Physics 249 Lecture 38, Dec 7th 2012
Reading: Chapter 12
HW 11: due Friday Dec 14th
1) Quantum field theory of the EM force. QED, quantum electro dynamics.
Among other things Quantum field theory:
a) Incorporates both quantum mechanics and relativity.
b) Explained why particles and light had similar wave and particle properties.
c) Explained the antiparticle and spin, which had been observed.
d) Explained conservation of charge
e) Explained the functional form of the EM field
Quantum field theories primary utility is in predicting highly relativistic interactions.
Though more static interactions like electromagnetic repulsion and attraction can also be
described. They are described as a sum of many simpler interactions.
In QED simple diagrams can be drawn to represent most interactions. 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. These
calculations will involve understanding how the initial state wave function is changed
into a final state wave function via the potential of the force that the photon transmites.
Example: Electromagnetic scattering:
1) An initial electron emits an intermediate state or virtual photon changing its
momentum to the final state value.
2) The photon is absorbed by a second initial state electron (or positron) changing its
momentum to a final state value.
The two scattering vertices can also be rotated to get interactions such as Compton
scattering, e-e-e-, or particle antiparticle annihilation, 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.
The interactions above follow a number of rules. They conserved electric charge number
and relativistic energy and momentum. Though the intermediate state, or virtual,
particles can have unusual or unphysical properties. For instance, an intermediate state
photon can be massive.
For properties we find that 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. We will find that these
properties actually derive from features of the force carrier such as the fact that it is
massless and does not carry electric charge.
2) With the details of the EM interaction largely understood (we will investigate some of
that work next), 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 also be understood as a
consequence of the properties of the force carriers and the charges involved.
3) A relativistic wave equation:
A key part of understanding particle physics interactions is to understand the wave
functions of the matter particles and the photon or other force carriers. Solving the
relativistic wave equations will lead to an understanding of the properties of the particles
we observe.
The simplest result to investigate using relativistic wave equations is the existence of
antiparticles. We will investigate this idea rigorously and discuss other results without
extensive derivation.
Consider Klein-Gordon wave equation solution
We want to do the same quantization for a relativistic theory starting with the relativistic
energy, momentum equation. Initially we will neglect any contribution from a potential.
E2 = p2c2 + m2c4
Note we will adopt a common particle physics set of units where c = 1 and
E2 - p2 = m2
Quantizing
pi  i ¶xi, ¶i = ¶/ ¶x …, E  i ¶ t,
results in the Klein-Gordon equation
=1
or additionally setting hbar to one
¶2
- 2 f +Ñ 2f = m 2f
¶t
With a solutions:
f (x,t) = Ne ip×x-iEt
f (x,t) = Ne-ip×x+iEt
Note that this is the one of the simplest set of solutions. A more general solution includes
a function of the four momentum u(p). Also there is an ambiguity or two solutions,
since there is a solution with an overall negative sign in the exponential corresponding to
the “negative” energy in solutions to the energy momentum equation, E=(p2-m2)1/2.
This solution also doesn’t include all the features we want for relativistic particles such as
spin. In fact it is the solution for a spin zero or scalar particle. However, it will let us
investigate the concept of antiparticles.
Consider multiplying the Klein Gordon equation by -i* and subtracting the complex
conjugate of the equation multiplied by -i. The mass terms don’t involve operators and
cancel out. Bringing out a time and space derivative
One can define a continuity equation as the change in time of the probability density in a
given volume is equal to the current of probability out of the volume:
This looks exactly like the equation above and allows us to interpret it.
The first term above is probability density  and the second the probability current j out
of the volume.
Calculating for  using the first wave function solution.  = 2E|N|2
Note the factor of 2E. The factor of 2E makes the probability density element d3x
Lorentz invariant. If you consider a particle in a Lorentz boosted from space contracts
since d3x d3x(1-v2)1/2. However, energy, the time like component dilates and the
probability goes up making the probability density element constant.
Now consider the wave functions with a negative sign in the exponential. These
solutions have negative probability density!
This issue can be interpreted and resolved by noting two things.
First, fundamentally the two solutions to the wave equations are independent.
y(x,t) = Ne ip× x-iEt and y (x, t) = Ne-ip×x+iEt .
It makes sense to identify them with independent particles both with energy E.
If you had one particle with probability density  = 2E|N|2 , E positive, and one
antiparticle with probability density  = -2E|N|2 , E also positive, then in if you consider
the total probability density of a system with one of each of these particles you can
annihilate them without changing the probability density. These equations predict that a
second solution to the wave equation should exist that corresponds to a particle that can
annihilate with the original equation. In fact we observe exactly this!
In electricity and magnetism the fundamental property that is associated with the particles
and antiparticle. The continuity equation implies that charge in conserved.
A more complex solution can be pursued by introducing the a general function u(p) into
the solution. If we do this we find that there are actually 4 independent solutions (6 if
you could the 2 we already found). Investigating the solutions using the same techniques
we find that 2 correspond to particle and two to antiparticles. Further investigating the
solutions by calculating the angular momentum we find that the two sets of two solutions
are subdivided into solutions with intrinsic angular momentum up ½ and intrinsic angular
momentum down -½.