$$

where the nabla-operator is spatial. Now the second time derivative on psi is neglected, and we get the Schrödinger equation back.

My question is: why do we neglect the second time derivative? Because it's not multiplied by m as the first order time derivative is? Or is there another reason?

Another question which is a little related is: obviously, the KG equation has a certain symmetry group, and the symmetry group of the Schrödinger equation is also known. So my guess would be that somehow we can obtain the Schrödinger symmetry group by the "Klein Gordon symmetry group" via a group contraction. Is this suspicion right?

And as a final question: does it make sense to look at the massless limit in this NR-limit? You can also perform a group contraction on the conformal algebra which describes massless particles to obtain a NR-limit called the "Galilean Conformal algebra". What does this exactly mean to describe "massless non-relativistic particles"?

Any thought will be appreciated :)

 

[haushofer]

Ok, I'm fairly convinced now that we're neglecting second order time derivatives with respect to the first order time derivative due to the mass term. But my other questions are still tickling me :)

 

[javierR]

These seminar slides may address your questions:
http://hep.physics.uoc.gr/mideast5/talks/Saturday/Gopakumar.pdf

In introductory quantum field theoretic situations, massless states such as photons need to be discussed relativistically.
However, note that the term "massless" is in quotes throughout the above slides when referring to the Gal. Conformal Group. We can imagine physical situations in condensed matter in which effectively massless objects exist and travel with a characteristic speed much less than the speed of light.

But also, as the authors of the above work point out, there is a corresponding group contraction in the AdS/CFT correspondence on the "AdS side": One can talk about "masslessness" in AdS space since there are different notions of mass in such a space; a physical one and a "Kaluza-Klein" type mass. There are states in which the latter can be zero while the former is non-zero (and vice-versa for that matter). The contraction in this context is to a "Galilean limit of AdS space". On the "CFT" side, it is conjectured that there should be a corresponding group contraction describing a "massless Galilean conformal field theory", in which the naive problem of non-relativistic massless states is somehow avoided (e.g. though a dynamical mass gap).

 

[haushofer]

Hey, thanks for the link. I'm already familiar with Gopakumar's work on this, but maybe this clearifies things. If not, I'll come back to things here later ;)