Physics 249 Lecture 37, Dec 5th 2012
Reading: Chapter 12
HW 10: due Friday Dec 7 th
1) Particle physics
Using quantum mechanics we had a tool that could be used to understand many small distance scale systems. This included a complete model of the electron orbitals in the atom that allowed us to understand atomic spectra, molecular bonding, and other properties such as magnetism.
At the same time we have an orthogonal theory, special relativity that was necessary to understand phenomena at or near the speed of light.
In addition, study of quantum systems had led to a large number of mysteries including the presence of two new forces, new properties such as spin, and new types of particles – anti particles.
We needed a new theory to explain these mysteries. This theory is known as quantum field theory: which was based on a combination of relativity and quantum mechanics.
The full solution to these mysteries will be a group of related quantum field theories, one for each force (except gravity), 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 propagated 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 duality between two things that had previously been thought of as separate, discrete particles and the fields caused by forces similar to the previous duality we had between particles and waves. 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.
The fields are said to be transmitted by virtual particles. 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. Further the existence of the photon and charge conservation can mathematically be shown to be a consequence of a symmetry principle. This introduces some key concepts in particle physics.

a) Only particles that carry the correct type of charge will interact by a given force. b) Conservation of the charges associated with a force. c) The idea that the properties of the force can be fully understood just starting from an underlying symmetry principle.
2) 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 sums of such interactions.
It is more difficult to conceptualize an attractive force. To understand this we invoke the
Heisenberg Uncertainty relationship. The standard Heisenberg uncertainty relationship is
 p
 x ~h. 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 can have negative momentum as long as the time period or distance the photon travels the momentum is within the Heisenberg uncertainty relationship. 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.
It is also difficult to conceptualize the concept of a field. To understand this we invoke the time energy Heisenberg Uncertainty,

E
 t~h. The virtual photons can have energy E during time t as long as they are consistent within the uncertainty relationship with having zero energy. Thus the field does not necessarily dissipate energy all the time.
Note that if you are uncomfortable with these ideas then you should go back and think about the concept of a field and realize that you should have been equally uncomfortable with the idea that you could can have action as a distance that is either attractive or repulsive from a single source and at infinite distances.
Enforcing the Heisenberg Uncertainty relationship put substantial constraints on the form of the force. Lets investigate those constraints to understand why we observe a 1/r
2
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 the force, F= dp/dt = dp/dr dr/dt ~ hbar c/r
2
. The functional dependence is a constant governing the strength over r
2
. The strength of the interaction depends on the charges involved. F=CqQ/r 2 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.