In particle physics, quantum electrodynamics
(QED) is the relativistic quantum field theory
of electrodynamics. In essence, it describes
how light and matter interact and is the first
theory where full agreement between quantum
mechanics and special relativity is achieved.
QED mathematically describes all phenomena
involving electrically charged particles interacting
by means of exchange of photons and represents
the quantum counterpart of classical electromagnetism
giving a complete account of matter and light
interaction.
In technical terms, QED can be described as
a perturbation theory of the electromagnetic
quantum vacuum. Richard Feynman called it
"the jewel of physics" for its extremely accurate
predictions of quantities like the anomalous
magnetic moment of the electron and the Lamb
shift of the energy levels of hydrogen.
== History ==
The first formulation of a quantum theory
describing radiation and matter interaction
is attributed to British scientist Paul Dirac,
who (during the 1920s) was able to compute
the coefficient of spontaneous emission of
an atom.Dirac described the quantization of
the electromagnetic field as an ensemble of
harmonic oscillators with the introduction
of the concept of creation and annihilation
operators of particles. In the following years,
with contributions from Wolfgang Pauli, Eugene
Wigner, Pascual Jordan, Werner Heisenberg
and an elegant formulation of quantum electrodynamics
due to Enrico Fermi, physicists came to believe
that, in principle, it would be possible to
perform any computation for any physical process
involving photons and charged particles. However,
further studies by Felix Bloch with Arnold
Nordsieck, and Victor Weisskopf, in 1937 and
1939, revealed that such computations were
reliable only at a first order of perturbation
theory, a problem already pointed out by Robert
Oppenheimer. At higher orders in the series
infinities emerged, making such computations
meaningless and casting serious doubts on
the internal consistency of the theory itself.
With no solution for this problem known at
the time, it appeared that a fundamental incompatibility
existed between special relativity and quantum
mechanics.
Difficulties with the theory increased through
the end of the 1940s. Improvements in microwave
technology made it possible to take more precise
measurements of the shift of the levels of
a hydrogen atom, now known as the Lamb shift
and magnetic moment of the electron. These
experiments exposed discrepancies which the
theory was unable to explain.
A first indication of a possible way out was
given by Hans Bethe in 1947, after attending
the Shelter Island Conference. While he was
traveling by train from the conference to
Schenectady he made the first non-relativistic
computation of the shift of the lines of the
hydrogen atom as measured by Lamb and Retherford.
Despite the limitations of the computation,
agreement was excellent. The idea was simply
to attach infinities to corrections of mass
and charge that were actually fixed to a finite
value by experiments. In this way, the infinities
get absorbed in those constants and yield
a finite result in good agreement with experiments.
This procedure was named renormalization.
Based on Bethe's intuition and fundamental
papers on the subject by Shin'ichirō Tomonaga,
Julian Schwinger, Richard Feynman and Freeman
Dyson, it was finally possible to get fully
covariant formulations that were finite at
any order in a perturbation series of quantum
electrodynamics. Shin'ichirō Tomonaga, Julian
Schwinger and Richard Feynman were jointly
awarded with a Nobel prize in physics in 1965
for their work in this area. Their contributions,
and those of Freeman Dyson, were about covariant
and gauge invariant formulations of quantum
electrodynamics that allow computations of
observables at any order of perturbation theory.
Feynman's mathematical technique, based on
his diagrams, initially seemed very different
from the field-theoretic, operator-based approach
of Schwinger and Tomonaga, but Freeman Dyson
later showed that the two approaches were
equivalent. Renormalization, the need to attach
a physical meaning at certain divergences
appearing in the theory through integrals,
has subsequently become one of the fundamental
aspects of quantum field theory and has come
to be seen as a criterion for a theory's general
acceptability. Even though renormalization
works very well in practice, Feynman was never
entirely comfortable with its mathematical
validity, even referring to renormalization
as a "shell game" and "hocus pocus".QED has
served as the model and template for all subsequent
quantum field theories. One such subsequent
theory is quantum chromodynamics, which began
in the early 1960s and attained its present
form in the 1970s work by H. David Politzer,
Sidney Coleman, David Gross and Frank Wilczek.
Building on the pioneering work of Schwinger,
Gerald Guralnik, Dick Hagen, and Tom Kibble,
Peter Higgs, Jeffrey Goldstone, and others,
Sheldon Glashow, Steven Weinberg and Abdus
Salam independently showed how the weak nuclear
force and quantum electrodynamics could be
merged into a single electroweak force.
== Feynman's view of quantum electrodynamics
==
=== 
Introduction ===
Near the end of his life, Richard P. Feynman
gave a series of lectures on QED intended
for the lay public. These lectures were transcribed
and published as Feynman (1985), QED: The
strange theory of light and matter, a classic
non-mathematical exposition of QED from the
point of view articulated below.
The key components of Feynman's presentation
of QED are three basic actions.
A photon goes from one place and time to another
place and time.
An electron goes from one place and time to
another place and time.
An electron emits or absorbs a photon at a
certain place and time.
These actions are represented in the form
of visual shorthand by the three basic elements
of Feynman diagrams: a wavy line for the photon,
a straight line for the electron and a junction
of two straight lines and a wavy one for a
vertex representing emission or absorption
of a photon by an electron. These can all
be seen in the adjacent diagram.
As well as the visual shorthand for the actions
Feynman introduces another kind of shorthand
for the numerical quantities called probability
amplitudes. The probability is the square
of the absolute value of total probability
amplitude,
probability
=
|
f
(
amplitude
)
|
2
{\displaystyle {\text{probability}}=|f({\text{amplitude}})|^{2}}
. If a photon moves from one place and time
A
{\displaystyle A}
to another place and time
B
{\displaystyle B}
, the associated quantity is written in Feynman's
shorthand as
P
(
A
to
B
)
{\displaystyle P(A{\text{ to }}B)}
. The similar quantity for an electron moving
from
C
{\displaystyle C}
to
D
{\displaystyle D}
is written
E
(
C
to
D
)
{\displaystyle E(C{\text{ to }}D)}
. The quantity that tells us about the probability
amplitude for the emission or absorption of
a photon he calls j. This is related to, but
not the same as, the measured electron charge
e.QED is based on the assumption that complex
interactions of many electrons and photons
can be represented by fitting together a suitable
collection of the above three building blocks
and then using the probability amplitudes
to calculate the probability of any such complex
interaction. It turns out that the basic idea
of QED can be communicated while assuming
that the square of the total of the probability
amplitudes mentioned above (P(A to B), E(C
to D) and j) acts just like our everyday probability
(a simplification made in Feynman's book).
Later on, this will be corrected to include
specifically quantum-style mathematics, following
Feynman.
The basic rules of probability amplitudes
that will be used are:
=== Basic constructions ===
Suppose, we start with one electron at a certain
place and time (this place and time being
given the arbitrary label A) and a photon
at another place and time (given the label
B). A typical question from a physical standpoint
is: "What is the probability of finding an
electron at C (another place and a later time)
and a photon at D (yet another place and time)?".
The simplest process to achieve this end is
for the electron to move from A to C (an elementary
action) and for the photon to move from B
to D (another elementary action). From a knowledge
of the probability amplitudes of each of these
sub-processes – E(A to C) and P(B to D)
– we would expect to calculate the probability
amplitude of both happening together by multiplying
them, using rule b) above. This gives a simple
estimated overall probability amplitude, which
is squared to give an estimated probability.
But there are other ways in which the end
result could come about. The electron might
move to a place and time E, where it absorbs
the photon; then move on before emitting another
photon at F; then move on to C, where it is
detected, while the new photon moves on to
D. The probability of this complex process
can again be calculated by knowing the probability
amplitudes of each of the individual actions:
three electron actions, two photon actions
and two vertexes – one emission and one
absorption. We would expect to find the total
probability amplitude by multiplying the probability
amplitudes of each of the actions, for any
chosen positions of E and F. We then, using
rule a) above, have to add up all these probability
amplitudes for all the alternatives for E
and F. (This is not elementary in practice
and involves integration.) But there is another
possibility, which is that the electron first
moves to G, where it emits a photon, which
goes on to D, while the electron moves on
to H, where it absorbs the first photon, before
moving on to C. Again, we can calculate the
probability amplitude of these possibilities
(for all points G and H). We then have a better
estimation for the total probability amplitude
by adding the probability amplitudes of these
two possibilities to our original simple estimate.
Incidentally, the name given to this process
of a photon interacting with an electron in
this way is Compton scattering.
There is an infinite number of other intermediate
processes in which more and more photons are
absorbed and/or emitted. For each of these
possibilities, there is a Feynman diagram
describing it. This implies a complex computation
for the resulting probability amplitudes,
but provided it is the case that the more
complicated the diagram, the less it contributes
to the result, it is only a matter of time
and effort to find as accurate an answer as
one wants to the original question. This is
the basic approach of QED. To calculate the
probability of any interactive process between
electrons and photons, it is a matter of first
noting, with Feynman diagrams, all the possible
ways in which the process can be constructed
from the three basic elements. Each diagram
involves some calculation involving definite
rules to find the associated probability amplitude.
That basic scaffolding remains when one moves
to a quantum description, but some conceptual
changes are needed. One is that whereas we
might expect in our everyday life that there
would be some constraints on the points to
which a particle can move, that is not true
in full quantum electrodynamics. There is
a possibility of an electron at A, or a photon
at B, moving as a basic action to any other
place and time in the universe. That includes
places that could only be reached at speeds
greater than that of light and also earlier
times. (An electron moving backwards in time
can be viewed as a positron moving forward
in time.)
=== 
Probability amplitudes ===
Quantum mechanics introduces an important
change in the way probabilities are computed.
Probabilities are still represented by the
usual real numbers we use for probabilities
in our everyday world, but probabilities are
computed as the square of probability amplitudes,
which are complex numbers.
Feynman avoids exposing the reader to the
mathematics of complex numbers by using a
simple but accurate representation of them
as arrows on a piece of paper or screen. (These
must not be confused with the arrows of Feynman
diagrams, which are simplified representations
in two dimensions of a relationship between
points in three dimensions of space and one
of time.) The amplitude arrows are fundamental
to the description of the world given by quantum
theory. They are related to our everyday ideas
of probability by the simple rule that the
probability of an event is the square of the
length of the corresponding amplitude arrow.
So, for a given process, if two probability
amplitudes, v and w, are involved, the probability
of the process will be given either by
P
=
|
v
+
w
|
2
{\displaystyle P=|\mathbf {v} +\mathbf {w}
|^{2}}
or
P
=
|
v
w
|
2
.
{\displaystyle P=|\mathbf {v} \,\mathbf {w}
|^{2}.}
The rules as regards adding or multiplying,
however, are the same as above. But where
you would expect to add or multiply probabilities,
instead you add or multiply probability amplitudes
that now are complex numbers.
Addition and multiplication are common operations
in the theory of complex numbers and are given
in the figures. The sum is found as follows.
Let the start of the second arrow be at the
end of the first. The sum is then a third
arrow that goes directly from the beginning
of the first to the end of the second. The
product of two arrows is an arrow whose length
is the product of the two lengths. The direction
of the product is found by adding the angles
that each of the two have been turned through
relative to a reference direction: that gives
the angle that the product is turned relative
to the reference direction.
That change, from probabilities to probability
amplitudes, complicates the mathematics without
changing the basic approach. But that change
is still not quite enough because it fails
to take into account the fact that both photons
and electrons can be polarized, which is to
say that their orientations in space and time
have to be taken into account. Therefore,
P(A to B) consists of 16 complex numbers,
or probability amplitude arrows. There are
also some minor changes to do with the quantity
j, which may have to be rotated by a multiple
of 90° for some polarizations, which is only
of interest for the detailed bookkeeping.
Associated with the fact that the electron
can be polarized is another small necessary
detail, which is connected with the fact that
an electron is a fermion and obeys Fermi–Dirac
statistics. The basic rule is that if we have
the probability amplitude for a given complex
process involving more than one electron,
then when we include (as we always must) the
complementary Feynman diagram in which we
exchange two electron events, the resulting
amplitude is the reverse – the negative
– of the first. The simplest case would
be two electrons starting at A and B ending
at C and D. The amplitude would be calculated
as the "difference", E(A to D) × E(B to C)
− E(A to C) × E(B to D), where we would
expect, from our everyday idea of probabilities,
that it would be a sum.
=== Propagators ===
Finally, one has to compute P(A to B) and
E(C to D) corresponding to the probability
amplitudes for the photon and the electron
respectively. These are essentially the solutions
of the Dirac equation, which describe the
behavior of the electron's probability amplitude
and the Klein–Gordon equation, which describes
the behavior of the photon's probability amplitude.
These are called Feynman propagators. The
translation to a notation commonly used in
the standard literature is as follows:
P
(
A
to
B
)
→
D
F
(
x
B
−
x
A
)
,
E
(
C
to
D
)
→
S
F
(
x
D
−
x
C
)
,
{\displaystyle P(A{\text{ to }}B)\to D_{F}(x_{B}-x_{A}),\quad
E(C{\text{ to }}D)\to S_{F}(x_{D}-x_{C}),}
where a shorthand symbol such as
x
A
{\displaystyle x_{A}}
stands for the four real numbers that give
the time and position in three dimensions
of the point labeled A.
=== 
Mass renormalization ===
A problem arose historically which held up
progress for twenty years: although we start
with the assumption of three basic "simple"
actions, the rules of the game say that if
we want to calculate the probability amplitude
for an electron to get from A to B, we must
take into account all the possible ways: all
possible Feynman diagrams with those endpoints.
Thus there will be a way in which the electron
travels to C, emits a photon there and then
absorbs it again at D before moving on to
B. Or it could do this kind of thing twice,
or more. In short, we have a fractal-like
situation in which if we look closely at a
line, it breaks up into a collection of "simple"
lines, each of which, if looked at closely,
are in turn composed of "simple" lines, and
so on ad infinitum. This is a challenging
situation to handle. If adding that detail
only altered things slightly, then it would
not have been too bad, but disaster struck
when it was found that the simple correction
mentioned above led to infinite probability
amplitudes. In time this problem was "fixed"
by the technique of renormalization. However,
Feynman himself remained unhappy about it,
calling it a "dippy process".
=== Conclusions ===
Within the above framework physicists were
then able to calculate to a high degree of
accuracy some of the properties of electrons,
such as the anomalous magnetic dipole moment.
However, as Feynman points out, it fails to
explain why particles such as the electron
have the masses they do. "There is no theory
that adequately explains these numbers. We
use the numbers in all our theories, but we
don't understand them – what they are, or
where they come from. I believe that from
a fundamental point of view, this is a very
interesting and serious problem."
== 
Mathematics ==
Mathematically, QED is an abelian gauge theory
with the symmetry group U(1). The gauge field,
which mediates the interaction between the
charged spin-1/2 fields, is the electromagnetic
field.
The QED Lagrangian for a spin-1/2 field interacting
with the electromagnetic field is given in
natural units by the real part of
where
γ
μ
{\displaystyle \gamma ^{\mu }}
are Dirac matrices;
ψ
{\displaystyle \psi }
a bispinor field of spin-1/2 particles (e.g.
electron–positron field);
ψ
¯
≡
ψ
†
γ
0
{\displaystyle {\bar {\psi }}\equiv \psi ^{\dagger
}\gamma ^{0}}
, called "psi-bar", is sometimes referred
to as the Dirac adjoint;
D
μ
≡
∂
μ
+
i
e
A
μ
+
i
e
B
μ
{\displaystyle D_{\mu }\equiv \partial _{\mu
}+ieA_{\mu }+ieB_{\mu }\,\!}
is the gauge covariant derivative;
e is the coupling constant, equal to the electric
charge of the bispinor field;
m is the mass of the electron or positron;
A
μ
{\displaystyle A_{\mu }}
is the covariant four-potential of the electromagnetic
field generated by the electron itself;
B
μ
{\displaystyle B_{\mu }}
is the external field imposed by external
source;
F
μ
ν
=
∂
μ
A
ν
−
∂
ν
A
μ
{\displaystyle F_{\mu \nu }=\partial _{\mu
}A_{\nu }-\partial _{\nu }A_{\mu }\,\!}
is the electromagnetic field tensor.
=== Equations of motion ===
Substituting the definition of D into the
Lagrangian gives
L
=
i
ψ
¯
γ
μ
∂
μ
ψ
−
e
ψ
¯
γ
μ
(
A
μ
+
B
μ
)
ψ
−
m
ψ
¯
ψ
−
1
4
F
μ
ν
F
μ
ν
.
{\displaystyle {\mathcal {L}}=i{\bar {\psi
}}\gamma ^{\mu }\partial _{\mu }\psi -e{\bar
{\psi }}\gamma _{\mu }(A^{\mu }+B^{\mu })\psi
-m{\bar {\psi }}\psi -{\frac {1}{4}}F_{\mu
\nu }F^{\mu \nu }.}
From this Lagrangian, the equations of motion
for the ψ and A fields can be obtained.
Using the field-theoretic Euler–Lagrange
equation for ψ,
The derivatives of the Lagrangian concerning
ψ are
∂
μ
(
∂
L
∂
(
∂
μ
ψ
)
)
=
∂
μ
(
i
ψ
¯
γ
μ
)
,
{\displaystyle \partial _{\mu }\left({\frac
{\partial {\mathcal {L}}}{\partial (\partial
_{\mu }\psi )}}\right)=\partial _{\mu }\left(i{\bar
{\psi }}\gamma ^{\mu }\right),}
∂
L
∂
ψ
=
−
e
ψ
¯
γ
μ
(
A
μ
+
B
μ
)
−
m
ψ
¯
.
{\displaystyle {\frac {\partial {\mathcal
{L}}}{\partial \psi }}=-e{\bar {\psi }}\gamma
_{\mu }(A^{\mu }+B^{\mu })-m{\bar {\psi }}.}
Inserting these into (2) results in
i
∂
μ
ψ
¯
γ
μ
+
e
ψ
¯
γ
μ
(
A
μ
+
B
μ
)
+
m
ψ
¯
=
0
,
{\displaystyle i\partial _{\mu }{\bar {\psi
}}\gamma ^{\mu }+e{\bar {\psi }}\gamma _{\mu
}(A^{\mu }+B^{\mu })+m{\bar {\psi }}=0,}
with Hermitian conjugate
i
γ
μ
∂
μ
ψ
−
e
γ
μ
(
A
μ
+
B
μ
)
ψ
−
m
ψ
=
0.
{\displaystyle i\gamma ^{\mu }\partial _{\mu
}\psi -e\gamma _{\mu }(A^{\mu }+B^{\mu })\psi
-m\psi =0.}
Bringing the middle term to the right-hand
side yields
The left-hand side is like the original Dirac
equation, and the right-hand side is the interaction
with the electromagnetic field.
Using the Euler–Lagrange equation for the
A field,
the derivatives this time are
∂
ν
(
∂
L
∂
(
∂
ν
A
μ
)
)
=
∂
ν
(
∂
μ
A
ν
−
∂
ν
A
μ
)
,
{\displaystyle \partial _{\nu }\left({\frac
{\partial {\mathcal {L}}}{\partial (\partial
_{\nu }A_{\mu })}}\right)=\partial _{\nu }\left(\partial
^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }\right),}
∂
L
∂
A
μ
=
−
e
ψ
¯
γ
μ
ψ
.
{\displaystyle {\frac {\partial {\mathcal
{L}}}{\partial A_{\mu }}}=-e{\bar {\psi }}\gamma
^{\mu }\psi .}
Substituting back into (3) leads to
Now, if we impose the Lorenz gauge condition
∂
μ
A
μ
=
0
,
{\displaystyle \partial _{\mu }A^{\mu }=0,}
the equations reduce to
◻
A
μ
=
e
ψ
¯
γ
μ
ψ
,
{\displaystyle \Box A^{\mu }=e{\bar {\psi
}}\gamma ^{\mu }\psi ,}
which is a wave equation for the four-potential,
the QED version of the classical Maxwell equations
in the Lorenz gauge. (The square represents
the D'Alembert operator,
◻
=
∂
α
∂
α
{\displaystyle \Box =\partial _{\alpha }\partial
^{\alpha }}
.)
=== Interaction picture ===
This theory can be straightforwardly quantized
by treating bosonic and fermionic sectors
as free. This permits us to build a set of
asymptotic states that can be used to start
computation of the probability amplitudes
for different processes. In order to do so,
we have to compute an evolution operator,
which for a given initial state
|
i
⟩
{\displaystyle |i\rangle }
will give a final state
⟨
f
|
{\displaystyle \langle f|}
in such a way to have
M
f
i
=
⟨
f
|
U
|
i
⟩
.
{\displaystyle M_{fi}=\langle f|U|i\rangle
.}
This technique is also known as the S-matrix.
The evolution operator is obtained in the
interaction picture, where time evolution
is given by the interaction Hamiltonian, which
is the integral over space of the second term
in the Lagrangian density given above:
V
=
e
∫
d
3
x
ψ
¯
γ
μ
ψ
A
μ
,
{\displaystyle V=e\int d^{3}x\,{\bar {\psi
}}\gamma ^{\mu }\psi A_{\mu },}
and so, one has
U
=
T
exp
⁡
[
−
i
ℏ
∫
t
0
t
d
t
′
V
(
t
′
)
]
,
{\displaystyle U=T\exp \left[-{\frac {i}{\hbar
}}\int _{t_{0}}^{t}dt'\,V(t')\right],}
where T is the time-ordering operator. This
evolution operator only has meaning as a series,
and what we get here is a perturbation series
with the fine-structure constant as the development
parameter. This series is called the Dyson
series.
=== Feynman diagrams ===
Despite the conceptual clarity of this Feynman
approach to QED, almost no early textbooks
follow him in their presentation. When performing
calculations, it is much easier to work with
the Fourier transforms of the propagators.
Experimental tests of quantum electrodynamics
are typically scattering experiments. In scattering
theory, particles momenta rather than their
positions are considered, and it is convenient
to think of particles as being created or
annihilated when they interact. Feynman diagrams
then look the same, but the lines have different
interpretations. The electron line represents
an electron with a given energy and momentum,
with a similar interpretation of the photon
line. A vertex diagram represents the annihilation
of one electron and the creation of another
together with the absorption or creation of
a photon, each having specified energies and
momenta.
Using Wick theorem on the terms of the Dyson
series, all the terms of the S-matrix for
quantum electrodynamics can be computed through
the technique of Feynman diagrams. In this
case, rules for drawing are the following
To these rules we must add a further one for
closed loops that implies an integration on
momenta
∫
d
4
p
/
(
2
π
)
4
{\displaystyle \int d^{4}p/(2\pi )^{4}}
, since these internal ("virtual") particles
are not constrained to any specific energy–momentum,
even that usually required by special relativity
(see Propagator for details).
From them, computations of probability amplitudes
are straightforwardly given. An example is
Compton scattering, with an electron and a
photon undergoing elastic scattering. Feynman
diagrams are in this case
and so we are able to get the corresponding
amplitude at the first order of a perturbation
series for the S-matrix:
M
f
i
=
(
i
e
)
2
u
¯
(
p
→
′
,
s
′
)
ϵ
/
′
(
k
→
′
,
λ
′
)
∗
p
/
+
k
/
+
m
e
(
p
+
k
)
2
−
m
e
2
ϵ
/
(
k
→
,
λ
)
u
(
p
→
,
s
)
+
(
i
e
)
2
u
¯
(
p
→
′
,
s
′
)
ϵ
/
(
k
→
,
λ
)
p
/
−
k
/
′
+
m
e
(
p
−
k
′
)
2
−
m
e
2
ϵ
/
′
(
k
→
′
,
λ
′
)
∗
u
(
p
→
,
s
)
,
{\displaystyle M_{fi}=(ie)^{2}{\overline {u}}({\vec
{p}}',s')\epsilon \!\!\!/\,'({\vec {k}}',\lambda
')^{*}{\frac {p\!\!\!/+k\!\!\!/+m_{e}}{(p+k)^{2}-m_{e}^{2}}}\epsilon
\!\!\!/({\vec {k}},\lambda )u({\vec {p}},s)+(ie)^{2}{\overline
{u}}({\vec {p}}',s')\epsilon \!\!\!/({\vec
{k}},\lambda ){\frac {p\!\!\!/-k\!\!\!/'+m_{e}}{(p-k')^{2}-m_{e}^{2}}}\epsilon
\!\!\!/\,'({\vec {k}}',\lambda ')^{*}u({\vec
{p}},s),}
from which we can compute the cross section
for this scattering.
== Renormalizability ==
Higher-order terms can be straightforwardly
computed for the evolution operator, but these
terms display diagrams containing the following
simpler ones
that, being closed loops, imply the presence
of diverging integrals having no mathematical
meaning. To overcome this difficulty, a technique
called renormalization has been devised, producing
finite results in very close agreement with
experiments. It is important to note that
a criterion for the theory being meaningful
after renormalization is that the number of
diverging diagrams is finite. In this case,
the theory is said to be "renormalizable".
The reason for this is that to get observables
renormalized, one needs a finite number of
constants to maintain the predictive value
of the theory untouched. This is exactly the
case of quantum electrodynamics displaying
just three diverging diagrams. This procedure
gives observables in very close agreement
with experiment as seen e.g. for electron
gyromagnetic ratio.
Renormalizability has become an essential
criterion for a quantum field theory to be
considered as a viable one. All the theories
describing fundamental interactions, except
gravitation, whose quantum counterpart is
presently under very active research, are
renormalizable theories.
== Nonconvergence of series ==
An argument by Freeman Dyson shows that the
radius of convergence of the perturbation
series in QED is zero. The basic argument
goes as follows: if the coupling constant
were negative, this would be equivalent to
the Coulomb force constant being negative.
This would "reverse" the electromagnetic interaction
so that like charges would attract and unlike
charges would repel. This would render the
vacuum unstable against decay into a cluster
of electrons on one side of the universe and
a cluster of positrons on the other side of
the universe. Because the theory is "sick"
for any negative value of the coupling constant,
the series does not converge but are at best
an asymptotic series.
From a modern perspective, we say that QED
is not well defined as a quantum field theory
to arbitrarily high energy. The coupling constant
runs to infinity at finite energy, signalling
a Landau pole. The problem is essentially
that QED appears to suffer from quantum triviality
issues. This is one of the motivations for
embedding QED within a Grand Unified Theory.
== See also
