Thank you to novium for supporting PBS. What does an electron really
look like? I mean if we zoom in all the way is it a sizeless speck of charge? Or
is it a multi-dimensional vortex of quantum strangeness? Is it the
boundary of a tiny universe with universe electrons of its own? Let's find out.
There isn't much that's simpler than the it's one of the elementary particles something not
made of smaller things at least according to the current theory. It conjures to mind a single point
or perhaps a tiny ball made of some electric charge and mass and not much else - like in
the cartoon description of the atom with electrons as orbiting balls. But if you've paid attention to
good pop-sci media you probably know that these pictures aren't accurate. Perhaps you know that
in quantum mechanics particles exist not as tiny nuggets of matter but rather as wave functions
fuzzy distributions that define the probability of where or what the particle might be when measured.
For example electrons in atoms are more accurately depicted as blurred shells than as orbiting balls
and a free electron might look like an undulating wave packet capable of interfering with itself and
doing other wave-like things. But are these wave functions really what an electron looks
like? These wavy structures seem more intuitive for something like the photon which is literally
a wave in the electromagnetic field and whose most important properties like frequency and wavelength
can be captured in wavy terms. But an electron has mass, it has electric charge, it has spin.
Where do these properties live in a fuzzy wave function? There are valiant efforts to represent
the non-spatial components of the electron wave function in particular the spin or wave function
which we talk about here but these are attempts to visualize a pretty abstract concept not really
what the electron looks like. So let's take a look. Let's really zoom in. We can never actually
see a fuzzy wave packet because the electron takes on precise properties when we measure it. Looking
at the electron means resolving its location and spatial structure as we measure more precisely the
wave function shrinks to something particle like. But how far can we zoom? Do we eventually see a
tiny ball or an infintesmal speck of charge and mass? And where does that charge and mass live?
What does an electron really look like up close? There's an agenda behind this thought experiment.
The very act of zooming in to see what an electron looks like will reveal some vexing contradictions
for example about why the electron has the properties that it has. These contradictions
foreshadow one of the biggest problems in physics today but that's for future episodes.
Today we're just looking at the electron. Let's try constructing an electron first using classical
physics. We'll do that by gathering enough electric charge from the surrounding universe
and sticking it together. Like charge repels via the Coulomb force and Coulomb's law tells us that the
closer we want to drag two bits of charge the more force it takes. That means it takes more energy
to assemble a more compact ball of charge than it does a more diffused one assuming the same amount
of charge. A bit of calculus gives us the energy needed to assemble a ball of any given radius and
this is the potential energy that that ball holds after assembled the smaller the ball the more
energy from Einstein's relativity we know that bound energy is equivalent to mass e =mc^2. So now
we can ask the question if we drag in one electron worth of electric charge until it's in a small
enough ball that its potential energy is equal to the known mass of the electron how big would
that ball be well around 2.8 by 10 ^ -5 M. This is the classical electron radius it's the size
an electron would have to be if its mass was all from the potential energy of it contain charge.
But it's not the real size of the electron. Our experiments tell us that the true size of
the electron is smaller than 10 ^ -7 M at least 100 times smaller than the classical electron
radius. But if we continue to crush our ball of charge down to that size the mass equivalence of
that potential energy is 10G electron volts which is 20,000 times larger than the actual measured
electron mass of 511 kilo electron volts and if our electron really is point like, having no size
then Coulumb's law gets a zero in the denominator and the energy summed all the way to that central
point is infinite. Now the electron isn't really a ball of electric charge mutually repelling itself.
Its charge does self- interact in complex ways manifested in the activity of quantum
field surrounding electron as we'll see but the equations and numbers I came up with do still give
the energy contained in the electromagnetic field that emanates from a bundle of charge of the same
size. And so this energy does still contribute to the electron observed mass. This issue of the
electron's electric field containing vastly more mass than the electron itself is or at least was
a real problem and one that persists even when we switch from classical to quantum mechanics.
But for now we get back to the electron what does our zoom in look like when we switch to
quantum mechanics or more precisely quantum field theory? In qft we describe electromagnetism in
terms of the activity of a quantum field and each elementary particle is an excitation in its own
quantum field. So the photon is an excitation of the EM field and the electron and positron are
excitations of the electron field. Quantum fields are sometimes described as these roiling oceans of
virtual particles appearing and vanishing. This isn't very accurate. Instead think of a quantum
field as having the potential for complex and random fluctuations when it encounters a
real particle whose fluctuations can be modeled using virtual particles. As a mathematical tool
we can model the coulomb force in terms of virtual photons in the EM field imparting momentum between
charged particles or we can model the spontaneous transfer of energy between the em and electron
fields in terms of a virtual photon turning into virtual matter anti pairs pair production. In this
case a virtual electron and positron. So if the EM field in quantum field theory is an ocean of
virtual photons at least mathematically. What does that look like? Well viewed from afar the activity
of these virtual photons is averaged over in into the smooth field described by classical
electromagnetism but as our zoom in reaches a certain point around 100 times larger than
the classical electron radius we transition to a regime where we can't ignore the quantum behavior
of the field. For example as the field strengthens and our view narrows we see that the interaction
between the em and electron fields strengthens. We can describe this as an increase in the rate
of pair production an increasingly active flickering of virtual photons and the virtual
electron positron pairs that they spawn this cloud of activity appears to grow stronger the closer we
get to the electron. It's not just that we're seeing the quantum properties of the stronger
em field - the very act of probing the electron this closely amplifies quantum fluctuations. The
Heisenberg uncertainty principle tells us that precision in spatial or temporal measurement
leads to larger uncertainty in momentum and energy respectively the more precise our snapshot of that
electron the more energetic quantum fluctuations we'll observe. We talk about this effect in more
detail in this video but the upshot is that the universe looks very different depending
on the scale that it's probed and so does the electron although somehow all scales knit together
into a unified whole. We are now at very small scales quite a bit smaller than the classical
radius here. It's clear that the bare electron appears to be dressed in this cloud of virtual
activity this dressing is the source of our self energy and the extra mass from the electron's EM
field. As we zoom ever closer to the electron the energy in the field approaches the rest
mass of the electron and a new interaction becomes important. Now the virtual electron positron pairs
have a decent likelihood of directly interacting with the central real electron. Sometimes a
virtual positron will actually annihilate with the real electron which then promotes its once
virtual partner electron to reality, it's as though the original electron shifted position.
There are two important implications to this. Effect one is that this effect actually recueses
the electron from having the colossal or even infinite mass predicted by classical physics.
More on that next time. Unfortunately the second implication of this self annihilation interaction
is that we're never actually going to get a clear view of the electron no matter how far we zoom in.
The closer we zoom the stronger the surrounding cloud of virtual fluctuations but also the more
frequent these interactions become. At some point the electron become this flickering blur that we
could never quite fix the location of the mass ends up smeared out both by the self energy mass
and by what is ultimately Heisenberg uncertainty. The electron's charge is also smeared out by the
latter and also smeared by something that I haven't told you about yet. Even before we got
close to the central electron virtual electron positron pairs we're smearing out that charge
those pairs interact with the ambient em field with the positrons tending to shift closer to the
center while virtual electrons shift further away. This so-called vacuum polarization happens all
around the central electron and has the effect of screening the external universe from that central
charge. In fact the observed charge should appear to rise as we get closer to the center past this
screening. If you calculate the effective charge screening all the way to infinitesimal distances
that central charge has to be infinite - that means the central electron would need to have
infinite charge itself to overcome that infinite charge screening. Of course we now know that we
can never probe the electron down to a perfectly localized point so that infinity is probably
meaningless. However if it is true that the strength of the electromagnetic force increases at
extremely small distances when charge screening is overcome even though we can't catch our electron
in our infinite zoom the elusive particle is extremely well understood at least mathematically.
We even know how to rescue the electron from having its mass and charge explode due to its
interaction with the surrounding quantum fields - although to understand that we need a tool called
renormalization which we'll explore soon. And we'll also see how this tricky issue with the
electron foreshadowed a much much bigger conundrum in physics - the hierarchy problem. And will meet
the particle that has yet to be rescued from this issue. But that is also another story. For now
let's admire the tiny electron outwardly simple but inwardly mercurial and it's a strangeness that
only compounds as we localize it in SpaceTime. Thank you to Novium for supporting PBS. Inspired
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