I'd like to expand a bit about a comment I did in the other subforum. I find it amusing that while Planck scale is a good cutoff for Gauge theories, it is not a good one for gravity?(adsbygoogle = window.adsbygoogle || []).push({});

The point is that for a cutoff [tex]\Delta x[/tex] in position space, we expect to use a cutoff [tex]\Delta p= \hbar /\Delta x[/tex] in momentum space. If the dynamics comes from a force with coupling constant about unity (ie, hc) at the cutoff scale, then both cutoffs are consistent.

This is because the change in momenta is equal to the force times the time interval. As space and time are treated in the same footing, we have that at a distance [tex]\Delta x[/tex] we expect momentum exchange of order [tex]F(\Delta x) {\Delta x \over c }[/tex]

Say otherwise, if we want to regulate with Planck scale, we will ask for space intervals up to Planck Length [tex]l_P[/tex] and forces up to Planck Force [tex]F_P \equiv {\hbar c \over l_P^2}[/tex]

Now, consider two particles at Planck Distance, this is, in the limit of the spatial cut-off. Its gravitational force is

[tex]

F_G= \hbar c {m_1 m_2 \over M_P^2} {1 \over l_P^2}

[/tex]

and it does not saturate the Force cut-off because elementary particles are far small than Planck mass. Thus consistency of the Planck length as regulator implies the existence of other forces beyond gravity: the Gauge forces. For instance if 1 and 2 have electromagnetism, then an extra force

[tex]

F_E= \alpha(\l_p) {1 \over l_P^2}

[/tex]

saturates the force cut-off.

Now, string theory automatically provides other forces (but when it provides them via Kaluza Klein it should have the same problem; I think modern strings use other techiques to induce the gauge group). As for LQG, it is sometimes touted as a "gravity only" theory, so I can not see how does it bypass this consistency problem.

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# QG self consistency and cut-offs

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