(March 4, 2017 at 7:36 am)Alex K Wrote: What we have here is the so-called path integral of the Standard Model of particles physics, embedded in a quantized version of Einstein's General relativity.
There's one symbol hidden in this very formula which indicates that this description is not complete. It's the upper case letter Lambda (looks like an A without the cross bar) right below the first integral sign (the big S).
k < Lambda tells you that we only take into account energies below the so-called cutoff energy scale Lambda, and ignore any goings-on higher than lambda. This is necessary because general relativity is what is called non-renormalizable - it contains infinities as you go to arbitrary large energies which cannot be gotten rid off in a consistent manner unless you introduce an infinite amount of parameters into the theory, which is not done in the above formula.
To a certain extent, the physics that goes on above this energy scale where we cut, can be represented as shifts in the known physical constants (this is made sure by the famed Appelquist-Carazzone-Theorem), but this is not an exact procedure:
strictly speaking, we need to include a whole infinite tail of additional field interaction terms into this formula to really capture the complete physics as observed at low energy processes.
As long as we study processes at energies far below the cutoff energy lambda, these additional terms contribute less and less to the physical goings-on and can be neglected to very good precision. But the necessity to have Lambda at all shows us the incompleteness of quantum gravity which is so often talked about.
If in the above formula you leave out the part labelled "Gravity", there is a consistent procedure to send Lambda to infinity and consistently get rid of this upper limit - this procedure is called renormalization and was introduced among others by Feynman, Schwinger and Tomonaga in the 1940s for the theory of quantum electrodynamics, which got them their Nobel. Gerardus 't Hooft and Martinus Veltman got their Nobel in 1999 for showing that renormalization also works for the full Standard Model of particle physics. Interestingly, they also showed that the Higgs Boson is essential theoretically in order to remove the cutoff energy scale from the theory. For Einsteinian gravity however, the standard renormalization procedure doesn't simply work because of the above-mentioned infinities. There are theorists who try to argue that there are renormalization prescriptions which do indeed work, but this issue is far from settled.
It's not my formula, Alex! But, thanks (again) for your very wonderful explanation!! Carroll, by the way, would say that k < lamba is for nearly everything that happens in our World, including, the stuff that is going on in our brains, and so, basically, the so-called Core Theory is, more or less, complete.
