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One of my professors told us this semester, that the 'infinities' that arise in QFT are partly due to the use of the $\delta$-distribution in the commutator relations which read (for fermions)

$\left\{\Psi(r'), \Psi^\dagger(r)\right\} = \delta(r-r')$

In reality we would not have such a $\delta$-distribution but an extended version of it.

Is this view correct? And if definitely yes, is my following view wrong?

As far as I understand it, the $\delta$-distribution is due to the fact that we deal with point particles. If e.g. the electron was an extended particle, then the $\delta$-distribution would be 'finite'.

Since experiments pin down the extension of a particle to $R < 10^{-18}m$ it is also likely that the $\delta$-distribution should really be there.

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  • $\begingroup$ The fact that $\delta (0)=\infty$ is really enough to see that this will yield some infinities when (anti-)commutation relations have to be employed. Since we don't measure $\infty$ in experiments, I think it's reasonable to say that this mathematical framework is an idealization. $\endgroup$ – Danu Mar 7 '14 at 10:10
  • $\begingroup$ I guess what my professor meant was infinities that we get when calculating cross sections. The $\delta$-distribution is of course infinit at $r = r'$, but it is well defined if we assume that we integrate over it to get the cross section. What I am more interested in is, what is the physics behind the $\delta$-distribution? Is it's occurrence really due to the concept of point particles in QFT? $\endgroup$ – physicsGuy Mar 7 '14 at 10:43
  • $\begingroup$ I'm afraid you will get things like $\delta^2(x)$ or even higher power to integrate, in such case if you calculate cross section it is not just that at certain point you get a infinite density, it is that the total cross section also become infinite, which is unphysical. $\endgroup$ – Jia Yiyang Mar 8 '14 at 2:07
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Yes -- they are the cause of ultraviolet (short distance) infinities in QFT. QFT could also have some other infinites in the infrared (large distance), which roughly come from massless degrees of freedom.

$10^{-18} m \sim 10 TeV$ (since $1000 nm \sim 10 eV$ ) is not that small for us, since we understand physics till almost one order of magnitude below that scale, and field theories seem like good working descriptions. So if the $\delta$-functions get resolved, it has to be at a much smaller length scale. Eg: String theory proposes one possible way of resolving them, by saying that if you zoomed in, you'd see vibrating extended objects (strings).

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