Welcome to Quantum Field Theory 1. Creation
and destruction. The first video in this series
on Quantum Field Theory.
Let�s start by reviewing a few results from
our quantum mechanics series. These provide
the context in which quantum field theory
was developed.
The birth of quantum mechanics was the concept
that the energy of the electromagnetic field
is quantized. At frequency nu these quanta
(which we call photons) have energy and momentum,
respectively, of �h� nu and �h� nu
over c, which is also �h� over lambda.
Here �h� is Planck�s constant, �c�
is the speed of light and lambda is wavelength.
This wave-particle duality, with the electromagnetic
field having both continuous, or wave, and
discrete, or particle, properties, is at the
heart of quantum weirdness.
Energy quantization is required to explain
the Planck formula for radiation intensity
versus frequency at a given temperature. This
correctly predicts that intensity increases
to a certain frequency then rapidly drops
off, unlike the classical prediction of an
ever-increasing intensity.
Because electromagnetic energy can be emitted
and absorbed, it follows that photons can
be created and destroyed. When this happens
their energy and momentum are transferred
from or to electrons, typically. This occurs
instantaneously in so-called quantum jumps.
For example, in the Bohr model of hydrogen,
the electron can only occupy a discrete set
of orbits. An electron in the third orbit.
Can suddenly transition to the second orbit
with the emission of a photon.
We might interpret this as three events occurring
simultaneously. Destroy an electron in the
third orbit. Create an electron in the second
orbit. And, create a photon, with the constraints
that total energy and momentum remain constant.
Eventually the wave-particle duality idea
was extended to electrons. These were found
to have wave properties described by a wavefunction,
with energy and momentum relations analogous
to those of a photon.
The quantum-jump process now applies to a
wavefunction. Instantaneously, an electron
in one quantum state.
transitions to a lower-energy state with the
emission of a photon, such that total energy
and momentum are conserved.
Finally, the development of relativistic quantum
mechanics predicted that all space is filled
with an invisible Dirac sea of negative-energy
electrons. Pulling out an electron appears
to create a visible electron and a visible
hole, which behaves like an antielectron,
what came to be called a positron.
In this cartoon representation, the Dirac
sea contains an infinite number of electrons
in all possible negative-energy states.
This forms the background of, quote, empty
space, and is assumed to be unobservable.
By adding energy we can pull an electron out
of the Dirac sea into a visible, positive-energy
state. This leaves behind a hole in the Dirac
sea.
What will be observed is the apparent creation
of an electron-positron pair. The process
can be reversed with the electron falling
back into the hole, so to speak, appearing
to an observer as annihilation of the electron-positron
pair.
As it stands, there are some loose ends and
rough edges in this theory. One is that although
we talk about photons being created and destroyed,
we don�t have a rigorous description of
the process and how it is coupled with a quantum
jump of an electron.
Another is that the Dirac sea concept is rather
strange. It�s hard to believe that empty
space is really filled with an infinite number
of invisible, negative-energy electrons. It
would be preferable to have a theory in which
electrons and positrons are actually created
and destroyed, just as we suppose photons
are, and corresponding more closely to what
is actually observed. These, and other considerations
motivated the development of quantum field
theory.
An early contribution was Einstein�s paper,
On the Quantum Theory of Radiation. This was
published in 1917, after development of Bohr�s
atomic model but before devel opment of the
Schr�dinger equation. Here we�ll consider
Einstein�s ideas, but in a different sequence
than given in his paper.
Imagine a box containing a number of atoms,
with each atom in either quantum state 1 or
the higher-energy quantum state 2.
The box also contains photons with a frequency
nu corresponding to the energy difference
of the two quantum states.
There are N-1 atoms in state 1 and N-2 atoms
in state 2, with respective energies E-1 and
E-2.
One possible radiation process is emission.
An atom in the higher energy state can spontaneously
emit a photon and transition to the lower
energy state. This will decrease N-2 and increase
N-1.
Another possibility is absorption. An atom
in the lower energy state can absorb a photon
and transition to the higher energy state.
This will decrease N-1 and increase N-2.
Let�s assume the entire system is in thermal
equilibrium at temperature Tee. Then emission
and absorption will be balanced in the sense
that there will be no net change in N-1 and
N-2.
Let�s write the probability that a single
atom in quantum state 2 will undergo emission
during time d t, as d W equals a constant
A, 2, 1 times d t . The probability is independent
of the presence of photons. It depends only
on the internal details of the quantum states
1 and 2.
Our picture is that an atom in state 2.
Can transition to state 1 with the emission
of a photon. And, the probability of this
occurring per unit time is A, 2, 1 .
Let�s write the probability that a single
atom in quantum state 1 will undergo absorption
during time d t as d W equals a constant B,
1, 2, times the radiation density u of nu,
times d t . The probability has to depend
on the radiation density because, unlike emission,
absorption cannot occur unless a photon is
present to be absorbed. And, the more photons
are present, the more likely absorption becomes.
Our picture is that a photon can encounter
an atom in state 1.
And be absorbed, causing the atom to transition
to state 2. The probability of this occurring
per unit time is B, 1, 2 times the radiation
density.
In thermal equilibrium N 1 and N 2 should
remain constant. So d N 2, the change in N
2 during a time d t, should be zero. Absorption
increases N 2. The number of absorptions is
the number of atoms in state 1, N 1, times
the probability of absorption per atom, B
1 2 times u of nu times d t.
Emission decreases N 2. The number of emissions
is the number of atoms in state 2, N 2, times
the probability of emission per atom, A 2
1 times d t. These two effects must cancel
out.
And we write the rate of change of N 2 with
respect to time is zero.
We can solve this equation for u of nu to
find that the radiation density is N 2 over
N 1 times A 2 1 over B 1 2.
In the first video of the quantum mechanics
series we described the Boltzmann distribution.
This tells us that at temperature T, the number
of atoms with energy E is proportional to
the exponential of minus E over k T, where
k is Boltzmann�s constant. Therefore, the
ratio N 2 over N 1 is the ratio of the corresponding
exponentials. We can write this as the exponential
of minus the quantity E 2 minus E 1 over k
T, which equals exponential of minus h nu
over k T.
Therefore, the radiation density is A 2 1
over B 1 2 times e to the minus h nu over
k T, or equivalently, 1 over e to the h nu
over k T.
However, this doesn�t agree with the observed
radiation density in thermal equilibrium,
given by Planck�s law, which we also described
in the first quantum mechanics video. That
has a denominator e to the h nu over k T minus
1.
While the expression we just obtained lacks
the minus 1. Let�s try starting with Planck�s
law and work backward to see what we may have
missed.
We replace e to the h nu over k T by N 1 over
N 2. We also identify the leading factor in
Planck�s law with A 2 1 over B 1 2.
After multiplying numerator and denominator
by N 2 we obtain A 2 1 over B 1 2 times N
2 over N 1 minus N 2.
This has an extra N 2 term relative to our
previous analysis, implying that there is
some additional emission process that we overlooked.
Multiplying through by B 1 2 times N 1 minus
N 2.
We get N 1 B 1 2 u of nu minus N 2 B 1 2 u
of nu equals N 2 A 2 1.
Moving all terms to the left-hand side and
combining the two N 2 terms.
We identify the expression for the equilibrium
of absorption and emission. The absorption
term is the same as before, but now the emission
term has N 2 multiplied by a sum of two terms.
For consistency of notation, let�s define
B 2 1 equal to B 1 2.
Then the terms in brackets imply that the
probability that an atom in state 2 will undergo
emission during a time d t has two contributions.
The first term we call spontaneous emission.
It�s independent of the radiation density,
that is, it doesn�t require the presence
of photons. It�s what we originally assumed
was the only emission process.
The second term we call stimulated emission.
It�s proportional to the radiation density,
so like absorption it only occurs in the presence
of photons. Photons are required to stimulate
this type of emission. We see that the form
of Planck�s law requires that such a process
must exist.
Stimulated emission implies that the following
process might be possible. An atom in state
2 could generate a photon through spontaneous
emission.
That photon could stimulate emission of a
second photon.
Those two photons could stimulate emission
of a third photon, and so on. As the number
of photons increases, the rate of stimulated
emission will also increase. Thus there should
be an exponential increase in the radiation
density. In this manner we might be able to
build a device that uses stimulated emission
to amplify light.
Such a device could be called a laser, for
light amplification by stimulated emission
of radiation. Thus Einstein�s 1917 paper
predicted the possibility of building lasers,
although due to technical challenges this
wasn�t accomplished until 1960. Furthermore,
any quantum field theory we develop must be
able to explain the existence and relative
strengths of these two emission processes.
