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## Is reverse time dilation posssible?

I agree with JesseM: the Casimir Effect proofs the possibility of less energy density than in a vacuum. However, the calculation of the vacuum energy density is not exactly a glorious chapter of modern physics. So I´d say it´s absolutely correct to say that there is no proof for negative energy.

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 Quote by Ich I agree with JesseM: the Casimir Effect proofs the possibility of less energy density than in a vacuum. However, the calculation of the vacuum energy density is not exactly a glorious chapter of modern physics. So I´d say it´s absolutely correct to say that there is no proof for negative energy.
John Baez has a good discussion of the problems with calculating the vacuum energy density here. He points out that although physicists have failed pretty badly in calculating the vacuum energy from quantum principles, an upper limit can be placed on it from cosmological observations, around 10^-9 joules per cubic meter. So I wonder, can the energy density between plates in the casimir effect go below the vacuum energy density by significantly more than 10^-9 joules per cubic meter? Does anyone know the equation for calculating the energy density between plates in terms of their size and separation? I found a bunch of pages that give an equation for the force (see here, for example), but I'm not sure how to translate this into energy density.

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 Quote by JesseM John Baez has a good discussion of the problems with calculating the vacuum energy density here.
Thanks for the reference. I am experiencing some sign confusion here, so I did some more reading, specifically

http://arxiv.org/PS_cache/gr-qc/pdf/0001/0001099.pdf

 (3) If you believe the recent observational data regarding the accelerating universe, then the SEC [strong energy condition] is violated on cosmological scales right now!
As nearly as I can figure out, the cosmological constant represents a positive energy density T_00 and (in a locally Minkowskian coordiante system) a negative pressure T_11 = T_22 = T_33 = -T_00.

[Any references otherwise from respectable sources will be gratefully accepted.]

In spite of the positivity of the actual energy density T_00 assigned to the cosmological constant $\Lambda$, the net effect is one of "effective" anti-gravity. Thus the over-simplification on my part is in thinking/describing that the strong energy condition is one of positive energy density - the energy density due to our positive cosmological constant is technically positive, but it causes the universal expansion to accelerate, not slow down, so it "acts" a lot like negative mass, and it meets the technical defintion of exotic matter in that it violates the strong energy condition.

Another way of thinking about it - in flat space-time, gravity couples to $\rho+3P$, so the negative pressure terms do give an effectively "negative" mass in spite of the positve density term

 Quote by Ich I agree with JesseM: the Casimir Effect proofs the possibility of less energy density than in a vacuum. However, the calculation of the vacuum energy density is not exactly a glorious chapter of modern physics. So I´d say it´s absolutely correct to say that there is no proof for negative energy.
Since we apparently live in an "open" universe which will keep on expanding to infinity, doesn't this violate conservation of energy in a rather big way?

Consider a thought experiment; assume there is a way to harness zero-point energy. Again, you would be violating conservation of energy because no matter how much energy you draw off, the Heisenberg Uncertainty Principle forbids us from ever having an energy state of exactly zero.

The Casimir Effect shows us that energy can indeed be harnessed for work (pressing two metal plates together), yet at the same time QM says that we can never use all of the energy (indeed, assuming zero point fluctuations are *just* powerful enough to prevent empty space from violating the Uncertainty Principle, we shouldn't be able to use any of it at all.)

In short, I don't see how we can explain away this creation of energy, unless we also allow the creation of negative energy.

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 Quote by LodeRunner Since we apparently live in an "open" universe which will keep on expanding to infinity, doesn't this violate conservation of energy in a rather big way?
Ich wasn't talking about conservation of energy, he/she was talking about negative energy, a separate issue. Energy is not conserved globally in general relativity, see here.
 Quote by LodeRunner The Casimir Effect shows us that energy can indeed be harnessed for work (pressing two metal plates together)
What do you mean by "harnessed", exactly? I don't know how you'd use the two plates to do work, and even if you could, I'd bet that you couldn't get out more energy than you put in...I did some quick googling and found there was a proposal by physicist Robert Forward to extract energy from the vacuum, but this page says:
 13. J. Maclay, "Unusual properties of conductive rectangular cavities in the zero point electromagnetic field: Resolving Forward's Casimir energy extraction cycle paradox," PROCEEDINGS of STAIF-99 (Space Technology and Applications International Forum-1999, Albuquerque, NM, January, 1999), edited by M.S. El-Genk, AIP Conference Proceedings 458, American Institute of Physics, New York 1999. Published in hardcopy and CD-ROM by AIP. This article presents the explanation of why Bob Forwards idea for extracting energy from the vacuum does not work. Usually we think that if the energy density in a box is greater than the energy density outside the box, that there will be an outward force on the walls of the box, which is what Bob’s idea was based on. However, for vacuum energy this it not true!

Negative energy is related to conservation of enegy because mathematically, one should be able to create energy out of nothing if one also creates an equal amount of negative energy.

My point, which you may or may not have refuted (unfortunately I have very little formal education, so many of the details of that link were lost on me), was that assuming all of empty space has a positive energy value would violate conservation of energy, whereas assuming that the value could go negative would not.

 Quote by JesseM What do you mean by "harnessed", exactly? I don't know how you'd use the two plates to do work, and even if you could, I'd bet that you couldn't get out more energy than you put in...I did some quick googling and found there was a proposal by physicist Robert Forward to extract energy from the vacuum, but this page says:
It's not a matter of extracting useful amounts of power; it's a matter of reducing the amount of zero-point energy in any region of space. If we accept the QM explanation, then it seems as though zero-point energy should be just strong enough to avoid breaking the Uncertainty Principle by allowing us to be too certain about the speed/position/value of a certain point in space.

My point is that if we reduce the amount of vacuum energy by any means, such as using it to push two small metal plates together (converting it into macroscopic kinetic energy), then that energy MUST be replenished somehow in order to prevent over-certainty. My assumption was that this was done by creating negative energy.
Perhaps there is another way it can be replenished, or perhaps the problem is altogether moot if GR and/or QM does allow for creation of energy in some circumstances.

Or perhaps I just don't know what I'm talking about 'cause I'm just a highschool dropout wannabe writer who reads up on physics in his spare time...

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 Quote by LodeRunner Negative energy is related to conservation of enegy because mathematically, one should be able to create energy out of nothing if one also creates an equal amount of negative energy.
But if you're creating an equal amount of negative energy, then you aren't actually violating conservation of energy, right?
 Quote by LodeRunner My point, which you may or may not have refuted (unfortunately I have very little formal education, so many of the details of that link were lost on me), was that assuming all of empty space has a positive energy value would violate conservation of energy
Why would a positive vacuum energy violate conservation of energy?
 Quote by LodeRunner It's not a matter of extracting useful amounts of power; it's a matter of reducing the amount of zero-point energy in any region of space.
Who said anything about useful? What I'm saying is that I don't think you can extract any energy from the vacuum using the casimir effect, no matter how tiny. I don't understand what connection you're making between reducing the energy density of a given region of empty space and extracting energy to do work from it.
 Quote by LodeRunner If we accept the QM explanation, then it seems as though zero-point energy should be just strong enough to avoid breaking the Uncertainty Principle by allowing us to be too certain about the speed/position/value of a certain point in space.
Huh? How would zero-point energy break the uncertainty principle if it were higher? What does it even mean to measure the speed/position of a "point in space", as opposed to a particle?
 Quote by LodeRunner My point is that if we reduce the amount of vacuum energy by any means, such as using it to push two small metal plates together (converting it into macroscopic kinetic energy), then that energy MUST be replenished somehow in order to prevent over-certainty. My assumption was that this was done by creating negative energy.
I'm not following at all. Are you basing your ideas about how the uncertainty principle relates to zero-point energy on something you've read somewhere, or is this your own idea? If it's something you've read, maybe you could provide a link or a reference so I could better understand your argument.

I do not know where I first read about the connection of the Casimir Force to the Uncertainty Principle, but a quick wikipedia lookup yields:
 In a quantum mechanical system such as the particle in a box or the quantum harmonic oscillator, the lowest possible energy is called the zero-point energy. According to classical physics, the kinetic energy of a particle in a box or the kinetic energy of the harmonic oscillator may be zero if the velocity is zero. Quantum mechanics with its uncertainty principle implies that if the velocity is measured with certainty to be exactly zero, the uncertainty of the position must be infinite. This either violates the condition that the particle remain in the box, or it brings a new potential energy in the case of the harmonic oscillator. To avoid this paradox, quantum mechanics dictates that the minimal velocity is never equal to zero, and hence the minimal energy is never equal to zero.
So it appears as though empty space in QM is subject to the same Uncertainty limitations as particles.

 But if you're creating an equal amount of negative energy, then you aren't actually violating conservation of energy, right?
This is exactly my point. Creation of positive and negative energy (or matter) in equal amounts does not violate conservation. Creation of positive-only energy, OTOH, does seem to violate conservation, and your explanation of vacuum energy (that all of empty space has some positive energy but never any negative energy) would seem to fall into this catagory.

 Why would a positive vacuum energy violate conservation of energy?
It wouldn't, provided you could prove that the total vacuum energy in the universe never changed, or that enough negative energy is created to counterbalance the positive.

 Who said anything about useful? What I'm saying is that I don't think you can extract any energy from the vacuum using the casimir effect, no matter how tiny. I don't understand what connection you're making between reducing the energy density of a given region of empty space and extracting energy to do work from it.
Well, I thought you said something about always having to "put in" more energy than you'd get out of vacuum energy. My point wasn't getting free energy out of vacuum energy; it was getting ANY energy out of it at all.

If one made microscopic and very light metal plates and put a small amount of pressure on them, not quite enough to move them on its own, but enough to move them after vacuum energy has exterted its influence, then wouldn't that be a case of someone turning vacuum energy into kinetic energy? Wouldn't that mean that we must have LESS vacuum energy than we started with?

My contention is as follows:

The Casimir Effect is a direct consequence of the Uncertainty Principle, and it proves we can convert vacuum energy into another form (kinetic.) BUT, the universe will not allow the reduction of vacuum energy in this manner because this would violate the Uncertainty Principle.

So, if the universe will not allow this reduction, then it must replace the energy lost to maintain Uncertainty. Empty space does not need to have an inherent energy; it just needs to have a non-zero value to preserve Uncertainty. This non-zero value does not need to be positive, so I believe that the most elegant explanation that preserves conservation of energy is to say that zero-point fluctuations include both positive and negative values.

I hope this will clear things up...

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 Quote by LodeRunner So it appears as though empty space in QM is subject to the same Uncertainty limitations as particles.
I still don't understand, that wikipedia quote was talking about a particle in a box, not empty space. I think the zero-point energy is related to uncertainty in the sense that the quantum fields in quantum field theory are in some sense subject to the uncertainty principle just like particles, but I don't think this has anything to do with measuring the position/speed of a point in space, which I don't think is even a meaningful concept.
 Quote by LodeRunner This is exactly my point. Creation of positive and negative energy (or matter) in equal amounts does not violate conservation. Creation of positive-only energy, OTOH, does seem to violate conservation, and your explanation of vacuum energy (that all of empty space has some positive energy but never any negative energy) would seem to fall into this catagory.
But this is just a static amount of energy associated with the vacuum, it isn't the "creation" of positive energy where none was previously. [quote=LodeRunner]It wouldn't, provided you could prove that the total vacuum energy in the universe never changed[/url] I think the vacuum energy is thought to have decreased at the end of the inflationary era, but I don't understand the details enough to say whether energy was conserved in this process...if the vacuum energy decreases, perhaps it would be made up for by the creation of a large number of energetic particles, so that the total energy of space + particles remains constant.
 Quote by LodeRunner Well, I thought you said something about always having to "put in" more energy than you'd get out of vacuum energy. My point wasn't getting free energy out of vacuum energy; it was getting ANY energy out of it at all. If one made microscopic and very light metal plates and put a small amount of pressure on them, not quite enough to move them on its own, but enough to move them after vacuum energy has exterted its influence then wouldn't that be a case of someone turning vacuum energy into kinetic energy? Wouldn't that mean that we must have LESS vacuum energy than we started with?
I don't know, I don't understand the casimir effect or quantum field theory well enough to address this. You may be right that the increase in kinetic energy of the plates as they are pulled together is balanced by the decreased vacuum energy between them.
 Quote by LodeRunner The Casimir Effect is a direct consequence of the Uncertainty Principle, and it proves we can convert vacuum energy into another form (kinetic.) BUT, the universe will not allow the reduction of vacuum energy in this manner because this would violate the Uncertainty Principle.
I still don't follow your reasoning. Why would it violate the uncertainty principle? From what I've read, the uncertainty principle's relation to vacuum energy has something to do with the different possible modes of the quantum field carrying energy and momentum. But by placing two plates near each other, you are reducing the number of allowable modes between the plates, because wavelengths that don't divide evenly into the spacing between the plates interfere with themselves. I think each individual mode that's present is still subject to the uncertainty principle in the same way as usual, but some modes are simply no longer present between the plates that are present in empty space, and this has something to do with why the vacuum energy is lower between the plates.

 To avoid this paradox, quantum mechanics dictates that the minimal velocity is never equal to zero, and hence the minimal energy is never equal to zero.
(emphasis mine.)

This means that empty space cannot exist, because empty space=0 energy. This is why we have "quantum foam" or zero-point fluctuations instead. And the Casimir Effect is a direct test of the existence these fluctuations. That is the relationship between the Casimir Effect and Uncertainty--zero-point fluctuations are a consequence of Uncertainty, and the Casimir Effect is a consequence of zero-point fluctuations.

Vacuum energy is present to preserve Uncertainty, and to convert it to another form (in essence taking it away from the vacuum) would allow us to be too certain about the value of a particular point in spacetime.

I can try to find some more links for you, but I'm a little pressed for time. I'm not saying anything that I haven't read in several pop science books (Hawking, etc.) I suppose we could take this to the QM forum and see if we can find a consensus there.

 You may be right that the increase in kinetic energy of the plates as they are pulled together is balanced by the decreased vacuum energy between them.
Where else is the kinetic energy coming from, if not a decrease in vacuum energy? Again, this is a Conservation problem. Assuming I'm not horribly wrong about Uncertainty applying to empty space, any vacuum energy converted to another form must be replaced or else we lose that Uncertainty. The idea of negative energy lets us seemlessly replace this lost energy, and no net energy will have been created--thus both Conservation and Uncertainty are preserved.

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 Quote by LodeRunner (emphasis mine.) This means that empty space cannot exist, because empty space=0 energy. This is why we have "quantum foam" or zero-point fluctuations instead. And the Casimir Effect is a direct test of the existence these fluctuations. That is the relationship between the Casimir Effect and Uncertainty--zero-point fluctuations are a consequence of Uncertainty, and the Casimir Effect is a consequence of zero-point fluctuations. Vacuum energy is present to preserve Uncertainty, and to convert it to another form (in essence taking it away from the vacuum) would allow us to be too certain about the value of a particular point in spacetime.
Ok, but you're not telling me what entity it is that you think is subject to uncertainty here--you said earlier that it was the position/speed of "points in space" which was uncertain, I still think that doesn't make any sense. What I've read in a few places is that different "modes" of the quantum fields which fill empty space carry momentum/energy and are therefore subject to uncertainty, and this is what's responsible for vacuum energy. In between the plates in the Casimir effect, there are fewer possible modes than in free space, therefore the minimum energy is less, without any violation of the uncertainty principle. Are you disagreeing with me about this?
 Quote by LodeRunner Where else is the kinetic energy coming from, if not a decrease in vacuum energy?
I'd guess you're probably right about this, but again, I don't have enough knowledge of this subject to say for sure. It's conceivable that it would instead be due to a decrease in some form of potential energy, for example.
 Quote by LodeRunner Assuming I'm not horribly wrong about Uncertainty applying to empty space, any vacuum energy converted to another form must be replaced or else we lose that Uncertainty. The idea of negative energy lets us seemlessly replace this lost energy, and no net energy will have been created--thus both Conservation and Uncertainty are preserved.
Just to be clear, do you think this is the opinion of mainstream physicists, or are you expressing your own ideas here? Like I said before, I don't think mainstream physicists would say there is any problem with the uncertainty principle being violated in the Casimir effect, in fact I think the lower energy density between the plates can be calculated from the uncertainty principle in exactly the same way that the energy of the ordinary vacuum is calculated from it (by looking at the minimum energy of all the allowable modes in each case, with more modes being allowed in empty space than between the plates).

I don't want anyone to get the impression that I'm pulling stuff out of my ***, so:

Mainstream physicists have said that the Casimir Effect arises as a consequence of Uncertainty, not that it violates it. Mainstream physicists have also brought up the possibility of negative energy or negative mass, and I have heard them use the Casimir Effect as one such example.

I have made the (possibly wrong) leap of logic to connect these two ideas and show why negative energy may be necessary in order to preserve Conservation while simultaneously preserving Uncertainty. I have not heard this idea from any source; it is just the result of me trying to make several different mainstream ideas fit together.

My claim that a reduction in energy would result in Uncertainty breaking is my own. Since I have always heard the Casimir Effect explained in terms of the Uncertainty Principle, I assume that the fluctuations are just strong enough to preserve Uncertainty. It may be that there is a margin there, and that it is possible to convert some vacuum energy into other forms but not enough to break Uncertainty, but because of the way it was explained to me, I have no reason to believe that such an excess of vacuum energy exists.

My claim that an increase in vacuum energy (e.g. to replace that loss through the Casimir Effect) violates Conservation is also my own, though I think it is a fairly straightforward assumption.

 Ok, but you're not telling me what entity it is that you think is subject to uncertainty here--you said earlier that it was the position/speed of "points in space" which was uncertain, I still think that doesn't make any sense. What I've read in a few places is that different "modes" of the quantum fields which fill empty space carry momentum/energy and are therefore subject to uncertainty, and this is what's responsible for vacuum energy. In between the plates in the Casimir effect, there are fewer possible modes than in free space, therefore the minimum energy is less, without any violation of the uncertainty principle. Are you disagreeing with me about this?
Interesting. I have never heard this explanation before. I have always read that it is the appearence and disappearence of virtual particles, the kind that are responsible for Hawking Radiation (one particle of the particle/antiparticle virtual pair appears just barely within the event horizon of a black hole and the other one appears just barely outside of it. The one outside escapes to freedom and becomes "real" whereas the other one falls into the black hole and reduces its mass through QM weirdness that I do not fully understand.) It was my understanding that all of empty space was composed of these virtual particle pairs, constantly appearing and annihilating one another--the quantum "foam". They have no mass and usually have no effect, except that Uncertainty commands that they MUST have a mass or energy value, at least sometimes.

And if all of my assumptions are correct, then they must also have an negative mass (or negative energy), at least sometimes.

Anyway, I really must be getting to bed now. It's been great debating with you... will come back again and check tonight or Friday night.
 Recognitions: Gold Member Staff Emeritus What are uncertain in quantum mechanics are complementary pairs of observables. There are several of these; the easiest to prove from Schroedinger's (or any other quantum) equation is momentum and position. The more accurately you know momentum the more uncertain you must be about position, and vice versa. If you multiply the uncertainties together you get Planck's constant, so if you were to plot the uncertainty on axes of position for x and momentum for y, the uncertainty curve would be a hyperbola with those two axes as asymptotes. Another set of complementary observables is spin around different axes. If you measure the spin around the z-axis, thus reducung its uncertainty to zero, the spins around the x and y-axes become completely uncertain. This is the physics behind the repeated Stern-Gerlach experiments described by Feynmann. The case of energy and time is a little different. It turns on they behave just like complementary observables, but because energy is an observable but time is not, the proof is different. I don't see any need for negative energy to balance conservation of energy in quantum mechanics. energy is conserved up to observation in quantum mechanics. And just as there is no preferred frame in relativity from which you can see "what really happens", so there is no opportunity in quantum mechanics to see the unobservable. For more on this, check out the threads on the quantum forum about hidden variables and Bell's theorem.

 Quote by pervect some popularized musings by Fenyman - don't appear to me to actually lead to any experimental predictions. I could be wrong, perhaps RandallB could cite some papers.
I wasn’t referring to his musings; Fenyman’s Diagrams have lead to improvements, discoveries, and understands in working out and planning partial collisions.
Work through some of the mildly complicated ones, you don’t need a cited papers to find them. Once you understand them & where they are useful, the hard part is to imagine a way to workout the issues without backwards time. Maybe with a MWI/HUP concept but I’d find that hard to implement usefully as the diagrams already have been.

As to the Casimir force—“ it is an actual observed force that is predicted and explained by QM as negative energy.” As it’s only explanation!
Therefore by using that explanation:
“we KNOW exotic energy & matter exists, period”.
Therefore by implication we know:
Negative gravitation even Wormholes must be real!
This is great! with ‘proof’ like this we should be able to show that HUP and thus QM is just as exclusively correct as that theory claims to be!
– except that was the assumption that started all these 'proofs'!

The point is; mind numbing "circular logic" like this may have some use in science (I’m not sure where), but why use something that hard to follow in fiction, when you already know that the cheesy ‘Flux Capacitor’ style is proven as acceptable anyway.

Now if this thread has turned to finding what’s ‘correct’, I don’t think Casimir will do. Entanglement is the only recognized non-circular ‘proof’ for QM/HUP and we already have two or three active threads debating that issue.

Accepting backwards time as something real that needs to be figured out is just one thing.
QM – Casimir – wormhole – HUP is a spaghetti bowl. I’ll take Occam as my guide.

SO, for me my money’s on backwards time a la Fenyman’s Diagrams is shown to be real and not a trick before Negative Gravitation is. In fact I’d bet that a Theory proving so would also show QM/HUP to wormholes as wrong.
Since I seem to be in a minority on that at least I should get good odds, do ya’ suppose they have a line in Vegas?
RB

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 Quote by RandallB Now if this thread has turned to finding what’s ‘correct’, I don’t think Casimir will do. Entanglement is the only recognized non-circular ‘proof’ for QM/HUP and we already have two or three active threads debating that issue.
You are doubting QM/HUP?? These fall into the realm of very well-established physics. This board has a rule against trying to call into question established physics, so I wonder which active threads you're talking about.
 Quote by RandallB SO, for me my money’s on backwards time a la Fenyman’s Diagrams is shown to be real and not a trick before Negative Gravitation is. In fact I’d bet that a Theory proving so would also show QM/HUP to wormholes as wrong.
What does "shown to be real" mean here? What possible test could distinguish backwards-time-as-something-real from backwards-time-as-a-useful-mathematical-trick, since it's clear that quantum field theory does respect causality and thus cannot be used to actually send messages backwards in time? I still think you're confusing physics with philosophy--unless a particular idea leads to testable consequences, it ain't physics.

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 Quote by LodeRunner I don't want anyone to get the impression that I'm pulling stuff out of my ***, so: Mainstream physicists have said that the Casimir Effect arises as a consequence of Uncertainty, not that it violates it. Mainstream physicists have also brought up the possibility of negative energy or negative mass, and I have heard them use the Casimir Effect as one such example.
Agreed (although by 'negative energy' I think they just mean something like 'energy lower than that of the vacuum')
 Quote by LodeRunner My claim that a reduction in energy would result in Uncertainty breaking is my own. Since I have always heard the Casimir Effect explained in terms of the Uncertainty Principle, I assume that the fluctuations are just strong enough to preserve Uncertainty. It may be that there is a margin there, and that it is possible to convert some vacuum energy into other forms but not enough to break Uncertainty, but because of the way it was explained to me, I have no reason to believe that such an excess of vacuum energy exists.
I don't understand what you mean by "the fluctuations are just strong enough to preserve uncertainty", or why converting vacuum energy into other forms (like the kinetic energy of the plates) would "break" uncertainty.
 Quote by LodeRunner My claim that an increase in vacuum energy (e.g. to replace that loss through the Casimir Effect) violates Conservation is also my own, though I think it is a fairly straightforward assumption.
I don't understand this either. In the Casimir effect, the vacuum energy between the plates decreases, it doesn't increase. Are you saying the vacuum energy would increase in some other region to compensate? I'm pretty sure that is not the case...like you suggested earlier, energy conservation may just have to do with the fact that the kinetic energy of the plates balances the decrease in vacuum energy in the space between them, or it may have something to do with creation/destruction of energetic particles, I'm not sure.
 Quote by LodeRunner Interesting. I have never heard this explanation before. I have always read that it is the appearence and disappearence of virtual particles, the kind that are responsible for Hawking Radiation (one particle of the particle/antiparticle virtual pair appears just barely within the event horizon of a black hole and the other one appears just barely outside of it. The one outside escapes to freedom and becomes "real" whereas the other one falls into the black hole and reduces its mass through QM weirdness that I do not fully understand.) It was my understanding that all of empty space was composed of these virtual particle pairs, constantly appearing and annihilating one another--the quantum "foam". They have no mass and usually have no effect, except that Uncertainty commands that they MUST have a mass or energy value, at least sometimes.
Sure, in quantum field theory I think it's true that to understand the properties of empty space you have to do a sum of Feynman diagrams in which virtual particles appear out of nothing and then annihilate each other. But what's your understanding of how this would explain the Casimir effect? Presumably the virtual particles would have to be behaving differently between the plates than they do in empty space, but why? Reading a little more, I think the usual explanation in terms of virtual particles is probably equivalent to what I was saying about the different "modes" of the quantum field--virtual particles have wavelengths just like modes, and in both cases only wavelengths that divide evenly into the separation between plates can avoid cancelling themselves out, so only these wavelengths contribute to the properties of the vacuum between the plates. See John Baez's page on the Casimir effect, for example:
 The Casimir effect is a small attractive force which acts between two close parallel uncharged conducting plates. It is due to quantum vacuum fluctuations of the electromagnetic field. The effect was predicted by the Dutch physicist Hendrick Casimir in 1948. According to the quantum theory, the vacuum contains virtual particles which are in a continuous state of fluctuation (see physics FAQ article on virtual particles). Casimir realised that between two plates, only those virtual photons whose wavelengths fit a whole number of times into the gap should be counted when calculating the vacuum energy. The energy density decreases as the plates are moved closer which implies there is a small force drawing them together.
And here's an article from "Physics World" which gives a longer explanation:
 Understanding the Casimir force Although the Casimir force seems completely counterintuitive, it is actually well understood. In the old days of classical mechanics the idea of a vacuum was simple. The vacuum was what remained if you emptied a container of all its particles and lowered the temperature down to absolute zero. The arrival of quantum mechanics, however, completely changed our notion of a vacuum. All fields - in particular electromagnetic fields - have fluctuations. In other words at any given moment their actual value varies around a constant, mean value. Even a perfect vacuum at absolute zero has fluctuating fields known as "vacuum fluctuations", the mean energy of which corresponds to half the energy of a photon. However, vacuum fluctuations are not some abstraction of a physicist's mind. They have observable consequences that can be directly visualized in experiments on a microscopic scale. For example, an atom in an excited state will not remain there infinitely long, but will return to its ground state by spontaneously emitting a photon. This phenomenon is a consequence of vacuum fluctuations. Imagine trying to hold a pencil upright on the end of your finger. It will stay there if your hand is perfectly stable and nothing perturbs the equilibrium. But the slightest perturbation will make the pencil fall into a more stable equilibrium position. Similarly, vacuum fluctuations cause an excited atom to fall into its ground state. The Casimir force is the most famous mechanical effect of vacuum fluctuations. Consider the gap between two plane mirrors as a cavity (figure 1). All electromagnetic fields have a characteristic "spectrum" containing many different frequencies. In a free vacuum all of the frequencies are of equal importance. But inside a cavity, where the field is reflected back and forth between the mirrors, the situation is different. The field is amplified if integer multiples of half a wavelength can fit exactly inside the cavity. This wavelength corresponds to a "cavity resonance". At other wavelengths, in contrast, the field is suppressed. Vacuum fluctuations are suppressed or enhanced depending on whether their frequency corresponds to a cavity resonance or not. An important physical quantity when discussing the Casimir force is the "field radiation pressure". Every field - even the vacuum field - carries energy. As all electromagnetic fields can propagate in space they also exert pressure on surfaces, just as a flowing river pushes on a floodgate. This radiation pressure increases with the energy - and hence the frequency - of the electromagnetic field. At a cavity-resonance frequency the radiation pressure inside the cavity is stronger than outside and the mirrors are therefore pushed apart. Out of resonance, in contrast, the radiation pressure inside the cavity is smaller than outside and the mirrors are drawn towards each other. It turns out that, on balance, the attractive components have a slightly stronger impact than the repulsive ones. For two perfect, plane, parallel mirrors the Casimir force is therefore attractive and the mirrors are pulled together. The force, F, is proportional to the cross-sectional area, A, of the mirrors and increases 16-fold every time the distance, d, between the mirrors is halved: F ~ A/d^4. Apart from these geometrical quantities the force depends only on fundamental values - Planck's constant and the speed of light.
Finally, this site talks about the connection to the uncertainty principle:
 The basis of zero-point energy is the Heisenberg uncertainty principle, one of the fundamental laws of quantum physics. According to this principle, the more precisely one measures the position of a moving particle, such as an electron, the less exact the best possible measurement of momentum (mass times velocity) will be, and vice versa. The least possible uncertainty of position times momentum is specified by Planck's constant, h. A parallel uncertainty exists between measurements involving time and energy. This minimum uncertainty is not due to any correctable flaws in measurement, but rather reflects an intrinsic quantum fuzziness in the very nature of energy and matter. A useful calculational tool in physics is the ideal harmonic oscillator: a hypothetical mass on a perfect spring moving back and forth. The Heisenberg uncertainty principle dictates that such an ideal harmonic oscillator -- one small enough to be subject to quantum laws -- can never come entirely to rest, since that would be a state of exactly zero energy, which is forbidden. In this case the average minimum energy is one-half h times the frequency, hf/2. Radio waves, light, X-rays, and gamma rays are all forms of electromagnetic radiation. Classically, electromagnetic radiation can be pictured as waves flowing through space at the speed of light. The waves are not waves of anything substantive, but are in fact ripples in a state of a field. These waves do carry energy, and each wave has a specific direction, frequency and polarization state. This is called a "propagating mode of the electromagnetic field." Each mode is subject to the Heisenberg uncertainty principle. To understand the meaning of this, the theory of electromagnetic radiation is quantized by treating each mode as an equivalent harmonic oscillator. From this analogy, every mode of the field must have hf/2 as its average minimum energy. That is a tiny amount of energy, but the number of modes is enormous, and indeed increases as the square of the frequency. The product of the tiny energy per mode times the huge spatial density of modes yields a very high theoretical energy density per cubic centimeter. From this line of reasoning, quantum physics predicts that all of space must be filled with electromagnetic zero-point fluctuations (also called the zero-point field) creating a universal sea of zero-point energy.
So again, I think that somehow the uncertainty principle can be used to find each individual wavelength's contribution to the vacuum energy, and the explanation for the lower energy between the plates is just that fewer wavelengths are allowed. I may be misunderstanding since I'm certainly no expert on quantum field theory, but that's the impression I get from articles like these.

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 Quote by JesseM So I wonder, can the energy density between plates in the casimir effect go below the vacuum energy density by significantly more than 10^-9 joules per cubic meter? Does anyone know the equation for calculating the energy density between plates in terms of their size and separation? I found a bunch of pages that give an equation for the force (see here, for example), but I'm not sure how to translate this into energy density.
The force is inversely proportional to the fourth power of the distance between the Casimir plates:

$$F \sim \frac{hc}{d^4}$$

You can consider the energy as a negative work done by the vacuum (W ~ F d):

$$\frac{F}{A} = \frac{E}{V} = \rho_{casimir} \sim - \frac{hc}{d^4}$$

$$\rho_{\Lambda} \sim 10^{-120}$$
$$\rho \sim \rho_{\Lambda} - \frac{hc}{d^4}$$
$$\rho \sim 10^{-120} - \frac{1}{d^4}$$