Quantum electrodynamics
From Wikipedia, the free encyclopedia
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.
[1]:Ch1
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 first able to compute the coefficient of spontaneous emission of an
atom.
[2]
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,
[3]
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,
[4] and
Victor Weisskopf,
[5] 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.
[6]
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 1940. Improvements in
microwave technology made it possible to take more precise measurements of the shift of the levels of a
hydrogen atom,
[7] now known as the
Lamb shift and
magnetic moment of the electron.
[8] These experiments unequivocally exposed discrepancies which the theory was unable to explain.
A first indication of a possible way out was given by
Hans Bethe. In 1947, while he was traveling by train to reach
Schenectady from
New York,
[9] after giving a talk at the
conference at Shelter Island
on the subject, Bethe completed the first non-relativistic computation
of the shift of the lines of the hydrogen atom as measured by Lamb and
Retherford.
[10]
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
Sin-Itiro Tomonaga,
[11] Julian Schwinger,
[12][13] Richard Feynman[14][15][16] and
Freeman Dyson,
[17][18] it was finally possible to get fully
covariant formulations that were finite at any order in a perturbation series of quantum electrodynamics.
Sin-Itiro Tomonaga,
Julian Schwinger and
Richard Feynman were jointly awarded with a
Nobel prize in physics in 1965 for their work in this area.
[19] 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.
[17] 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".
[1]:128
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 1975 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,
[20][21] 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,
[1] 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.
[1]:85
- 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 a 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.
It is important not to over-interpret these diagrams. Nothing is implied about
how a particle gets from one point to another. The diagrams do
not imply that the particles are moving in straight or curved lines. They do
not
imply that the particles are moving with fixed speeds. The fact that
the photon is often represented, by convention, by a wavy line and not a
straight one does
not imply that it is thought that it is more
wavelike than is an electron. The images are just symbols to represent
the actions above: photons and electrons do, somehow, move from point to
point and electrons, somehow, emit and absorb photons. We do not know
how these things happen, but the theory tells us about the probabilities
of these things happening.
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 total probability amplitude. If a
photon moves from one place and time—in shorthand, A—to another place
and time—in shorthand, B—the associated quantity is written in Feynman's
shorthand as P(A to B). The similar quantity for an electron moving
from C to D is written E(C to D). The quantity which 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'.
[1]:91
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 making the assumption that the square of the total of the
probability amplitudes mentioned above (P(A to B), E(A to B) and 'j') is
just our everyday
probability.
(A simplification of Feynman's book.) Later on this will be corrected
to include specifically quantum mathematics, following Feynman.
The basic rules of probability amplitudes that will be used are that
a) if an event can happen in a variety of different ways then its
probability amplitude is the
sum of the probability amplitudes of
the possible ways and b) if a process involves a number of independent
sub-processes then its probability amplitude is the
product of the component probability amplitudes.
[1]:93
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) – then 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 are 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.)
[1]:89, 98–99
Probability amplitudes
Feynman replaces complex numbers with spinning
arrows, which start at emission and end at detection of a particle. The
sum of all resulting arrows represents the total probability of the
event. In this diagram, light emitted by the source S bounces off a few segments of the mirror (in blue) before reaching the detector at P.
The sum of all paths must be taken into account. The graph below
depicts the total time spent to traverse each of the paths above.
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.
Probability amplitudes 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 actually 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. No satisfactory reason
has been given for
why they are needed. But pragmatically we
have to accept that they are an essential part of our description of all
quantum phenomena. 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
or
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 of probability amplitudes as complex numbers
Multiplication of probability amplitudes as complex numbers
Addition and multiplication are familiar 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 start 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) actually consists of 16 complex
numbers, or probability amplitude arrows.
[1]:120–121
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 just
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.
[1]:112–113
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 describes 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:
where a shorthand symbol such as
stands for the four real numbers which 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 end points.
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 very difficult 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".
[1]:128
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 totally 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."
[1]:152
Quantum field theory
Theory that brings
quantum mechanics and
special relativity together to account for subatomic theory.
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 by the real part of
[22]:78
-
where
- are Dirac matrices;
- a bispinor field of spin-1/2 particles (e.g. electron–positron field);
- , called "psi-bar", is sometimes referred to as the Dirac adjoint;
- is the gauge covariant derivative;
- e is the coupling constant, equal to the electric charge of the bispinor field;
- Aμ is the covariant four-potential of the electromagnetic field generated by the electron itself;
- Bμ is the external field imposed by external source;
- is the electromagnetic field tensor.
Equations of motion
To begin, substituting the definition of
D into the Lagrangian gives us
Next, we can substitute this Lagrangian into the
Euler–Lagrange equation of motion for a field:
-
to find the field equations for QED.
The two terms from this Lagrangian are then
Substituting these two back into the Euler–Lagrange equation (
2) results in
with complex conjugate
Bringing the middle term to the right-hand side transforms this second equation into
-
The left-hand side is like the original
Dirac equation and the right-hand side is the interaction with the electromagnetic field.
One further important equation can be found by substituting the
Lagrangian into another Euler–Lagrange equation, this time for the
field,
Aμ:
-
The two terms this time are
and these two terms, when substituted back into (
3) give us
-
Now, if we impose the
Lorenz gauge condition, that the divergence of the four potential vanishes
then we get
which is a
wave equation
for the four potential, the QED version of the classical Maxwell
equations in the Lorenz gauge. (In the above equation, the square
represents the
D'Alembert operator.)
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 which can be used to start a computation of the
probability amplitudes for different processes. In order to do so, we
have to compute an
evolution operator that, for a given initial state
, will give a final state
in such a way to have
[22]:5
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:
[22]:123
and so, one has
[22]:86
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. Quantum physics considers particle's
momenta
rather than their positions, 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
[22]:801-802
To these rules we must add a further one for closed loops that implies an integration on momenta
,
since these internal ("virtual") particles are not constrained to any
specific energy–momentum – even that usually required by special
relativity (see
this article 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
[22]:158-159
and so we are able to get the corresponding amplitude at the first order of a
perturbation series for the
S-matrix:
from which we are able to 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
[22]:ch 10
-
-
One-loop contribution to the electron
self-energy function
-
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 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.
[23] 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 do not converge, but
are an
asymptotic series.
From a modern perspective, we say that QED is not well defined as a QFT to arbitrarily high energy.
[24] The coupling constant runs to infinity at finite energy, signalling a
Landau pole. The problem is essentially that QED is not
asymptotically free. This is one of the motivations for embedding QED within a
Grand Unified Theory.