25-year update on the “Millennium problems” in physics

Back in the year 2000, physicists gathered with an unusual purpose: to choose the 10 greatest unsolved problems in fundamental physics for the new millennium. At that time, we had:

• discovered most of the particles of the Standard Model, but not yet the Higgs boson,
• a strong idea that gravitational waves existed and carried energy, but no direct detection of their existence,
• robust evidence for the existence of dark matter and strongly suggestive evidence for the existence of dark energy, but no direct detection of either,
• and it was also a time where physicists placed a lot of hope in speculative ideas — such as supersymmetry, grand unification, extra dimensions, and string theory — for driving physics forward.

The limitations were that, in order to be considered, the problem must be deemed important, well-defined, and articulated in a clear way, that each participant could submit one and only one question, and that duplicate entries would be rolled together into one. At the end of the conference, after all submissions were tallied, three legendary physicists, Michael Duff, David Gross, and Ed Witten, selected ten problems as the Millennium Problems in physics, designed to inspire and challenge physicists in the 21st century and beyond.

It’s now been a full quarter-century since those Millennium Problems were introduced, and while we’ve made some progress on a select few of them, resolutions to many of them seem as far away as ever. Here’s the current status of where we are on each one.

This chart of particles and interactions details how the particles of the Standard Model interact according to the three fundamental forces that quantum field theory describes. When gravity is added into the mix, we obtain the observable Universe that we see, with the laws, parameters, and constants that we know of governing it. However, many of the parameters that nature obeys cannot be predicted by theory, they must be measured to be known, and those are “constants” that our Universe requires, to the best of our knowledge.

1.) Are all the (measurable) dimensionless parameters that characterize the physical universe calculable in principle or are some merely determined by historical or quantum mechanical accident and uncalculable? (submitted by David Gross)

This is a vital question about the nature of reality. The Standard Model provides us with a framework for reality: it tells us how many and what species of fundamental particles should exist, and it also tells us how those particles relate to (i.e., interact with) one another. However, there are aspects of the Standard Model, such as:

• the strengths of the various interactions/forces,
• the rest masses/energies of the fundamental particles,
• and how particles with the same quantum numbers mix together,

that are not predicted and aren’t even predictable, in principle, within the Standard Model. Instead, they must be measured, empirically, in order to be known.

The “dimensionless parameters” are also known as “fundamental constants,” and one of the goals of physics is to learn where — if anywhere — they come from. Certain extensions of the Standard Model may offer hope, and a proposed formula known as the Koide formula appears to offer approximate relationships between some of them, but the 21st century, so far, has not only not offered compelling answers, but has presented us with evidence that there are even more unaccounted-for dimensionless parameters than were known 25 years ago. Not only haven’t we made substantial progress on this front, but the problem presently looms larger than ever.

It’s generally assumed that at some level, gravity will be quantum, just like the other forces. While the semi-classical approximation for computing the decay of black holes involves performing quantum calculations in the classical background of Einstein’s curved space, that approach might not be valid for capturing the full suite of physical behavior that the outgoing radiation possesses, particularly as far as information is concerned.

2.) How can quantum gravity help explain the origin of the universe? (submitted by Edward Witten)

This question isn’t just profound, but it’s a much more complex question than this deceptively simple phrasing of it implies. We know that our two pictures of reality — quantum field theory for the electromagnetic, weak and strong nuclear, and Higgs forces, plus general relativity for gravitation — are fundamentally incompatible. We also know that, at extremely early times, the Universe was in a very high-energy state, and that quantum effects are expected to become important everywhere, even for gravitation.

So how can you “marry” these two pictures together? The assumption is that a quantum theory of gravity is required, and that therefore quantum particles that carry the gravitational force (the graviton) must exist, just as photons and gluons exist.

Presumably, at some early stage — before the start of the hot Big Bang, and even before the onset of cosmic inflation — there was an initial event that started everything off. However, this is not firmly established; it is just one viable (although strongly favored by many) possibility. Presumably, if-and-when this occurred, quantum gravitational effects were important. And potentially, these two (speculative, unknown) aspects of the Universe are related to one another. 25 years after this question was formulated, some progress has been made on ruling out possible origins of our Universe that avoid a non-singular beginning, but the rest of it is just as uncertain as ever. Although many who work on it will disagree, I would go as far as to assert that no substantial progress has been made concerning the question of quantum gravity since this question was first formulated.

Two possible pathways for proton decay are spelled out in terms of the transformations of its fundamental constituent particles. These processes have never been observed, but are theoretically permitted in many extensions of the Standard Model, such as SU(5) grand unification theories.

3.) What is the lifetime of the proton and how do we understand it? (submitted by Steve Gubser)

This question may surprise the non-physicists among you out there, as you might ask, “Wait a minute, isn’t the proton stable, and isn’t its lifetime therefore infinite?” And the answer is that yes, the proton is stable: as far as we’ve observed it. As the lightest species of baryon (a particle made up of three valence quarks), there is no known pathway for a proton to decay, as any decay (such as into pions and leptons) would violate the conservation of baryon number.

However, there are two reasons to consider that the proton could decay.

• In the Standard Model, “baryon number” is not an explicitly conserved quantity; it’s possible to violate baryon number through a set of interactions (known as sphaleron interactions) so long as the difference of “baryon number minus lepton number” is conserved.
• And in extensions to the Standard Model, including in nearly all grand unified theories and in all versions of string theory, superheavy bosons that facilitate the allowable decay of protons are mandatory.

The fact that we not only haven’t observed proton decay, but have constrained that if proton decay occurs, it occurs with a mean lifetime that’s greater than 2 × 10³⁴ years, which rules out the simplest type [Georgi-Glashow SU(5)] of grand unified theory. This constraint is about a factor of 10 greater than it was 25 years ago, but we are no closer to understanding the proton’s stability than we were back in the year 2000.

The Standard Model particles and their supersymmetric counterparts. Slightly under 50% of these particles have been discovered, and just over 50% have never shown a trace that they exist. Supersymmetry is an idea that hopes to improve on the Standard Model, but it has yet to achieve the all-important step for supplanting the prevailing scientific theory: having its new predictions borne out by experiment.

4.) Is Nature supersymmetric, and if so, how is supersymmetry broken? (submitted by Sergio Ferrara & Gordon Kane)

The notion of symmetries in physics is incredibly powerful, as there’s a fundamental connection between symmetries in our physics theories and conservation laws within our Universe. Of course, our Universe isn’t perfectly symmetrical in many ways: we have electric charges but not magnetic ones, the weak interactions fundamentally violate mirror-symmetry, matter-antimatter symmetry, and time-reversal symmetry, and all neutrinos appear to be left-handed while all antineutrinos appear to be right-handed.

Nevertheless, it’s possible that there are yet-undiscovered symmetries in nature, with one of the most explored and compelling physical scenarios being supersymmetry. Most compellingly, supersymmetry — which hypothesizes the existence of at least one “superpartner” particle for each of the particles in the Standard Model — offers a potential solution to the dark matter puzzle, the puzzle of high-energy unification of the strong force, and the hierarchy problem (which appears as number 9 on this list) all at once.

Unfortunately, if supersymmetry were “the solution” to the hierarchy problem, it would have already appeared in existing Large Hadron Collider (LHC) data. (See the last section of this article for more detail.) 25 years ago, many assumed that not only was nature fundamentally supersymmetric, but that supersymmetric particles were certain to appear at the LHC. Instead, the LHC has shown us that those assumptions were, in fact, erroneous, and are not borne out by our physical reality. Nature may still be supersymmetric at some much higher energy scale, but not only is there no experimental evidence favoring that scenario, it’s the case that even if nature is supersymmetric at some point, it won’t solve the one problem (the hierarchy problem) that provided the theoretical motivation for proposing it in the first place. This question, which presumed the answer to the first part would be “yes,” has yielded no hints that point to anything other than “no.”

In theory, there could be more than three spatial dimensions to our Universe, so long as those “extra” dimensions are below a certain critical size that our experiments have already probed. There is a range of sizes in between ~10^-19 and 10^-35 meters that are still allowed for a fourth (or more) spatial dimension, but nothing that physically occurs in the Universe can be allowed to rely on that fifth (or more) dimension.

5.) Why does the universe appear to have one time and three space dimensions? (submitted by Shamit Kachru, Sunil Mukhi, & Hiroshi Ooguri)

Here in our Universe, it’s verifiably measurable that there are three dimensions of space and one (and only one) dimension of time. Forces, like electromagnetism and gravity, spread out in three dimensions as you move farther away from the sources (i.e., charges) that generate those forces, which explains why they obey an inverse-square relation. However, it seems that many other options could have been possible, and that if there are further unifications to the forces, including a potential theory of everything, it would imply that our Universe once existed with several “extra dimensions” within it: a key prediction of string theory.

From a theoretical point of view, we have no idea what the dynamics would be that would take us from a full string theory — which predicts (at least) a 10-dimensional spacetime governed by a Brans-Dicke-like (scalar + tensor) theory of gravity, along with an enormous spectrum of particles and interactions — to the Universe we find ourselves in today: with only a 4-dimensional spacetime with no scalar contributions to gravity, with the restricted (Standard Model-only) spectrum of particles and interactions we observe. I once likened this process to that of an unlikely broken box, and despite a further 25 years of work on this puzzle, the best answer physicists have to offer is to mutter something qualitative about “compactification” without any known mechanism or quantitative process that can be tested.

In other words, this is another question that we are no closer to answering in 2025 than we were 25 years ago.

If dark energy is not a constant over time, then it shouldn’t be described by a single parameter, w, but rather by two or more parameters that allow an evolution with time. In this figure, the intersection of the dotted lines correspond to a simple cosmological constant for dark energy; the DESI data, when combined with CMB and/or supernova data, strongly favors an evolving form of dark energy that weakens over time. The case where w0 = -1 and wa (or w’) = 0 corresponds to constant dark energy.

6.) Why does the cosmological constant have the value that it has, is it zero and is it really constant? (submitted by Andrew Chamblin & Renata Kallosh)

It’s hard to believe, but back in the year 2000, the “cosmological constant” that we observe within the Universe — i.e., the evidence for dark energy — was brand new. First published in 1998, the evidence for not only an expanding Universe, but an accelerating universe, was slowly taking hold in the field of physics, although many still resisted the idea given the (then-substantial) uncertainties surrounding the initial data.

• Was the Universe truly accelerating?
• Was it accelerating as though the Universe had a cosmological constant?
• And was that cosmological constant — appearing in Einstein’s general relativity — the same as the vacuum expectation value that should arise as the zero-point energy of space in our quantum field theories?

25 years later, we have partial answers. The Universe is definitely accelerating, and its observed acceleration is consistent with being caused by a cosmological constant, although further work needs to be done to establish whether dark energy’s “energy density” is truly constant, or whether it changes with time, as some recent data suggests. However, that third question — about whether (and if so, how) the observed acceleration of the Universe relates to the zero-point energy of space from quantum field theory — remains as elusive as ever, as our calculations yield predictions for the energy density that are enormous and definitively ruled out.

There remains the possibility that the cosmological constant’s value will prove to be small, non-zero, and in line with what’s predicted by our quantum field theories, but that’s not a calculation we’ve been able to do, nor is it one that we’ve made any meaningful progress on over the past 25 years. The cosmological constant appears to be a real feature of the Universe, one that makes up most of the Universe’s energy, but we are no closer to understanding its value.

The difference between a Lie algebra based on the E(8) group (left) and the Standard Model (right). The Lie algebra that defines the Standard Model is mathematically a 12-dimensional entity; the E(8) group is fundamentally a 248-dimensional entity. There is a lot that has to go away to get back the Standard Model from string theories as we know them; with fewer parameters and fewer assumptions, our presently understood Universe is much more simply described by the Standard Model and general relativity than any other description.

7.) What are the fundamental degrees of freedom of M-theory (the theory whose low-energy limit is eleven-dimensional supergravity and which subsumes the five consistent superstring theories) and does the theory describe Nature? (submitted by Louise Dolan, Annamaria Sinkovics, & Billy & Linda Rose)

I think there were a lot of people who were anticipating that progress would have been made on these fronts, as it was a huge revelation to uncover that the different superstring theories were all identical in some sense: they were different formulations of the same underlying theory, known as M-theory. The Standard Model is often written out in terms of its Lie algebra: as SU(3) × SU(2) × U(1), and if you’ve ever seen groups like E(8) × E(8) or SO(32), you know that those are two (of the five) examples of superstring theory that are shown to be equivalent through M-theory.

The big problem is: these superstring theories are huge, enormous, complicated, and contain many many things — extra dimensions, extra particles, extra symmetries, extra relationships, etc. — that must be completely eliminated, somehow, in order to recover the Universe we observe. Despite an enormous amount of effort by a great many very smart physicists, we both:

• have no idea what the fundamental degrees of freedom of M-theory are,
• and we have no idea whether M-theory describes “Nature,” or our reality, at all.

I would again contend that no substantial progress has been made on this question at all over the last 25 years.

Encoded on the surface of the black hole can be bits (or quantum bits, i.e., qubits) of information, proportional to the event horizon’s surface area. When the black hole decays, it decays to a state of thermal radiation. As matter and radiation fall into the black hole, the surface area grows, enabling that information to be successfully encoded. When the black hole decays, entropy will not decrease, but rather will remain constant, as Hawking radiation is an entropy-conserving (adiabatic) process. How or if that information is encoded into the outgoing radiation is not yet determined.

8.) What is the resolution of the black hole information paradox? (submitted by Tibra Ali & Samir Mathur)

This, at least, is one that people have worked on extensively, and have uncovered a number of interesting aspects about over the last 25 years, even if the answer is ultimately, “It’s still unresolved.” The black hole information paradox, very simply, states that when matter either forms or falls into a black hole, it has properties, or information, associated with it. This information includes:

• the quantum numbers of the particles that fall in,
• the bonds and entanglements between the particles that fall in,
• and the types and properties of the particles that fall in.

The “paradox” arises because these black holes are not fundamentally stable, and will decay, over time, through a process known as Hawking radiation: where energy, emitted mostly in the form of photons, is carried away from the black hole until after a long amount of time, ~10⁶⁷ years or more, the black hole has evaporated away entirely.

So where does the “information” about the particles that went into making the black hole wind up? Is it lost? Is it conserved, and somehow encoded in the outgoing radiation? Or is there some other resolution to the paradox?

While most people favor the “conserved and somehow encoded” option, and while many interesting investigations into black hole firewalls and other phenomena have been conducted, the truthful answer is, “We still don’t know the answer.” I would say that some progress has been made here, but a final resolution still seems very far away, just as was the case 25 years ago.

This to-scale diagram shows the relative masses of the quarks and leptons, with neutrinos being the lightest particles and the top quark being the heaviest. No explanation, within the Standard Model alone, can account for these mass values.

9.) What physics explains the enormous disparity between the gravitational scale and the typical mass scale of the elementary particles? (submitted by Matt Strassler)

Here it is: the hierarchy problem. If you look for a “natural” mass scale in particle physics, you’ll find one: the Planck mass, which is around ~10²² times greater than the mass of the electron. Even the heaviest particles in the Standard Model, the top quark and the Higgs boson, are a factor of ~10¹⁷ lighter than the Planck mass, which itself is a measure of the gravitational scale.

There have been many proposals to attempt to explain this disparity, just as there are many proposals to attempt to explain #6: why the cosmological constant is so weak compared to the energy scale predicted by quantum field theories. Unfortunately, despite everything we’ve learned, all we can say is a long list of things that can’t be responsible for it.

What explains these tremendous disparities? How can we understand what the masses of the fundamental particles are? Are the “cosmological constant” problem and the “hierarchy problem,” both about the great difference between (naively) predicted values and observed values, related? And is there anything to the observation that if — instead of the Planck mass, we put something like a “neutrino mass” into the cosmological constant problem — we get an answer that aligns with reality?

This puzzle remains unresolved, but we have made progress on constraining what the solution can’t be, with “supersymmetry” being ruled out as the solution marking perhaps the most substantial advance.

This depiction of a lattice QCD method shows that space and time are discretized into a set of grid-like points on a lattice. As the spacing between points decreases and the overall size of the lattice tends toward infinity, the true value for QCD calculations is approached more and more accurately.

10.) Can we quantitatively understand quark and gluon confinement in Quantum Chromodynamics and the existence of a mass gap? (submitted by Igor Klebanov & Oyvind Tafjord)

You might get to the end of this list and feel a little bit of despair. Of the nine prior entries, we have six that I would declare, “No progress” as the verdict, two with, “Okay, it’s not supersymmetry” as the verdict, and one, “Okay, we’ve established some observational facts about dark energy, but don’t understand its value theoretically,” as the verdict.

But this one changes the story substantially, because this final question on the list has actually seen tremendous progress made thanks to a novel technique that has truly come into its own here in the 21st century: the technique of Lattice QCD. Unlike quantum electrodynamics, which is a theory where we can calculate things perturbatively — with greater numbers of exchanged particles contributing less and less to interaction strengths — quantum chromodynamics (QCD) is non-perturbative.

Advances in computing power and in calculational technique for Lattice QCD have begun to change this story over the past 25 years. The answer to this question is now known to be “yes” for certain: yes it is possible to understand confinement, as well as the presence (or absence) and magnitude of a mass gap, and the way to do it is through Lattice QCD. Lattice QCD turns out to (very likely) hold the solution to the longstanding muon g-2 puzzle as well, as has been shown only over the past couple of years.

The fact that even one of these “Millennium Problems” has fallen in just the first 25 years of the 21st century should give us hope for pursuing the rest of them. It’s often by “attempting the impossible” that we wind up achieving the remarkable, and in that regard, fundamental physics is no different than any other human endeavor.