Our universe’s fine tuning may be why the Standard Model is so mathematically ugly
String theory seems to be on its last legs, though its proponents may not admit it.
String theory is the only real contender for a theory of everything these days. In a recent interview, string theory guru Michio Kaku suggested that the Standard Model, the theory that has been confirmed over and over as our best understanding of how matter and forces behave, just “pops out” of string theory. Kaku went on to say that the Standard Model is “one of the ugliest theories proposed so far”.
Statements like this, which might have only caused some grumbling in past decades, are downright incendiary among many physicists. String theory critic Sabine Hossenfelder said in a tweet that Kaku’s statement was “just wrong” and that
if string theory was actually really simpler than the Standard Model, then physicists would actually USE IT to make predictions for the LHC [Large Hadron Collider]. They don’t. Why? Because it’s useless.
While there is no real consensus on the merits of string theory, for now it remains a mathematically beautiful theory that is indeed “useless” for making predictions. It also makes so many unobserved predictions that you could call it an ugly theory, but ugly in the way that the Standard Model is beautiful. (It’s what’s on the inside that counts after all.)
Indeed, the whole Theory of Everything program may be more a reflection of human ideals than humble submission to reality, akin to geocentrism, complete with unseen angels hiding behind the heavenly spheres, rather than the refinement of heliocentrism from Copernicus to Kepler to Newton.
I say string theory is ugly in the sense that mathematical elegance and physical elegance are two different things. A mathematically elegant theory is simple on paper, but it may lead to complex and confusing predictions that aren’t observed in real experiments. A physically elegant theory is one that explains the data in as simple a way as possible no matter how complex the math needs to be and no matter how many experimental parameters have to be inserted. (I say explains the data not predicts the data because I don’t claim to be a logical positivist. A theory that makes good empirical predictions but tells you nothing deeper is only a useful tool waiting for an explanation. I do not see the Standard Model as such a theory, albeit it is abstract.)
From this perspective, the Standard Model is mathematically troubling but physically very elegant. It explains exactly what we see in experiments, and, while it is confounding to many physicists that it predicts no more, there is some elegance in that. After all, it does explain the world we live in. Since the Higgs boson was discovered in 2012, the Standard Model’s overall predictions have been confirmed to astounding degree of accuracy.
General Relativity isn’t quite so successful at this but I’ll get to that later.
The theory that pops out of string theory is not the Standard Model but a massive theory with a huge number of undiscovered particles and extra dimensions. Theorists have also so far failed to show that it replicates Einstein’s general relativity, our best theory for how gravity and the universe behave at large scales. This is despite the fact that string theory originally caused a lot of excitement precisely because it looked as if it could explain gravity at small scales.
String theory has been, in many ways, a dream for a mathematically elegant description of nature where experimental parameters are predicted rather than measured, and the laws of the world we live in become, not one set of many possibilities, but the only possible ones.
To me this desire is strange, as if we hoped that all planets were like the Earth and not, instead, the result of peculiar circumstances. The fact that we have to insert experimentally determined parameters suggests, rather, that ours is but one of many possible universes with just one of many possible sets of laws.
I often say that string theory probably explains some universe, just not this one.
One of the core features of string theory, particularly supersymmetric string theory, is its symmetry. This is what makes it so simple and mathematically elegant. Yet, this is also what makes it a problem theory because the universe is not symmetrical. We have more matter than antimatter. We have fermions like electrons and quarks of certain types and bosons like photons and gluons of certain types but not vice versa as predicted by supersymmetry.
Indeed, the lack of symmetry in the observed universe is one of the big problems with grand unification theories. The standard model fits a U(1)xSU(2)xSU(3) group theory structure, where the U(1)xSU(2) represents electroweak forces (the unification of electromagnetic and the weak force responsible for radioactive decay) and SU(3) representing the force that binds the cores of atoms together. You can unify this theory is various ways such as SU(5) or E8xE8 or SO(10). All of these are different ways of manipulating either complex or real numbers in rotational or other spaces.
Group theory is a mathematical description of a very familiar physical concept of rotation, translation, or, for complex numbers, phase symmetry. And so the Standard Model is largely a way of describing the phase symmetries of force carrying waves. U(1) is the simplest and the one we are familiar with if you’ve ever played with an oscilloscope and noticed how, in a complex demodulation of a signal, you can rotate the overall phase and still have the same signal.
An even simpler illustration of U(1) is to look at a fixed spring that bounces up and down. If you represent its position, which is one dimensional, as a real number, and its momentum, also one dimensional, as an imaginary number, and you add them together, you get a complex representation of the spring’s state in its two dimensional “phase space”. For an undamped and unforced (read: ideal) spring, if you use the right units and if you apply a rotation in the complex plane to the spring’s position/momentum state at any given moment, you end up with a new state that is in the same phase space trajectory as the original one. This is a U(1) symmetry of the spring’s trajectory over time and so arises, in this case, from time translation symmetry. Going to higher groups like SU(2) and SU(3) are similar, but more complex multidimensional rotations of more complex phase space trajectories.
Meanwhile, gravity has a symmetry that often confounds intuition. A common misconception (that I fell into in my younger days) is that gravity has what is called a Poincaré symmetry, which includes four translations and six rotations in space and time (three of the rotations are space-time plane rotations or Lorentz boosts which manifests as linear acceleration). This isn’t quite true, however, because, while Einstein’s theory of general relativity has a lot of coordinate symmetry, it doesn’t have any physical symmetry at all. The Poincaré group arises when you mistakenly equate coordinate systems with physical measurements, a concept that general relativity does away with (but it hangs around in special relativity). Only scalar values are true measurements in general relativity, and they have no symmetry under coordinate transformations because they are invariant. Since only scalars exist as standalone measurable objects, nothing measurable has symmetry in the theory.
This is why Kretschmann, shortly after Einstein published his theory, claimed that general relativity’s symmetry was the “identity” symmetry, meaning it had none at all. The manifold of spacetime is fixed and immutable and the gravitational field for a given manifold has different coordinate representations but no equivalent physical ones. Thus, while coordinate systems are an artificial way of representing different observers, they have no physical significance.
Many theories of quantum gravity add symmetry such as conformal invariance to Einstein’s equations. Conformal symmetry essentially means multiplying your gravitational field by some function over your coordinate system. In Einstein’s theory this produces a new matter term based on the conformal function that wasn’t there before, so Einstein’s is not conformally invariant. In a conformal gravity theory it does not. Unfortunately such conformal theories are, for physicists, “weird” with four-derivative equations unlike the usual two for most physical theories. Four derivative theories often have stability problems. There are ways around this such as conformal symmetry breaking where you only get the four derivative theory at high energies, but so far there’s no compelling reason to accept conformal gravity as a replacement for general relativity.
While these ideas are mathematically interesting, physically what observations and experiments demand is very different. General relativity has matched observations exceptionally well provided that there is a large quantity of matter and energy in the universe that we can’t see electromagnetically. While there are also modified theories of general relativity that explain these observations without additional matter and energy, these come with their own issues, and, because we can “see” dark matter and energy gravitationally in various ways, it is difficult to explain them away with modified forces. In fact, attempts to do so such as Tensor-Vector-Scalar theory, bimetric theory, and so on have all been getting beaten down by experiment in the last few years, and so have not caught on since, both physically and mathematically, dark matter and energy are pretty elegant solutions.
Studying quantum gravity experimentally is another issue altogether because the energies involved in unification are fantastic, but we can nonetheless study some quantum phenomena under gravity’s influence such as falling entangled particles and gain some insight into how gravity affects such things.
Other problems with general relativity such as our ability to “renormalize” it (control infinities in the equations) into a quantum theory are less of a problem than you might think. The so called “incompatibility” between General Relativity and quantum theory that string theorists like to point out might only require mathematics to provide the right tools rather than a new physical theory. We have been here before. Newton and Leibniz had to invent calculus to get theories of motion right. Greek geometry simply was not up to the task. Riemann and others had to invent geometry on curved spaces to help Einstein create his theory. Where would gravity be without that?
Theories such as asymptotic safety suggest that gravity can be renormalized, although we aren’t quite there yet. The kind of renormalization that particle theorists use is a very simple kind (perturbative renormalization) that works well for the Standard Model but not for many theories that are nonetheless renormalizable. Once a wild stallion of theoretical physics, mathematicians have made considerable progress taming uncontrollable infinities and showing that they are artifacts of how we mathematically represent theories rather than problems with the theories themselves.
Unlike many GUTs, string theory also wants to unify matter with forces. “Matter” is made up of various “Dirac” or spinor fields. Spinors are so strange that it is difficult to have an intuitive understanding of them at all. While their mathematical description is clear and has many experimental confirmations, physically one can only understand them by various analogies. Spinors obey a group called the Spin group. One of the better analogies is that of twisting and untwisting a ribbon called the “plate or belt trick”, which requires not a 360 degree rotation but a 720 degree rotation. (This is fun to do at home.) This equates with fermions being spin-1/2. All matter appears to obey the Dirac equation save maybe the neutrino.
I always think of Dirac fields and the spin group as being like the “square root” of the more intuitive scalar and vector field matter and rotational groups. Given that matter tends to come in matter anti-matter pairs this makes a lot of sense. An electron and a positron form a complete set, like two pieces of a friendship bracelet. What doesn’t make sense is why there is so much matter and it did not all annihilate at the Big Bang.
One idea is that all the anti matter was thrown back in time from the Big Bang while the matter was thrown forward in time (from our perspective). I came up with this idea on my own once and thought it was crazy but then found that that didn’t stop theoretical physicists from exploring it. It is actually peer reviewed published research (I cannot find the citation unfortunately) with of course no experimental confirmation. But I find it compelling if only because we cannot find any symmetry violations between matter and anti-matter that would make anti-matter preferred.
Then there’s the problem of left-handed neutrinos, so maybe we actually need four universes to come from the Big Bang, anti-matter/matter and left and right handed. But now I’m getting facetious.
The universe is under no obligation to be symmetrical. If our laws describing it are, then perhaps we just haven’t found the violations yet.
Grand Unification in the Physical World
All these group theories present a mathematical picture of what grand unification means in that we somehow want to find one group to rule them all, but what does it mean physically? There are different levels of unification. String theory attempts to unify everything, including matter and forces. Other unification theories only try to unify the Standard Model forces into a single group theory since the Dirac equation unifies matter.
Largely, unification physically has to do with coupling constants. All of the forces in the Standard Model couple to matter and one another with different strengths, so they can be held to be distinct forces from one another. You cannot perform a U(1)xSU(2)xSU(3) operation that allows the different forces to interchange with one another. You have to combine the groups into some larger group with a compatible coupling or set of couplings so that any group operation will create a valid description of the same experiment. Right now this is not the case. In other words, I cannot perform a group operation and turn a gluon into a photon because their physical behavior is different.
But we know from the theory of renormalization group flow, which describes coupling constants, that these are not really constant. Rather, they are functions of what energy you are measuring an interaction at. As we measure experiments at higher and higher energies (read this as shorter and shorter wavelengths), we find that coupling constants change.
Renormalization group flow also describes a cutoff for each part of the Standard Model at which that theory is no longer valid. While we do not know exactly what these cutoffs are, in Grand Unification theory, we believe that certain theories combine at certain energies.
Hints of unification are apparent in mathematical models of how coupling constants for the various theories change:
At the points where the lines cross, the coupling constants are the same. Beyond those crossings, we believe that the linear extrapolation fails and the constants remain the same. First electromagnetism and weak theory combine to form electroweak theory. Then these combine with the strong force.
So the coupling constants might obey some sort of inward spiral:
This suggests that all the forces may combine at some point in the sense that they are no longer distinguishable from one another. What symmetry they would obey then is anybody’s guess.
If we could confirm any kind of GUT, that would be a phenomenal achievement, but also one that implies, almost by default, that there are many undiscovered particles waiting for us to find to account for all the broken symmetries.
Then again, there may be no GUT. After all, our universe may be one that has the laws it has because of the circumstances under which it began, much as the Earth has the climate and life it does because of how it began.
The universe may be part of a much larger one with external influences that we cannot even detect because they occurred “before” or during the Big Bang or are otherwise outside our particle horizon now. Without observing other universes, it will be difficult to understand what is possible. There may be universes out there that are so vastly different from ours that they point to laws of physics that we can’t even fathom.
While I think we have a ways to go before we hit any wall to our ability to understand and refine the laws of the universe, such a wall may exist such that our universe simply provides no evidence for the larger reality in which it exists. While that is a scary thought, it is also humbling and points to the need to find better data before we know what direction GUT needs to go.