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.
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.
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.
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.
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
- the energy of the fluctuations of boson fields (the fields for the photon, the gluons, the W, the Z and the Higgs, and even the graviton) is positive
- the energy of the fluctuations of fermion fields (the fields for the electron, muon, tau, 3 neutrinos and 6 quarks) is negative!
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.]