Vacuum energy cutoff and Lorentz invariance....

[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?

 

[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)

 

[A. Neumaier]

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

It displays on p.5 (and elsewhere) an instance of the vacuum fluctuation myth:

A vacuum is not empty but seethes with transient particles of matter and antimatter, which bubble in and out of existence.

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.

 

[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.

 

[asimov42]

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.

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...)

 

[A. Neumaier]

concerns about infinite vacuum energy are also solved?

The vacuum energy is exactly zero after renormalization.

 

[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.

 

[A. Neumaier]

the picture of the vacuum as containing a quantized oscillator at every point?

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.

is this expected to hold at all energy scales?

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.

 

[asimov42]

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.

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...

 

[A. Neumaier]

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?

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.

But renormalization results in a state where none of the oscillators are excited? Apologies, as I'm not being precise here...

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.

 

[asimov42]

No. for a single oscillator, one can shift the ground state energy to an artbitrary point as only energy differences have a physical interpretation. Shifting it to zero or something nozero 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.

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?

 

[A. Neumaier]

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?

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

 

[bhobba]

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?

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

 

[asimov42]

Ah, ok, one more question (Hope thread isn't too long already):

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...)

 

[A. Neumaier]

If Lorentz invariance is broken in e.g. whatever theory of quantum gravity turns out to be correct

Local Lorentz invariance should not be broken in any consistent theory of quantum gravity.

for observers moving at constant velocity (so Unruh effect aside), would the vacuum 'look different'?

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.

 

[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?

 

[A. Neumaier]

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?

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

 

[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?

 

[bhobba]

I'd like to ask something with regards to this thread. We know an interaction exists between this chaotic virtual particles and physical matter.

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

 

[oquen]

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.

Thanks
Bill

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
https://www.physicsforums.com/threads/vacuum-values.900484/

 

[A. Neumaier]

We know an interaction exists between this chaotic virtual particles and physical matter.

No. We know there are interactions between various fields (as encoded in the action or Hamiltonian). Virtual particles appear only in Feynman diagrams, and their alleged interaction is just a spot on the diagram.

Is the value of the ground state random

The ground state is not a variable that could have a value. It is a special state of the whole system, in which nothing happens.

 

[asimov42]

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

Are the differences due to different field content? I thought the QFT vacuum was the best approximation we had to the true vacuum state?

Also, previously, Prof. Neumaier, you mentioned that a consistent theory of quantum gravity should be locally Lorentz invariant. However there are a range of theories now (outlined, e.g., here: arXiv:hep-ph/9812418v3, as provided by Haelfix) that break local Lorentz invariance, at least slightly (with very significant fine tuning, granted). One question that seems to not have a clear answer is, if Lorentz invariance is violated, whether this would lead to a different vacuum energy (VEV) in different directions in spacetime.

Hoping anyone can shed light on this, or point me to where I might look.

 

[A. Neumaier]

I thought the QFT vacuum was the best approximation we had to the true vacuum state?

THe QFT vacuum describes the state of a universe in which nothing exists, in the sense that all elementary (renormalized) quantum fields vanish. This is the only true vacuum state. once there is matter, it induces gravitational and electromagnetic fields everywhere in space-time (as far as causally reachable), resulting even locally in a state that is no longer a vacuum.

if Lorentz invariance is violated

Yes, there are studies looking at what happens if lorentz iunvariance is violated. But there are no experimental signs that it is, so this possibility can well be ignored at present.

In any case, it has nothing to do with vacuum energy. It makes no sense fishing in the dark, suspecting vacuum energy everywhere ...

 

[asimov42]

Thanks Prof. Neumaier - I think this is starting to make sense... just for my own, final clarification, from previous posts:

The vacuum energy is exactly zero after renormalization.

So zero vacuum energy after renormalization in QFT.

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.

Yes, there are studies looking at what happens if Lorentz invariance is violated. But there are no experimental signs that it is, so this possibility can well be ignored at present.

In any case, it has nothing to do with vacuum energy. It makes no sense fishing in the dark, suspecting vacuum energy everywhere ...

So cases of Lorentz violation, as currently studied, have nothing to do with vacuum energy; referring to the quote just above (about changes at the Planck scale), nothing can be said about whether vacuum energy changes at all, as the energy scale increases (i.e., up, down, the same, something entirely different - who knows)? (as making any kind of assumption about quantum gravity doesn't make sense, at this stage)

 

[A. Neumaier]

[1] So zero vacuum energy after renormalization in QFT.

[2] So cases of Lorentz violation, as currently studied, have nothing to do with vacuum energy; referring to the quote just above (about changes at the Planck scale), nothing can be said about whether vacuum energy changes at all, as the energy scale increases (i.e., up, down, the same, something entirely different - who knows)? (as making any kind of assumption about quantum gravity doesn't make sense, at this stage)

[1] Yes.

[2] Yes. If Lorentz symmetry is violated, it must involve fundamental changes in the basic theories. The resulting alternative theories might predict many things in many directions, depending on who makes suggestions. Nothing of this is reliable since it is speculation about something so far unrelated to experiment and hence uncheckable.

 

[_PJ_]

Can quantum fluctuations be renormalised not with cutoff per se, but as consideration that on short scales, there are negative contributions and positive that average and balance? This average may be zero, nonzero or even dynamic?

 

[A. Neumaier]

Can quantum fluctuations be renormalised

One renormalizes the theory, the fields, and the coupling constants, not the fluctuations.

 

[_PJ_]

One renormalizes the theory, the fields, and the coupling constants, not the fluctuations.

Indeed.
Sorry I worded that badly for efficacy. I meant the uncertainty in energy (i.e. Hamiltonian) for the relevant field.

 

[A. Neumaier]

Indeed.
Sorry I worded that badly for efficacy. I meant the uncertainty in energy (i.e. Hamiltonian) for the relevant field.

The energy is certain in eigenstates of $H$, in particular in the vacuum state, and uncertain in all other states. This is completely independent of renormalization.

 

[asimov42]

Prof. Neumaier, thanks for your answer - and also very much for your willingness to share your expertise on the forums here (it's much appreciated). Hopefully, one last curious question - I peeked at a previous StackExchange post, where you'd commented on a related question (topic: "Effects of a non-Lorentz-invariant vacuum state"),

Indeed, in quantum gravity, there doesn't seem to be an observer-independent notion of a vacuum state, due to the Unruh effect. Instead one has a family of Hadamard states that, taken together, replace the vacuum state of flat QFT. This family as a whole is indeed Poincare invariant, even invariant under the group of volume-preserving diffeomorphisms.

In this case, is each Hadamard state Poincare invariant? If the above is correct, does this, in any way 'fix' the problem of Lorentz violation and whatever happens to vacuum energy?