String Theory & ZPE: Is There a Connection?

[Kittel Knight]

Is there any relationship between String Theory and ZPE?
I mean, does ST offer anything new, any new reason to ZPE?
Thanks!

 

[Careful]

Is there any relationship between String Theory and ZPE?
I mean, does ST offer anything new, any new reason to ZPE?
Thanks!

Are you hoping that ZPE is something *physical* ?

 

[mitchell porter]

Supersymmetry, which is a feature of string theory, makes the zero-point energies of particles and their superpartners cancel.

But first let's talk a little about the meaning of ZPE. In popular culture, people want to use ZPE as an energy source. In modern physics, however, the ZPE of fields in a vacuum was an experimental problem because all that energy ought to be generating a huge gravitational force that isn't there.

One of the attractions of supersymmetry is that it gets rid of this force, because the ZPE of a boson field can cancel the ZPE of a fermion field, and supersymmetry gives every fermion a boson partner, and vice versa. But if you have exact supersymmetry, these superpartner pairings should have the same mass, which should make the superpartners of the known particles already detectable, and they're not. So usually it is supposed (by string theorists) that supersymmetry is broken slightly, and the undetected superpartners have big masses after all, and that they are the dark matter detected in astronomy.

In astronomy, there's also the "dark energy" which is accelerating the expansion of the universe. We know almost nothing about it except its overall effect, but many models would explain it as due to the energy of some extra fields. I don't know much about this, but I think in some models this energy comes from the vacuum energy of those fields, which is the same as their ZPE. (In other models, the fields have a "nonzero vacuum expectation value", so they're not in their lowest energy state, and the dark energy is coming from field excitations.) So it's possible that residual ZPE from supersymmetry-breaking is having enormous effects on cosmological scales and timescales.

Whether this can help anyone utilize ZPE technologically is another question. Another example of an enormous force active everywhere in the universe is ordinary gravity. But that doesn't mean you can build a "gravity generator". Gravity is everywhere because matter is everywhere and matter gravitates. If you want to create a one-Earth-gravity force on your spaceship, you'll need one Earth's amount of mass to do it! Similar considerations apply to tapping into the dark energy. If ZPE can be used at all, it's going to be the ZPE of fields that we already have access to, like electromagnetism, and it's probably only going to be a minute extra factor in physical situations where other forces dominate.

 

[Kittel Knight]

Supersymmetry, which is a feature of string theory, makes ...

Thanks, Mitchell !

 

[Kittel Knight]

Are you hoping that ZPE is something *physical* ?

Did anyone say that?!

 

[Careful]

Did anyone say that?!

No, but in 99% of the cases on public fora, people mean that By the way, Mitchell is correct in mentioning that supersymmetry does not solve the vacuum energy problem: an elementary reason is that it is broken in nature and one has to be careful that the supersymmetry breaking mechanisms do not destroy the cancellation of infinities by large numbers (which all known mechanisms appear to do). I don't know about superpartners as dark matter but it seems like a preposterous idea : one would have to squeeze all this mass precisely between distance scales of 10^{20} and say 10^{30} meters from centers of galaxies in order to fit the galaxy rotation curves and not destroy ordinary Newtonian laws at smaller scales. I am not sure either wether the ideas of such massive particles do not run into further probems either : would one not expect such massive particles to form, at a much faster rate than ordinary matter does, planets and stars ? Indeed, all atoms of such matter would be much tighter packed than ordinary matter does and densities would go sky high. Therefore, gravity would take over very quickly, stellar collapse would happen way before ordinary stars do and supermassive black holes would start radiating away. Therefore, if so, this cannot solve the dark matter puzzle ... anyway, the argument may be wrong but I see no obvious mistake. I think MOND is still by far the best idea in that direction.

Oh well, I see these dark matter scenario's only take neutralino's into account ... hhhmmm isn't that replacing one problem with another? Namely, why are so few supersymmetric particles present?

 

[Haelfix]

If we are talking about weak scale supersymmetry, then indeed the lightest superpartner is a candidate term for dark matter and happens to be around the right range by dimensional analysis. It is of course the only possible particle in that scenario, since on general grounds the other superpartners decay away much too rapidly and you just wouldn't expect for there to be many around so late in the universe's history.

There are basically only two well motivated particle physics explanations for darkmatter. The first either has a weakly interacting particle somewhere around LHC energies, or alternatively you can have a much lighter particle in a sort of condensate state that can escape direct detection (eg an axion). Everything in between is of course possible, but then there is not much of a plausible theory in there.

 

[Careful]

If we are talking about weak scale supersymmetry, then indeed the lightest superpartner is a candidate term for dark matter and happens to be around the right range by dimensional analysis. It is of course the only possible particle in that scenario, since on general grounds the other superpartners decay away much too rapidly and you just wouldn't expect for there to be many around so late in the universe's history.

Well, you make ''another'' argument here, but your conclusion is the same as mine (I didn't give those scenario's much thought anyway). But still, you kind of did not respond to my first comment because somehow, these neutralino's shouldn't disturb Newton's law where it is supposed to hold. That's quite unlikely no? Or am I missing something, is there a reason why the neutralino's should be where they are supposed to be ?

What disturbs me in general about these particle theory attempts is that people didn't learn anything for the last century. Indeed, precisely 100 years ago people where looking for invisible matter in the neighborhood of the solar system to explain away anomalies in Newtonian physics (which required huge fine tuning). Einstein put away such travesty by his theory of general relativity: now, people do the same thing again, but this time on a gigantic scale (and with some kind of 'ghost' matter which resembles Feynman's angles) ! What can I say?

 

[lumidek]

Are you hoping that ZPE is something *physical* ?

ZPE is physical whenever it couples to gravity; and whenever one may compare the total energy between two configurations with different net values of ZPE. In other words, ZPE is almost always physical.

In particular, the cosmological constant - which is physical because it makes the expansion of the Universe accelerate - has contributions from ZPEs from all known fields in Nature. The expected largeness of these terms - so much larger than the observed cosmological constant - is called the cosmological constant problem.

But there are also "non-problematic" examples of the ZPE. ZPE of the electromagnetic field (coming from all the allowed standing wave modes) between two metallic plates depends on the plates' distance. That's why the gradient of this total energy will be demonstrated as the Casimir force - an effect that's been experimentally verified to agree with the predictions fully based on ZPEs.

After all, the negative binding energy of the Hydrogen atom, any other atom, or any other quantum physical system is nothing else than a generalization of ZPE for different-than-quadratic potentials. All these things are totally physical and observable.

One must be very careful about ZPEs in string theory because they're the simplest players to demonstrate many effects.

For example, the very critical dimension D=26 of the bosonic string and D=10 of the superstring may be derived from the ZPEs. For example, in the bosonic case, there are D-2 transverse bosonic oscillators of a bosonic string. Each of them gives a hbar.omega/2 ZPE. However, the frequencies depend on the wave number along the string and one must sum over them. So the total ZPE of these vibrations is proportional to

(D-2) x sum (n/2)

The sum goes over n from 1 to infinity.

Here, D-2 comes from the number of transverse directions, n comes from the wave number (in frequency), and 1/2 is from hf/2. The first excited state must be massless because we only have (D-2) states at this level - from D-2 transverse oscillators - instead of (D-1) that would be needed for the little group of a massive particle. That means that

(D-2) x sum (n/2) + 1 = 0.

However, the sum of positive integers is -1/12, so we have

(D-2) x (-1/24) = -1,
D-2 = 24,
D = 26,

the right critical dimension. This calculation is not just a funny dirty trick, it's the actual conformally correct calculation that may be phrased in a more formal regulated framework but the essence of the calculation will be the calculation above, anyway.

There exist equivalent ZPE-based calculations of the critical dimension. In the covariant formalism, the bc-ghosts actually have the central charge c=-26, requiring D=26 bosons to cancel the conformal anomaly on the world sheet. The c=-26 result may also be reduced to some kinds of ZPE terms.

In supersymmetric theories, ZPEs typically tend to cancel.

At any rate, string theory makes these features of quantum theory more important, not less important or more disputable. Any hope that string theory would "undo" some of the key insights of the quantum revolution or relativistic revolution is totally misguided. String theory is another step in the progress towards more accurate and more complete physical theories - perhaps the last step. It is surely not a step to return physics to the 19th, 18th, or 17th century. ;-)

 

[Careful]

ZPE is physical whenever it couples to gravity; and whenever one may compare the total energy between two configurations with different net values of ZPE. In other words, ZPE is almost always physical.

In particular, the cosmological constant - which is physical because it makes the expansion of the Universe accelerate - has contributions from ZPEs from all known fields in Nature. The expected largeness of these terms - so much larger than the observed cosmological constant - is called the cosmological constant problem.

But there are also "non-problematic" examples of the ZPE. ZPE of the electromagnetic field (coming from all the allowed standing wave modes) between two metallic plates depends on the plates' distance. That's why the gradient of this total energy will be demonstrated as the Casimir force - an effect that's been experimentally verified to agree with the predictions fully based on ZPEs.

After all, the negative binding energy of the Hydrogen atom, any other atom, or any other quantum physical system is nothing else than a generalization of ZPE for different-than-quadratic potentials. All these things are totally physical and observable.

One must be very careful about ZPEs in string theory because they're the simplest players to demonstrate many effects.

For example, the very critical dimension D=26 of the bosonic string and D=10 of the superstring may be derived from the ZPEs. For example, in the bosonic case, there are D-2 transverse bosonic oscillators of a bosonic string. Each of them gives a hbar.omega/2 ZPE. However, the frequencies depend on the wave number along the string and one must sum over them. So the total ZPE of these vibrations is proportional to

(D-2) x sum (n/2)

The sum goes over n from 1 to infinity.

Here, D-2 comes from the number of transverse directions, n comes from the wave number (in frequency), and 1/2 is from hf/2. The first excited state must be massless because we only have (D-2) states at this level - from D-2 transverse oscillators - instead of (D-1) that would be needed for the little group of a massive particle. That means that

(D-2) x sum (n/2) + 1 = 0.

However, the sum of positive integers is -1/12, so we have

(D-2) x (-1/24) = -1,
D-2 = 24,
D = 26,

the right critical dimension. This calculation is not just a funny dirty trick, it's the actual conformally correct calculation that may be phrased in a more formal regulated framework but the essence of the calculation will be the calculation above, anyway.

There exist equivalent ZPE-based calculations of the critical dimension. In the covariant formalism, the bc-ghosts actually have the central charge c=-26, requiring D=26 bosons to cancel the conformal anomaly on the world sheet. The c=-26 result may also be reduced to some kinds of ZPE terms.

In supersymmetric theories, ZPEs typically tend to cancel.

At any rate, string theory makes these features of quantum theory more important, not less important or more disputable. Any hope that string theory would "undo" some of the key insights of the quantum revolution or relativistic revolution is totally misguided. String theory is another step in the progress towards more accurate and more complete physical theories - perhaps the last step. It is surely not a step to return physics to the 19th, 18th, or 17th century. ;-)

I will respond later, have to go and entertain my kids now

 

[lumidek]

I will respond later, have to go and entertain my kids now

I hope that you're not educating them to think that ZPE is unphysical. That would be unacceptable even for an arrogant sub-par scientist such as Leslie Winkle. ;-)

 

[Careful]

I hope that you're not educating them to think that ZPE is unphysical. That would be unacceptable even for an arrogant sub-par scientist such as Leslie Winkle. ;-)

Haha, no they immediately learn the distinction between physical principles and particular mathematical representations of them so that they are not fooled into thinking than the conclusions of one particular choice of representation lead to an inescapable *physical* paradox

To start with, you are master in writing an entire epistle without actually answering the question, you have political talent ! Concerning your arguments about the reality of vacuum energy, needless to say that I disagree with everything you say *physically*. Of course, you are correct within the particular representation class spanned by QFT and (supersymmetric) string theories. However, the complete inability to solve the cosmological constant problem within those theories deeply indicates that our notion of what the physical vacuum is in those theories is wrong. If you have something intelligent to say which contradicts that, please go ahead and you may go to Copenhagen. Second, there is no known phenomenon even in flat spacetime QFT which requires those vacuum fluctuations, the Casimir effect has been safely explained without them.

Of course, you may protest now, that we don't even know the correct physical principles behind even
flat spacetime QFT (which is indicated by the failure so far of axiomatic QFT) and therefore you have no idea for what other representations you should look. That is correct, but that is NO argument against my position, neither is it an argument for your position (which simply is to declare yourself ignorant).

So, not every scientist disagreeing with Sheldon is a Leslie Winkle, but it might even be a super Sheldon

Don't get me wrong, on my personal scale ST still scores more than twice as much as LQG does, but still then, it is only 6,5 on 10 which is not bad, but not something which induces an orgasm either.

 

[haushofer]

A related question which never became clear to me: in string theory you have multiple ways of regularize. From the CFT side of the story you can understand the -1/12 as ZPE quite good I would say, but one can also use exponential regularization or analytic continuation. All these methods yield the same -1/12, while naively these methods differ in plausibility. Why do these methods all yield the same answer?

 

[arivero]

A related question which never became clear to me: in string theory you have multiple ways of regularize. From the CFT side of the story you can understand the -1/12 as ZPE quite good I would say, but one can also use exponential regularization or analytic continuation.

I am even more intrigued when going to the supersymmetric case. If you use Riemann zeta function regularization, the process of cancelling bosonic and fermionic excitations amounts to the traditional process of removing the pole of the zeta function to obtain Dirichlet's eta and so an analytic extension in the whole half plane. So it seems that very powerful magic (or mathematics) is going on, and the zeta function is not just a regularization trick.

 

[lumidek]

A related question which never became clear to me: in string theory you have multiple ways of regularize. From the CFT side of the story you can understand the -1/12 as ZPE quite good I would say, but one can also use exponential regularization or analytic continuation. All these methods yield the same -1/12, while naively these methods differ in plausibility. Why do these methods all yield the same answer?

Dear Haushofer, it's because -1/12 is the correct answer so any sufficiently consistent regularization technique is guaranteed to end up with the correct result.

More concretely, -1/12 is the value that is needed for modular invariance. Imagine a path integral over the torus. The partition sum has to invariant under the exchange of the a,b cycles of the torus. This is only achieved if the ground level energy is counted so that the sum of positive integers is -1/12.

The regularizations that produce -1/12 "easily" are those that respect the conformal symmetry, e.g. the invariance of the torus partition function under the rotation of the rectangle by 90 degrees. For the zeta-function regularization, this can be shown to be the case. In the same way, one can add the exponentially decreasing regulator to multiply the integers, send the regulator to "being invisible", and subtract the 1/epsilon terms. This method also produces the correct conformal answers because the "epsilon" is dimensionful and we may subtract the negative powers of epsilon by counterterms. These counterterms may be interpreted as a "worldsheet vacuum energy" - and by dimensional analysis, they're only able to change the coefficient of the right power of epsilon, usually 1/epsilon^2. That's why this counterterm doesn't influence the finite, epsilon^0, piece, even though it makes the result finite.

The sum of positive integers "really is" equal to -1/12 and any professionally enough done regularization inevitably has to arrive to the same result. The result is the result that has "consistent" physical properties that may be reduced to symmetries, e.g. the symmetry exchanging the cycles a,b of the torus.