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.