← Feynman Diagrams ↑ →
The accepted wisdom is that Feynman diagrams should not be taken literally, but should merely be treated as aids to calculation. That is not how Feynman regarded them, as described in, for example,
QED: The Strange Theory of Light and Matter». In the interpretation of quantum mechanics used here, they are understood as statements in quantum logic. Since we cannot say what combination of particle interactions takes place between measurements, we must sum the possibilities using quantum logical
OR. In Feynman diagrams only lines and vertices have meaning. They represent structures of matter as configurations of particle interactions in the absence of background of space or spacetime.
The Time Ordered Vertex for QED
The interaction density for qed is
where photon creation and annihilation operators are distinguished by the vector index,
a.
I(x) is the sum of eight terms, each of which can be represented can be represented diagrammatically as a time ordered
vertex» or node. Lines above the node correspond to creation operators, and those below the node correspond to annihilation operators. The photon is represented by a wavy line, electrons by a upward arrow and positrons by a downward arrow.
Wick’s Theorem
The
perturbation expansion
is a sum of terms representing
n interactions. In quantum logic a sum stands for disjunction. So, the meaning of the perturbation expansion is that we cannot say how many interactions take place in any given physical process.
Wick’s theorem» can be used to replace the time ordered product with a normal ordered product by (anti)commuting annihilation operators to the right and creation operators to the left. Let
φ be the field operator,
If
x is before
y,
x0 < y0, the
Feynman propagator»,
D(x − y), gives the amplitude for the creation of an antiparticle at
x and its annihilation at
y. If
x is after
y,
x0 > y0 it gives the amplitude for creation of a particle at
y and its annihilation at
x.
Definition: The Feynman propagator, or contraction of φ†(y) and φ(x) is
where plus is used for Bosons and minus for Fermions, and Θ is the step function,
Θ(x) = 0 if x ≤ 0,
Θ(x) = 1 if x > 0.
Wick’s Theorem:
For two field operators,
For n field operators,
where 1 ≤ i, j, k ≤ n and contracted pairs are omitted in the normal ordered product.
Proof:
where plus is used for Bosons and minus for Fermions. This is Wick’s theorem for two field operators.
A detailed proof by induction can be carried out for
n field operators, but the proof is no more evident than the theorem itself, which just means that we do the normal ordering by carrying out the contractions.
The S-matrix
Initial and final states can be expressed as sums of plane wave states by using the resolution of unity in momentum space. The time evolution between
t0 and
t1 is given by a matrix in momentum space
In the case of scattering, the initial state (generated by a
particle accelerator»), and the final state (typically measured by
bubble chamber»,
wire chamber» or
silicon detector» are well represented as pure momentum states. In this case the interesting interaction takes place at the scattering event, and
t0 and
t1 are not important.
Definition: The S-matrix» (or scattering matrix) is
The
S-matrix is found from the perturbation expansion by first normal ordering the terms using Wick’s theorem. Then, for the interaction density at
x, the creation operator acting on the initial state
gives, for a photon,
for a Dirac particle,
and for an antiparticle,
Similarly, the annihilation operators in the interaction density acting on the final state
gives, for a photon,
for a Dirac particle,
and for an antiparticle,
| To keep track of the contractions in normal ordering the perturbation expansion, the terms are represented by graphs». A particle created at x0 may be annihilated at a later time y0. An antiparticle created at x0 may be annihilated at an earlier time y0. Each contraction is represented by connecting the corresponding lines between vertices, and, at the same time, removing time ordering. |  |
After carrying out the contractions, all topoligically equivalent time ordered diagrams are combined into a single diagram with no time ordering between the nodes. There are n! diagrams with n nodes. So, removing the ordering of nodes generates a factor n! and cancels the factor 1 ⁄ n! in the perturbation expanson, leaving an integration for a diagram with n vertices,
|  |
The Feynman Propagator
Provided that the integrals converge as
, the photon propagator can be
evaluated,
where
is a dummy variable,
is a non-vector and
is 3-momentum from
j to
i;
if
and
if
.
g is the metric tensor. Similarly, provided that the integrals converge as
, the propagator for a Dirac particle can be
evaluated,
These results are
demonstrated by combining the integrations for the advanced (
) and retarded (
) parts. In practice, we do not have convergence at
in certain diagrams with internals loops. Treating the resulting divergence is
regularisation» and can be carried out in a number of ways.
Conservation of Energy and Momentum
Gather all the exponential terms from internal and external lines with
xi in the exponent. Provided the time from
t0 to
t1 is large, the result is a
representation of a delta function
where
refer to the arrowed line coming from vertex, the arrowed line going into the vertex, and the photon line. The delta function shows that the tilda’d quantities are conserved. For internal lines,
, are the dummy variables introduced in the contour integration. For external lines
. Energy,
p0, was originally defined to be the zero component of a vector. This is not a conserved quantity. Vectors are products of measurement, and only have real meaning in measurement. By definition, internal lines do not correspond to measured states. So,
p0 has no meaning on internal lines in a Feynman diagram. The conserved tilda’d quantities are of more interest than vector quantities and it is usual to
redefine energy.
Redefinition: Energy is the conserved quantity, 
, which appears on the lines of a Feynman diagram.
With this definition, energy-momentum,
, is conserved, but is not a vector, and does not obey the
mass shell condition on internal lines in Feynman diagrams. Particles are said to be
off shell» on internal lines. On external lines, representing measured states, this definition of energy coincides with the original definition for measured states, as the time component of a vector. Particles are said to be
on shell» on external lines, meaning that the mass shell condition is obeyed.
Feynman Rules
After using the delta functions to carry out the integrals over tilda’d quantities, and imposing the rule that energy-momentum is conserved at each vertex, there remains an integral for each independent internal loop,
Each vertex contributes a factor
For external lines in the initial state we have, for a photon,
for a Dirac particle,
and for an antiparticle,
For external lines in the final state we have, for a photon,
for a Dirac particle,
and for an antiparticle,
For internal arrowed lines we have
and for internal photon lines we have
In addition there is a minus sign if an odd number of commutations of Fermion creation and annilation operators is required to put the diagram into normal order. The limit
ε → 0 should be taken after evaluation of integrals for loops and for the initial and final states. If
|p| > 0 then the photon propagator can be replaced with
Certain diagrams contain a divergence when photon energy goes to zero. In this case
ε2 should be retained until after evaluation of the integral to control the infrared divergence (
ε2 plays the role of the small photon mass commonly used for this purpose).
Feynman Diagrams ↑ Scattering →
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