Quantum electrodynamics

Near the end of his life, Richard 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.

Feynman diagram elements

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, ${\displaystyle {\text{probability}}=|f({\text{amplitude}})|^{2}}$. If a photon moves from one place and time ${\displaystyle A}$ to another place and time ${\displaystyle B}$, the associated quantity is written in Feynman's shorthand as ${\displaystyle P(A{\text{ to }}B)}$. The similar quantity for an electron moving from ${\displaystyle C}$ to ${\displaystyle D}$ is written ${\displaystyle E(C{\text{ to }}D)}$. 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.[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 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:[1]^:93

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.

Compton scattering

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.)[1]^{:89, 98–99}

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 modulus 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

${\displaystyle P=|\mathbf {v} +\mathbf {w} |^{2}}$

or

${\displaystyle P=|\mathbf {v} \,\mathbf {w} |^{2}.}$

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 of probability amplitudes as complex numbers

Multiplication of probability amplitudes as 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.[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 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

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 Maxwell's equations, 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:

${\displaystyle P(A{\text{ to }}B)\to D_{F}(x_{B}-x_{A}),\quad E(C{\text{ to }}D)\to S_{F}(x_{D}-x_{C}),}$

where a shorthand symbol such as ${\displaystyle x_{A}}$ stands for the four real numbers that give the time and position in three dimensions of the point labeled A.

Electron self-energy loop

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".[1]^:128

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."[1]^:152

Substituting the definition of D into the Lagrangian gives

${\displaystyle {\mathcal {L}}=i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -e{\bar {\psi }}\gamma ^{\mu }(A_{\mu }+B_{\mu })\psi -m{\bar {\psi }}\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }.}$

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,

${\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)-{\frac {\partial {\mathcal {L}}}{\partial \psi }}=0.}$

(2)

The derivatives of the Lagrangian concerning ψ are

${\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\psi )}}\right)=\partial _{\mu }\left(i{\bar {\psi }}\gamma ^{\mu }\right),}$

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \psi }}=-e{\bar {\psi }}\gamma ^{\mu }(A_{\mu }+B_{\mu })-m{\bar {\psi }}.}$

Inserting these into (2) results in

${\displaystyle i\partial _{\mu }{\bar {\psi }}\gamma ^{\mu }+e{\bar {\psi }}\gamma ^{\mu }(A_{\mu }+B_{\mu })+m{\bar {\psi }}=0,}$

with Hermitian conjugate

${\displaystyle i\gamma ^{\mu }\partial _{\mu }\psi -e\gamma ^{\mu }(A_{\mu }+B_{\mu })\psi -m\psi =0.}$

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,

${\displaystyle \partial _{\nu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }A_{\mu })}}\right)-{\frac {\partial {\mathcal {L}}}{\partial A_{\mu }}}=0,}$

(3)

the derivatives this time are

${\displaystyle \partial _{\nu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }A_{\mu })}}\right)=\partial _{\nu }\left(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }\right),}$

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial A_{\mu }}}=-e{\bar {\psi }}\gamma ^{\mu }\psi .}$

Substituting back into (3) leads to

Now, if we impose the Lorenz gauge condition

${\displaystyle \partial _{\mu }A^{\mu }=0,}$

the equations reduce to

${\displaystyle \Box A^{\mu }=e{\bar {\psi }}\gamma ^{\mu }\psi ,}$

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, ${\displaystyle \Box =\partial _{\alpha }\partial ^{\alpha }}$.)

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 ${\displaystyle |i\rangle }$ will give a final state ${\displaystyle \langle f|}$ in such a way to have[22]^:5

${\displaystyle M_{fi}=\langle f|U|i\rangle .}$

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:[22]^:123

${\displaystyle V=e\int d^{3}x\,{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu },}$

and so, one has[22]^:86

${\displaystyle U=T\exp \left[-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'\,V(t')\right],}$

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.

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's 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 ${\displaystyle \int d^{4}p/(2\pi )^{4}}$, 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[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:

${\displaystyle M_{fi}=(ie)^{2}{\overline {u}}({\vec {p}}',s')\epsilon \!\!\!/\,'({\vec {k}}',\lambda ')^{*}{\frac {p\!\!\!/+k\!\!\!/+m_{e}}{(p+k)^{2}-m_{e}^{2}}}\epsilon \!\!\!/({\vec {k}},\lambda )u({\vec {p}},s)+(ie)^{2}{\overline {u}}({\vec {p}}',s')\epsilon \!\!\!/({\vec {k}},\lambda ){\frac {p\!\!\!/-k\!\!\!/'+m_{e}}{(p-k')^{2}-m_{e}^{2}}}\epsilon \!\!\!/\,'({\vec {k}}',\lambda ')^{*}u({\vec {p}},s),}$

from which we can compute the cross section for this scattering.

The predictive success of quantum electrodynamics largely rests on the use of perturbation theory, expressed in Feynman diagrams. However, quantum electrodynamics also leads to predictions beyond perturbation theory. In the presence of very strong electric fields, it predicts that electrons and positrons will be spontaneously produced, so causing the decay of the field. This process, called the Schwinger effect,[23] cannot be understood in terms of any finite number of Feynman diagrams and hence is described as nonperturbative. Mathematically, it can be derived by a semiclassical approximation to the path integral of quantum electrodynamics.

• De Broglie, Louis (1925). Recherches sur la theorie des quanta [Research on quantum theory]. France: Wiley-Interscience.
• Feynman, Richard Phillips (1998). Quantum Electrodynamics (New ed.). Westview Press. ISBN 978-0-201-36075-2.
• Jauch, J.M.; Rohrlich, F. (1980). The Theory of Photons and Electrons. Springer-Verlag. ISBN 978-0-387-07295-1.
• Greiner, Walter; Bromley, D.A.; Müller, Berndt (2000). Gauge Theory of Weak Interactions. Springer. ISBN 978-3-540-67672-0.
• Kane, Gordon, L. (1993). Modern Elementary Particle Physics. Westview Press. ISBN 978-0-201-62460-1.
• Miller, Arthur I. (1995). Early Quantum Electrodynamics: A Sourcebook. Cambridge University Press. ISBN 978-0-521-56891-3.
• Milonni, Peter W. (1994). The Quantum Vacuum: An Introduction to Quantum Electrodynamics. Boston: Academic Press. ISBN 0124980805. LCCN 93029780. OCLC 422797902.
• Schweber, Silvan S. (1994). QED and the Men Who Made It. Princeton University Press. ISBN 978-0-691-03327-3.
• Schwinger, Julian (1958). Selected Papers on Quantum Electrodynamics. Dover Publications. ISBN 978-0-486-60444-2.
• Tannoudji-Cohen, Claude; Dupont-Roc, Jacques; Grynberg, Gilbert (1997). Photons and Atoms: Introduction to Quantum Electrodynamics. Wiley-Interscience. ISBN 978-0-471-18433-1.

• Dudley, J.M.; Kwan, A.M. (1996). "Richard Feynman's popular lectures on quantum electrodynamics: The 1979 Robb Lectures at Auckland University". American Journal of Physics. 64 (6): 694–98. Bibcode:1996AmJPh..64..694D. doi:10.1119/1.18234.