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)(adsbygoogle = window.adsbygoogle || []).push({});

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

[tex]

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

[/tex]

Then we consider the fact that [itex] \phi \sim e^{-iEt}[/itex]. In the NR-limit we write

[itex]E = m + \epsilon[/itex] and note that m is much bigger than epsilon.

So, Zee sees this as a motivation to write

[tex]

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

[/tex]

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

[tex]

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

[/tex]

where the nabla-operator is spatial. Now the second time derivative on psi is neglected, and we get the Schrodinger 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 Schrodinger equation is also known. So my guess would be that somehow we can obtain the Schrodinger 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 :)

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

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