# I Vacuum energy cutoff and Lorentz invariance...

1. Dec 24, 2016

### asimov42

One more question before Santa comes. There are a number of different related threads, so hopefully I'm not repeating this - however, I haven't found a crisp answer yet.

If one introduces a UV cutoff in the vacuum energy (in an attempt to avoid having infinite vacuum energy), is it possible at all to preserve Lorentz invariance? I thought at one time I'd read an abstract about this, but can't find it now. In Minkowski spacetime, any UV cutoff would lead to Lorentz violation, correct?

Since there are strong constraints on Lorentz invariance violation from, e.g., astrophysical data (Fermi, etc.), do we have to look to new physics (quantum gravity) for a solution to this problem?

2. Dec 25, 2016

### vanhees71

It is not possible to preserve Lorentz invariance and introduce an energy cut-off in a naive way. It's bad old-fashioned practice. Rather one uses normal ordering of the operators describing energy and momentum in terms of non-interacting fields (in the interaction picture, which only exists in a formal sense due to Haag's (in)famous theorem) and for the radiative corrections (depicted by Feynman diagrams) with loops one has to renormalize. The most straightforward but technically a bit cumbersome way is to not regularize at all but just do the necessary subtractions of the 1PI diagrams a la BPHZ. A very elegant and convenient method is to regularize using some scheme preserving as many symmetries as possible. One of the most elegant ones among these is Dimensional Regularization. Dim. Reg. brought QFT to a new level of success with the PhD work by 't Hooft (supervised by Veltman) and established the Standard Model of elementary particle physics. For a very nice book on the corresponding history (almost a suspense thriller and definitely a "page turner" :-))

Frank Close, The Infinity Puzzle, Basic Books (2011)

3. Dec 25, 2016

### A. Neumaier

It displays on p.5 (and elsewhere) an instance of the vacuum fluctuation myth:
For my account of the infinity puzzle see my tutorial paper Renormalization without infinities - a tutorial, which discusses renormalization on a much simpler level than quantum field theory.

4. Dec 25, 2016

### vanhees71

Yes, Close's book is a popular-science book, and he falls into the trap of this traded fairy tail about "vacuum fluctuations". Nevertheless the history about what I'd call the third period of renormalization history (the first is Feynman, Schwinger, Tomonaga, and the poor Dyson who'd have deserved the Nobel prize as much as the other three; the second Bololiubov, Parasiuk, Hepp, and Zimmermann solving the puzzle about "overlapping divergences", and the third is what was started by Veltman and 't Hooft leading to the proof of the renormalizability of superficially renormalizable gauge models) is told accurately, as far as I know the original papers (I didn't carefully check all the details, but I trust Close here, whose a particle theorist himself).

Your renormalization script is great. I've seen it before.

Another nice example at the same level (showing examples for a simple function, from classical and finally non-relativistic quantum mechanics) can be found here:

Bhattacharjee, J. K.; Ray, D. S., Time-dependent perturbation theory in quantum mechanics and the renormalization group, American Journal of Physics, Volume 84, Issue 6, p.434-442 (2016)
http://dx.doi.org/10.1119/1.4944701

Unfortunately there seems not to be an arXive preprint version.

5. Dec 26, 2016

### asimov42

If one renormalizes in this way, Lorentz invariance is then preserved and concerns about infinite vacuum energy are also solved? This seems quite amazing! Is there a catch? (seems like there must be...)

6. Dec 26, 2016

### A. Neumaier

The vacuum energy is exactly zero after renormalization.

Last edited: Dec 26, 2016
7. Dec 26, 2016

### asimov42

Ah, ok, thanks Arnold! Regarding renormalization, what, then, happens to the picture of the vacuum as containing a quantized oscillator at every point? (hopefully this is not just a philosophical issue)

I know there have been historical objections to the renormalization approach (although less so recently). If renormalization is applied and we end up with a vacuum energy of zero, we have a Lorentz invariant result, which is great - but is this expected to hold at all energy scales? This really goes back to the question immediately above, although through renormalization, sensible, physical results are obtained.

8. Dec 26, 2016

### A. Neumaier

The quantum field has a quantized oscillator at every point. The vacuum is just the state of the quantum field where none of these oscillators is excited.

At all scales currently accessible to experiment. Probably something nontrivial changes at the Planck scale, but what and how is a matter of a not yet existing theory of quantum gravity. But we know already that in curved space, the vacuum is an observer-dependent notion.

9. Dec 26, 2016

### asimov42

Thanks Arnold - I'll ask one more (probably naive) question: in QFT, before renormalization, you have an oscillator at every point, which does have a zero-point energy ... and so is excited, correct? But renormalization results in a state where none of the oscillators are excited? Apologies, as I'm not being precise here...

10. Dec 26, 2016

### A. Neumaier

No. For a single oscillator, one can shift the ground state energy to an arbitrary point as only energy differences have a physical interpretation. Shifting it to zero or something nonzero has no consequences at all - the ground state is always unexcited. But one has to shift the ground state energy to zero if one wants to use oscillators in QFT.
Actually, after renormalization, the oscillators are gone; the Hilbert space of an interacting theory is no longer a standard Fock space - in 4D, what it is is not even known (this is one of the big unsolved problems). There is still a vacuum state, and one continues to use the free notions (for lack of a better terminology), but everything has to be interpreted with caution. Even the conventional particle picture is valid only asymptotically, at times $t\to\pm\infty$ - for the calculation of scattering amplitudes.

Last edited: Dec 27, 2016
11. Dec 26, 2016

### asimov42

Ah, I see - thanks! Since there are (in theory) an infinite number of oscillators, is it possible to shift the ground state of each independently (by a different amount), such that the ground state of the ensemble (not correct terminology) is zero energy? Or is this necessary?

12. Dec 27, 2016

### A. Neumaier

Each must be shifted to whatever it takes to make the ground state have energy zero.

13. Dec 27, 2016

### Staff: Mentor

No. But like many things in QFT you start with a fiction and proceed to the correct explanation. At first you simply introduce a cutoff and not worry about Lorentz invariance. Then you learn the correct solution - normal ordering.

It's the same with virtual particles etc etc in QFT - to get a bit of an early grip and develop intuition you are told these fictions - but later you learn the truth.

Thanks
Bill

14. Jan 13, 2017

### asimov42

If Lorentz invariance is broken in e.g. whatever theory of quantum gravity turns out to be correct, what effect would this have (if one can speculate) on the physical vacuum? That is, for observers moving at constant velocity (so Unruh effect aside), would the vacuum 'look different'? (perhaps different numbers of particles detected...)

15. Jan 14, 2017

### A. Neumaier

Local Lorentz invariance should not be broken in any consistent theory of quantum gravity.
In view of the preceding, an arbitrary answer to this question would be correct, but noninformative.

Due to the Unruh effect, which states that different observers perceive different states as vacuum if they are accelerated with respect to each other, the notion of vacuum effectively loses its objective meaning in quantum gravity. Since the universe is not a vacuum this does not really matter, however.

16. Jan 14, 2017

### asimov42

Ah, interesting! So, for now, we do know that at all accessible energy levels probed to date, our data is consistent with the physical vacuum containing no physical particles - is that correct?

17. Jan 15, 2017

### A. Neumaier

There is no physical vacuum (in the sense of quantum field theory) realized in Nature, so your question cannot be answered.

18. Jan 16, 2017

### oquen

I'd like to ask something with regards to this thread. We know an interaction exists between this chaotic virtual particles and physical matter. It is this fundamental interaction that determines the ground state energies of all the atoms and thus all the molecules and all the condensed matter present in the universe. Is the value of the ground state random (like constants of nature being random).. is it like the Higgs expectation value. What I'm saying is, could the value be different and hence affecting the fundamental interactions depending on the initial condition (or value) of the vacuum?

19. Jan 16, 2017

### Staff: Mentor

Who told you that?

Virtual particles do not exist, they are just a heuristic to get a feeling for some things in QFT. Most certainly what you wrote above is NOT true.

In regards to this thread one simple way to get around the naive infinite energy of the vacuum is a cutoff, but its wrong because its not Lorentz invariant. Its just another 'falsehood' told to beginning students that is later corrected. Some even go as far as to say its infinite and we measure differences so its OK to be infinite. Bollocks. Normal ordering is the correct answer - but more advanced.

Thanks
Bill

20. Jan 16, 2017

### oquen

oh.. then let's not talk about virtual particles.. please go to this thread which I started in order to avoid hijacking this thread, thank you