Lecture 1: Jan 24th 2012
Reading, Griffiths Introduction and Chapter one (through page 48)
1) State of knowledge before particle physics: early 20th century.
The idea of the wave and particle nature of matter and light was being established. Both
light and particles had properties such as interference and diffraction that could be
understood in terms of superposition of waves. Both also were shown to have particle
like and quantized properties as was shown in the case of light by the photoelectric effect
experiments for photons.
We already had knowledge of the atom and its constituents, the proton, and neutron and
orbiting electrons. A simple quantum model that included wave like properties for the
electron was necessary to describe the electron energy and angular momentum levels of
the atom. In more detail the electron energy and angular momentum states are derived by
solving a quantum wave equation for a particle in the presence of a electric potential.
The wave functions of electron, which described their properties when bound to an atom
including quantized energy and angular momentum levels, were the solutions to that
wave equation.
These quantum properties governed the understanding of small distance scale physics
such as the atom. The atom was composed of discrete particles that obeyed wave
equations with discreet, quantized solutions.
An orthogonal theory of relativity was developed by Einstein to understand the
physics of objects traveling near or at the speed of light and gravity. Relativity also had
experimental proofs such as the observation of light bending in a gravitational field
around the sun and later the measurement of time dilation effects.
Notice also that there are integral relationships between electromagnetic theory and
relativity. The photon, which can be described as a combination of changing electric and
magnetic fields, must moves at precisely the speed of light. In Maxwell’s equations the
speed of light is an integral constant.
Particle physics will be described a by a fully relativistic version of quantum mechanics.
One of the first true particle physics experiments was the Compton scattering experiment
where a photon is scattered off an electron with a shift in the angle and energy/frequency
or momentum of the photon. The experiment can be understood by treating the light as a
massless, discrete particle with a wavelength E=p=h and applying conservation of
relativistic energy and momentum. These will be the exactly way we will treat problems
in particle physics where we all particles will be governed by relativistic wave equations
and obey conservation of relativistic energy and momentum.
2) The mysteries of the time.
The fact that light and particles such as electrons both obeyed quantum mechanics and
relativity, but the two theories were not well related.
The discovery of antiparticles: The positron a particle with the electrons mass which
could be seen bending with the same radius of curvature but the opposite way in a
magnetic field. F=qvxB. Q had to be opposite to have a force in the other direction.
Spin. The quantum mechanical property of spin has no classical analog.
Antiparticles and spin were not fully understood until the particles were treated using
relativistic wave equations where both properties are integral to the solutions of the wave
equations.
Weak decays. Changed a particle from one type to another. Didn’t obey conservation of
momentum and energy. n->p+eWhat held the protons together in the nucleus?
3) We needed a new theory to explain these mysteries. Quantum field theory: which was
based on relativity and quantum mechanics. The full solution to these mysteries will be a
group of related quantum field theories, one for each force, comprising the standard
model if particle physics.
In quantum field theory the electric and magnetic forces or electromagnetic force are
described by fields. However, the electromagnetic field is caused by many photons being
exchanged. The information of the field is transmitted through the interactions of
individual particles making it quantized.
Also in quantum field theory the electron is also a field. This is easy to conceptualize
since in a bound state it can be spread out in a probability distribution around the atom.
This creates a symmetry between two things that had previously been thought of as
separate, discrete particles and the fields caused by forces. We will see in quantum field
theory the forces and particles are treated using very similar techniques. They will be
described by solutions to the same types of relativistic wave equations.
There is also a real quantum or excitation of each field possible: the photon in the case of
the electromagnetic field. These real excitations of the field can be produced around
charged particles like an electron. The real excitation of the electron field is simply the
electron.
The electromagnetic interactions only happen to objects that have electric charge. Finally
we find that charge is conserved in quantum electrodynamics and further the existence of
the photon can mathematically be shown to be a consequence of charge conservation.
This introduces a key concept in particle physics. Only particles that carry the correct
type of charge will interact by a given force.
4) Forces in particle physics
It is easy to imagine a repulsive force as photons being exchanged from one electron to
another as they approach each other. The smooth force we observe is from a large
number of photons being exchanged.
If they pass by each other very quickly the interaction can be modeled as just one photon
being exchanged. Though there is a quantum probabilistic nature leading to a
distribution of different outcomes. For instance various possible scattering angles
governed by a probability distribution. In fact all processes in particle physics can be
modeled by particle exchanges or, where appropriate, sums of such interactions in a
perterbative expansion.
It is more difficult to conceptualize an attractive force. To understand this we invoke the
Heisenberg Uncertainty. The usual Heisenberg relationship is px ~h. You can also
write a relationship with time and energy Et~h, which makes sense because relativity
mixes time and space and energy and momentum. In quantum field theory we typically
say that the photons that carry in the information of the field are virtual. By virtual we
mean that they can have properties that are unphysical or at least non typical. For
instance a photon, which is typically massless, can have a mass or have negative
momentum as long as the time period or distance the photon travels is small enough that
the energy of the mass or the momentum is within the Heisenberg uncertainty
relatioionship. Though note that even in these interactions with virtual particles energy
and momentum is strictly conserved at every step.
Since for the virtual photon the properties can be unphysical in the brief exchange the
force can be repulsive by exchanging a negative momentum photon.
5) EM Force
As of the early 20th century physicists had understood the atom, and properties such as
interference and diffraction using QM.
We had experimentally seen that light and particles such as electrons both had wave,
particle and field properties. We knew that fast moving particles obeyed relativity.
However, initial attempts to create relativistic quantum theories led to problems like
negative energies.
We needed a theory that:
a) Incorporated both quantum mechanics and relativity.
b) Explained the antiparticle and spin, which had been observed.
c) Explained why particles and light had both wave and particle properties.
d) Explained the quantization of light, which before had been thought of as an
electromagnetic wave.
e) Conservation of charge
The quantum field theory, Quantum Electrodynamics and could explain all these features.
In quantum field theory the electric and magnetic forces or electromagnetic force are
fields (as is the electron). However the field is caused by many photons being
exchanged, which carry the information of the field. At the basic level everything is
governed by single exchanges of particles. The photon and the electron are the quanta of
these fields and the field can be excited such that the quanta are observable as real
particles. Note that the electromagnetic field can be excited to produce a photon in the
presence of electric charge or in other words electric charge is what the photon couples
to.
Diagrams of the interactions can be drawn. These diagrams will always have a photon
interacting with two particle lines at a vertex. The three particle vertex is the basic
interaction element in particle physics. Calculation in particle physics will involve
determining the probability of each three particle vertex.
For Compton scattering:
1) An initial state photon is absorbed by 2) an initial state electron resulting in 3) an
intermediate state electron. Then the 1) intermediate state electron emits 2) a final state
photon and becomes 3) a final state electron with a different momentum.
The two Compton scattering vertices can also be rotated to get interactions such that
e-e-e- becomes e-e-e-e- or e-e+e-e+. We find a related, though not
identical, probabilities for that “crossed” interactions. e-e-e- is related to e-ee-e- and e-e+e-e+ and since all three processes involve vertices with a single
photon and 2 electron/positron objects. Depending on the orientation these diagrams can
either have absorption or emission of a photon, electron scattering, or particle antiparticle
pair annihilation or creation to or from a photon.
How does this give a 1/r2 force? The virtual photons have energy maximum E and exist
for a maximum time of t such that Et~hbar. Or you can say they have maximum
momentum p and cross a maximum distance r such that pr~hbar where r=tc. Under the
Heisenberg uncertainty principle their physics properties are totally uncertain and
consistent with zero within uncertainty. If they transfer a smaller amount of momentum
then they can cross a larger distance or vice-versa. Then for force is F= dp/dt = dp/dr
dr/dt ~ hbar c/r2. The functional dependence is a constant governing the strength over r2.
The strength of the interaction depends on the charges involved. F=CqQ/r2 or U=-CQ/r.
The photons are excited by the presence of charges so their number is going to be
proportional to qQ times some constant that governs the strength of individual
electromagnetic interactions.
The interactions above follow a number of rules. They conserved electric charge number
and relativistic energy and momentum. Also the interaction is long range and high
enough probability to be easily observed, results in decays in short time scales, and also
strong enough to form bound states between particles like electron and protons.
6) With the details of the EM interaction largely understood, weak interactions and the
how protons were held together in the nucleus remained mysteries.
We had to propose new interactions that would have their own quantum field theories.
Weak interaction. This interaction had much smaller probabilities or strengths of
interaction than the EM interaction. Also needed to propose a new particle to conserve
momentum and energy. n->p+e- Neutrino: electrically neutral and almost undetectable.
Strong interaction. This interaction had to have a much larger probability or strength of
interaction to hold protons together in the nucleus but very short range.
The properties of these interactions will be understood as we start to study the qualitative
aspects of each interaction.
We also observed a number of new mystery particles in cosmic ray experiments
muon: first seen in cosmic rays this particle had all the properties of the electron except it
was ~200 times more massive. Underwent decay interactions with time scales indicative
of weak interactions.
Pion: also >200 times more massive but appeared to interact with strong strength in
matter. Also came in neutral and charged versions. Charged versions decayed on weak
times scales. Neutral version decayed to photons with electromagnetic times scales.
Kaon: more massive than two pions and seen to decay to multiple pions with weak time
scales. Also interacted in matter with strong strength.
Charged muons, pions and kaons also interacted with the electromagnetic strength but at
a much smaller level then the electron since the interaction effect (acceleration) is
inversely proportional to the mass.