Fields don’t exist.
I mean, a field with grass in it, that kind of field does exist.
But a field in physics?
A gravitational field? An electric field? A magnetic field? A quantum field?
No such thing.
I’m not knocking the physicists who came up with these fields.
These fictions can be convenient.
But sometimes, these fictions can blind us to the underlying reality.
And that’s what’s happening right now in physics.
Our long-time love affair with fields is blinding us to the true nature of space and everything in it.
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Let’s start by getting straight on what a field is.
A field is a number at every point in space.
That’s it.
I mean, it doesn’t have to be a single number at every point in space.
Technically, it can be:
- a scalar, represented by a single number;
- or a vector, represented by three numbers in three-dimensional space;
- or a matrix, represented by nine numbers in three-dimensional space;
- or a tensor, represented by 27 numbers, or 81 numbers, or 243 numbers, or...
OK, I’ll stop there.
All these different tensors may be daunting if you’re not mathematically minded, but they don’t change the fundamentals.
A field is a number – or a set of numbers – at every point in space.
These numbers can change over time, because, well, the universe isn’t static, it evolves.
And really, that’s it.
That’s what a field is: a number – or a set of numbers – at every point in space, that can change over time.
Phantom fields
Now you may have noticed that when you look at space, you don’t see a little number inscribed at every point, no matter how closely you look.
In a sense, fields exist only in our imagination.
We imagine that we can measure a number – a temperature, a force, a probability – at every point in space.
Then we imagine that there’s a continuous field – of temperature, of force, of probability – stretching out across space.
In fact, as we’ll see, we can’t measure any of those quantities at every point in space.
A field – a gravitational field, an electric field, a magnetic field, a quantum field – is more an expression of a particular theory of physics than of our ability to actually measure a particular quantity at every point in space.
And that’s fine. A theory of physics is a model of reality, rather than reality itself.
So sure, in a sense, fields exist only in our imagination.
But there’s a much deeper sense in which fields don’t exist.
Too hot to handle
Let’s start with temperature.
We can certainly come up with an approximate number for the average temperature of the air in a cabin.
It might be 25°C, according to a thermometer on the wall.
If we move that thermometer around the cabin, we might even be able to come up with different numbers for the temperature of the air in different parts of the cabin.
It might be 40°C around the wood stove, 30°C at the peak of the ceiling, but only 10°C on the floor near the corners.
We can certainly imagine a number of degrees at every point in the cabin, in other words, a temperature field.
But as every anthropologist knows, you can’t observe a system without disturbing it. As we move the thermometer around the cabin, we inevitable disturb the air, mixing the hot air around the wood stove with the cold air on the floor near the corners, changing the temperature field we’re trying to measure.
And that’s not the worst of it.
We’re large-scale creatures. What matters to us is that it’s warmer near the wood stove than on the floor near the corners, a few metres away. We’re not so concerned with what it might mean for the temperature to be different a few nanometres away.
If we were, we’d find that the concept of temperature simply breaks down at such a small scale.
It’s not that it’s hard to measure the temperature of the air in the cabin at scales of few nanometres.
It’s that there’s no such thing as temperature at these scales.
Temperature is a large-scale measure of the aggregate motion of molecules.
At scales of a few nanometres, there’s no such thing as the temperature of the air in the cabin, there are only individual molecules of nitrogen, oxygen and water, each moving at its own speed, faster, on average, near the wood stove, slower, on average, on the floor near the corners.
I’m not just being pedantic here.
Let’s go smaller still, to scales of a few picometres, or a few femtometres, or a few attometres. Actually, let’s go to scale of a few zeptometres. At such a small scale, if you pick a random point in the cabin, not only is there no such thing as temperature, chances are, there’s not even a molecule at this point.
How can we hold on to a meaningful concept of the aggregate motion of molecules at every point in space when, at most points in space, there are no molecules?
I’m not saying that the concept of a temperature field isn’t helpful. It is helpful, if you want to warm your toes by the wood stove.
I’m just saying that the concept of a temperature field isn’t fundamental.
There’s a deeper level of reality, to which the temperature field is a mere approximation, a coarse-graining, a macroscopic model of what’s really going on at a microscopic scale.
The danger is that the concept of a temperature field, varying continuously over space, might blind us to the underlying reality of discrete molecules.
Which way is down?
Let’s leave the temperature field. After all, no one – except me – talks about temperature fields; everyone knows that temperature is just an aggregation of the motion of individual molecules.
Let’s move on to a field that was popular in physics for centuries: Newton’s gravitational field.
Here, the idea is that there’s a number at every point in space that represents the gravitational force at that point.
Actually, in this case, it’s a vector – three numbers in three-dimensional space – representing the magnitude and direction of the gravitational force.
Take this point in space, for example, where I’m standing right now. The gravitational field is 9.81 Newtons per kilogram towards the centre of the Earth. That means that if I let go of this one-kilogram stone, a gravitational force of 9.81 Newtons will act to accelerate it towards the centre of the Earth. In other words, if I let go of the stone, it falls down, accelerating at 9.81 metres per second squared.
That’s at this point in space.
If I go 1,000 miles up, the gravitational field decreases to 6.25 Newtons per kilogram, again, towards the centre of the Earth, near enough.
If I go to the surface of the Sun, the gravitational field increases to 274 Newtons per kilogram, towards the centre of the Sun.
This concept of a gravitational field is powerful and precise: powerful and precise enough to land a spacecraft on the moon.
But the concept of a gravitational field masks a more powerful, more precise underlying reality.
Einstein reframed gravity. Instead of a gravitational field – a gravitational force at every point in space – he imagined gravity as a consequence of the curvature of space-time.
It’s not that this stone, when I let it go, is acted on by a mysterious force towards the centre of the Earth. It’s that the Earth distorts the space around it in such a way that this stone accelerates towards the centre of the Earth simply by following the shortest possible path through space-time.
In Einstein’s conception, there’s no such thing as a gravitational force, no such thing as a gravitational field. There’s just matter – particles, stones, planets – distorting space-time and following the shortest possible path through space-time.
General Relativity is not only a more beautiful vision of gravity, it’s also more powerful and more precise. It explains astronomical anomolies, from the precession of the perihelion of the planet Mercury to the existence of black holes.
Again, I’m not saying that the concept of a gravitational field isn’t helpful. It is helpful, if you want to land a spacecraft on the moon.
I’m just saying that the concept of a gravitational field isn’t fundamental.
There’s a deeper level of reality, to which the gravitational field is a mere approximation.
Again, the danger is that the concept of a gravitational field might blind us to the underlying reality of space-time curvature.
Maybe...
None of this is controversial.
So let’s get controversial.
Schrödinger’s equation describes a probability field.
What’s a probability field? Well, it’s a number at every point in space representing the probability that, say, there’s an electron there.
If that sounds slightly abstract – actually, let’s be honest – if that sounds absurdly abstract to you, well, it does to me too.
With a temperature field, we can’t actually measure a temperature at every point in space, because moving the thermometer around changes the temperature at each point in space, and because, at smaller scales, there’s no such thing as temperature.
With a probability field, it’s way worse.
We can’t actually measure a probability at every point in space because we can’t actually measure probability.
There’s no such thing as a probabilitometer.
Sure, we can take a look at a particular point in space and see whether there’s an electron there, but that gives us a binary answer, not a probability. When we make such an observation, we never find a 33% probability of an electron or a 75% probability of an electron. Either there’s an electron there, or there’s not.
It gets even worse.
When we move a thermometer around, we disturb the air whose temperature we’re trying to measure, in the straightforward sense that we’re mixing slow-moving molecules with fast-moving molecules.
When we observe whether or not there’s an electron at a particular point in space, we’re certainly going to disturb that electron.
More than that, according to Heisenberg’s uncertainty principle, we’re going to render certain attributes of the electron, such as its momentum, unknowable.
More than that, according to the Copenhagen interpretation of quantum mechanics, we’re going to collapse the wavefunction, destroying the probability field for the electron.
This is simply weird.
No physicist loves this idea: a probability field that defines a probability we can’t measure and disappears the moment we take look at it.
But almost every physicist apologizes for this idea. They try not to talk about it too much. They try not to think about it too much. Regardless, all of physics, at the smallest scales, is founded on Quantum Field Theory, which yields probabilities, in much the same way as Schrödinger’s equation.
So you might be forgiven for wondering...
...if the temperature field masks the underlying reality of discrete molecules...
...and the gravitational field masks the underlying reality of space-time curvature...
...then maybe this probability field, too, is blinding us to some underlying reality?
Convenient fictions
Actually, there may an even deeper sense in which fields don’t exist.
I’ve said that a field is a number at every point in space.
But what does that mean?
If space is continuous, as physicists have assumed for centuries, a number at every point in space means an infinity of numbers across the universe.
Indeed, it means an infinity of numbers in every cubic metre of space, an infinity of numbers in every cubic nanometre of space, an infinity of numbers in every cubic attometre of space.
We’re so used to thinking of space as continuous that the infinities inherent in fields don’t bother us. After all, we can come up with a continuous equation to model this infinite number of numbers, right?
But maybe the universe isn’t so profligate with its infinities.
If space is discrete, as physicists have been speculating for the last century or so, then there’s not an infinite number of points in space, there’s a finite number.
There’s no such thing as a field that varies smoothly over space because space itself isn’t smooth.
Indeed, if space – and everything in it – is a discrete hypergraph, as in the Wolfram model, then not only is there a finite number of points in space at which to define a field, there’s no need for fields at all.
Let me explain.
Take the gravitational field.
According to Newton’s laws of motion and gravitation, we can define a gravitational field at every point in continuous, three-dimensional space. These things exist: space, time, matter and a gravitational field at every point in space that determines how matter accelarates through space over time.
According to Einstein’s general theory of relativity, we can define a curvature at every point in continuous, four-dimensional space-time. These things exist: space-time, matter and a curvature at every point in space-time that determines the shortest path for matter through space-time.
According to the Wolfram model, there’s no need to define a gravitational field, or a space-time curvature, or any other number at every point in space or time. All that exists is the hypergraph.
Sure, matter is tangling of the nodes and edges of the hypergraph.
Sure, that matter, that tangling of the nodes and edges of the hypergraph, influences the motion of other matter, other tangling of the nodes and edges of the hypergraph, in a way that’s consistent with Einstein’s theories and Newton’s laws.
And sure, we can, if we like, define a number at every node of the hypergraph that’s equivalent to Einstein’s space-time curvature or Newton’s gravitational field.
But those numbers – those fields – don’t really exist.
They’re convenient fictions, aggregate measures of the sub-Planck-scale evolution of nodes and edges, inventions of large-scale creatures like ourselves to make sense of super-Planck-scale space, time and motion.
All that really exists is the hypergraph.
Fields don’t exist
Fields don’t exist.
They’re figments of physicists’ fertile imaginations.
That’s not to say that they’re not helpful. They are. They’re a pretty powerful and pretty precise way of envisioning phenomena that can be represented by a number – or a set of numbers – at every point in space.
The danger is that these fictions can blind us to the underlying reality.
And that’s what’s happening right now in physics.
Our long-time love affair with fields is blinding us to the true nature of space and everything in it.
It’s blinding us to the hypergraph.