Quantum Fluctuations and Their Energy

Matt Strassler [August 29, 2013]

In this article I am going to tell you something about how quantum mechanics works, specifically the fascinating phenomenon known as “quantum fluctuations”, and how it applies in a quantum field theory, of which the Standard Model (the equations that we use to predict the behavior of the known elementary particles and forces) is an example.  A deep understanding of this phenomenon, and the energy associated with it, will lead us directly to confront what is certainly one of the most dramatic unsolved problem in science: the cosmological constant problem.  It will also lead us to the puzzle known as naturalness or the hierarchy problem, though I’ll explain that elsewhere.

An aside: in quantum field theory, quantum fluctuations are sometimes called, or attributed to, the “appearance and disappearance of two (or more) `virtual particles‘ “. This technical bit of jargon is unfortunate, as these things (whatever we choose to call them) are certainly not particles — for instance, they don’t have a definite mass — and also, more technically, because the notion of “a virtual particle” is only precisely defined in the  presence of relatively weak forces.

Cl_vs_Qu_BowlMarble

Fig. 1: Normal intuition would lead us to expect that anything like a marble sitting in anything like a bowl would sit quietly at the bottom. But a quantum mechanical particle in a trap of some sort will have a position and motion that are constantly fluctuating. These fluctuations have energy; the motion-energy of a quantum particle in a trap is never zero.

Quantum fluctuations are deeply tied to Heisenberg’s uncertainty principle. Here’s the classic, simplest example (Figure 1): if you put a marble at the bottom of a bowl, it will stay there indefinitely, as far as you can tell. That is what you’d expect, from daily experience.  And it would be true, in the absence of quantum mechanics. But if you put a very lightweight particle in a tiny bowl, or some other type of trap, you will discover it doesn’t sit at the bottom. If it did sit motionless at the bottom, it would violate the uncertainty principle — a principle which assures you can’t know exactly where the particle is (i.e., at the bottom) and how it’s moving (in this case, not moving at all) at the same moment.  You may think of this, imperfectly but usefully, as due to a sort of jitter which afflicts the particle and prevents it from settling down the way your intuition from marbles in bowls leads you to expect.  One useful aspect of this imperfect picture is that it gives you an intuitive idea that there might be energy associated with this jitter.

In quantum field theory — the quantum equations for fields, such as the electric field, there is a similar effect. Let me now explain it.

Fluctuations of Quantum Fields

Every elementary particle (I speak of real particles now) in our universe is a ripple — a small wave, the wave of smallest possible intensity — in a corresponding elementary quantum field (Figure 2). A W particle is a ripple in a W field; a photon [a particle of light, which you may think of as the dimmest possible flash] is a ripple in the electric field; an up quark is a ripple in the up quark field.

And if there are no particles around? Even in what we consider empty space, the fields are still there, sitting quietly in empty space, much as there’s water in the pond even if no wind or pebbles are making ripples on its surface, and there’s still air in the room even if there’s no sound.

Fig. 2: (Left) A particle may be viewed as a little ripple in a field, which away from any particles just sits quietly -- naively -- like the marble in the bottom of the bowl.  (Right) But in fact the value of the field at every point in space is constantly fluctuating, just as the particle in the bowl is always jittering.

Fig. 2: (Left) A particle may be viewed as a little ripple in a field, which away from any particles just sits quietly — naively — like the marble in the bottom of the bowl. (Right) But in fact the value of the field at every point in space is constantly fluctuating, just as the particle in the bowl is always jittering.

But here’s the thing: those fields are never entirely quiet. Quantum fields never quite maintain a constant value; their value at any point in space is always jittering around a bit. This jitter is called “quantum fluctuations”, and just as for the particle in the tiny bowl, it is a consequence of the famous “uncertainty principle” of Heisenberg.  (You can’t know a field’s value, and how it’s changing, at exactly the same time; your knowledge of at least one, and typically both, must inevitably be imperfect.)  Again, these quantum fluctuations are sometimes described as being due to two or more “virtual particles”, but this name really reflects a technical issue (i.e., how you can calculate the fluctuations’ properties using Feynman’s famous diagrams) more than it guides you as to how you should really think about them.

Obvious question: are you sure there are really quantum fluctuations for fields? Answer: Yes, though I won’t explain it now. One example: quantum fluctuations are known to cause the strengths of forces to drift as you measure them at shorter and shorter distances, and not only do we observe such drift in data, what we observe matches, to high precision, with what we calculate using the Standard Model.  This success confirms not only the presence of quantum fluctuations but also the detailed structure of the Standard Model, down to distances of about a millionth of a millionth of a millionth of a meter. Another example: the response of an electron to a magnetic field can be measured to about one part in a trillion; it can also be calculated, using the Standard Model, to about one part in a trillion, assuming the existence of these fluctuations of the known fields of nature. Amazingly, the measurement agrees with the Standard Model calculation.

Importantly, that jitter creates a certain amount of energy — a lot of energy. How much? The better your microscope (or particle accelerator), the more jitter you can detect, and the more energy you discover the jitter has. If you’d like to see how we estimate the amount of this energy, click here (sorry, page not yet written). If not, or if you’d just like to get the main point and then come back to study this estimate, just accept what I’m about to tell you.

The Energy of These Fluctuations and the Cosmological Constant

Let’s consider a box of size one meter by one meter by one meter, and ask: how much energy, roughly, do we calculate is inside the box due to the jitter in a single elementary field?  (See Figure 3.)

Calculation 1: Suppose, as our experimental measurements at the Large Hadron Collider [LHC]  suggest, that the Standard Model is a valid description of all processes that occur at distances of larger a millionth of a millionth of a millionth of a meter — let’s call this the “LHC-ish distance”, about 1/1000 the radius of a proton, because that’s roughly the scale the experiments at the LHC can probe — and processes involving elementary particle collisions with energies smaller than about 1000 times the proton‘s mass-energy [i.e. it's E=mc² energy].  This energy is the typical mass-energy of the heaviest particle that we could hope to discover in the LHC’s proton-proton collisions, so let’s call it the “LHC-ish energy”.  Then the amount of energy in the fluctuations of each field in the Standard Model (say, for example, the electric field) is this: in every cube whose sides are an LHC-ish distance, there’s something like an LHC-ish energy inside.  In other words, the energy density is about one LHC-ish energy per LHC-ish volume.  Compare this with ordinary matter, whose energy density is a few proton or neutron mass-energies  (an atomic nucleus worth of mass-energy) for every atom, whose volume, since a proton or neutron is 100,000 times smaller in radius than an atom, is about 1,000,000,000,000,000 (a thousand million million) times larger than a proton’s volume.  (Remember the atom is emptier, relatively speaking, than the solar system.) That means the energy density of quantum fluctuations of the electric field is roughly a million million million times more than ordinary matter, and so the mass-energy in fluctuations of the electric field inside a cube one meter on a side is about a million million million times larger than the mass-energy stored in a cube of solid brick, one meter on each side. How much energy is that? Easily enough to blow up a planet, or even a star! In fact, it’s comparable to the total mass-energy  of the sun. (Egad!) Now, one can’t release this energy from the vacuum of space, for good or evil — so don’t worry about its presence, it’s not directly dangerous.  But this is already enough to raise the specter of the cosmological constant problem.

Calculation 2: Suppose, as is relevant for the question of the hierarchy problem and the naturalness of the universe, that the Standard Model describes all particle physics processes down to the length scale where gravity becomes a strong force — the so-called Planck length, which is another thousand million million times smaller than the distance considered in Calculation 1. Then the amount of energy from fluctuations of the electric field inside a cube a meter on all sides is larger than in Calculation 1 by

  • (1,000,000,000,000,000)4 = 1 with 60 zeroes after it.

If you take this number and multiply by the number given in Calculation 1, you get easily enough energy to blow up every star in every galaxy in the visible part of the universe… many many many many times over. And that’s how much energy there is in every single cubic meter — if the Standard Model is correct for physical processes of size all the way down to the Planck length.

Fig. 2:

Fig. 3: The amount of energy due to quantum fluctuations of any field is enormous. In the Standard Model, the total energy in a meter of empty space is vastly more (Calculation 1) than in a cubic meter of ordinary matter; and there’s even unimaginably more (Calculation 2) if the Standard Model is valid all the way down to the Planck length. But the universe’s slow expansion (Measurement 0) suggests the total energy of a cube empty space (often called the `dark energy’) is much much less than that stored in a cube of ordinary matter. This is the cosmological constant problem: a profound apparent failure of the otherwise highly successful equations used in particle physics and gravity.

More generally, if the Standard Model (or any typical quantum field theory without special symmetries) is valid down to a distance scale L, the energy of the fluctuations in a cube of size L³ is approximately hc/L (for each field), where h is Planck’s quantum mechanics constant and c is the universal speed limit, known usually as “the speed of light”.  That means the energy density is roughly hc/L4 — if L decreases by a factor of 10, the energy density goes up by a factor of 10,000! That’s why these numbers in Calculations 1 and 2 are so darn big.

These statements must really seem bizarre to you. They are bizarre, but hey — quantum physics is bizarre in many ways. Moreover, neither quantum mechanics in general, nor quantum field theory in particular, have previously led us astray. As I mentioned earlier, we have plenty of evidence that the very basic calculations like the ones required here work beautifully in quantum field theory. The fact that there are quantum fluctuations, with associated energy, is so deeply built into quantum mechanics that to declare it simply to be false requires you to explain a whole library of experimental results for which quantum mechanics gave correct predictions. So as scientists we have no choice but to take our calculation very seriously, and to try to understand it.

A couple of obvious questions you may ask: Why can’t we easily tell whether all that energy is there or not?  Why doesn’t all this vast amount of energy have an enormous effect on ordinary matter, including us?!   Answer, part 1: Because there’s the same amount of energy in every cubic meter of space (Figure 4), both inside and outside every box you can draw.  An analogy: there’s air pressure inside a house, but it doesn’t cause the house to explode as long as there’s equal air pressure outside the house. Similarly, the fact that this energy density of tiny quantum fluctuations is constant throughout space and time means that there’s no effect on objects that sit within it and move through it. Only changes in energy from place to place, or over time, will affect particles, and the atoms that are made from such particles, and people and planets made from such atoms. And indeed, this energy from quantum fluctuations is the same everywhere, always, so it’s impossible to feel it, or be pushed around by it, or release it for good or evil.

Fig. 3: The energy stored in empty space has no effect on ordinary objects because it's the same everywhere, at all times.  However, gravity (in Einstein's theory) does respond to a constant energy density; it changes how the universe expands.

Fig. 4: The energy stored in empty space has no effect on ordinary objects because it’s the same everywhere, at all times. However, gravity (in Einstein’s theory) does respond to a constant energy density; it changes how the universe expands.

However! Answer, part 2: While in Newton’s law of gravity, where gravity pulls on mass, this energy of empty space will have no effect, the same is not true in Einstein’s version, where gravity pulls on energy and momentum. Whether calculation 1 is right, or calculation 2 is right, or something in between, such a vast amount of energy in every cube of space — what is often called “dark energy” — would cause the universe to expand with extreme speed! (In fact, this is the mechanism behind “cosmic inflation”, which is a phase that the universe may have gone through long ago, making it the rather uniform place we see today.) The fact that the universe is not expanding at tremendous speed implies that the energy density of space should be vastly less than the mass-density of ordinary matter, instead of vastly greater. In every cubic meter of empty space there is only about one atom’s mass-energy, whereas in a cube of bricks the mass-energy is that of its huge number of atoms —  the number being about 1 with 30 zeroes after it.  The fact that there is apparently so little energy density in empty space, despite all the energy we calculate should be there from quantum fluctuations of the fields we already know about, is the mother and father of all great puzzles in particle physics: the cosmological constant problem.

Next obvious question: are you sure the quantum fluctuations really have energy, or is it possible they don’t, thereby eliminating the cosmological constant problem? Answer: Yes, I’m sure quantum fluctuations do have energy; it’s what’s called zero-point energy, and it’s completely fundamental to quantum mechanics, and due yet again to the uncertainty principle.  And this can be checked: n a clever experiment, the energy in a small region can be made to have a measurable impact called the “Casimir effect”, which was predicted in the 1940s, first observed in the 1970s and tested more carefully in the 1990s.  [There is some controversy about whether this is really relevant to the question, however.]

The cosmological constant problem is a very serious one. We know, experimentally, that the universe is not expanding at a spectacular rate; it’s expanding rather slowly; that’s Measurement 0 in Figure 3.  So

  • either this calculation (even calculation 1, which doesn’t assume anything that we don’t know experimentally about the Standard Model) is wrong, somehow, and the energy simply isn’t there, or
  • the effect of this energy on the universe’s expansion is not what we think, because our understanding of gravity is wrong, or
  • it’s a correct calculation, but it answers the wrong question in some way we don’t understand.

Nobody knows for sure.  I’ll talk about possible solutions to this problem in a separate article on the cosmological constant.  But let me mention one solution that is interesting but certainly doesn’t work, because it will be relevant elsewhere.

Could The Energy from Different Fields Cancel Out?

Now here’s a cute idea for getting rid of all that energy. It turns out that

So maybe, even though each field’s energy is huge, when you add up the energy from all the fields, the total energy is zero — or at least really tiny?

Well, you can do this calculation, and in the Standard Model you’ll see it doesn’t work; there are way too many fermions, and there should be a huge negative energy in empty space.

One cool thing about the speculative theory called “supersymmetry” is that it forces you to add exactly the right particles (a “superpartner particle” for every known type of particle) so that you get this cancellation automatically! In fact, it’s the only type of speculative theory currently known to humans in which this would happen.

Unfortunately, it doesn’t actually solve the cosmological constant problem. If supersymmetry isn’t explicitly manifest [and in our world it can't be -- the known particles would in this case have had identical masses to their hypothetical superpartner particles and would have been discovered long ago] then the cancellation is only partial.  And this partial cancellation, which could invalidate Calculation 2, still at best leaves you with the huge amount of energy density mentioned in Calculation 1.  As noted in Figure 3, that gigantic amount of energy density is still enough to make the universe behave very differently from what we observe, unless there’s something wrong with Einstein’s theory of gravity.

In short, at the present time, no one knows a clever way to automatically make the energy density from the fluctuations of different fields cancel out in a world that, down to LHC-ish distances, is described by the Standard Model. In fact, no one knows how to do it in any even slightly non-supersymmetric quantum field theory (and even then, combining supersymmetry with gravity tends to reintroduce the problem.)

To say this another way: even though it is possible that there is a special cancellation between the boson fields of nature and the fermion fields of nature, it appears that such a cancellation could only occur by accident, and in only a very tiny tiny tiny fraction of quantum field theories, or of quantum theories of any type (including string theory).  Thus, only a tiny tiny tiny fraction of imaginable universes would even vaguely resemble our own (or at least, the part of our own that we can observe with our eyes and telescopes).  In this sense, the cosmological constant is a problem of “naturalness”, as particle physicists and their colleagues use the term: because it has so little dark energy in it compared to what we’d expect, the universe we live in appears to be highly non-generic, non-typical one.

[As I mentioned at the beginning, there is a second big problem associated with quantum fluctuations which you may wish to read about.  It is known from different points of view as the Standard Model's naturalness problem or the hierarchy problem.  Since it is a bit more complicated to describe, and deserves its own discussion, I've written a special article about it here.]

123 responses to “Quantum Fluctuations and Their Energy

  1. Dear Matt, there might be a broken link here: ” If you’d like to see how we estimate the amount of this energy, click here.”
    As always, very VERY interesting post; as a physics enthusiast (but not physicist, unfortunately) I really do appreciate your work.
    Best,
    Piermatteo

  2. kashyap vasavada

    Nice article. Question for clarification : Are you saying that even if LHC finds SUZY particles at that energy, there will be only a partial cancellation and the major problem will still remain?

    • Correct. Supersymmetric particles at the LHC solve the hierarchy problem — the naturalness problem of the Standard Model, associated with the Higgs. They do not solve the cosmological constant problem.

  3. Could we posit that Dark Matter fields supply the necessary cancellation?

    • We can posit whatever we want; but then we need equations and good reasoning. There’s no known solution of this type in which dark matter fields do the cancellation automatically. Maybe that’s just because no one’s been clever enough… but it seems difficult.

      • “there are way too many fermions, and there should be a huge negative energy in empty space.” Isn’t dark energy that huge negative energy? This may sound obvious or outright stupid, but shouldn’t gravity cancel the energy of fermions?

        • No — the dark energy is a bug, not a feature, and this is called the cosmological constant problem. The observed dark energy is incredibly small, not huge, due itself to another unnatural cancellation. Yes, the dark energy may be more than half the energy density of our universe, but it is smaller (in magnitude) by a huge factor — about 10^40 even if vmax is only 500 GeV or so — compared to the energy density in Figure 5 or 6 from each fermion. See http://profmattstrassler.com/articles-and-posts/particle-physics-basics/quantum-fluctuations-and-their-energy/

          In other words, even if vmax = 500 GeV or so, then
          1) the Standard Model’s energy density is something near -(500 GeV)^4/(hc)^3 = -62,500,000,000 GeV^4/(hc)^3
          2) that of gravity (and other things) is near +(500 GeV)^4/(hc)^3 = +62,500,000,000 GeV^4/(hc)^3
          3) and the sum of the two equals the dark energy = 0.000,000,000,000,000,000,000,000,000,006 GeV^4/(hc)^3.
          If vmax is larger, the cancellation is even more spectacular.

  4. Matt, isn’t a Planck-scale cutoff somewhat of a copout? Namely, when you calculate the mass of an electron, you don’t put a cutoff at the Planck scale — instead, you take a bare mass (which is infinite), and subtract all quantum corrections (which are also infinite), and postulate that the result is finite and equal to the observed electron mass. If you trust the renormalization idea, the similar calculation should be done in this case as well — start from a bare infinite cosmological constant, subtract the infinite contribution of the zero-point fluctuations, and require the result to be the finite (and small) experimentally observed value. The only problem is that we don’t actually know how to perform the renormalization procedure in curved-space QFT.

    On the other hand, if you don’t trust the renormalization idea, you should put a Planck-scale cutoff to *all* loop-corrections. In particular, when you calculate the running of, say, electron mass, you start from a *finite* bare Planck-scale mass, calculate the *finite* loop corrections (they are finite because of the imposed cutoff), and then subtract them to obtain the renormalized value for the electron mass. This would probably require as much fine-tuning as it does for the cosmological constant (calculate huge zero-point energy as you did, postulate equally huge bare cosmological constant, but such that their difference is a very small observed value). What I am saying is that — if you take the route of the cutoff at the Planck scale — the fine-tuning problem you have is not exclusive to the cosmological constant, but happens for virtually all SM parameters.

    One should not treat the cosmological constant differently from other SM parameters. If you accept that there is no problem with the renormalization of, say, fermion masses, then there is no problem with the renormalization of the cosmological constant either. The only issue is that we don’t know how to actually perform the renormalization on non-Minkowski backgrounds. Btw, this is not just my own argument, see for example arXiv:1002.3966 (section IV).

    Best, :-)
    Marko

    • Regarding arXiv:1002.3966, it is worth pointing out that some theorists consider the identification of the cosmological constant with the vacuum energy density to be an upfront mistake.

    • Marcel van Velzen

      Marko,

      Many thanks for the link. What a great article.

      First of all, the article correctly explains why Einstein called the cosmological constant his “greatest blunder”, not for adding the cosmological constant to his equation (which is correct) but for the reason he added the cosmological constant to his equation (static universe) and even that “cure” was mathematically incorrect.

      Secondly, the article explains the current situation in physics as it needs to be explained.
      Not by natural/unnatural but understood/not (yet) understood!

      • We can discuss Bianchi and Rovelli later. Very few people who’ve spent a lifetime on this project agree with them. At this point, let us simply note there is no consensus.

        • In my view, it would be helpful to highlight where the differences in opinion about the cosmological constant problem come from. For example, what evidence there is that quantum fluctuations and the zero-point energy are physically relevant on very large scales, where decoherence and dissipation erase the contribution of quantum phases? Are large statistical ensembles interacting on macroscopic scales classical objects or quantum objects?

          • It’s actually easy to see where Rovelli goes wrong, and it’s a trivial point. The world is derived from quantum mechanics, not the other way around. They think that they can protect the cosmological constant term by quantum corrections if they hide it on the left hand side of Einsteins equation, but that is incorrect. Whether it is on the left or right, it will undergo renormalization unless they keep the geometry classical through all scales (which is forbidden for other reasons). Now, they might argue that the quantum corrections (which is actually a large sum of many contributions from different physics through many different scales) is identically zero, and then the cc is simply akin to an integration constant and can be anything you want. Well yes, that would work, but they need to actually show this. For instance, global supersymmetry does exactly this by exact cancellations as Matt described above. But no such mechanism is proposed, therefore they have merely assumed the think that they are trying to prove.

            One of the real and deep problems with the cc problem, is that there probably won’t ever be an exact direct calculation as an answer, as whatever the answer actually is, necessarily involves physical processes that we have no hope to ever directly measure or at least, not to measure with the sort of precision necessary for an exact answer.

    • I’m sorry, Marko, but I don’t even know where to begin.

      Re your first paragraph (renormalization) – you are missing the point about renormalization and naturalness. This is far too technical for this website, and I’m simply going to send you back to modern field theory textbooks about why this is NOT a problem in the case of the electron mass and why it is a HUGE problem in the case of the cosmological constant.

      Re your second paragraph: wrong, wrong, wrong.

      Re your third paragraph: wrong again. h\

      [One specific point to show this: in the case of a weakly-interacting, weakly-broken N=4 supersymmetric Yang Mills theory with a Z_4 symmetry preserved, there are no infinities at all, everything can be calculated, all the fermion masses remain small and the gauge boson masses remain zero; but the vacuum energy is still huge, as are the scalar masses. So it has nothing to do with infinities, or cutoffs... and it's not true that determining fermion masses is just as bad as vacuum energy or scalar masses. Let me say that again: no infinities, no cutoff, huge problem. You're just missing the issue entirely.]

      I don’t know where you learned field theory, but you need to go back and read basic technical issues in the subject: about relevant and marginal and irrelevant operators, about renormalization in the Wilson’s sense, and about how chiral symmetry protects the electron’s mass while no symmetry protects the cosmological constant; about the difference between additive renormalization and multiplicative renormalization.

      I’d like to make a request. In the last couple of days, you’ve been making a lot of definitive, confident, and totally wrong statements here. This is not helpful to the project of this website, which is to educate the public — correctly. Please try to get your facts straight, or make less definitive statements, or refrain from these types of comments. Shorter comments, with a smaller set of issues and fewer bald errors, would be helpful.

      • Marcel van Velzen

        Matt, Thanks for your example between square brackets (on the special supersymmetric Yang Mills theory). That example (together with your explanations above) shows nicely why the vacuum energy and the Higgs mass are the first places to look for a deeper understanding of quantum field theory or BSM physics.

      • Matt,

        I am sorry if my comments came across as definitive, confident, bald in errors or such. I certainly did not intend them to be interpreted in such a way. My point of view on certain things (renormalization in particular) is obviously different than yours. This is understandable, not because of my level of education, but because of the fact that I am coming from the quantum gravity side. In general relativity the idea of renormalization (as formulated in QFT) fails in a very spectacular way, so I don’t have such level of confidence in renormalization. (And I won’t be surprised if you now reply something on the lines of “no, general relativity is just a wrong theory and should be changed with something renormalizable”.)

        I was just stating my opinion on the topic, inviting your comment (if only implicitly). And I’m certainly not afraid to admit that I am wrong about something, if you convince me with arguments. But I didn’t expect you to just keep repeating the word “wrong” and to judge my level of of knowledge on QFT. I expected relevant argumentation about the topic.

        If there is no possibility for two-way communication here, no problem, I’ll go away. Sorry for intruding on your blog.

        Best, :-)
        Marko

  5. I was going to ask a much less knowledgeable version of the same question. Would it be possible to have an explanation of renormalisation which is a bit better that the (completely meaningless but far too common) “subtract one infinity from another to get a finite answer”?

    • In fact, defining renormalization as “subtracting one infinity from another” is wrong. One has to do renormalization even in theories that have no infinities at all. So a lot of this discussion simply reflects people who are not really field theory experts trying to make smart statements without a complete understanding of the issues.

      Renormalization is, however, complicated and subtle. (That’s why people who know some, but not lots, of field theory so often get it wrong.) I cannot explain it in a few words. It would require a long article.

      But if anyone tells you that renormalization is simply about removing infinities, you can be sure that that person does not fully understand field theory. Even if that person is long-dead Richard Feynman.

  6. Thank you for this article! I think it helped me to roughly understand the argument for the first time.

    So the point is, even if some new physics shows up at an energy scale M that is similar to the LHC-ish energy scale L, and even if the new physics has some nice symmetries so that what happens above M does at least not make the problem worse, it is extremely hard to believe that the total contribution to the zero-point energy from new physics above M almost perfectly cancels the contributions from known physics, but not quite, just so that the tiny cosmological constant comes out at the end.

    It would be a bit like that: There is a newspaper lying in your driveway, then a tsunami washes it away, then there is a hurricane and a comet strike, and in the end the newspaper winds up exactly in your driveway just one millimeter from where it was before. (Actually such unnatural coincidences are often used for comic effect. I hope the universe is not a giant practical joke or “hidden camera” show. ;) )

    Is it correct to say that for the hierarchy problem there is more hope, as the values that result at the end (e.g. the vacuum expectation value of the Higgs field), are roughly on the order of L, so it could be plausible that the cancellations from new physics work out right if M is not too far above L?

    • Your hurricane and tsunami and comet strike analogy is excellent. I may use it again. (It’s also like the guy who gets punched and blasted across a room and stands up with his hair perfectly combed.)

      Yes, it is easier to solve the hierarchy problem — there are many proposed solutions. There are few proposed solutions for the cosmological constant problem… indeed, except for a selection bias (i.e. weak anthropic arguments), almost none that have more than a handful of adherents.

      • Let’s keep in mind, however, that the number of adherents to an unsettled problem has nothing to do with the objective truth. Many spent a great deal of their careers arguing for the inevitability of low-scale SUSY above few TeV and, based on what we know now, it looks like they were wrong.

        • I wouldn’t say “has nothing to do with”. I’d like to think the opinions of experts on a subject are at least *slightly* better than a wild guess. How about just “is not an infallible predictor of”?

  7. I think this is really the same question as vmarko asked, but I’ll ask it anyway: If the zero-point energy you calculate for a field is scale-dependent, then it isn’t really a density at all, because density means amount per volume. By imposing a cut-off you turn it into a density, but what is the justification for imposing a cut-off? Is it just in order to make the density well-defined?

    • I did not impose a cutoff. All I said was that the Standard Model is valid down to a certain distance scale. We could do the calculation in string theory, where there is no cutoff, field theory is valid down to the string scale, and there are no infinities anywhere. It would not help: we would have exactly the same problem.

      • I don’t think I understand the QFT concept of “scale”. If we assume that the SM is valid down to a certain scale, we can calculate the field energy due to jitters at that scale. But wouldn’t whatever is valid beyond that scale make its own contribution to the total energy? Can we assume it is zero?

        Or is it an assumption of QFT or the SM that at some scale, the fields become discrete, and you can go no further?

  8. Tienzen (Jeh-Tween) Gong

    “it appears that such a cancellation could only occur by *accident*, …”

    Every *accident* is still an event which can still be described (with some languages, equations). What is the *description* of this accident?

  9. Awesome! This from a lay person, probably very simple to dismiss… We learned in school that these virtual particles arise as matter-antimatter pairs that then self-annihilate (still considered a useful model??). If so, is it considered possible that anti-matter reacts different to gravity than matter (ie opposite) and thus even in Einstein’s model of gravity, this background energy cancels out gravitationally? Seems I read that someone respectable was trying to confirm whether antimatter was affected normally by gravity??

    • As a general remark (I’m not qualified to answer your specific question): When scientists do experiments to find out whether a certain effect exists, this does by no means imply that they believe (or even consider it plausible) that the effect exists. It would in fact make science much less credible if people would only try to prove things they strongly believe, as that could lead to all kinds of self-deception. The most impressive conclusions are often reached when people try to rule out possibilities or to confirm conservative theories, and then they find that it does not work and that the facts force them to change their minds.

    • wtwo,

      Somewhat late to chime in here, but I will anyway. Yes, if half the virtual particles had antigravity this might go a long way to solving the problem at hand. However, Edwin Steiner and Columbia are right in their comments. Antimatter antigravity simply does not square with Einstein’s General Relativity (Einstein’s theory of gravity). Antimatter antigravity would represent a clear violation of Einstien’s Principle of Equivalence, the very foundation of GR. (You can look up the Principal of Equivalence. It is not hard to understand).

      Bottom Line: If antimatter does have antigravity, then GR cannot be the correct theory of gravity.

      Physicists don’t have new gravity theories lying around that fit the known data as GR does, so you should not expect them to part ways with GR unless they are forced to by experiment. You are correct in noting: “…that someone respectable was trying to confirm whether antimatter was affected normally by gravity.” CERN is going to conduct just such an experiment on a beam antihydrogen in the 2015 timeframe when the LHC beams power up again. See AEgIS experiment. However, Edwin Steiner is correct; the vast majority of physicists (I would guess 99.9%) believe that antimatter will fall in complete accordance with GR. Even the physicists involved in the upcoming AEgIS experiment, I suspect, expect so. However, the effect of gravity on antimatter is such simple experiment in concept, yet so complicated in practice that it is a hard one for experimentalists to resist. Of course if by some slim chance antimatter falls up all bets are off, we will have a scientific revolution on our hands! But then pity the poor theoretical physicists who will have to come up with a theory that accounts for antimatter antigravity and everything that GR has so successfully accounted for.

  10. You really must be one of the best teachers ever. These articles of yours are just great. In the end though, I’d say this problem just means we are clearly missing something fundamental, and hopefully someone will make a breakthrough in my lifetime.

  11. Dear Prof Strassler: I think that the most natural response to this problem is to declare: “Well, vacuum energy just doesn’t gravitate”. However, that’s wrong, and I think it would be very useful if you could include a detailed discussion as to why it is wrong.

    • Yes, but I mean we should perhaps not necessarily put too much on the plate eiter.

      I feel bad for Matt, the undertaking here is always so large. Every time someone spends time to explain a physics concept on the internet some wise guy instantly comes up with a counterargument (usually a well known counter argument that is known to be wrong) which requires a lengthy explanation. And on and on it goes.

      Suffice it to say, explaining why DE doesn’t gravitate requires explanation of the equivalence principle as well as getting into some subletiies regarding weakly coupled physics coupling to gravity.

  12. Professor Matt.:
    If QF jittering produce that much energy , and jitterings are activity , how the UP provide causal power to fuel that activity ?

  13. As QF are not substance or medium then in a field without particles what is ( field value ) ? Value of what ? Are values to exist in space ? What does these value physically represent ? If it is the appearance and vanishing of random jittering in values then we are Back to , values of what?
    Thanks.

  14. Interesting stuff. IMHO the thing that seems to be missing from this problem is the strong force. It’s there in the proton and the bag model, where it’s a kind of elastic tension. But after low-energy proton-antiproton annihilation to gamma photons, it’s allegedly disappeared. It can’t have. It’s fundamental. You need an elastic tension to make a ripple propagate through a rubber mat, so surely you need it for a ripple to propagate through space? Regardless of any calculation issues with the vacuum catastrophe, conservation of energy suggests to me that the increasing expansion of space isn’t actually down to an increasing dark-energy “pressure”, but instead is down to the “strong-force tension” getting weaker as the energy-density of space reduces.

  15. Very nice article.

    So the first obvious solution was that maybe fluctuations aren’t really there and you covered it in the article – we know from experiments they are there. But what about a different variant of this idea – what if fluctuations are not always there but rather are associated with ordinary particles? For the sake of the argument imagine that each peak of the field ripple/particle is really a sum of two Gaussians centered on the same spot – one very sharp defining what we usually associate with particles and another one extremely broad. In this picture the fluctuations would result from overlaps of those broad Gaussians, the reason we observe them here on Earth is that we are in the middle of a large concentration of ordinary matter, but in deep space they would be very weak to nonexistent.

    In this picture dark matter and dark energy might be manifestations of the energy of those fluctuations.

  16. Sorry, maybe I misunderstood your Calculation 1 and 2, but I think that you are just saying that the energy density is roughly hc/L^4 with a couple of examples, based on what is typically measured at the LHC and what is an intrinsic limit of the theory.
    When you write that the Standard Model is valid “down” to a distance scale L, my understanding is that it is valid “up” to that distance… (again sorry if I am confused or my translation from the English to my language is wrong :-) )
    I dare not ask: is the whole library of experimental results for which quantum mechanics gave correct predictions all done at about a “LHC-ish distance” with about a “LHC-ish energy” inside? How many LHC-like laboratories are there? How many scientists have a complete knowledge of all the required underlying technology? It is relatively easy to build up a bowl, we have some thousands of years of experience and there are so many bowls in so many places… as long as the bowl metaphor adequately communicates the essence of the concept.

    • Hi Giulio! In this context energy is inversely proportional to distance. That means in order to “see” physics at half the size , you need twice the energy. (Google for “Planck relation”, e.g. http://en.wikipedia.org/wiki/Planck_constant ). We know the standard model is valid at large distances (say one inch for example) down to very very small distances which Prof. Strassler calls “LHC-ish” here. That means it is valid from very low energies (corresponding to a wave length of one inch, say) up to very high “LHC-ish” energies.
      Quantum mechanics has most definitely been confirmed on an impressively wide range of energies. Over the centuries and decades, experimental physicists have worked up from molecular and atomic energies of a few eV (electron volts) up to the current frontier at the LHC with energies in the TeV range (that is 1.000.000.000.000 times more energy!). Cosmic rays have been observed with even much higher energies. It is worthwhile to think about how amazing it is that one theoretical framework explains observations over such a vast range of energies and sizes.
      There are several other particle accelerators and at the LHC itself there are several independent sites of experiments. Just google if you are interested.

      • P.S. For quantum mechanics itself “centuries” is of course exaggerated (quantum physics is roughly one century old), but for large distances and objects quantum mechanics nicely blends into classical physics, which has been experimentally confirmed literally for centuries.

        • Ok, thanks, now I see the derivation: L is the wavelength and L^3 the volume. It was clearly explained also in the article.
          So QM does not blend only with Einstein’s version of gravity… And in fact Einstein did not believe in QM ;-)

          • Yes, how to unify QM and general relativity (Einstein’s theory of gravity) is an open question on which there is a lot of research. However, there is overwhelming experimental confirmation for both of these theories, so the answer cannot simply be that one of them is “wrong”, rather the answer will likely be a much more interesting insight about how nature behaves on very small distance scales and very large energy scales.

            To say that Einstein “did not believe in QM” is an oversimplification, I think. Actually Einstein was one of the first physicists back then who fully acknowledged the necessity of a quantum theory of physics. On March 16 in 1910 he wrote in a letter to Laub, for example: “I consider the quantum theory certain. My predictions with respect to specific heats seem to be strikingly confirmed.” (Quantum theory in 1910 was not yet QM as we know it today, however.) Genius that he was, Einstein saw with fully clarity the deep and unsettling implications that the developing theory of QM had for our understanding of physical reality. He tried without success to find a classical underpinning for QM. Today we have proof that an explanation of QM by an underlying classical theory is impossible (see “Bell’s theorem”), ironically based in part on work by Einstein (see EPR paradox). Today we also have much more experimental confirmations of QM (and general relativity) than in Einstein’s days.

  17. @Columbia at 4:31 AM

    “…unless they keep the geometry classical through all scales (which is forbidden for other reasons)”.

    Could you please spell out these reasons and point us to experimental evidence that backs up your assertion?

    • The brief sketch/idea is that if you look at Einstein’s equation, you know that the right hand side at the very least is quantum mechanical on small scales (matter is quantized!), therefore you must promote it to an operator in the usual way etc. But then you can’t solve for the classical metric field (on the left), without violating the linearity of quantum mechanics, therefore it too has to be promoted to an operator. Therefore gravity must be quantized. (There are about ten different thought experiments along simiilar lines, for instance in the Feynman lectures on gravitation). Related to this, we know that at the very least, quantum mechanical matter also gravitates in the usual way (neutron interferometry experiments in the presence of a gravitational field).

      • “Therefore gravity must be quantized”.

        The trouble is that nobody knows how to build a consistent quantum theory of gravity. This is precisely why the cosmological constant problem (and other open issues as well) are non-trivial challenges with no clear cut consensus in sight.

        • I agree. In fact, a consistent quantum theory of gravity might not even exist. It might be that a UV completion is required, where new degrees of freedom replace variables or operators like the metric field. However, what my above thought experiment does show, is that it is inconsistent to leave gravity unquantized and classical.

          • Ervin Goldfain

            ” However, what my above thought experiment does show, is that it is inconsistent to leave gravity unquantized and classical.”

            I see two weak points with this line of reasoning:

            1) if a viable quantum theory of gravity is not found, then gravity remains by default a classical theory and your argument becomes irrelevant.

            2) if a viable quantum theory of gravity is eventually found, then one must prove that it stays immune to classicalization via decoherence and dissipation on macroscopic to cosmic scales. Note that this is far from being trivial, as the loss of coherence and dissipation have strong experimental support in all open quantum systems .

          • Hi Ervin
            1) The choice is between a theory that is manifestly inconsistent (classical gravity + quantum matter) and something else. Something else tends to win, even if we don’t know what it is.
            2) There is nothing inconsistent in the above scenario with decoherence. Indeed you expect the emergence of the classical world at large distances (large here means a few orders of magnitude away from the Planck scale).

  18. Hi Matt,

    As an interested amateur, here’s my simplistic question…

    You said ” The better your microscope (or particle accelerator), the more jitter you can detect, and the more energy you discover the jitter has.”

    This seems similar to the ‘Coastline Problem’ in which the apparent length of a coastline approaches infinity as your measuring-stick’s length approaches zero – the coast’s length is whatever you want it to be, determined by your choice of length of the measuring-stick.

    Given that the circumference of, say, the Isle of Wight, could exceed the diameter of the observable universe (using an appropriately small measure), is this similar to the energy of your metre cube approaching infinity as your distance scale approaches zero?

    …or have I just got the wrong end of the stick? :-)

  19. @ Columbia

    We’re obviously not on the same page. Let’s agree to disagree.

  20. Interesting that you mentioned the holographic principal in a recent post. Could the quantum fields and their jitters just be an illusion, a mere book keeping device, which allows us to reason about particle motions in 3-d space when the underlying reality is 2-d space.

    So independent quantum jitters ( i.e in a vacuum and independent of particles ) would not contribute to dark energy at all.

    • Ah i was thinking since the amount of dark energy was comparable with the amount of matter ( including dark ) then the quatum jitters associated with particles ( average one atom per cubic meter ) should provide the dark energy rather than the quatum jitters in the vacuum itself ( in the absence of particles ).

      But I understand dark energy is measured to be smooth across the universe so that rules the above thought out.

      Curious why the amount of dark energy is comparable to that of matter ( + dark ). Would that imply some thermal equilibrium in the early universe?

  21. ~what Jim said~
    ” . . .seems similar to the ‘Coastline Problem’ . . . ”

  22. I read that virtual particles ( borrow) energy from vacum , this is related to my previous question about the source of jittering energy …. Would you please clarify this vague point.
    Thanks

  23. As a particle physics outsider, I very much appreciate the effort you put into these tutorials.

    You state that every elementary particle is a ripple of the smallest possible intensity in its quantum field. Wouldn’t this imply that quantum fields are governed by nonlinear equations, since linear systems have no minimum intensity? What is the name of the nonlinear equations that produce this minimum intensity (so I can look them up)?

    In basic quantum mechanics, I understand that wave functions are basically linear, so superposition applies. Wouldn’t an alternative explanation be that quantum fields have no minimum value, but you see an observable particle only when the amplitude of the field exceeds a certain minimum value?

    Intuition (admitedly a poor guide in the quantum world) would say that Explanation 1 is the correct way out of the cosmological constant dilemma: The vacuum energy simply isn’t there. The fields do not increase in amplitude as the scale is decreased. A particle known to be in a small box must (because of the uncertainty principle) have a higher energy than one known to be in a larger box. However, the probability of finding a particle in a box gets smaller and smaller as the size of the box is decreased, so the average energy does not blow up. Any comments?

    • I’m not an expert but I can give you some pointers: The deeper reason for the “smallest possible intensity” is not non-linearity but rather the canonical commutation relations which encode the fact that you can never know at the same time the value of a field at a point and its rate of change at the same point. It goes roughly like this: You find that you can describe each mode of the field (i.e. each independent way the field can “vibrate”) as a quantum mechanical harmonic oscillator (google this term for lots of info). Such an oscillator has a zero-point energy proportional to its (fixed) oscillation frequency and you also can increase its energy only in steps proportional to that frequency. (These steps are the reason that we never see “half a particle”.)

      The “ripple in a field” analogy for particles is good on a non-technical level, but you should not take it too literally. Quantum fields are much stranger than that if you dive into the details.

    • Torbjörn Larsson, OM

      If you go through Matt’s articles on the subject, you’ll see he has already described all of this. http://profmattstrassler.com/articles-and-posts/particle-physics-basics/fields-and-their-particles-with-math/ Good reading!

  24. From what I remember of quantum field theory (I’m talking massless spinless particles here, real basic stuff) you can eliminate the zero-point energy by fiat, simply by subtracting the right amount from the Hamiltonian density. Does this not work properly in the more complicated cases, or is it just that it is too obviously artificial, or is there an even subtler reason why it doesn’t address the problem?

  25. kashyap vasavada

    Matt, now that we understand the CC problem we would like to know about what efforts are being made to solve this. Is it possible that the enormously large value of CC you are getting is valid only in the early universe (near big bang inflation)? For some reason the large quantum fluctuations are not present in the current universe. After all LHC is just one little place in the whole universe!! So the real problem may be that CC is not really a constant but has varied by a large amount over the history of universe. I understand that it is not possible to introduce a time varying function in GR and still maintain covariance. Any thoughts on this?

  26. Point of reflection: we measure effects of quantum fluctuations then we say they are caused by UP , what about UP caused by QF ?

  27. What activate the jittering? Are jitters consumers or producers of energy? DeltaE.delta T ~ h/2pi is relational not causal.
    Please clarify.

  28. Pingback: Quantum Fluctuations and Their Energy | pocket2mail

  29. Can I add that IMHO vacuum fluctuation can be likened to the ripplets on the surface of the sea, they aren’t the same as virtual particles, the Casimir effect is very weak, and a photon is like an oceanic swell wave without the trough. The energy density of “empty” space is something like the depth of the ocean, and you don’t know how deep it is, though you can be reasonably sure the depth varies. Interesting stuff is space and energy. At the fundamental level I just can’t separate them.

  30. The maximum I got is : the virtual particles activity causes the vacum zero-point energy then the later activates the former in a closed loop , I am not convinced , may Matt. Help ?

  31. There is one thing I’d like to have explained more thoroughly: what is the mechanism how is vacuum energy supposed to act on space? Why should any amount of energy stored in quantum fluctuations cause the space to expand?

    • From what I found, the answer to this is not so straight-forward that it can be put in one sentence. I’ll try it in a few paragraphs. Beware that I’m just learning myself so my answer is not authoritative. Also a verbal description cannot faithfully express these things – for the real thing you need the equations.

      The main part of the connection is general relativity (Einstein’s theory of gravity). This theory connects the geometry of spacetime (the “shape of the development of the universe” in a vague sense) to the energy and momentum of all the stuff in the universe. To put it simply, the Einstein equations tell us:

      (curvature of spacetime) = (density and flow of energy and momentum)

      (Both sides are of course precise mathematical objects in the real theory.)
      This law is universal, so it applies to any form of energy, be it vacuum or not.

      The left side of this equation tells us about the expansion of space, among other things. However, we are not done yet: If space expands (or contracts) this has an effect on the stuff in the universe that makes up the right side of the equation. This effect is not universal, it depends on which stuff we are talking about: dust, radiation, vacuum energy, etc. react all differently. So for each kind of stuff in the universe we need to know how its energy density changes when space changes. This is expressed in an “equation of state” that relates energy density and pressure.

      Vacuum energy has a very particular equation of state, because its energy density is constant. (After all, expanded vacuum is still vacuum.) For other stuff the energy density decreases when space expands. It turns out that (positive) vacuum energy has negative pressure, and enough negative pressure that the net effect of it will be expansive. (This can only be understood with the Einstein equations. It is not at all an effect like the positive pressure of a gas causing a balloon to expand!)

      Ok, it’s getting complicated. The Einstein equations are a pain to solve and we need to couple them to several equations of state, etc. In order to sort all this out, physicists make some simplifying assumptions: For example one assumes that there are only a few different kinds of stuff in the universe (one of them being the vacuum energy). One also assumes that the universe looks the same everywhere and in any direction. (This may sound strange, as it does not look like this in our closest neighborhood. But observations show that if you look at all the visible universe, it looks pretty much the same everywhere on average.)

      With these assumptions in place you can then actually work out whether the universe is going to expand or to contract, and how the rate of this expansion/contraction will change. You find that it depends on how much of each kind of stuff there is in the universe, and in particular you find that if the relative amount of positive vacuum energy is big enough, it will cause an accelerated expansion of the universe.

      I’m sorry, this almost became an article. I don’t know if this explanation was useful to you, I hope so. If anyone knows a more direct way to explain this, I’d be interested.

      • Edwin, here’s my two cents: I think there’s an issue with the vacuum energy density remaining constant. Conservation of energy is not something we dismiss lightly, see my comment above dated August 31. And since the dimensionality of energy can be expressed as pressure x volume and space has a positive volume, how can positive vacuum energy possibly have a negative pressure? I think there’s another issue here in that gravity is caused by a concentration of positive energy, and gravity attracts. For energy to repel, people flip a sign. But IMHO what they’re missing is that it isn’t energy per se that attracts, it’s a concentration of energy, the delta energy. And it doesn’t suck space in, it merely alters the motion of light (and matter) through space. Light “veers” towards the centre over time, and we call it curved spacetime. The concentration of energy is typically bound up as matter in the guise of a star, because it’s bound it cannot expand, and around it is an energy-pressure gradient in the surrounding space. But imagine you were to smooth this all out into unbound spatial energy in a universe that cannot be infinite because it started small 13.8 billion years ago. What you then have is space with an innate energy-pressure. See the \ diagonal in the stress-energy-momentum tensor. Given that this isn’t adequately balanced by some opposing tension, space just has to expand. It can’t do anything other than expand. Also see the shear stress term. Shear stress! That says to me that space is somehow elastic. Like it expands like a squeezed-down stress-ball when you open your fist. Gravity doesn’t suck space in so it was never going to contract, and here we are. But it’s like plumber’s foam too, wherein the bubble-walls get thinner and are less able to confine the pressure. Only space is space, there are no bubbles in it. Matt will doubtless give me short shrift for this, but nobody has explained to me why this outline picture is wrong.

        • Energy (non-)conservation gets more complicated when you take general relativity into account. I do not yet understand it well enough to give a precise argument here.

          Regarding the pressure: You cannot get the sign of a quantity from dimensional analysis like this. The negative sign is clear when you look at it this way: If you compress – that is decrease the volume of – something that has positive pressure, you put work into it, so its energy increases. If you decrease the volume of vacuum with positive vacuum energy density, the energy decreases, corresponding to negative pressure.

          One connection I did not make so well in my reply: Pressure can also be seen as a flow of momentum. For example: flow of x-momentum in the x-direction. This is how the pressure enters the right side of the Einstein field equations. Therefore pressure gravitates, and in the case of positive vacuum energy it is the large negative pressure that dominates over the contracting effect of the positive energy density and causes the expansion.

          Regarding shear stress: In this context I think it is more natural to think of it as flow of momentum, e.g. flow of x-momentum in the y-direction. I wouldn’t say it has something to do with elasticity here.

          • You know how you said “It is not at all an effect like the positive pressure of a gas causing a balloon to expand”? I think it is, provided you see it right. Imagine you have a small amount of C4 explosive in a flaccid but necktied balloon. You somehow detonate it whereupon the C4 turns into gas, the gas expands, and the system reaches an equilibrium whereby the gas pressure is balanced by the tension (=negative pressure) in the balloon skin. You didn’t add any energy to expand the balloon, you just released it. Now imagine that the gas is some way corrosive, and degrades the balloon skin turning it into something like silly putty. The balloon gets bigger, and you still aren’t adding any energy. The longer you wait the weaker the balloon skin gets, and the more the gas expands. It’s in a vacuum rather than ambient air pressure. I think it’s something like that, but with no distinction between the balloon skin and the gas. Both are space. As for what happens in the end I don’t know, but I find myself drawn to this:

            http://www.slate.com/blogs/bad_astronomy/2013/06/01/prince_rupert_s_drops_glass_droplets_exploding.html

          • The puzzling part is that a negative pressure still results in an outwards push. Could this be considered analogous to the fact that a gas with a negative temperature will heat, rather than cooling, whatever it is in contact with?

          • @harryjohnston: It is not a mechanical effect like an “outward push”, it is truly a gravitational effect. We do not have an intuitive understanding for that because all the materials we usually experience, including the planet Earth, have tiny pressures when you compare them to the density of the mass energy, so we do not notice the gravitational effect of pressure.

          • kashyap vasavada

            This is addressed to Edwin Steiner, harryjohnston and duffieldjohn.
            I think, one can understand negative pressure of vacuum energy from thermodynamics. I was waiting for Matt to answer the question. But he did not get around to answer this one. I hope, he does not mind my trying to answer this one. in any case he is welcome to correct me! This is his blog! From thermodynamics,
            total (heat) energy going into system = change in internal energy +pressure x change in volume. For an isolated system, L.H.S. =0 .As universe expands with constant vacuum energy density ,internal energy has to increase because of increase in volume (there is more vacuum!) The only way, above equation works is with negative pressure. Let me know if you guys are convinced with this argument.

          • Kashyap: I’m not convinced. Conservation of energy isn’t something I discard lightly. Negative pressure is tension. See my balloon analogy, and see http://arxiv.org/abs/0912.2678. Look at page 5 and note this: “We see that the modification of GR entailed by MOND does not enter here by modifying the ‘elasticity’ of spacetime (except perhaps its strength), as is done in f (R) theories and the like”.

          • Edwin –

            Sean Carroll discusses energy-momentum conservation in this post

            http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

            which is probably a good stepping-off point for you. (I am not sure it will necessarily help you explain things to the others in this thread).

            Your mathless description of pressure is pretty good; it is hard to describe without resorting to T_{\mu\nu} and cases where T_{xx} = T_{yy} = T_{zz} = p. Your logic that the cosmological constant is effectively the same as a stress-energy tensor where p = \-rho is fine, I think; certainly in the case with a uniform matter density that would deform Minkowski space into de Sitter space.

            It’s useful to understand that pressure is a source of gravitation, but is usually negligibly small. It’s important in the negative pressure cosmological context (where it is large), but also in gravitational collapse — positive pressure (arising from the thermal motion of material in a self-gravitating object, e.g. the gas in a star) balances the tendency to collapse, however because pressure is itself a source of gravitation, it can become a significant contributor to the collapse itself, which causes a sufficiently high pressure to accelerate rather than forestall collapse. Likewise, when negative pressure is large, it becomes a significant negative contributor to gravitational collapse. In the absence of significant contributions from other sources of gravity, a large negative pressure results in gravitational expansion. And in the cosmological constant case, or in its EFT equivalents, that results in more space with the same inherent negative pressure. Finally, the negative pressure component is *everywhere* but is still very small compared to contributions from other sources in and around galaxy clusters.

          • @Brody: Thanks for the link. The article and some of the discussion there is interesting. Also the pages by John Baez you linked to in another comment ( http://math.ucr.edu/home/baez/vacuum.html ) were helpful.

        • The Cosmological Constant does not violate Conservation of Energy. Your attempt to gain a non-mathematical understanding via analogy to ordinary mechanical forces you are familiar with is insufficient to show otherwise. Space is like elastic only is some very superficial ways that can allow you to gain some intuition about how General Relativity works but inferring meaningful conclusions about space-time from an analogy is fallacious. The full answer to “why is this wrorg?” is in the maths, not simplistic pictures and analogies to things you already understand. Sorry.

          • A partial answer to “why is this wrong?” is that he is not considering that all the components of the symmetric tensor can transform into one another by Lorentz transformations. For carefully chosen observers and a suitable set of coordinates, he could be right in principle, however being right does not justify abandoning gauge freedom.

            By way of example:

            ‘Shear stress! That says to me that space is somehow elastic.’

            but the often-useful Newtonian gauge is shear-free.

        • Your last reply did not have a reply button. May be webmaster is telling us to cut it out! As far as I know MOND has been ruled out experimentally. Also Newton’s law is so fundamental and has worked so well in classical physics that MOND seems to be a very unlikely explanation of dark matter etc.

          • kashyap vasavada

            Sorry. The above reply on MOND was meant for duffieldjohn. I forgot to write that.

          • I don’t think MOND is right, kashyap. Sorry I didn’t make that clear. I only referred to Milgrom’s paper because it referred to f(R) gravity along with elasticity and the strength of space. This harps back to what I said in my previous comments. The shear stress in the stress-energy tensor suggests that space has an elastic nature. IMHO the expansion of the universe is increasing because the strong-force tension is reducing, not because spatial/dark energy/pressure is being spontaneously created out of nothing. It’s like that balloon analogy. The total energy of the universe is conserved. As it expands the pressure drops. Energy = pressure x volume. Bigger volume, lower pressure, same energy. The pressure is reducing but the tension is reducing faster, so the expansion goes runaway. It’s something like drooping silly putty combined with expanding plumber’s foam without any bubbles.

          • The “Reply” buttons are just missing due to a limit on the depth of comment nesting, I think.

      • In regards to the cosmological constant problem, some of what you said seems to be my line of thought- though I am not a physics major at all, haha.

        To compare this empty box of one elementary particle to dark energy might be misleading us (which may be why we get this CC problem), because perhaps this dark energy/empty box is different when acting with dark matter and with many particles and in an expanding space itself. So 3 factors are different.

        It seems to me (without any knowledge of what you wrote above or what Einsteins equation said) that the expansion of space itself would play a factor in the energy.

        Also- if dark energy has “potential” energy which is kinda like these quantum disturbances, then perhaps it is used up, stored, used up, stored as it is expanding space.

        I’m just saying dark energy doesn’t necessarily follow a constant flow that doesn’t change.

        But I really have no degree in this, these are just my thoughts as an outsider.

  32. Pingback: A New Academic Year | Of Particular Significance

  33. Matt. : you only mentioned the energy due to particles jitter , what about fields with no particles ? Gravity for ex.

    • @aa.sh.: although we don’t really know for sure, it is thought that quantum gravity does have associated particles called gravitons. In any case, the zero-point energy the professor is talking about is due to jitter in the fields, not in the particles. If you re-read the section titled “Fluctuations of Quantum Fields” I think you’ll find this was made quite clear.

  34. Matt,
    Q1)How did you calculate the values (for each case) for the energy stored in empty space? and, how can you “experimentally (accurately and directly) measure” the energy in empty space?
    Q2)Does gravity change the energy density of empty space? For example, does an massive object like the sun increase the energy density of its nearby empty space? If so (or not so), why?
    Q)Does the energy stored in empty space (=quantum fluctuations energy) create repulsive or attractive effects(force)? What mechanisms explain such force(=repulsive or attractive effects)? If quantum fluctuations energy is not zero, it should be positive value and should have attractive effect like any attractive effect between objects/systems with non-zero energy(=assuming momentum and energy of any system is always “positive”). I do not understand what you are talking about here:”the energy of the fluctuations of fermion fields (the fields for the electron, muon, tau, 3 neutrinos and 6 quarks) is negative!” How could “momentum(and energy)” associated with the fluctuations of fermion fields be “negative.”?? Momentum should always be positive!? Is this not the case? If not, what is the convincing evidence that it can be negative?

    • @elm: Your questions are fairly basic so I’ll take a stab at them. I’m sure one of the professionals will step in if I make any important mistakes. (Q1) The details are too complex for a blog comment, but the energy density can be calculated using the mathematical description of the Standard Model. The naive version is to treat each frequency at which a particle can exist as a quantum harmonic oscillator, and sum up the zero-point energies. In reality you have to allow for interactions as well. (Q2) The conventional answer in GR is no, but it depends a bit on how you look at it. In the best way of describing GR, the existing curvature of space appears on the left-hand side of the equation, while the energy/momentum density appears on the right-hand side. You could rearrange it so as to treat the curvature as having an energy/momentum density, but the equations are much less elegant that way. (I’m not sure what happens in quantum gravity theories.) (Q3) I’m less certain of the fine details of this one, but see Edwin’s comments above. From a GR point of view, it isn’t a force at all, it’s more like a sponge expanding as it fills with water [bad analogy alert!]. However, it “looks” like a repulsive effect to a naive observer. In GR, what we call gravity isn’t simply an attractive force like in Newton’s theory, it can do all sorts of stuff. Only the attraction is easily visible under normal circumstances, but delicate experiments in orbital satellites have explicitly confirmed some of these other effects.

      • @harryjohnston, you sound like you do not know what you are talking about. as you could not answer any of these questions at any sufficiently deep or fundamental level. you words sound like copying and repeating what others or texts / books said before and elsewhere instead of showing clear deep understanding of the subject.

        • I believe my answer provided an appropriate level of detail for the original poster. My PhD was in quantum optics, not the Standard Model, but I understand the basics well enough. In any case, to answer these questions in the level of detail you seem to be wanting would require a textbook, not a blog comment!

    • Matt Strassler, too bad you did not answer any of these questions, and you did not write a page about how to estimate the amount of this energy. Matt Strassler said “If you’d like to see how we estimate the amount of this energy, click here (sorry, page not yet written)” Hopefully, you will answer these questions in the near future and write a page you have not yet done.

  35. Hi Professor.

    Congratulation for your blog, it is something needed for many years. And your job is excellent! After finding your blog, I started to look to you as the “Carl Sagan” of this century beginning.

    I have one question about the energy of vacuum. Doesn’t matter what you consider to be the right answer for the level of its energy, my problem is the same.
    The accepted cosmology theory (The Big Bang Theory) assert that the space is continually created from the beginning of the Universe. The expansion is not happening in a pre-existing space but the “new” space is created all the time. If this new space is in fact a vacuum space and it have an intrinsic energy … where this energy is coming from?
    Even for the measured energy of the vacuum (unbelievable small compared with the calculated one, as you explain above) the total amount of “new” energy for the Observable Universe must be tremendous.

    Maybe the theory doesn’t accept this idea of additional energy during the Universe expansion, but I never read any argument against it. How the quantum theory explain this “total energy increasing”? Or maybe the question it is not in its research “area”?

    Thank you.

    • @Adrian: your question is fairly basic so I’ll take a first stab at it. No doubt the professionals will correct me if needed, and perhaps add some much needed subtlety. The short, standard answer is that energy/momentum is not a conserved quantity in General Relativity. I’m told that there is a somewhat more complicated sense in which you *can* still treat energy/momentum as conserved, but only within a finite volume (and allowing for flow in and out, of course). As far as I know, there is no sense in which the Universe as a whole is obliged to conserve the total energy/momentum. (Also, note that if the Universe is in fact infinite the total wouldn’t be a well-defined concept anyway.)

      • Hi Harry.
        Thank you for your answer. I am aware of the idea of non conservative total energy of Universe or even of large parts of it. But for me, such “explanation” is not an explanation at all. It is very similar with the way a priest respond when you ask him a real hard question: “God works in mysterious ways!”. It is simply an other way to say “I don’t know.”

        An Universe that not conserve its total energy is illogical, because it is impossible to establish ONLY ONE WAY this can happen, with other words if Universe is non-conservative it must be so in all imaginable ways. That would mean that it is not possible to create a coherent science. And I simply can’t accept this idea.
        An infinite Universe it is also illogical, because physical infinity has the bad habit to create paradoxes. All the time.

        Yes, maybe the Universe is illogical and full of paradoxes when we try to explain it (because we have a limited mind and use limited tools, like math). It is a possibility. But I can’t accept this either.

        So, for me, the energy of vacuum must come from somewhere but we just don’t know where from. Yet. I think this is a more elegant and more honest answer. No offense!

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  37. Hi Matt, Thanks for the great article. I have always found the CC problem bothersome. One reason is the renormalization issues alluded to above, and the counter is go read a text book … ok. The other is also alluded to above “I think that the most natural response to this problem is to declare: “Well, vacuum energy just doesn’t gravitate””. Let me rephrase this:
    For vacuum fluctuations = 0 , but the RMS energy: sqrt() = big. So perhaps gravity couples to but not to sqrt().

    • That’s usually the first thing people try. To try to degravitate the quantum vacuum (note this is what supersymmetry and conformal symmetry accomplishes quite well). Unfortunately, this program hits major snags. The first is that we know that at least part of this stuff really does gravitate. When you weigh yourself, you are measuring a nonzero fraction of your mass due to quantum effects within nuclei. In the lab, we would expect departures from the equivalence principle from quantum effects like the lamb shift. We don’t see this, therefore you have to arrange your explanation piecemeal, knowing about all the different physical effects that must cancel for this idea to work and you quickly end up back at the beginning.

      • My understanding is that most of my mass is from kinetic and binding energy inside nucleons. Both of the these contribute expectation(E). My point is that quantum fluctuations don’t contribute to expectation(E), they only contribute to expectation(E^2).

  38. It took out my expectation values:
    For vacuum fluctuations \= 0 , but the RMS energy: sqrt(\) = big. So perhaps gravity couples to \ but not to sqrt(\).

    • I can’t post certain symbols it seems:
      For vacuum fluctuations expectation(E)= 0 , but the RMS energy: sqrt(expectation(E^2)) = big. So perhaps gravity couples to expectation(E) but not to sqrt(expectation(E^2)).

  39. So, I don’t know much about QM and I’m a little confused by this article. I think that maybe the issues are more technical than you’re willing to get into and so it’s a little like when adults try to explain why certain things when you’re a child, but they’re not willing to get into explaining sex, so they make all sort of non-explanations that only leave you more confused.

    From reading explanations in the past, I’ve gotten the impression that higher energies and shorter length scales are necessarily linked, and that new physics can manifest itself at higher energies/shorter length scales, which is the whole reason for building the LHC. So, then, why would it be surprising that calculation 1 and calculation 2 differ? If they didn’t, would the LHC be necessary in the first place?

    And if seeing different phenomena at different length/energy scales is expected, then why is it suddenly unexpected that physics at the planck scale don’t match up with physics at the cosmological scale?

    In the past (i.e. the 90s and 00s) I used to see lots of articles talking about the need for a “quantum theory of gravity” and that GR couldn’t be applied to quantum mechanics because naive theories of quantum gravity were non-renormalizable – but you seem to be ignoring that “issue” by simply taking the plank-scale zero-point energy and plugging it into the GR equation – can you explain what’s going on here? Did we never need a quantum theory of gravity, or is this simply another manifestation of the same issue?

  40. Great article. One slight quibble. You state that a mass energy of one million million million times that of a cubic meter of brick is comparable to that of the sun. I get order of 10 ^ 21 kg for the bricks pile times the million million million, but the sun is of order 10 ^ 30. So it seems there is a difference of a factor of a billion. Not quite “comparable” in my book.

  41. @Matt Why do we expect that we will get all the answers from a single mathematical framework.
    Let us imagine an ant living on a nylon thread. Assuming the ant has the intelligence to figure out the physics of the oscillations of the thread and figures out the relationships between energy, velocity and the amplitude of the oscillations. Now an ant physicist asks the question, why is the velocity of the wave on the nylon thread so much and not any other value. Since we are outside the system (of the nylon thread and the ant) as an external omniscient observer, we know that the physics of the nylon thread world are NOT solely determined by the mathematics of the sine wave but also by the properties of the nylon itself.
    Now taking this analogy of the anthropic,sorry ant principle, we can consider several possible worlds that ants can imagine. Worlds with different kinds of threads (of varying thickness) from the same nylon, or different kinds of nylon itself or worlds made of entirely different kinds of materials such as steel (again of different kinds) or kevlar or other composite materials. Now if there was a string theorist among the ant, he will be despairing at the all the possible theoretical solutions he can come up wondering why he can’t get a single unique solution.
    Now coming back to our situation, it could also be that there is only one material from which our universe is made of and the physics of the world is limited as much by the properties of that substrate as by the degrees of freedoms allowed by the mathematics that is used to describe the system.
    The solutions to the question of multiple universes will be restricted by factors (materials) that allowed to be used in the making of the universes and NOT by the mathematics itself. Within each of the different universes there will limitations that allow different kinds of curves and local minima.
    So our quest for physics should start asking what kind of materials substrates can allow the existence of our rich and dynamic universe(s) and try to imagine doing experiments to look beyond the ephemeral fields and start scratching under the surface of the substrates

  42. I’m not qualified to comment on your physics but I can point out a minor typo:
    “And this can be checked: n a clever experiment.”

  43. Hi, great article! Hay I was thinking can we use calculations to determine what time period would be necessary to exist before a quantum fluctuation produced an electron-positron pair?

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  45. OK, I’m starting to get use to this idea that Higgs field is very important, that physicists don’t know if it is a composite
    field or elementary. It has non-zero average value, which is constant throughout the space and time and non-dependant of the Higgs particle/s. It’s predicted by Standard Model theory because without it the theory doesn’t make sense. The Higgs particle is a disturbance in the Higgs field that dies out/ decays, but is different from virtual particles that don’t have a definite mass. This particle is a boson, so it’s a carrier of some force, (what force?) but not gravity because gravitons should have 0 mass and spin 2, so Higgs boson is not a graviton but something else. OK professor, you have helped me to understand Higgs field a lot better. Got this from The Higgs FAQ 2.0
    Matt Strassler [October 12, 2012]
    Did I say thank you? Well, thank you indeed.

  46. If quantum fields are always there, pervading space, then how are they not an ether ?

    This has bugged me for a while !

    Cheers professor

  47. Is it possible that dark energy was at one time stronger than it is now? that perhaps over time as the universe expands, its energy lessens? Would this help explain the acceleration of the universe while the dark energy is too weak to do such a thing?

    • In fact, inflation is the theory that dark energy was once much, much stronger than it is now, and caused the universe (at least our patch of it) to inflate to an immense size. Then the dark energy dropped enormously in size. We know from the success of Big Bang nucleosynthesis and many other measurements that dark energy was not an important component of the universe shortly after the hot part of the Big Bang got going; energy first in relativistic and then in non-relativstic particles was far more important than dark energy for the next billion years.

  48. In realty- the space quantum vacuum has many particles- right? and the space is expanding, right? So, If we take that “box” of a single elementary particle with its quantum fluctuations -and then add another elementary particle and start to expand the space, would the energy lessen? Seems counterintuitive, but maybe there is something about expansion of space that plays a role on the dark energy and makes it weaker?

    Also- perhaps dark energy is not what is inside this empty box at all? Perhaps we are comparing apples and oranges? Could dark energy be a different kind of energy or non-energy altogether that what we would see from the quantum fluctuations of an elementary particle?

    • We do have mathematics behind our words, and the math helps us make precise predictions that allow us to compare our ideas with observations and experiments. If we didn’t, we wouldn’t be doing physics.

      *By definition*, dark energy is something that does not “weaken” as space expands. We can predict the impact of such a thing on the expansion history of the universe, and we find that the prediction agrees with data. What you are suggesting would predict an expansion history of the universe that disagrees with data — so independent of whether it’s a nice idea or a consistent idea, it’s ruled out.

      • But what about the dynamic dark energy idea? ( I read some papers on this yesterday and this sounded very similar to what I was saying about dark energy …. that it is changeable.

        And this is a basic question: If in fact this is a tweaking that needs to be done to the theory of gravity…what happens to the 70% dark energy? I mean, this 70% is space -the dark space we see between everything? – but we are calling it dark energy?

        I saw a lecture where they said dark energy had energy and effects of gravity (something to that effect)…what else do we know about it?

  49. OH-
    Is it possible we have two kinds of pressure: gravity pressure and dark pressure (which we call dark energy) and that we only have 3 forces and that gravity is not a force to be unified?
    And in the early universe we had 1 force (those 3) and 1 pressure (those 2) which then spread to 3 and 2?

    Also- When people call dark energy, energy…isn’t that wrong? And when space expands and more dark energy is now formed in the space….isn’t it not energy that is being created….but pressure?

    And isn’t pressure, forces and energy all different?

    • IS IT POSSIBLE WE COULD BE LOOKING AT IT LIKE THIS?
      THE EARLY UNIVERSE HAD:

      1 PRESSURE? DARK GRAVITY …WHICH DIVIDED INTO GRAVITY AND DARK PRESSURE
      1 FORCE? UNIFIED THEORY… WHICH DIVIDED INTO 3 FORCES
      1 ENERGY SOURCE OF QUANTUM FLUCTUATIONS… WHICH DIVIDED INTO MATTER AND DARK MATTER

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  52. Umm…….
    In calculation 1, if the number in Fig.3 would right, 10^27, I think there is something wrong. The 5th line in Calculation 1→ and processes involving elementary particle collisions with energies SMALLER than about 1000 times the proton‘s mass-energy. I think it is not smaller but larger.
    Indeed, the LHC energy couldn’t so small as 1/1000 smaller than a proton mass-energy. And if I correct it into LARGER, then the result in Calculation 1 is right, ie 10^27.
    Maybe i’am wrong or whatever…….. I poor in math.

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