How many times can I half the distance between my hands? Assuming perfect coordination and the ability
to localize my palms to the quantum level. 15 halvings gets them to within a cell’s width.
33 to within a single atom, 50 and they’re a proton’s width apart. Half the distance 115
times and they’re a single Planck-length apart - 1.6x10^-35 meters. Surely we can keep going - .8,
.4, .2 x10^-35 m? Bizarrely, those distances might not even exist in any meaningful way.
There’s no limit to the number of times you can half a number,
but the same might not be true of space. The planck length is thought to represent the
minimum length for which the concept of length is even meaningful. Here,
the illusion that space is smooth and continuous breaks down. But what happens when you go smaller?
Does space break into discrete chunks? Does space even exist as we know it? Before we zoom in
on space, let’s zoom back in time to see where the idea of the Planck length even came from.
In the final year of the 19th century, Max Planck ushered in the quantum age by
thinking about hot pokers. He found the long-sought mathematical description of
blackbody, or thermal radiation - by requiring that the energy of light in this heat-glow was
not infinitely divisible. Rather came in quanta - chunks of energy that we now call photons.
Planck’s discovery hinges on a single number that appears in his equation - the
Planck constant. It represents the chunkiness of thermal radiation.
Multiply the frequency of the light by this number and you get the energy of a single photon.
Max Planck had introduced the whole quantized light thing as a mathematical trick. He expected
the value of the Planck constant to be zero, which would mean that energy could be infinitely
divided. But the constant remained stubbornly non-zero, even if it’s very, very small. That
simple fact revealed the fundamental blockiness of the subatomic world. We now see it everywhere
in quantum mechanics - it’s a fundamental constant of nature that defines the scale of the quantum.
From the Planck constant comes the Planck length.
It’s the length you get when you combine the gravitational constant, the speed of light,
and Planck’s constant in just the right way to give units of length - the square root
of G times h-bar over c^3. Where h-bar is just the Planck constant divided by 2 pi.
As you now know, the number comes out pretty small - around 10^-35 m. But why is this random-seeming
combination of constants so important? Well, it represents the scale at which space itself
is thought to “become quantum”. I say “thought” because we’ve never been able
to conduct experiments at that tiny scale. So to understand why physicists believe this length is
so important, we’ll have to do some experiments in our brains. Follow me through a couple of thought
experiments - or gedankenexperiments as a good German physicist like Max Planck would call them.
To start, let’s say you’re trying to measure the distance to an object. You shine a laser beam at
it, which reflects back to a detector that records the light travel time, which also gives you the
distance because you know the speed of light. But that distance measure has an uncertainty because
you can only clock the instant of the return to within one wave-cycle of the electromagnetic wave.
That gives a distance uncertainty of around one wavelength of whatever type of light you’re
using. OK, easy enough - just use a very short wavelength to get a very precise distance measure.
But now you have a new problem. Light carries energy and momentum - and the shorter the
wavelength, the more it carries. If you bombard your object with a powerful short-wavelength X-ray
laser and you may get an amazing distance measure, but you’ll transfer a lot of momentum. You can try
turning down the power of your beam, but remember that the minimum intensity is when you have only a
single photon. And because you can’t know exactly how the momentum transfer will happen,
you’ll always have an uncertainty in the particle’s final momentum that’s roughly equal
to the momentum of the single photon. You may see where this is going - we’re about to discover
the Heisenberg uncertainty principle. Bare with me though with just a touch of mathematics.
A photon’s momentum is the Planck constant divided by its wavelength.
So just replace photon momentum with the uncertainty momentum of the measured object,
and replace wavelength with uncertainty in its position. Rearrange and voila,
Heisenberg. Or close enough - A proper derivation gets you a factor of 1 over 4 pi.
This was the line of reasoning that Heisenberg himself went through, and we call this thought
experiment the Heisenberg microscope. We now know that the uncertainty principle is far more
fundamental than just the effect of disturbing the system that you’re trying to measure - and we’ve
explored that previously. It also applies to other pairs of variables besides position and momentum.
The point I want to focus on today is that through the uncertainty principle we see that the Planck
constant represents the limit to which we can measure the universe. We recently talked about
how with a bit of clever physicsing it’s possible to stretch the uncertainty principle to the limit.
For example, we should be able to measure location in space down to any conceivable precision,
perhaps even infinite precision - as long as we’re happy to have infinite uncertainty in
momentum. But it turns out that the Planck constant defines a new source of quantum
uncertainty that you can’t ever physics away. And we hit that uncertainty at the Planck length.
Let’s continue our gerdankenexperiment, now adding two key ideas from Einstein:
first that mass and energy are equivalent, as expressed by the most famous equation ever, E=mc2;
and second, that mass and energy warp the fabric of spacetime. So back to Heisenberg’s microscope.
Let’s say we’re trying to measure our distance with perfect precision, and who cares about
momentum. We keep decreasing the wavelength of our measuring photon - ultraviolet - X-ray -
gamma-ray - which also increases the photon’s energy and momentum. As we crank up the energy
even further we start to notice something. The photon is starting to produce an observable
gravitational field. Even though photons are massless, if enclosed in a system a photon
creates what we call effective mass, according to Einstein’s famous equation. The resulting
gravitational field changes the distance to the object, adding a new uncertainty to the distance.
Okay, one last dip into the math to see how big this uncertainty is. Space is stretched
by a factor equal to the effective mass times the gravitational constant divided by c^2.
Let’s replace the mass with the effective mass of our photon - its energy over c^2,
and the energy of a photon is Planck’s constant times c^2 over the wavelength.
We have this thing that’s full of our wonderful fundamental constants - in fact,
in the exact form as the Planck length squared divided by the wavelength.
And this is where, I hope, it gets interesting. As you pump up the energy of your photon,
reducing its wavelength also reduces the regular Heisenberg uncertainty, but at the same time this
new uncertainty increases. You’re still winning the uncertainty game - up to a point. When the
wavelength of the photon reaches exactly the Planck length these two uncertainty terms become
the same, and any further decrease in photon wavelength actually increases your measurement
uncertainty due to the warping of space. So this is one way of thinking about it - the
Planck length represents the best possible resolution that any distance can be measured.
It also represents the minimum size that you can meaningfully ascribe to anything.
Imagine now that you’re trying to measure the distance across a one-Planck-length
object. You need a photon with a wavelength smaller than one-Planck-length.
But that photon has enough effective mass to produce a black hole with a Planck-length event
horizon - so any attempt to measure something that small swallows it in a black hole.
Another way to think about that is that if you try to measure a size smaller than the Planck length,
the warped geometry changes that size to give 100% uncertainty. This explains why the Planck length
is this combination of fundamental constants - it’s the wavelength of a photon that creates a
black hole of the same size, and so represents the fundamental limit of the measurability of space.
To convince you that this really is fundamental,
let me give you one more example that doesn’t involve light. Imagine we want to precisely
define the position of an elementary particle - let’s say an electron. To define its position
we need to be able to say that all of the electron’s mass is within a certain volume.
So defining position means shrinking that volume down as much as possible.
But the Heisenberg uncertainty principle says that when we do that we increase the uncertainty
in momentum of that bundle of electron-ness. That also means increasing the uncertainty
in energy density in that volume. By the time we localize the electron to within one Planck length,
the uncertainty in energy is about equal to the entire mass-energy of the electron. At that point,
you can’t say there’s just one electron. The uncertainty in the energy induces a process called
pair production. The act of localiuzing the electron causes pairs of virtual electrons
and positrons to appear around it. The positron will annihilate with either the new electron
or the original. This happens continuously, so it looks like the electron flits around in position,
thwarting any attempt to fix its location. For any particle, this uncertainty due to
this pair production occurs when you localize its energy to within a Planck-length-diameter volume.
So the universe seems to be conspiring to stop us measuring distances or sizes smaller than the
Planck length. But now we get to the big question: does that mean that smaller sizes/lengths/chunks
of space don’t exist? Does that mean space is not continuous on the tiniest scales?
That might be true, but everything I’ve said today doesn’t quite get us that far. What we can
say is that distances are undefined on the Planck scale. At that scale,
Heisenberg’s principle tells us that the curvature of space is fundamentally uncertain.
In the same way that you get virtual particles on subatomic scales, on the Planck scale you
get virtual spacetime fluctuations, and even virtual black holes and wormholes - a
fluctuating roil of spacetime that John Archibald Wheeler called the spacetime foam. At least,
that’s the case if you try to define a continuous fabric of space beyond that Planck Length.
Now we don’t really know how well this picture represents space on this scale. The nature of
space is defined by Einstein’s general relativity, and GR breaks down at the Planck scale because
we can’t sensibly define distances. We think that space AND time - spacetime - “go quantum” at that
scale - but we just don’t know in what way. We need a theory of quantum gravity to answer this.
We’ll come back to what the contending theories say about the true nature of space another time.
So is there a smallest length? Well there’s a smallest meaningful length, at least for any
intuitive conception of space. Quantum uncertainty thwarts our attempt to understand the universe
by simply splitting it into smaller parts. You’ll have to stay tuned to the show, and to the future
of physics, to find out what might lie beneath the smallest possible scale of measurable spacetime.