Infinities in QFT come from high momenta. I sometimes hear that is equivalent to coming from short distances, but I'm not sure I see the connection.(adsbygoogle = window.adsbygoogle || []).push({});

The free propagator G(x-y) which I think goes like 1/|x-y|^2 is singular for short distances (when x=y). In momentum space G(x)= ∫d^4k exp[ikx] /(k^2+m^2). If x=0, then clearly ∫d^4k exp[0] /(k^2+m^2) diverges for large momenta k. However, if x is not equal to zero, then the expression is perfectly finite even when k is allowed to be really large in the integral. So it seems distance is even more important than momenta in determining the divergence of the free propagator.

Also, for a phi^4 theory it's true that the self-energy at one loop is G(z-z)=G(0) where z is the vertex coordinate that is integrated over. This leads to infinity. So you can say short distances leads to infinities.

However, for phi^3 theory, it is this term which leads to infinities: G(y-z)G(y-z) where y and z are the vertex coordinates that are integrated over. You have an infinity here that's caused by repetition of the Green's functions rather than y=z.

So for the phi^3 case, the infinities aren't caused by short distances between the vertices.

So are infinities really explained by short distances?

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# Infinities and short distances in QFT

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