INTRODUCTION TO FEYNMAN DIAGRAMS - PART 1
By Stephen Speicher
This is Part 1 of a simplistic multi-part introduction to Feynman
diagrams. The goal is to provide the non-technical reader with
some understanding of Feynman diagrams, enough to better
appreciate their use in and application to both the standard
theory, and the TEW.
I use the standard terminology in the hope that the reader will
eventually recognize and understand these concepts as presented
in articles and books, at least on a basic level. No attempt is
made to provide a mathematically rigorous definition of the
technical jargon; instead I characterize the terminology in a
manner more easily understood by the non-technical, while not
overly abusing what would be acceptable to the physicist or
mathematician. Part 1 outlines the background context for Feynman
diagrams, with the following Parts actually presenting the
diagrams themselves.
The mathematical cornerstone of quantum mechanics is the
Schroedinger wave equation for a free particle. This equation is
linear, and obeys the superposition principle, i.e., linear
combinations of solutions to the equation are themselves also
solutions. If we limit motion to the x direction only, the
Schroedinger equation will be satisfied by a general complex
function of position x and time t -- the wavefunction called
psi(x,t) -- which can be formed by a superposition of any number
of waves which are of a certain general complex form. (By
'complex' is meant a number representation having a real and an
imaginary part, imaginary by virtue of being denoted along with
i, the square root of minus one)
The general form of this Schroedinger equation is:
i h
---- psi(x,t)' = H psi(x,t)
2 PI
where i = square root of -1,
PI = the usual ratio of the circumference to the diameter
of a circle (3.14...),
h = Planck's constant (6.626*10^(-34) Joule seconds),
H = Hamiltonian operator,
' = partial differentiation with respect to time, t.
The Hamiltonian describes the total energy of the system, and its
name is derived from Hamilton's formulation of Newtonian
mechanics. Just as Newton's second law is the equation of motion
in classical mechanics, this Schroedinger equation is the time
dependent equation of motion for quantum mechanics. The
Hamiltonian in the equation describes the energy, much like the
Hamiltonian corresponds to force in Newtonian mechanics.
In Hamiltonian mechanics the state of a classical system is
described by a series of coordinates with corresponding momenta,
while the state of a quantum system is described by the state
vector psi(x,t). For a free particle the specification of H is
rather simple, but for a particle with interactions, H can become
quite complex. Consequently, exact solutions of this
Schroedinger equation are limited to simple forms of the
Hamiltonian, and most times one needs to resort to approximate
methods of solution.
The most powerful, and most widely used approximation method in
quantum mechanics is perturbation theory. The general approach is
to separate the system to be solved into two parts, with the
first part having a known solution. For the time dependent
Schroedinger equation, a known Hamiltonian is found, or at least
one which differs little from a known Hamiltonian. The
perturbation is the difference between the known and the actual
Hamiltonian, and with the gradual inclusion of more and more of
the perturbation, the known solution is more and more perturbed
into the unknown solution.
For instance, let us say that the Hamiltonian is of the form
H = H_0 + V,
where the states and energies of H_0 are known exactly, and we
consider V as a small correction to H_0. If we parameterize V by
introducing a coupling constant, e, then the Hamiltonian looks
like
H = H_0 + eV,
and we expect that as we vary e from zero to one, each state of
H_0 will become a state of the total system, and the energy will
smoothly become the energy of the entire state.
This perturbation approach will often approximate by considering
beginning terms of an infinite expansion. This method is not just
a practical approach to solving the problem, but it represents a
theoretical approach which highlights the whole idea of Feynman
diagrams. The Feynman diagrams are a graphical technique to
picture the interactions which occur in pertubation theory. Each
picture is a concretization, and precise mathematical elements
can be derived from the picture following very specific rules.
These Feynman rules permit precise calculations, depending on the
theory employed. Whether the standard theory, or quantum
electrodynamics, or quantum chromodynamics, the rules calculate
everything from decay rates to relativistic cross sections from
the relativistic scattering matrix. Our purpose here is not to
calculate differential decay rates or integrals over momenta, so
our focus will be on the qualitative analysis of Feynman
diagrams, not the quantitative analysis which is much more
complex. However, do not underestimate the value of the
qualitative approach, since the Feynman diagrams are a very
powerful tool for understanding the physics involved.
TO BE CONTINUED.
© 2001 Stephen Speicher
INTRODUCTION TO FEYNMAN DIAGRAMS - PART 2
By Stephen Speicher
This is Part 2 of a simple introduction to Feynman diagrams.
Before we delve into the diagrams, a quick review of particles in
the standard theory.
There exist four fundamental types of interactions, and for each
interaction there are one or more associated boson particles
which mediate the accompanying force. In order of increasing
strength:
Interaction Particle
---------- --------
Gravity Graviton
Weak W+, W-, Z0
Electromagnetic Photon
Nuclear Gluon
The relative strength between gravity and the nuclear force is 39
orders of magnitude! The nuclear force holds the nucleons
together, and is very short range (~10^-15m.), while the
gravitational force has an unlimited range. These boson particles
are usually referred to as field particles, or exchange
particles.
Aside from field particles, all other particles can be grouped
either as hadrons or leptons. There are two classes of hadrons
which differ by their mass and spin: mesons, such as the pion or
the kaon, and baryons, such as the proton and the neutron.
Whereas hadrons are associated with the nuclear force (this used
to be called the strong force, but now that term is more usually
applied to quarks), leptons are not. Leptons, such as the
electron and the muon, are considered to be elementary particles,
lacking any structure. The hadrons, however, are composite,
composed of constituent particles called quarks. Quarks and
leptons together are referred to as fermions.
Each particle has a corresponding anti-particle, which is
identical in mass and spin, but opposite in charge or color
(Color should not be taken literally -- it is a quantum number
asscociated with quarks. There are many cutesy terms used, such
as the flavors of quarks, quantum numbers called strangeness and
charm.) For instance, the anti-particle of an electron is a
positron. When an electron and a positron collide, they
annihilate each other, their mass converted to electromagnetic
radiation. In reverse, there is pair creation, where an electron
and a positron can be created from electromagnetic radiation.
There are many other particles -- the Particle Data Group (an
international collaboration of high-energy physics groups) lists
more than 150 different particles in their year 2000 Review of
Particle Physics. But according to the standard theory, the
intervening space between matter particles -- the vacuum -- is
teeming with particle pairs which go into and out of existence
too fast for them to be observed. This can be seen as a direct
consequence of the Heisenberg Uncertainty Principle.
One of the several uncertainty relationships is between energy
and time, such that * > h/2PI (a
very, very small number). So, even if there is nothing in the
vacuum, there is an uncertainty which allows a very small amount
of energy to exist for a very short duration of time. For
instance, the pair creation of our electron and its
anti-particle, the positron, can appear and disappear within one
billionth-trillionth of a second, without violating energy
conservation. These particle pairs are referred to as virtual
particles (as distinguished from real particles) because they
cannot be observed. A real photon in a free state can be
observed, but a virtual photon pops in and out of existence too
fast for observation to be made.
As mentioned in Part 1, Feynman diagrams are a graphical
technique to picture the interactions which occur in pertubation
theory. We will soon see that there is a one-to-one
correspondence between virtual particles and certain lines of
Feynman diagrams, just as real particles are pictorially
represented by others. The quantitative analysis of Feynman
diagrams is beyond the scope of this brief article, but the
pictorial representations will help us to understand more about
the particle interactions.
Feynman diagrams are space-time graphs, with, say, the x
(horizontal) axis representing spacial motion, and the y
(vertical) axis representing time. There is no attempt to
quantify time in the graph, nor any attempt to literally quantify
the speed or direction taken -- this is strictly a qualitative
pictorial.
There are four elementary processes for these particle
interactions:
1) emission of a photon by a charged particle,
2) absorption of a photon by a charged particle,
3) creation of a particle-anti-particle pair,
4) annihilation of a particle-anti-particle pair.
The fermions, such as the electron, are shown with solid lines
and an accompanying arrow in the direction of time movement.
Bosons, such as the photon, are drawn with squiggly lines.
Interactions between particles occur at the intersection of these
lines, and that point of intersection is called the vertex.
(Note: An electron will be denoted, as usual, by an e-, and the
positron as an e+. However, because of ascii limitations, I will
not be using the standard print notation for many particles. The
vertex will be denoted as o, and squiggly lines for the boson
will be broken but should be taken to be continuous in the
overall direction given. The photon will be denoted by ph. Also
note that anti-particles often have arrows pointing opposite the
direction of time because anti-particles are considered to
travel backwards in time. For simplicity, we will ignore that
convention here.)
So, the emission of a photon by a charged particle (1), would
look like this.
e-
\
\
^ ~~~ ph
\ ~~~
\ ~~~
o ~~~ (1)
/
/
^
/
/
e-
In this diagram we read time as going from bottom to top, so the
lower electron is the intial state and the upper electron is the
final state after the photon is emitted.
The absorption of a photon by a charged particle (2), looks like
this.
e-
\
\
^
\
\
o ~~ (2)
/ ~~
/ ~~
^ ~~
/ ~~
/ ~~
e- ph
Again we read time as flowing from bottom to top, so the photon
is absorbed by the electron, and the electron then exists in its
final state.
The creation of a particle-antiparticle pair (3) looks like this.
e- e+
\ /
\ /
^ ^
\ /
\ /
o (3)
~
~
~
~
~
~
ph
Note that the photon creates a pair, where the e- is the electron
and the e+ is the anti-particle, the positron.
The annihilation of a particle-anti-particle pair (4) looks like
this.
ph
~
~
~
~
~
~
o (4)
/ \
/ \
^ ^
/ \
/ \
e- e+
Here the electron and the positron are annihilated and a photon
is created.
With these elementary processes we can now start to picture the
interactions which occur between various particles. Here is the
Feynman diagram for an electron-proton interaction (5).
e- p
\ /
\ /
^ ^
\ /
\ /
o~~~~~~~~~~~~~o (5)
/ | \
/ | \
^ ph ^
/ \
/ \
e- p
We read this as time going from bottom to top, so the electron
and the proton interact, but they do so through the exchange
particle, the boson photon. This photon is a _virtual_ particle.
The one-to-one correspondence which I mentioned previously is
that the inner lines of a diagram represent virtual particles,
and the outer lines represent real particles.
We are all familar with the elementary school experiments which
demonstrate the repulsion between charged particles. We can use
the Feynman diagram to represent how that repulsion interaction
takes place in the standard theory (6).
e- e-
| /
\ /
| /
\ ^
^ /
\ /
| /
\ ~~~o
| ~~~ |
\ ~~~ |
| ~~~ |
\ ~~~ |
o~~~ | ^ (6)
/ | |
/ ph |
^ |
/ |
/ |
e- e-
This illustrates how one electron (on the left) emits a photon
and recoils, while the second electron (on the right) absorbs the
photon and acquires its momentum. (Again, the broken lines are
just due to the limitations of ascii.) This makes clear,
pictorially, the standard theory idea of the virtual photon being
the carrier of the electromagnetic force.
TO BE CONTINUED IN PART 3.
© 2001 Stephen Speicher
|