# I Vacuum cutoff, normal ordering, Planck scale and 10^120...

1. Mar 6, 2017

### asimov42

Hi all,

This is a followup to a question from a couple of months ago regarding vacuum energy and cutoffs, basically to clarify some ideas.

Given the usual picture of the vacuum as containing quantized harmonic oscillators at every point, it is not possible to apply a 'fixed' energy cutoff (at, for example, the Planck scale) and maintain Lorentz invariance. One must then deal with the problem of infinite vacuum energy - however, the solution to this is to apply normal ordering in the context of renormalization, which results in a 'physical' vacuum energy (vacuum expectation value) of zero after renormalization, and this is the approach in QFT. Hoping this is correct so far.

Based on general relativity, an infinite vacuum energy should result in infinite curvature (given the energy density), as the ground state energy can't be ignored. However, I often see the quote that the vacuum energy, based on measures of spacetime curvature (in association with the cosmological constant), is 120 order of magnitude too small (or smaller than expected). That is, one would expect a value 120 orders of magnitude larger.

Now my question: the quoted value of 120 orders of magnitude is based on a Planck-scale cutoff (is must be based on a cutoff, otherwise the difference would be infinite). But we know that in QFT a fixed energy cutoff will result in Lorentz violation. Since the idea that a cutoff must be imposed is at odds with QFT and Lorentz invariance, is there some way to square this issue that I'm not aware of (Lorentz invariance with a cutoff somehow - which from previous posts I believe is not), or when one sees the quote of 120 order of magnitude, is the assumption simply that some type of limit will be necessary, and that at the Planck scale 'new physics' from a theory of quantum gravity will explain the issue?

Upshot: why quote a difference of 120 order of magnitude?

Thanks all!

2. Mar 6, 2017

### haushofer

Well, for one thing: QFT assumes a continues spacetime background, so if you send the cut-off to infinity, you explicitly assume this continuous background. If spacetime is quantized, this would necessarily introduce a cut-off like the Planck-length; in taking the cut-off to infinity, you effectively integrate out the quantum-gravity effects.. A finite cut-off seems to violate Lorentz invariance, but since the whole notion of continuous symmetries like those of the Lorentz algebra breaks down in a quantized spacetime, I'm not sure how big of a problem such a cut-off would be.

Related to that I wonder whether a quantized spacetime would somehow give us insight into the whole notion of renormalizability, since naively it would seem that every theory becomes finite. But this is all hand-waving.

By the way, I think this topic should belong in the Beyond the Standard Model subforum :)

3. Mar 6, 2017

### asimov42

But wait, from a GR perspective, isn't taking the cutoff to infinity the most problematic case? In that the vacuum energy at each point in the continuum is infinite (and so the difference is not 120 order of magnitude, it's infinite). When you say taking the cutoff to infinity integrates out quantum-gravity effects, do you mean purely on the QFT side?

4. Mar 7, 2017

Yes.