 
Hello.
I'm Peter Boyle from the
Quantum Physics Research Group
in the School of
Physics and Astronomy
at the University of Edinburgh.
In my research, I use computers
to simulate the strong force.
What is mass?
Isaac Newton told us it is so
hard we have to push something
to change its speed.
 
At the quantum
level, particles are
described by ripples and fields.
We have plane waves.
 
We can think of theta as a
kind of angle, the phase angle.
The amount of energy in each
wavelength is quantised.
de Broglie told us
that the momentum
is related to how tight
these ripples are in space.
 
Einstein told us
energy is related
to how fast the wave
oscillates in time.
For a given particle
the energy depends
on the momentum
in a specific way.
For massless particles,
the square of the energy
is proportional to the
square of the momentum.
 
But for massive
particles, there's
rest energy, or rest mass.
E equals MC squared.
 
The speed is given by
the slope of E with p.
For massive
particles, this is 1.
Massive particles
have the familiar p
equals mv in the
non-relativistic limit.
For massive particles
this separation
is M squared C to the fourth.
This beautiful
simplicity gets broken,
because space still contains
stuff, like water, glass,
metals, or you and I.
When massless particles, like
light, travel through a medium,
they slow down.
We see this as bending.
When we dip a paint
brush in water,
light couples to the
charges in normal matter.
Neutrinos only couple
to matter very weakly.
They usually pass straight
through the earth unaffected.
We know glass splits light
into separate wavelengths.
The speed in glass
depends on the momentum,
and light no longer follows
its massless energy momentum
relation.
But the question is, does
a medium give light mass?
Well, glass and water
are complicated,
because the charges
are bound in molecules.
Conductors are simpler, because
the charges are free to move.
We've all seen
that light branches
off silvered glass, mirrors.
Simpler, still, are
superconductors,
where there is no
resistance at all
below a critical temperature.
The origins of the
Higgs mechanism
lie in the understanding of
superconductivity stemming
from Ginzburg and
Landau in 1950.
Electrons pair, spin
up, and spin down.
And below a critical
temperature,
the density of the Cooper
pairs, phi, becomes non-zero.
We explain in more detail,
later in this section.
For now, when the
material superconducts,
the magnetic field
becomes massive
inside the superconductor.
The superconductor spits
out the magnetic field lines
and will levitate
above a magnet,
in what is known as
the Meissner effect.
Now we're going to look
at the Higgs mechanism.
Let's try to understand
some of the pieces
- minus 1/4 F mu nu F mu nu.
That's shorthand
for electromagnetism
and Maxwell's equations.
 
This is equal to
E squared minus B
squared, which is
the energy density
of the electric and
magnetic fields.
You probably know E from the
parallel plate capacitor,
where we have a
potential difference
between these plates.
We have an electric field in the
middle, where the separation is
D. Note that the 0 is arbitrary.
That is we can add a constant
to the electrostatic potential
without changing any physics.
 
The D mu phi squared term here,
phi is a scalar complex field.
That means it has
two components.
Phi equals phi 1, phi 2.
Professor Higgs has drawn
the two components, here,
as Cartesian coordinates.
 
This term, D mu phi,
couples the scalar field
to the electromagnetism.
 
We'll draw phi 1 and phi
2 as Cartesian coordinates
on a plane.
If we take a value
of the field, we
can take this angle, theta, as
a phase angle for the field.
Now we have our two
Cartesian coordinates,
and we can draw the
energy density associated
with the potential term, D
phi, as a function of these two
coordinates.
And we see we a Mexican
hat-shaped potential.
 
The Cooper-pair field
of superconductivity
develops just such
a potential when
the Meissner effect occurs.
 
If we ignore electromagnetism
and the potential term for now,
a massless Higgs field
satisfies our wave equation.
This is really easy to solve.
We just need help from
a flexible friend.
Recall, phi has two
components, phi 1 and phi 2.
 
We'll represent all of space
as the perpendicular direction.
 
And time will be time.
With the Slinky, we can
represent phi 1 and phi 2
by the position of the Slinky.
Space is the vertical
direction, and time is time.
We see that phi 1 can oscillate.
Phi 2 can oscillate.
The Slinky is solving
the wave equation.
Of course, both directions can
oscillate at the same time,
either clockwise, which we
identify with particle modes,
or anti-clockwise, which we
identify with anti-particles.
The amplitude of these
modes are quantised.
You can write down the
particle solutions.
 
Phi 1 equals A cos px minus Et.
And Phi 2 equals A
sine px minus Et.
And we can think of these as
being associated with the phase
angle, theta of x and t, which
is equal to this term, theta px
minus Et.
 
So what about mass?
Well, for that we have
to look and introduce
our potential term Now it's
quite hard with a slinky,
so we'll do a
thought experiment.
We'll consider this linking as
a set of masses living on discs.
The position of each mass
represents phi 1 and phi 2.
These little masses will be
connected by massless springs.
 
There would be a tower of
these discs going all the way
to plus and minus
infinity in space.
If the masses are small,
and the distance is small,
these will still satisfy
the same wave equation.
 
Note also, for later reference,
that even if we shift phi
by a constant, it will still
satisfy the wave equation
and still have the same
type of oscillations.
 
Now however, we can
introduce a potential
by sticking extra springs
to each mass on each disc.
 
This will have the shape
of a quadratic potential,
and a bit like a
skateboarder in roller park,
phi will develop its own
oscillations, as well as,
because of the wave equation,
from this potential.
We'll call these
springs wave springs.
 
And the springs living on
each disc potential springs.
 
And we can see, if we
consider our Slinky,
and pretend that it is
also being acted upon
by these potential springs, that
we acquire a restoring force
for our circular waves
both from the wave springs
and from the potential springs.
In particular, if you consider
an infinite wavelength
constant zero momentum
mode, and displace it,
it will still oscillate
solely because
of these potential springs.
 
That means the
particle has energy
when it's at rest, rest mass.
Another point to note, is
that by introducing this mass,
we have lost the freedom to
shift the field into any value
and still have a wave solution.
 
The field prefers
to be at value zero.
We've seen that a zero,
or constant potential,
corresponds to
massless particles -
and that the quadratic,
or bowl-shaped potential,
corresponds to
massive particles.
The question now is, does the
Mexican hat-shaped potential
correspond to massive
or massless particles?
And the answer, as
we'll see, is both.
 
So we'll continue out
thought experiment,
imagining that this potential
applies to our Slinky.
First of all, the field phi must
choose a value for the vacuum
from where we start.
It has to lie somewhere on
the minimum of this energy.
 
There are many values
around this minimum light.
And the field must choose
one of these spontaneously.
We'll call its value V for
vacuum expectation value.
 
Because the field
spontaneously chooses this,
the process is called
spontaneous symmetry breaking.
If we align the Slinky with
this vacuum expectation value,
we can analyse how its
fluctuations around this
behave.
Fluctuations in the
radial direction
receive a restoring force
from this potential.
 
Fluctuations along
the minimum do not.
If we again think of the
infinite wavelength zero
momentum mode, we will
see that displacements
in the radial direction
continue to oscillate,
receiving a restoring
force from this potential.
These are massive modes.
The mass is determined
by the curvature
of this potential
at the minimum.
 
Constant, infinite wavelength,
zero momentum modes,
which are displaced
around the minimum
did not receive any restoring
force from the potential.
And therefore, they remain
massless and do not oscillate.
In fact, we see that the
number of massless modes
is the dimension
of this minimum.
In our case, it's simple.
The dimension is when we have
a single line, admittedly bent,
but the circle is a single line.
In more complicated situations,
the dimension can be bigger.
This is Goldstone's theorem.
The massless modes
are Goldstone bosons.
Goldstone's theorem
was thought to rule out
spontaneous symmetry breaking
and a relativistic field
theory.
This is because
the massless modes,
which have taken
new energy to create
and should have
already been seen.
The Higgs mechanism provides an
escape to Goldstone's theorem.
 
