James O'Brien for Quanta Magazine

James O'Brien for Quanta Magazine

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Richard Feynman looked tired when he wandered into my office. It was the end of a long, exhausting day in Santa Barbara, sometime around 1982. Events had included a seminar that was also a performance, lunchtime grilling by eager postdocs, and lively discussions with senior researchers. The life of a celebrated physicist is always intense. But our visitor still wanted to talk physics. We had a couple of hours to fill before dinner.

I described to Feynman what I thought were exciting if speculative new ideas such as fractional spin and anyons. Feynman was unimpressed, saying: “Wilczek, you should work on something real.” (Anyons are real, but that’s a topic for another post.)


A monthly column in which top researchers explore the process of discovery. This month’s columnist, Frank Wilczek, is a Nobel Prize-winning physicist at the Massachusetts Institute of Technology.

Looking to break the awkward silence that followed, I asked Feynman the most disturbing question in physics, then as now: “There’s something else I’ve been thinking a lot about: Why doesn’t empty space weigh anything?”

Feynman, normally as quick and lively as they come, went silent. It was the only time I’ve ever seen him look wistful. Finally he said dreamily, “I once thought I had that one figured out. It was beautiful.” And then, excited, he began an explanation that crescendoed in a near shout: “The reason space doesn’t weigh anything, I thought, is because there’s nothing there!”

To appreciate that surreal monologue, you need to know some backstory. It involves the distinction between vacuum and void.

Vacuum, in modern usage, is what you get when you remove everything that you can, whether practically or in principle. We say a region of space “realizes vacuum” if it is free of all the different kinds of particles and radiation we know about (including, for this purpose, dark matter — which we know about in a general way, though not in detail). Alternatively, vacuum is the state of minimum energy.

Intergalactic space is a good approximation to a vacuum.

Void, on the other hand, is a theoretical idealization. It means nothingness: space without independent properties, whose only role, we might say, is to keep everything from happening in the same place. Void gives particles addresses, nothing more.

Aristotle famously claimed that “Nature abhors a vacuum,” but I’m pretty sure a more correct translation would be “Nature abhors a void.” Isaac Newton appeared to agree when he wrote:

… that one Body may act upon another at a Distance thro’ a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity, that I believe no Man who has in philosophical Matters a competent Faculty of thinking, can ever fall into it.

But in Newton’s masterpiece, the Principia, the players are bodies that exert forces on one another. Space, the stage, is an empty receptacle. It has no life of its own. In Newtonian physics, vacuum is a void.

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Frank Wilczek

That Newtonian framework worked brilliantly for nearly two centuries, as Newton’s equations for gravity went from triumph to triumph, and (at first) the analogous ones for electric and magnetic forces seemed to do so as well. But in the 19th century, as people investigated the phenomena of electricity and magnetism more closely, Newton-style equations proved inadequate. In James Clerk Maxwell’s equations, the fruit of that work, electromagnetic fields — not separated bodies — are the primary objects of reality.

Quantum theory amplified Maxwell’s revolution. According to quantum theory, particles are merely bubbles of froth, kicked up by underlying fields. Photons, for example, are disturbances in electromagnetic fields.

As a young scientist, Feynman found that view too artificial. He wanted to bring back Newton’s approach and work directly with the particles we actually perceive. In doing so, he hoped to challenge hidden assumptions and reach a simpler description of nature — and to avoid a big problem that the switch to quantum fields had created.


In quantum theory, fields have a lot of spontaneous activity. They fluctuate in intensity and direction. And while the average value of the electric field in a vacuum is zero, the average value of its square is not zero. That’s significant because the energy density in an electric field is proportional to the field’s square. The energy density value, in fact, is infinite.

The spontaneous activity of quantum fields goes by several different names: quantum fluctuations, virtual particles, or zero-point motion. There are subtle differences in the connotations of these expressions, but they all refer to the same phenomenon. Whatever you call it, the activity involves energy. Lots of energy — in fact, an infinite amount.

For most purposes we can leave that disturbing infinity out of consideration. Only changes in energy are observable. And because zero-point motion is an intrinsic characteristic of quantum fields, changes in energy, in response to external events, are generally finite. We can calculate them. They give rise to some very interesting effects, such as the Lamb shift of atomic spectral lines and the Casimir force between neutral conducting plates, which have been observed experimentally. Far from being problematic, those effects are triumphs for quantum field theory.

The exception is gravity. Gravity responds to all kinds of energy, whatever form that energy may take. So the infinite energy density associated with the activity of quantum fields, present even in a vacuum, becomes a big problem when we consider its effect on gravity.

In principle, those quantum fields should make the vacuum heavy. Yet experiments tell us that the gravitational pull of the vacuum is quite small. Until recently — see more on this below — we thought it was zero.

Perhaps Feynman’s conceptual switch from fields to particles would avoid the problem.


Feynman started from scratch, drawing pictures whose stick-figure lines show links of influence between particles. The first published Feynman diagram appeared in Physical Review in 1949:

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Two electrons exchange a photon.

To understand how one electron influences another, using Feynman diagrams, you have to imagine that the electrons, as they move through space and evolve in time, exchange a photon, here labeled “virtual quantum.” This is the simplest possibility. It is also possible to exchange two or more photons, and Feynman made similar diagrams for that. Those diagrams contribute another piece to the answer, modifying the classical Coulomb force law. By sprouting another squiggle, and letting it extend freely into the future, you represent how an electron radiates a photon. And so, step by step, you can describe complex physical processes, assembled like Tinkertoys from very simple ingredients.

Feynman diagrams look to be pictures of processes that happen in space and time, and in a sense they are, but they should not be interpreted too literally. What they show are not rigid geometric trajectories, but more flexible, “topological” constructions, reflecting quantum uncertainty. In other words, you can be quite sloppy about the shape and configuration of the lines and squiggles, as long as you get the connections right.

Feynman found that he could attach a simple mathematical formula to each diagram. The formula expresses the likelihood of the process the diagram depicts. He found that in simple cases he got the same answers that people had obtained much more laboriously using fields when they let froth interact with froth.

That’s what Feynman meant when he said, “There’s nothing there.” By removing the fields, he’d gotten rid of their contribution to gravity, which had led to absurdities. He thought he’d found a new approach to fundamental interactions that was not only simpler than the conventional one, but also sounder. It was a beautiful new way to think about fundamental processes.


Sadly, first appearances proved deceptive. As he worked things out further, Feynman discovered that his approach had a similar problem to the one it was supposed to solve. You can see this in the pictures below. We can draw Feynman diagrams that are completely self-contained, without particles to initiate the events (or to flow out from them). These so-called disconnected graphs, or vacuum bubbles, are the Feynman diagram analogue of zero-point motion. You can draw diagrams for how virtual quanta affect gravitons, and thereby rediscover the morbid obesity of “empty” space.

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A graviton encounters a quantum fluctuation.

More generally, as he worked things out further, Feynman gradually realized — and then proved — that his diagram method is not a true alternative to the field approach, but rather an approximation to it. To Feynman, that came as a bitter disappointment.

Yet Feynman diagrams remain a treasured asset in physics, because they often provide good approximations to reality. Plus, they’re easy (and fun) to work with. They help us bring our powers of visual imagination to bear on worlds we can’t actually see.

The calculations that eventually got me a Nobel Prize in 2004 would have been literally unthinkable without Feynman diagrams, as would my calculations that established a route to production and observation of the Higgs particle.

Olena Shmahalo/Quanta Magazine

One way that the Higgs particle can be produced and then decay into daughter particles.

On that day in Santa Barbara, citing those examples, I told Feynman how important his diagrams had been to me in my work. He seemed pleased, though he could hardly have been surprised at his diagrams’ importance. “Yeah, that’s the good part, seeing people use them, seeing them everywhere,” he replied with a wink.


The Feynman diagram representation of a process is most useful when a few relatively simple diagrams supply most of the answer. That is the regime physicists call “weak coupling,” where each additional complicating line is relatively rare. That is almost always the case for photons in quantum electrodynamics (QED), the application Feynman originally had in mind. QED covers most of atomic physics, chemistry and materials science, so it’s an amazing achievement to capture its essence in a few squiggles.

As an approach to the strong nuclear force, however, this strategy fails. Here the governing theory is quantum chromodynamics (QCD). The QCD analogues of photons are particles called color gluons, and their coupling is not weak. Usually, when we do a calculation in QCD, a host of complicated Feynman diagrams — festooned with many gluon lines — make important contributions to the answer. It’s impractical (and probably impossible) to add them all up.

On the other hand, with modern computers we can go back to the truly fundamental field equations and calculate fluctuations in the quark and gluon fields directly. This approach gives beautiful pictures of another kind:

Animation courtesy Derek Leinweber.

Gluon activity in a vacuum.

In recent years this direct approach, carried out on banks of supercomputers, has led to successful calculations of the masses of protons and neutrons. In the coming years it will revolutionize our quantitative understanding of nuclear physics over a broad front.


The puzzle Feynman thought he’d solved is still with us, though it has evolved in many ways.

The biggest change is that people have now measured the density of vacuum more precisely, and discovered that it does not vanish. It is the so-called “dark energy.” (Dark energy is essentially — up to a numerical factor — the same thing Einstein called the “cosmological constant.”) If you average it over the entire universe, you find that dark energy contributes about 70 percent of the total mass in the universe.

That sounds impressive, but for physicists the big puzzle that remains is why its density is as small as it is. For one thing, you’ll remember, it was supposed to be infinite, due to the contribution of fluctuating fields. One bit of possible progress is that now we know a way to escape that infinity. It turns out that for one class of fields — technically, the fields associated with particles called bosons — the energy density is positive infinity, while for another class of fields — those associated with particles called fermions — the energy density is negative infinity. So if the universe contains an artfully balanced mix of bosons and fermions, the infinities can cancel. Supersymmetric theories, which also have several other attractive features, achieve that cancellation.

Another thing we’ve learned is that in addition to fluctuating fields, the vacuum contains non-fluctuating fields, often called “condensates.” One such condensate is the so-called sigma condensate; another is the Higgs condensate. Those two are firmly established; there may be many others yet to be discovered. If you want to think of a familiar analogue, imagine Earth’s magnetic or gravitational field, elevated to cosmic proportions (and freed of Earth). These condensates should also weigh something. Indeed, simple estimates of their density give values far larger than that of the observed dark energy.

We’re left with an estimate of the dark energy that is finite (maybe), but poorly determined theoretically and, on the face of it, much too big. Presumably there are additional cancellations we don’t know about. The most popular idea, at present, is that the smallness of the dark energy is a kind of rare accident, which happens to occur in our particular corner of the multiverse. Though unlikely a priori, it is necessary for our existence, and therefore what we are fated to observe.

That story, I’m afraid, is not nearly so elegant as Feynman’s “There’s nothing there!” Let’s hope we can find a better one.


For anyone interested in learning more about Feynman diagrams and quantum electrodynamics, the author recommends Feynman’s book QED: The Strange Theory of Light and Matter.

This article was reprinted on Wired.com.







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  • The ground state energy is certain and does not fluctuate. Look at the Hydrogen atom: what fluctuate there are coordinates and momenta of constituent particles, not the ground state energy. The same statement is valid in QFT.

  • Dear Mr. Wilczek,
    I would say Thanks for your text.
    I willing to say that I love physics. Quanta have been done a great work on this scenario.

    So, that said, please keep the texts flowing !!! This is a privilege and a great pleasure.
    Best wishes,

  • Most fascinating. It is unfortunate that while the principles and ideas are fascinating and logical that the math used to represent them are accessible only to well trained physicists and mathematicians. In my studies of Feynman,Maxwell, Bohm, Kaku and Close etc…there is always a barrier to full comprehension because the underlying principles are expressed n this heavy math. That said there is always possibilities for an insightful thought experiment. Discover explored Einstein's thoughts in a recent issue. That said; it appears that whenever the equations and calculations derive an infinity that some error or omission must exist. Consider the quantum foam which emerged from calculations of infinitely tiny space and how it led to the acceptance of a Plank Length limit. In calculating the energy of space the idea of balancing negative and positive energies which reduce any individual measurement without effecting the average appears most logical and perhaps there are further offsets to Dark Energy. I imagine space this way; when a virtual particle is created out of a zero value space a void is created to offset it (the way a a particle and anti-particle form) One perhaps is real, the other is merely a whole in the background from where it came. If the two collide the whole is refilled in an energetic interaction. Perhaps our energy, fields and particles are linked to opposite fields in an non-observable connected universe. I would like to see more study of particle-anti-particle collisions because the very existence of such doppelgangers suggests a more complex universe than even general relativity or dark energy studies predict. Please keep working and posting your work as there are many world citizens who value these explorations .

  • I was going to say something, but Eduardo made the perfect comment before me. Thank you very much mr Wilczek.

  • Awesome stuff, I have a question: The Dark Energy equations take in consideration that the universe is closed? Would a leaking universe deal with the problem of infinite density?

    I think Quanta should have more posts in a week, Always bringing interesting issues. 😀

  • Extremely well written … but could we expect anything less from Prof. Wilczek? No! Can't see how you can find the the weight of empty space without doing dimensional translations of (name your choice of) free energy … which I am informed are expressly forbidden in GR. How can that be? How can everything be so right and yet have some basic element be so out of place?

  • Thank you, Prof. Wilczek, for such a good read, and a reminder that an understanding of the world is a hard struggle won, one step at a time. And we still don't know where it all will lead…

  • Feynman lived in Altadena CA and drove a van with his diagrams painted on the sides.

  • Thank you for giving me the opportunity to peak into the mind blowing world of physics. You have a beautiful talent for explaining very complicated

  • sorry for the blip. I touched my computer screen and it sent my message. Thank you for giving someone who has very little knowledge of physics a glimpse into your world and understand what you are saying. I have always wondered how differently a physicist looks at the world in which we live.

  • For years I have thought these diagrams were over done. I think Feynman would have been much better off learning some number theory and topology. It would have served him better in the long haul. Please comments from physics grad students. FRD

  • A great article that conceptually explains the modern view on how fields are fundamental and not particles!!

  • I hope Professor Wilczek, or anyone else, can answer my questions about the void. Can there be math in it the void if there is nothing in it? If there is nothing in it, and there is no math, how can we even conceive of it? Yet we do.

  • FW didn't mention Feynman's one observation in his book Quantum Electrodynamics (or some such title) in about chapter 25 (essay 25?). There he "proves" the no photons have zero mass. (The idea is that no emission or absorption is instantaneous. So when one averages over all the events in those 2 lifetimes, one gets a virtual mass since the average time between emission and absorption is less than the average separation.) So now if all photons have mass (are off-shell in techno-parlance) that means that Feynman's own propagator is problematic. And ultimately it is that propagator which causes the infinities.
    I might add that I have another proof that all photons have mass that doesn't require RF's averaging process which is ultimately derived from another Feynman analysis with JWheeler–the Lagrangian of the Wheeler-Feynman universe. Heck, that Lagrangian itself proves also that all photons have mass if one thinks deeply enough about it.

  • How does current physics view the relationship between the Casimir effect and gravity? It seems to me that the same wavelength elimination between "mirrors" should happen between any nearby atoms, and accretion would happen–positive feedback thanks to the relative reduction of the range of gluon frequencies. Forgive my intuitive physics, but I'd love to know how this is considered. No gravitons in my cartoon… maybe clumps perturb the gluon activity pattern with regular probabilities… It's been a long time since I've looked into this level of reality's fabric…so cool. Thank you! Matthew

  • "The ground state energy is certain and does not fluctuate"

    In a perturbed system, it can.

  • Great primer on Feynman diagrams, sir.

    Lol. Space is what keeps everything happening at the same place. Very clever. Are your going to be asserting the literal equality of special dimensions with that of time in the future, I wonder?

    Also better to know something about that great man.

  • Two questions: is it possible for the result of the integral of a function when taken to infinity to be finite. Are there asymptotic equations that satisfy this (finite area over infinite length)? Also, have you yet managed to divorce the mathematical principle from its literal quantum expression on the issue of enticement? Knowing this principle and associated equations would help in what I'm working o these days. Thanks in advance.

  • Thank you for an excellent summary of one of the greatest problems – or should I say the fundamental problem, really – in our understanding of the basic structure of the universe. Reading these columns, inevitably followed by thinking for days afterwards about the issues raised in them, is always the highlight of my week. Please know that they are *much* appreciated, and we are enormously fortunate to have them produced by someone so talented at discussing them clearly!

  • https://www.youtube.com/watch?v=-EtHF5ND3_s describes Riemann's Paradox.
    It explains how, under certain conditions, two infinite series, when subtracted from one another, can produce any answer you wish! This is supremely non-intuitive of course, but I wonder if it bears any relation to this odd paradox with vacuum energy density.

  • The answer to D. McFarlane's question is yes. The episode is "The Bat Jar Conjecture" (Season 1, Episode 13). The janitor, a former Russian physicist, supplied the correct value for the given diagram, but Sheldon refused to submit it because he didn't believe that a janitor could possibly answer a question that he could not. The brown and tan van mentioned by Kenneth Clark was featured in recent episode (Season 9, Episode 3, "The Bachelor Party Corrosion") of BBT. The van was destroyed in the story (but not in reality). Another episode (Season 1, Episode 5, "The Hamburger Postulate") is interesting because it had two important "vacuum polarization" diagrams (like the one in Frank's article about the interaction of a graviton with a vacuum fluctuation). Sheldon got an answer for that one, but it was wrong. The bottom line said "beta > 0", which Leslie corrected in the middle of the night. Sheldon lamented the fact that he would have to share the Nobel Prize with Leslie. The fact that this particular beta is negative demonstrates the asymptotic freedom of QCD, for which Frank along with David Gross and David Politzer were awarded the Noble Prize. Both Sheldon and Leslie were over 35 years too late for that. That the episode was only five years after the award was a mystery to many physicists – not because the episode came so early :o). Now you know how the show's Science Consultant, David Salzburg, earns his money.

  • I apologize for the incorrect spelling of Prof. Saltzburg's name. The spell checker made me do it.

  • By the definition provided:
    "…a region of space “realizes vacuum” if it is free of all the different kinds of particles and radiation we know about…"

    This statement is factually false:
    "Intergalactic space is a good approximation to a vacuum."

    There is no location in intergalactic space that is free of electromagnetic radiation. In fact, all of intergalactic space is suffused with electromagnetic radiation (EMR). This is not a theoretical claim, it is an assertion of fact based on incontrovertible evidence. That evidence is readily accessible on the internet; simply search on "hubble deep field". Here is one of the many images and links you will find:


    What this image shows us are thousands of galaxies in a very small patch of the sky extending out as far as the Hubble telescope can see. All of those galaxies are emitting EMR omnidirectionally. It then follows that everywhere in that image where there is no matter (here defined as anything with rest mass) there must be EMR. Since we have good reason to believe that the cosmos looks more or less the same in all directions, it then follows that everywhere there is not matter there must be EMR.

    By straightforward observation we have established that "empty space" does not exist in the observable cosmos, which obviates the need for the very concept of a "space" possessing an independent physical reality. On the cosmological scale, observed reality consists of matter, at least some of which radiates EMR, and EMR itself, that is all. "Space" is unobserved and therefore unnecessary to any realistic model of the cosmos.

    Because theoreticians choose, for whatever reason, to ignore observational reality and assume, as the author does here, that "Intergalactic space is a good approximation to a vacuum", they wind up having to add back into their model fictitious elements like vacuum energy and virtual particles in order to get the model to bear some semblance to observed reality. The model nonetheless retains the "empty space" concept which means that it cannot be said to actually resemble observed reality. In this way the theoretician's penchant for oversimplification leads inexorably to overcomplexification.

    This is a backwards way to do science. Theory has to be grounded in observational reality if it is to avoid the thicket of mathematical fantasias that characterizes the current standard model of cosmology.


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