During the past few centuries scientists have probed physical reality at ever shorter scales.
This incredible endeavor has given us a tower of scientific theories, where biology stands on
the science of chemistry, which in turn stands on atomic physics, which stands on nuclear physics,
which stands on particle physics. Particle physics is the science of
elementary particles, like the up and down quarks inside the protons and neutrons,
and the electrons orbiting them. These are the basic building blocks of the atom.
Thousands of scientists have dedicated their lives to uncover this incredible intellectual tower.
The question I want to know the answer to is how tall is this tower? Will this tower end somewhere?
Does a final theory, that ends this reductionist probing into ever shorter distances,
exist? Or will we find layer upon layer of theories without ever finding an endpoint?
Is this tower truly infinite, or will we be able, at some point, to conclude
that indeed our scientific voyage to understand the true nature of reality has a destination?
At least one theory attempts to show that indeed such a reductionist journey has an end point.
What is this theory? And what does it have to say
about how reality actually works? That’s coming up right now!
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We have very good reasons to believe that this tower of scientific theories must be finite.
In other words, a final theory must exist. To understand this let’s look at it from
the perspective of making a measurement. Any measurement must involve some kind of interaction.
We need a probe to observe an object. The probe can be a photon, an atom,
an electron or even a tennis ball. Whatever it is, we need to throw
something at an object so that we can get a signal back, which is then our observation.
Now, any probe will carry energy and that energy will cause spacetime to curve.
At low energies, this curvature of space and time is inconsequential,
but when we begin to observe very short distances then we need to use probes of higher and higher
energies. This is because as things get smaller, we have to use shorter wavelength objects like
high frequency photons or higher energy electrons in order to measure them because
larger wavelengths will not reflect any signal back. And this energy will begin to curve space
and time more and more. And at some point, if the object of our measurement gets small enough,
the high energy of our probe will cause such a large curvature that it will create a black hole.
And the key feature of a black hole is that no signal can ever escape it. This means that there
must exist a shortest distance beyond which it is impossible to perform a measurement.
That distance is the Planck length, because in order to measure something at this Planck length
scale, we have to put so much energy into this length that it creates a black hole.
This Planck energy of Planck mass equal to 1.2 X 10^19 GeV.
This is an enormous amount of energy in a very small volume. The Planck length is 1.6 x10^-35 m.
It is 10^20 times smaller than a proton, which is itself 100,000 times smaller than an atom.
This argument suggests that the tower of scientific theories probably has to end here. That
a final theory will exist, because if you cannot measure distances beyond a certain length scale,
then it makes no sense to have theories beyond that point. You would never be able to test them.
You may say, well, just because we can’t measure something, it doesn’t mean something doesn’t
exist beyond this scale. You may be right about that. But our theoretical models can only go
as far as the information accessible to us. In the last video about theories of everything
we discussed how science has already had a lot of success at unifying seemingly dissimilar theories
to a single greater theory. Two quick examples are the unification of the electric and magnetic force
into the electromagnetic force, and then later the unification of the electromagnetic force
with the weak force – the combination being called the Electroweak force.
We have thus seen many times how seemingly very different forces actually come from a
unified theory. So, this seems to suggest that we might be able to unify the rest of the theories
into a single theory of everything. To continue this reductionism,
we would want a theory that unifies all the forces. Today we have two overarching
theories for forces: Einstein’s Theory of General relativity for gravity, and The standard model
for the electromagnetic, weak and strong force An obvious problem is that the standard model
is a quantum field theory, but general relativity is a classical field theory.
Quantum theories are generally what we consider more fundamental because they explain
more phenomena than classical physics. In fact, many processes
only work if we use quantum theories – like our theory about the structure of the atom.
The problem is that gravity doesn’t seem to want to be treated at a quantum level.
We actually have no evidence that requires gravity to be quantum. Nevertheless,
many people are searching for theories of quantum gravity in order to find a theory of everything.
A theory of everything should not only unite forces, but also unite the theory for
all matter, in other words everything we see and experience around us.
The question is then what this theory of everything should look like?
We have talked about string theory in a few previous videos. This is the idea that all
matter and forces are fundamentally made of one dimensional strings vibrating in 9-10 dimensions.
The graviton emerges from its mathematics. But its mathematics is so flexible that this
theory could potentially describe a near endless number of universes, not just the one we live in.
So the problem is that it is not able to produce falsifiable statements. It is simply too flexible.
According to string theory the standard model of particle physics is merely a coincidence. It could
just as well have been completely different. We have also discussed Loop Quantum Gravity or
LQG which attempts to quantize space-time itself. But LQG only deals with gravity,
not the other particles or forces. There exists, however, a third newer
and lesser known candidate for a final theory which is called Quantum Holonomy Theory or QHT.
This was pioneered by two Danish scientists, physicist Jesper Grimstrup and mathematician
Johannes Aastrup. It begins by asking question, what would it take for a theory to be a final
theory. How can a theory be immune to further scientific reductions? How does reductionism end?
Although this seems almost impossible. The presumptive idea is that the final theory
must be based on something incredibly simple. The more structure there is at the macro scale,
and the deeper the theory needs to go, the simpler the building blocks need to be. Grimstrup and
Aastrup therefore believe that for a theory to be immune to further scientific reductions, that
is, for a theory to be final, it must be based on a principle that is extremely simple. What does that
actually mean? Well, in quantum holonomy theory, holonomy refers to geometry and curvature, it is
based on the mathematics of empty 3-dimensional space. Just Space, not even time, and no stuff,
no things within that space. Just empty space. Let us look at that. What is a fundamental
quality of space? Well, one thing is that you can move things around in space.
So the starting point of quantum holonomy theory is the mathematics of moving stuff around. And
again, the stuff itself does not matter here, it is simply the action of moving stuff around.
How can we describe the action of moving an arbitrary object from any point A
to any point B? Well, there are many ways of doing this, for instance,
you can rotate the object a bit to the right or you can rotate it forwards 45 degrees.
You can also flip it around some skewed axis. Or you rotate the object several times to the
left while you move it from point A to point B. There are clearly infinite many ways you can move
an arbitrary object between points in space. Let’s call any one of these combination of
movements from point A to point B, a recipe. A recipe for a combination of
movements in physics is called a gauge field. A gauge field is the recipe of how to move
one particle from point A to point B. Gauge fields are what makes up the forces
in the standard model. Since they are recipes of moving things around
in space, they actually represent how things interact with each other, or how forces work.
The sum of all mathematical recipes is called the “Configuration space”
of these recipes. So, the configuration space of ALL these recipes is all the different ways
you can move objects between points. This is infinite.
The key insight in quantum holonomy theory is that the geometry
of this space stores a lot of information. What do I mean when I say geometry of this space?
Basically, it means that two different recipes for moving stuff around can be
said have a relationship between each other. For example, if we have one recipe R that tells
us to rotate an arbitrary object slightly to the left when we move them, and we have
a second recipe S that tells us to rotate the same object just a tiny bit more to the left,
and finally we have a third recipe T that rotates the same object 2 times to the right,
does it make mathematical sense to say that R is closer to S than to T? In other words,
can we say that R, S and T are related, that there is a geometry between R, S and T?
This is a complicated mathematical problem that some scientists worked on for several years. What
they found is that yes, this enormous space of all possible recipes for moving things around
DOES have a geometry, meaning that the recipes are related.
But an even more interesting thing they found is that this geometry
results in mathematics that looks almost identical to the mathematics that’s almost identical to what
we already know from quantum field theory – this includes the mathematics of the Standard model.
But QHT mathematics gives us more than just the standard model, which is what you would expect
if this theory is to be a candidate for a theory of everything. Let’s look at the overall picture:
In QHT you start with an empty 3D space, and you consider within this space, an
infinite number of “recipes” which are basically combinations of movements. These are gauge fields.
The gauge fields represent the mathematics of how things interact with each other. This space
of recipes is infinite, but more importantly the recipes are mathematically related to each other.
They have a geometry. Now, here’s the keys
to the kingdom -- from this geometry you can obtain a what’s called a Bott-Dirac operator.
Don’t worry about what this is. But just know the square of this operator gives us the Hamiltonian
for both matter particles and force carrying particles. The Hamiltonian represents
the formula for all the energy in a system. This Hamiltonian can be written as the following:
D squared is the Bott-Dirac operator, H1 is the bosonic Hamiltonian, or in simpler words it tells
us how the energies of the forces evolve in time, H2 is the fermionic Hamiltonian which tells us
how the energy of the matter fields evolve in time. Omega represents some extra corrections
that’s relevant only at high energies, for example, close to the moment of the big bang.
Once you have a description of the energies of all the matter and forces in the universe,
that’s all you need to need to understand how matter interacts in the universe. And that’s what
you can get from the mathematics of QHT. This is essentially everything we would need to describe
the universe, once all the math is worked out. The remarkable aspect of this theory is that from
simply considering the movements of objects in empty space,
all this rich mathematics that appears to resemble the known mathematics of the universe comes out.
And if QHT is correct, it has some astonishing implications. For example,
QHT actually offers an explanation as to why the universe is quantum mechanical.
If you buy that the simplest way you can describe the universe is moving stuff around in 3D space,
then the math shows that this movement has to have a quantum description.
In other words, the math of the simplest description
of the universe dictates that the universe must be quantized. This is a central result of QHT.
QHT also predicts a few other surprising things, for example, Gravity is not quantized – that a
solution to the problem of reconciling gravity with quantum theory, does not
require a quantization of general relativity. Maybe the reason why physicists haven’t found
a theory of quantum gravity is because such a theory does not actually exist.
Perhaps gravity due to the curvature of spacetime — is simply not quantum.
In addition, no singularities exist. There can be no infinite density because QHT
gives an upper limit to how close you can pack matter together. This would be close to the
Planck volume. If there is a limit, then the mass density in any part of space cannot be infinite.
This also means that there are no singularities, that is no infinite curvature or hole in
space-time inside a black hole. Light still can’t escape because the curvature is too high,
but it is not an infinite curvature. And since singularities can’t exist,
the Big bang could not have come out of it. It must have come from some finite density universe.
It could not have come from nothing. It was more like a big bounce according to QHT. This is the
same prediction as in Loop quantum gravity. But don’t get too excited because QHT still would
not tell us what existed prior to the bounce, only that the current universe must have come from it.
So QHT seems to solve several major problems in physics. Unlike string theory, there is no need
for new hypothetical objects like strings, or 6-7 extra dimensions, or supersymmetric particles
that we have never been found. We only need the mathematics of moving stuff in 3D space.
Now, I want warn you that I simplified this greatly. Although the concept is super simple,
the mathematics is super complex. It’s non-commutative geometry.
And very few physicists have expertise in it, so it remains relatively obscure.
I’d like to give a big shout-out to physicist Jesper Grimstrup
for helping me understand this fascinating theory. If you want to learn more about it, check out his
brand new book on Amazon. The link is in the description. It’s written in simple language,
so I think you’re going to enjoy it. And if you have a question, post it in
the comments, and I will try to answer it. I will see you in the next video my friend!