# The Vacuum Energy is fictitious.

• Tyger
In summary, the conversation discusses the idea of vacuum energy, which is considered to be fictitious and an artifact of the way the Casimir Effect is described. The Casimir Effect is described as an attractive force between two perfectly reflective plates, which is dependent on the area and separation between the plates. The conversation also discusses the concept of virtual particles/waves and how they contribute to the Casimir Effect. There is debate over whether the energy involved in the effect belongs to the vacuum or the plates themselves. Some argue that the concept of vacuum energy is not necessary to explain the Casimir Effect. Others believe that the agreement between prediction and observation is a strong argument for the existence of vacuum energy. The conversation also includes links to websites with more information and
Tyger
The "Vacuum Energy" is fictitious.

I have bad news for the nice folks who thought that by clapping our hands together we could have free energy from the vacuum. The vacuum energy is an artifact of the way the Casimir Effect is described.

Here are three websites that have good descriptions of the effect.

http://physicsweb.org/article/world/15/9/6

http://focus.aps.org/story/v2/st28

They describe an attractive force between two perfectly reflective plates that goes as:

F=pi2h-bar*CA/240r4

where A is the area and r is the separation between the plates.

The problem with assuming that the energy involved is "in the vacuum" is that it completely omits the question of the energy used to make the surfaces of the plates. Note that while the force varies as the inverse fourth power, the energy varies as the inverse third power. So we should think of the energy as asociated with a field of virtual photons emanating from the surface of the plates, and the Casimir Effect as being due to a kind of distance dependent surface tension, whereas ordinary surface tension, whch has the dimensions of energy divided by area, is independent of distance.

Last edited:

Originally posted by Tyger
The vacuum energy is an artifact of the way the Casimir Effect is described.

Just to be sure, it's your understanding that vacuum energy doesn't exist. Correct? If so, where'd you get this from?

Originally posted by jeff
Just to be sure, it's your understanding that vacuum energy doesn't exist. Correct? If so, where'd you get this from?

From the realization that the people who described the experiment had failed to take into account the possibility that the putative "vacuum energy" may actually have been expended in making the surface of the plates. When you consider that the force and energy go as r&minus;4 and r&minus;3 respectively, that pretty much clinches the notion that the energy belongs to the plates and not the vacuum.

Alas, the fuzzy thinking of the people describing the experiment has fed the fuzzy thinking of the people who want to get free energy out of the vacuum!

Last edited:
The Casimir Effect is indeed a result of vacuum energy. It is, to put it more correctly, a result of virtual particles/waves. By the laws of quantum mechanics, waves must be quantisized, and between two plates (non-conducting I note), only certain wavelenghts are allowed between them. Outside the plates, there are infinitely more virtual ones allowed. Hence you get a virtual energy density difference. Result: The plates are pushed (or pulled) together. Nothing coming from the plates themselves.

As for the powers of the various items in the equation, the dimensions still come out correct, and it would make sense considering a normal inverse square drop off of force combined with the fact it is a force related purely to surface area.

Side: Yes, it is a symmetric force, hence you can't get energy from it to power things.

When you consider that the force and energy go as r?4 and r?3 respectively, that pretty much clinches the notion that the energy belongs to the plates and not the vacuum.

I'm sorry, I don't see why that would follow.

If you want to "prove"

or otherwise demonstrate that there is such a thing as vacuum energy you will need something besides the Casimir Effect to do it. The concept isn't necessary to describe the CE. All that is necessary is that we introduce the idea that it takes a certain amount of energy per unit area to form the surface, over and above the normal surface tension, and this energy is associated with a field which is distance dependent.

There's a pretty simple test which can be applied to anything electromagnetic, and that is scale invariance. Which interpretation (and that is what we're talking about here) is scale invariant.

Originally posted by Tyger
From the realization that the people who described the experiment had failed to take into account the possibility that the putative "vacuum energy" may actually have been expended in making the surface of the plates...

So this is your invention. Correct?

if it's correct that The "Vacuum Energy" is fictitious than there should be modification to theories such as string theory loop quantum gravity won't it?

Originally posted by Tyger
or otherwise demonstrate that there is such a thing as vacuum energy you will need something besides the Casimir Effect to do it. The concept isn't necessary to describe the CE.

True, vacuum energy is not the only possible explanation for the Effect. However, the CE was predicted as a result of vacuum energy. The agreement between prediction and observation is a very strong argument for the correctness of a theory. It is, in fact, the only test of correctness or incorrectness the scientific comunity can accept.

Originally posted by jeff
So this is your invention. Correct?

You got it. Be back in a while with more on the subject.

Here's some stuff I found

on a web search a few minutes ago.

http://alsystems.algroup.co.uk/casimir/VED2.html

http://www.xs4all.nl/~mke/vacuum.htm

On the SciAm page I especially liked the comments of John Baez which are pretty much the way I see it, that at least for the CE they are an artifact of our bookkeeping.

The second page is an interesting but not very convincing derivation of the idea that the VE doesn't really belong to the vacuum at all. I'm afraid he really lost me on the "space curvature" part.

The third page is one of the kookier pages on the subject I ran across. Consideraring that cosmic rays millions of times more energetic than anything the particle accelerators put out srike the Earth every day I think we're safe.

So the basic idea I have is that the putative VE is really just energy associated with a kind of surface tension, perhaps the ordinary surface tension, or the material work function, and is not associated with any ambient vacuum energy.

If anyone knows of any convincing paper on the subject I would appreciate knowing about it.

Last edited by a moderator:
Originally posted by Tyger
on a web search a few minutes ago.

...

On the SciAm page I especially liked the comments of John Baez which are pretty much the way I see it, that at least for the CE they are an artifact of our bookkeeping.
...

BTW Rutwig posted this in another thread discussing Casimir Effect. It is the 1996 experimental report of measuring the effect:

Originally posted by rutwig
You can find the announcement of the first measure in here:

http://www.aip.org/enews/physnews/1996/split/pnu300-3.htm

Since your view is close to those expressed by Baez, as a convenience, here are Baez comments on the SciAm board:

<<John Baez is a member of the mathematics faculty at the University of California at Riverside and one of the moderators of the on-line sci.physics.research newsgroup. He adds some context:

"The concept of vacuum energy shows up in certain computations in quantum field theory, which is the tool we use to conduct modern particle physics. In reality, particles interact with one another through a variety of forces. This is a complicated business, so in quantum-field theory we start by studying an idealized model in which particles do not interact at all. This is called a 'free-field theory.' Then we use this free-field theory as the basis for studying the 'interacting-field theory' we are really interested in.

"In quantum-field theory, the vacuum state is defined to be the state having the least energy density. Something funny happens when we use a free-field theory to study an interacting-field theory: the vacuum state of the free-field theory is different from vacuum state of the interacting-field theory. The vacuum state of the interacting-field theory may have more or less energy than that of the free-field theory; the difference is called the vacuum energy.

"One should not take this vacuum energy too literally, however, because the free-field theory is just a mathematical tool to help us understand what we are really interested in: the interacting theory. Only the interacting theory is supposed to correspond directly to reality. Because the vacuum state of the interacting theory is the state of least energy in reality, there is no way to extract the vacuum energy and use it for anything.

"It is a bit like this: say a bank found it more convenient (for some strange reason) to start counting at 1,000, so that even when you had no money in the bank, your account read $1,000. You might get excited and try to spend this$1,000, but the bank would say, 'Sorry, that \$1,000 is just an artifact of how we do our bookkeeping: you're actually flat broke.'

"Similarly, one should not get one's hope up when people talk about vacuum energy. It is just how we do our bookkeeping in quantum field theory. There is much more to say about why we do our bookkeeping this funny way, but I will stop here." >>

Tyger, I noticed at the beginning of the thread that you have a link to
the Usenet FAQ explanation of the Casimir Effect----your first link is to a Physics FAQ mirror site in IIRC Australia.
The same explanation of the CE is at the John Baez site.

this explanation does not depend on the actual position of the zero (energy zeros are often just a bookkeeping point of reference and what matters are the differenences, as you know well)

the explanatation, which you posted at the start, does not depend on saying there is some amount of energy density in the vacuum. it depends on saying that this VARIES depending on how far the conducting plates are apart.
(i.e. with whatever zero in whatever theoretical framework in whatever units, the energy density varies as IIRC the inverse fouth power of separation)

this IIRC is what an experimental test should show is true or not true

in principle, I think, this fourthpower variation can be proven true or false experimentally even if the energy scale is adjusted so that the energy in the generic vacuum is zero (the position of the
zero does not seem to matter). My guess is that this is obvious to you and that I am just making it explicit.

I take no position on this. Rutwig posted a link to the 1996 experiment by someone at Los Alamos. It don't know how solid this experiment was or to what extent it verified the fourth power
dependence on the distance. I thought it was an odd detail
that in the actual experiment he used a gold sphere---it was
not two parallel plates after all.

Last edited by a moderator:
Writing the ground or vacuum state of a quantum system as |0>, we can calculate the VEV (vacuum expectation value) <0|H|0> of the hamiltonian. This is just the ground state energy of the system. However, although the vacuum is a writhing ocean of quantum fluctuations, because we are generally interested only in energies relative to the vacuum, we conventionally perform a "subtraction" in the hamiltonian by setting <0|H|0> = 0. However, we can detect the vacuum by inducing a shift &Delta;&epsilon; in it's energy through the introduction of a source and sink where particles would be created and annihilated with the interceding propagation of particles between source and sink creating a force between them. Although the VEV of the energy is not observable, the force resulting from it's shift created by disturbing the vacuum in this way is. In the Casimir effect the source and sink are thin parallel conducting plates with the Casimir force between them given roughly by &Delta;&epsilon;/d where d is the separation between the plates.

Last edited:
Tyger, do you happen to know if Lamoureux or whoever
measured the force in 1996 actually found a recipr. fourthpower
dependence on the separation?

that would be evidence of the same recipr. fourthpower
dependence of the energy density----wouldnt clinch it
but would fit the common picture people have

if he just found some attractive force at some distance
and then did not vary the distance it wouldn't seem to
be very meaningful

also, any idea why he used a gold sphere and a plate, instead
of the two plates that people frequently think of as schematic
for the Casimir effect?

So it would seem

that the real cognizenti already realize that vacuum energy is fictitious, that it's a sort of "urban myth" in physics.

I'm going to make some general comments, mostly relating to Marcus's and Jeff's posts.

I really liked Baez' description, something meaningful to a physicist and the man in the street.

Now energy is both conserved and localizable (conservation at all vertexes, etc.), so this phony bookkeeping will eventually catch up with you. In this case there really is energy in the vacuum, but it is associated with virtual photons from the plates, not ambient vacuum energy. What's more, to stretch the plates out, increase their area, you would have to do work, so indeed it does represent a surface tension. Whether it repressents all the plates surface tension is a very good question. After all, if you let the plates come together, all that energy will disappear and their area will be halved. You may remember that I used the surface tension of the electron gas to explain why metals were strong in the physics puzzle. Everything fits together in one way or another.

The fellow used a sphere because a thin plate tends to warp and throw off the measurements. And yes, the fourth power law held.

This is something Jeff should be interested in, which is the best way to look at electromagnetism/electrodynamics. People often say that the photon has rest mass zero but this is somewhat misleading. In reality the photon can have any rest mass squared from minus to plus a very large value. Till the point where it merges with the weak interaction, to be precise. This leads to a very simple but important fact, the electromagnetic interaction is scale invariant, par excellence. You can deduce many things from this scale invariance, from the law of black body radiation to atomic radiation moments. You can use it to check your equations, in much the same way people use dimensional analysis, before you begin to make you calculations. If scale invariance applys in a situation, you know it is electromagnetic, if it doesn't apply, then something is involved too, or else. None of the other interaction are fully scale invariant.

Last edited:
Originally posted by Tyger
...energy is both conserved and localizable (conservation at all vertexes, etc.), so this phony bookkeeping will eventually catch up with you. In this case there really is energy in the vacuum, but it is associated with virtual photons from the plates, not ambient vacuum energy.

Actually the origin of the shift arises from changes in the boundary conditions for the fields. In fact such effects may even be viewed as a result of the local breaking of the translational invariance of the vacuum by the sinks and sources.

Originally posted by Tyger
People often say that the photon has rest mass zero but this is somewhat misleading. In reality the photon can have any rest mass squared from minus to plus a very large value.

Recall the classical relation p&mu;p&mu; = -m2 between a particle's 4-momentum and rest mass. In the case of the photon m=0. But the time-energy uncertainty principle allows photons (and other massless field quanta) to have energies other than those allowed by the relation p&mu;p&mu; = 0 as long as their lifetimes are sufficiently brief. Such photons are called "virtual" and their 4-momenta are said to be "off the mass shell", i.e. p&mu;p&mu; has a nonzero value which I'm guessing you're thinking of as a sort of (square root of) virtual photon "rest mass", but it really shouldn't be thought of that way and no one does.

Originally posted by Tyger
...the electromagnetic interaction is scale invariant...

An interaction is scale-invariant when it's coupling has no dependence on the energies at which a given interaction occurs. Mathematically, it's beta function vanishes. This is not the case with the electromagnetic or any other known fundamental interaction.

You along with everyone else here need to learn QFT.

Last edited:
A quick read of the first paper on your ref. list yields the following -
Casimir realized 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,
Surely this means that the the Casimir force is a quantity of vacuum determined by the number of virtual photons. As photons are fields of electromagnetic force they create a greater vacuum than gravitons hence the difference in the formula i.e. fourth power in place of second power.
I wonder why Casimir ever thuoght it possible to get less than a whole wave length?

Originally posted by jeff
Actually the origin of the shift arises from changes in the boundary conditions for the fields. In fact such effects may even be viewed as a result of the local breaking of the translational invariance of the vacuum by the sinks and sources.

Of course the translational invariance of the vacuum is broken, it has virtual photons being emitted and absorbed by the sinks and sources.

Recall the classical relation p&mu;p&mu; = -m2 between a particle's 4-momentum and rest mass. In the case of the photon m=0. But the time-energy uncertainty principle allows photons (and other massless field quanta) to have energies other than those allowed by the relation p&mu;p&mu; = 0 as long as their lifetimes are sufficiently brief. Such photons are called "virtual" and their 4-momenta are said to be "off the mass shell", i.e. p&mu;p&mu; has a nonzero value which I'm guessing you're thinking of as a sort of (square root of) virtual photon "rest mass", but it really shouldn't be thought of that way and no one does.

If you look at it my way, with scale invariance as a defining property of the field, you can get rid of the "uncertainty principle" justification and still get the right result. You replace all these other rules with three:

The photon is a vector particle.

It has a certain coupling strength.

It's interaction is scale invariant.

Add in the spins and masses of the charged particles and you have something to work with.

An interaction is scale-invariant when it's coupling has no dependence on the energies at which a given interaction occurs. Mathematically, it's beta function vanishes. This is not the case with the electromagnetic or any other known fundamental interaction.

I occurs that my definition of scale invariance, which depends on all scale quantities in a physical system, may be different from the QFT definitiion, creating a certain misunderstanding here. I'll have to put together an explanation of how I use the term and apply it to physical problems.

You along with everyone else here need to learn QFT.

It's entirely possible that QFT could be greatly simplified by a little rethinking of viewpoints. In particular the "renormalization problem" might be solved. Which mouse will be brave enough to bell the cat?

Originally posted by elas
...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

In other words we have a quantization in the space between parallel plates analogous to the standard particle in a box scenario familiar from quantum mechanics.

Yep

Originally posted by jeff
In other words we have a quantization in the space between parallel plates analogous to the standard particle in a box scenario familiar from quantum mechanics.

That right, that's why the total energy is less when the plates are brought closer together. It's a continuous spectrum when the plates are isolated, and all possible energies are represented by the virtual photons. Very simple picture, easy to understand without confusing people with the idea that the vacuum has instrinsic energy. Nice illustration of the reality of virtual photons and their significance in my view.

Originally posted by Tyger
It's entirely possible that QFT could be greatly simplified by a little rethinking of viewpoints.

Have you studied QFT?

Originally posted by Tyger
...the "renormalization problem" might be solved.

What do you think renormalization is?

Last edited:

Originally posted by Tyger
That right

Yes, I know.

It seems to me that there is a fundamental question to be asked .
Do you believe that creation starts from absolute nothing or from some force or energy that always existed and therefore does not require an explanation regarding its origin?
If creation starts from 'nothing' then the vacuum force is the source of all creation. Given that it is not possible to create a greater force (or quantity of energy) than that which existed at the beginning, then vacuum must remain the primary force. This leads to two questions 'What carries the vacuum force?' and 'Are the secondary forces (electromagnetic, strong and weak) neccessary, if so how are they created?'
If you subscribe to 'The existing or permament force' camp then original creation is outside your field of reference, it becomes an act of magic (such as the Big Bang or Steady State) and is not subject to scientific questioning.
Just as we "cannot clap our hands and get free energy from vacuum" neither can we assemble numbers into formula to give free energy; there also has to be an underlying reality that can be explained in words.
On my website I have used vacuum to propose a none mathematical explanation of force that can be applied to any existing numerical theory. I have yet to see any non mathematical explantion of the nature (or structure) of force in any of the quantum theories. Indeed the authors of those books written specifically for laymen, stress that quantum theory does not deal with how or why but only with event prediction. It is my belief that by altering the words used to explain the numbers we can add to quantum theory an explanation of 'how and why' and that begins with the acceptance of vacuum as the only fundamental force.

Elas, my own opinion

Originally posted by elas
It seems to me that there is a fundamental question to be asked .
Do you believe that creation starts from absolute nothing or from some force or energy that always existed and therefore does not require an explanation regarding its origin?
If creation starts from 'nothing' then the vacuum force is the source of all creation. Given that it is not possible to create a greater force (or quantity of energy) than that which existed at the beginning, then vacuum must remain the primary force. This leads to two questions 'What carries the vacuum force?' and 'Are the secondary forces (electromagnetic, strong and weak) neccessary, if so how are they created?'
If you subscribe to 'The existing or permament force' camp then original creation is outside your field of reference, it becomes an act of magic (such as the Big Bang or Steady State) and is not subject to scientific questioning.
Just as we "cannot clap our hands and get free energy from vacuum" neither can we assemble numbers into formula to give free energy; there also has to be an underlying reality that can be explained in words.
On my website I have used vacuum to propose a none mathematical explanation of force that can be applied to any existing numerical theory. I have yet to see any non mathematical explantion of the nature (or structure) of force in any of the quantum theories. Indeed the authors of those books written specifically for laymen, stress that quantum theory does not deal with how or why but only with event prediction. It is my belief that by altering the words used to explain the numbers we can add to quantum theory an explanation of 'how and why' and that begins with the acceptance of vacuum as the only fundamental force.

arrived at after years of studying Nature, is that in some very deep sense the world is made of "mathematical things", that it is mathematical to the core. So I don't see anything wrong with a mathematical description of the world. And I don't know if any meaning can be given to "what existed before the beginning?"

In this topic I'm trying to sharpen up discussion about a very misused and abused notion, that of vacuum energy.

Originally posted by Tyger
...I'm trying to sharpen up discussion about a very misused and abused notion, that of vacuum energy.

Hmm...

Last edited:

Originally posted by jeff

Hardly. Let's hear your description of the Casimir Effect so we can judge how much sense it makes! I saw your description of the inverse square law for electricity, you could have invoked scale invariance (not Bjorken scaling BTW) to say the same thing much more simply.

And I've yet to see an original idea come from you.

Originally posted by Tyger
Hardly. Let's hear your description of the Casimir Effect

I've been describing the casimir effect throughout my posts to this thread.

Originally posted by Tyger
I saw your description of the inverse square law for electricity, you could have invoked scale invariance...to say the same thing much more simply.

Alright einstein, put your money where your mouth is and show us the math.

Originally posted by Tyger
And I've yet to see an original idea come from you.

Any crackpot can spin bullsh*t theories. You don't even understand the theories you're criticizing. Why not reflect on QFT after you've mastered it? It's very hard to come up with better ideas when you don't understand the ones your aiming to replace.

Clarification

I need some clarification on current theory.

In high school physics I was taught that the expansion of the universe was slowing down. This leads to a big crunch after it slows down enough to collapse back in on itself.
I really liked this concept because of the cyclical aspect to nature it implied.

Recently on some mexican TV station, I saw some guy from Plank institute in Munich (if i remember correctly). He was saying the expansion of the universe is accelerating due to this mighty vacuum force. Is this the same vacuum this thread is reffering to?

Which of these is favored currently?
IS this a result of outdated information or conflictiong theory?

Originally posted by HunterSThompson
...the expansion of the universe is accelerating due to this mighty vacuum force. Is this the same vacuum this thread is reffering to?

We're discussing the electromagnetic vacuum which isn't the same. It's generally accepted that the universe is undergoing accelerated expansion, but next to nothing is known about the underlying physics.

Originally posted by jeff
I've been describing the casimir effect throughout my posts to this thread.

Alright einstein, put your money where your mouth is and show us the math.

Any crackpot can spin bullsh*t theories. You don't even understand the theories you're criticizing. Why not reflect on QFT after you've mastered it? It's very hard to come up with better ideas when you don't understand the ones your aiming to replace.

Sounds like abuse.

Jeff, old boy

Originally posted by jeff
Writing the ground or vacuum state of a quantum system as |0>, we can calculate the VEV (vacuum expectation value) <0|H|0> of the hamiltonian. This is just the ground state energy of the system. However, although the vacuum is a writhing ocean of quantum fluctuations, because we are generally interested only in energies relative to the vacuum, we conventionally perform a "subtraction" in the hamiltonian by setting <0|H|0> = 0. However, we can detect the vacuum by inducing a shift &Delta;&epsilon; in it's energy through the introduction of a source and sink where particles would be created and annihilated with the interceding propagation of particles between source and sink creating a force between them. Although the VEV of the energy is not observable, the force resulting from it's shift created by disturbing the vacuum in this way is. In the Casimir effect the source and sink are thin parallel conducting plates with the Casimir force between them given roughly by &Delta;&epsilon;/d where d is the separation between the plates.

this is the only description of the CE you've given. You're saying that the vacuum energy can be arbitrarily set to zero or some other value. I'm saying that it is observable in the form of the surface tension of the material involved. A specific physical prediction, no crackpot idea involved. What's more there may be very real and practical ways of observing the VEV, such as scattering electons off the virtual photons between the plates, so who's theory is more real, mine which makes specific predictions, or yours which uses a little mathematical trickery to make the fields vanish? Do you know how shocked people were when Bohm and Aharanov showed that the vector potential was real and observable? There were plenty who thought they could subtract it out and disregard it.

And you're right, I don't understand QFT the way I want to, but that doesn't mean I don't know about its very real shortcomings. And what do you plan to do to reduce the twenty six free parameters in the Standard Model?

Last edited:

Originally posted by HunterSThompson
Recently on some mexican TV station, I saw some guy from Plank institute in Munich (if i remember correctly). He was saying the expansion of the universe is accelerating due to this mighty vacuum force. Is this the same vacuum this thread is reffering to?
in principle, yes, it is the same vacuum. however, the vacuum is not well understood enough to actually predict the properties of the vacuum that would yield the observed acceleration of the cosmos.

Which of these is favored currently?
IS this a result of outdated information or conflictiong theory?

acceleration is a rather new observation. but it is well supported and most cosmologists believe that the universe is currently accelerating and therefore no big crunch. instead the universe will end a cold and desolate wasteland.

Originally posted by marcus
Sounds like abuse.

It isn't.

Last edited:

Originally posted by Tyger
this is the only description of the CE you've given.

I also posted,

...the origin of the shift arises from changes in the boundary conditions for the fields. In fact such effects may even be viewed as a result of the local breaking of the translational invariance of the vacuum by the sinks and sources.

...we have a quantization in the space between parallel plates analogous to the standard particle in a box scenario familiar from quantum mechanics.

Originally posted by Tyger
You're saying that the vacuum energy can be arbitrarily set to zero or some other value.

In the absence of gravity, yes.

Originally posted by Tyger
I'm saying that it is observable in the form of the surface tension of the material involved. A specific physical prediction...

Hmm...surface tension. We'll, with respect to this wording, I think the closest we can get to each other on this issue is this: Since causality precludes perfectly rigid bodies, like any force, the casimir force will - however slightly - deform the objects on which it acts.

Is your view experimentally differentiable from the standard one?

Last edited:

Originally posted by lethe
...the universe will end a cold and desolate wasteland.

• Other Physics Topics
Replies
3
Views
1K
• Quantum Physics
Replies
46
Views
5K
• Mechanics
Replies
34
Views
2K
• Other Physics Topics
Replies
1
Views
3K
• Quantum Physics
Replies
11
Views
1K
• Quantum Physics
Replies
46
Views
2K
• Cosmology
Replies
3
Views
2K
• Electrical Engineering
Replies
8
Views
3K
• Quantum Physics
Replies
5
Views
2K
• Cosmology
Replies
2
Views
3K