### The Diversity of Grand Unified Theories

The grand unified theories of everything in physics are supposed to explain, well, everything. They are just one step shy of theories of Life, the Universe, and Everything.

So how are grand unified theories of everything doing these days? This link from Quanta Magazine gives a beautiful interactive map of the situation:

https://www.quantamagazine.org/20150803-physics-theories-map/

The diversity is amazing. If you were hoping for an answer with the brevity and simplicity of 42, however, you will be sorely disappointed.

### Lessons from Alchemy

Historically, if a discipline suffers from an overabundance of theories it is likely that something is amiss with the overall approach, not just with the individual theories.

Alchemy is instructive. Alchemy is what chemistry was before a unifying theory of chemical elements and their properties was finally uncovered. After that, the fascinating diversity and strangeness of the mostly directionless field of alchemy disappeared.

Alchemy’s single largest impediment against progress was not a lack of experimentation, but a stubborn bias towards an idea that simply did not work: The belief that there were only four or five elements. To the alchemists, the idea that the worlds had, say, about a hundred elements was offensive in a way that is surprisingly familiar to modern science: They just could not believe that the deeper fabric of the universe was that large and ayrbitrary. Consequently, the four or five elements that had been proposed centuries ago always seemed like a good launching point for almost every new experiment or attempt at exploration.

The great irony of alchemy is that the alchemists who believed in simple fundamentals were essentially correct in their assumptions, but had failed miserably in practice because they were looking at wrong level of how the universe is constructed.

All of the compounds at which they were looking were are in fact composed of just three primary elements, those being protons, neutrons, and electrons. The alchemists, alas, did not know that those three elements operated at levels and energies so far beyond their reach that they would not be identified unambiguously until centuries later.

### Grand Unification Theories as Alchemy

Sadly, the diversity and lack of convergence of grand unification theories in physics suggests that it is at present an endeavor that is more closely aligned with alchemy than chemistry.

### The Problem: Particle Parochialism

So if there is some kind of bias or misconception that is keeping particle physics from converging to a single well-defined path and set of grand unification concepts, what might it be?

Here is a simple suggestion: Particle parochialism, by which I mean the tendency to isolate and separate the problem of why particles exist from the medium in which they exist, that is, from space and time.

The converse approach is to assume that despite their seeming simplicity in comparison to the wild zoo of particles that we have uncovered, space and time as we know them are at some deeper level just as structured and arbitrary as the zoo of particles we find within them. More importantly, the deeper structures of space and time are structured in ways that require the existence of those particles, and vice-versa. That is, to emerge from its current alchemy-like status grand unification will need to look for some deeper and odder set of fundamental ideas that link particles, space, and time (p+s+t) into a single theory from which properties such as distance, direction, rotation, charge, particle spin, quantization, quantum uncertainty, and classical certainty all emerge on an equal basis.

### Why Symmetry Theories Keep Failing

Like fish who accept water as a given, physicists tend to accept certain concepts as givens that require no further breakdown or explanation, such as distance (space), time (change), charge, spin, and the rules of quantum mechanics. They then try to combine some subset of these givens, usually the more particle-focused ones such as charge and spin, into a broader structure that combines them in diverse ways to create our universe. Those broader structures are called symmetry theories.

It won’t work. The problem is not symmetry theory *per se*, but that fact that the pieces that physicists and mathematicians are feeding into it are too complex and need to broken down into more fundamental and likely stranger properties. What’s being done now it like trying to build a Ford Fiesta from GM and Nissan motors and frames. With enough creativity and hacking the result may sort-of work and sort-of look like a Fiesta. But it will always end up hokey and not very satisfying.

Such is the current state of symmetry theories in physics. Just as someone trying to understand how to build a Ford Fiesta must abandon prefab motors and frames in favor of more primitive alloys, electronics, and plastics to create an actual Fiesta, someone seeking to explain the particles and fields we see must go farther down into the fabric of the universe, and there seek out ideas that precede and contribute as much to space and time as they do to particles within space and time.

### The Siren’s Lure of Mathematical Infinities

So how carefully do physicists and mathematicians presently pick apart the properties of space, for example? Do they respect limits known for real space as also implying limits in their abstractions of that space, many of which are extend deeply into the structure of mathematics?

Most physicists accept the concept of a point particle, but there is flatly no evidence from the physical world that true point-like objects exist in space. Quite the contrary: Quantum physics requires that no true point-sized objects exist, since such a point would require infinite energy to overcome quantum space-momentum uncertainty.

So what if the only way to create a fully self-consistent theory for describing particles+space+time is to apply that same principle to mathematics? That is, what if ideal points in mathematics are no more real than they are in physics, and so cannot be used to describe the real universe without creating paradoxes? That would mean that even the simple number line we were taught in elementary school is just a classical approximation that only works if you don’t push it too far.

That happens to be true. A ruler contains an infinite number of irrational points. Attempting to express just one of those points as a literal real number, one that a computer could process for example, would require a storage device capable of storing an infinite number of digits. Such a storage device cannot be constructed, since it would require infinitely more mass and space than exists in the universe.

Rotation is another remarkable property of space, one that is extremely complex when expressed as executable mathematics. Yet in both relativity and particle physics, it is accepted as “given” in trying to explain how physics works.

### It’s Time to Dive Deeper

The bottom line is that when it comes to space, we are fish, and space is the water we swim in every day. That makes us accept its properties so deeply that we don’t realize how complex, weird, oddly specific, and non-intuitive they really are.

So when we try to create a grand theory of particles, we can’t help leaning towards treating it mainly as an issue of why particles exist. But the deeper problem is this: Why do particles+space+time exist? How do they help create each other? Which of the deepest properties of those three are deeply interconnected at some level we are currently missing? In that approach, no part of the particles+space+time triad would be isolated from the other two, or any more fundamental.

### The Trickiest Part: Creating Boundary-Aware Mathematics

Trickiest of all, mathematics cannot keep assuming non-existent ideas if it is to describe such deeper relationships correctly. Thus p+s+t physics would no longer be able to use Dirac deltas, which are fundamental to much of quantum physics math, without simultaneously recognizing that every such invocation in effect blows up the universe by creating a point of infinite energy. At present, mathematical physics gets an almost universal free pass on such highly convenient usages. That will need to change, since otherwise the infinities get shifted off into the math and enable statements that are non-physical, such as point-like particles.

What is needed instead are boundary-aware expansions of mathematics that incorporate the mathematical equivalent of quantum uncertainty deeply into their fabric and formulas. How this would work is unclear, since it would require a rethinking of the entire concept of cost-indifferent use of infinitesimals, replacing it with a formulation where uncertainty is the norm and specificity emerges only after the application of sufficient resources. Computer science, with its very much real-world representations of otherwise idealized number concepts likely would be an instructive starting point.

An indication of progress would be formulations in which the uncertainty of quantum mechanics becomes very natural and, quite likely, much more computationally efficient. The latter property would emerge due to the omission of the false levels of precision that approximations of infinitely precise numbers imply when performing such calculations.

### Hints and Allegations

But what about the physics itself? It’s easy to assert that there exist deeper “links” between space and fermions, for example, but what would the nature of those links even be?

The lesson from chemistry is that unexpected patterns are sometimes important… and sometimes not. Mendeleev, using little more than cards with notes and a lot of persistence, managed to ferret out patterns that in time would prove deeply related to the later worlds of both nuclear physics and quantum mechanics. He could not even begin to explain why eight was such an important number in the patterns he saw, but it didn’t really matter to him. His objective was simply to find those patterns, whatever they might be. A similar logic applied over a century later with recognition of the eight-fold way in particle physics.

The data for linking such disparate concepts as space and particles is far sparser, and so is also much more treacherous to interpret. One simple example that occasionally makes its way into peer-reviewed physics papers is the curious coincidence of 3 spatial dimensions and the SU(3) gauge group that defines the color force. A link between the color force and the number of spatial dimensions would not necessarily be direct, however. In fact, as with the examples from chemistry, the occurrence of the number 3 in both of these cases would be more likely an emergence from an underlying common source that at present is not even recognized as existing, let alone specified in detail.

Quantum theory is rich with opportunities deeper exploration, especially since issues such as boundaries between quantum and classical physics remain so oddly ill-defined. Quantum uncertainty in particular forces a deeper relationship between particles and space, since it is this uncertainty that makes the concept of a point particle in real space into a volume limit that can never be reached in practice. If the lure of infinitesimals is discarded, this seemingly minor aspect of uncertainty is transformed into something deeper and a good deal stranger, a problem of how to define what a particle even is if it is not a point hidden away by uncertainty.

### The Challenge

If particles, space, and time are in fact far more deeply and fundamentally tied into each other in ways we do not yet understand, the implication is that attempts too unify all of physics through particle-focused symmetry theories will necessarily fail. Like cars made from motors and frames from the wrong sources, many such failures will occur in odd, “almost right” ways that are close enough to seem promising.

But the very breadth and diversity of such theories argues that such promises are simply mirages in a desert of misdirection. It is more likely that Grand Unification needs to dive deeper… much, much deeper. Such dives are likely to prove very uncomfortable, since the infinity-studded mathematical tools used now are likely to fail or, worse, misdirection when applied to a world . It is likely that real and continuum forms of mathematics will require serious reexamination, extension, and even outright repair. The resulting extended formalisns will provide both the improved vision and structural support needed to explore those hidden depths safely.