# Quantum Electrodynamics

# QUANTUM ELECTRODYNAMICS

Quantum electrodynamics, also known by its acronym, QED, is a relativistic quantum field theory that describes at a fundamental level the electro-magnetic interactions among electrically charged elementary particles such as electrons, positrons, muons, and quarks. Remarkably simple in form, it nevertheless respects the principles of special relativity and quantum mechanics, two of the great scientific revelations of the twentieth century. In addition, QED is essentially a complete theory of the electron's electromagnetic interactions and therefore provides a dynamical basis for atomic physics and all natural phenomena that spring from it, including chemistry, biology, and technology. At the quantitative level, QED predictions, as later briefly surveyed, have been tested to nearly one part in 100 billion, making it the most successful physical theory ever devised.

In its simplest form, which will primarily be discussed here, QED combines James Maxwell's equations for electric and magnetic fields with Paul Dirac's quantum theory of electrons. When introduced, its novelty was to provide a quantization of electromagnetic fields that provided a particle interpretation in terms of massless quanta called photons. (That idea actually had its origin in Max Planck's quantum light theory that was invented to explain blackbody radiation in 1900.) Also, the relativistic quantum description of the electron requires, as shown by Dirac, that it have an antiparticle partner called the positron (given that name because of its opposite sign "positive" electric charge) which can annihilate or be pair produced with electrons. Thus, pure QED can be viewed as a fundamental theory of interacting electrons, positrons, and photons. Easily extended to other heavier charged particles such as muons and quarks, it can also be applied in the nonrelativistic (low-velocity) limit that is often more appropriate for many-body condensed-matter or quantum optics systems.

QED is a special kind of quantum field theory referred to as renormalizable. That means short-distance quantum fluctuations (called quantum loop corrections) which are common to all QED processes and are, in fact, infinite (ultraviolet divergent) can be absorbed into the definition of the measured electron mass *m*_{e} and electric charge *e.* This "renormalization" of charge and mass renders the predictions of the theory finite and unique.

Understandably, the occurrence of infinite quantum effects caused much concern in the early days of QED. To some, renormalization appeared to be a way of sweeping a fundamental defect in QED under the rug rather than confronting it. In more recent times, however, QED has been viewed as an effective theory that must break down at very short distances (high energies) and be subsumed by a larger more complete theory. The infinities are expected to be artifacts of not including that still largely unknown, new short-distance physics. Renormalization of QED absorbs all those unknown short-distance physics effects into the measured value of *m*_{e} and *e.* Predictions from QED in terms of those renormalized parameters are then insensitive to such unknowns. The validity and power of this approach are confirmed by the quantitative success of QED as a stand-alone theory.

In contemporary elementary particle physics, QED is actually only part of a more complete theory called the Standard Model that describes strong, weak, and electromagnetic interactions. It, like QED, is based on symmetry considerations and the principle of local gauge invariance that will be illustrated via QED.

To appreciate the origins of gauge symmetries as fundamental descriptions of nature, it is instructive to consider how one introduces electromagnetic interactions into Dirac's theory of electrons. Consider the electron field *ψ* (*x* ) that depends on the space-time coordinate *x.**ψ* (*x* ) itself is called a four-component spinor field which indicates that under relativistic Lorentz transformations (i.e., space-time translations, rotations, and velocity boosts), it acts like a field with spin ½, where spin is the intrinsic angular momentum in units of *ħ* , Planck's constant divided by 2π, carried by the electron. With no interactions the four-component field *ψ* (*x* ) should satisfy Dirac's free field equation (hence, natural units are employed with *ħ* and the speed of light *c* set equal to 1): where ℏ is the bare electron mass (i.e., before interactions are turned on),γ^{μ},μ 0, 1, 2, 3 are 4 × 4 Dirac matrices, and the repeated index μ is summed over. Solutions to that equation represent the evolution of a free (noninteracting) electron field as a function of the space-time coordinate. Similarly, the electromagnetic field potential *A _{μ}* (

*x*) has four components labeled by the index μ. Under Lorentz transformations it acts like a spin-1 field. It satisfies the free field Maxwell equations.

To go from a theory of classical fields to a quantum field theory, one second quantizes the theory. That entails replacing *ψ* (*x* ) and *A _{μ}* (

*x*) with anticommuting (for spin ½) and commuting (for spin 1) operators that can create or annihilate electron (positron) and photon states.

So far, the equations given above describe noninteracting particles. An elegant formalism for introducing interactions between electrons and photons is the principle of local gauge invariance. One notes that the free theory is invariant under what are called U(1) global phase rotations: where 0 ≤ θ < 2π, and *e*_{0} is the bare electron's charge. If *ψ* (*x* ) is a solution to the Dirac equation, then e^{i}e_{0}Θ*ψ* (*x* ) is also a solution. Physics is invariant under such phase changes. The set of possible phase transformations form a U(1) group. That symmetry is associated with the conservation of electric charge (carried by the electron). It is a global symmetry because the same phase change is made at all spacetime points *x,* that is, θ is a constant.

The free Dirac equation is not invariant under local transformations of the more general form where *θ* (*x* ) now varies with *x.* To render the theory invariant under local phase transformations, one requires the simultaneous change called a gauge transformation on the electromagnetic potential and replaces the derivative in the Dirac equation by the so-called covariant derivative (Maxwell's free equations are already invariant under gauge transformations.) That change makes the full combined theory gauge invariant, at the expense of introducing a coupling or interaction term between the photon and the electron field. When the fields are quantized, that important term causes an interaction that allows quantum excitations of electrons, positrons, and photons characterized by coupling strength *e*_{0} between those particles. It is responsible for all electromagnetic interactions of electrons. The space-time dependent U(1) symmetry is called a local gauge symmetry, and *A*_{μ}(*x* ) is called a gauge field. It can be thought of as a connection field that allows invariance under local electron field rephasing or changing of the gauge.

Treating the interaction in (6) as a perturbation on an otherwise free set of fields, one can compute QED interaction effects as quantum loop expansions in the small fine-structure constant where *e* is the renormalized electric charge in which infinite (and finite) short-distance vacuum polarization quantum corrections from *e*^{+}*e*^{-} virtual pairs have been absorbed. Similarly, the bare mass m^{0}_{e}; is replaced by a renormalized measurable physical mass *m*_{e} that eliminates the remaining short-distance infinities from electron self-energy quantum corrections.

Although the roots of QED date back much earlier, a systematic program of renormalization and calculational formalism was developed by Sinitiro Tomonaga, Richard Feynman, and Julian Schwinger in the 1940s. In particular, the Feynman diagram approach gave a simple systematic method of calculation. Quantum loop corrections were then carried out for various atomic physics properties, scattering processes, static electron properties, etc. Those calculations were compared with very precise experimental measurements such as the Lamb shift and hyperfine atomic structure, which confirmed the validity of QED with spectacular success. Such measurements also helped advance the state of the art in atomic physics experimentation with developments in laser and other sophisticated technologies. As an illustration of experimental progress in QED, consider the determination

**TABLE 1**

Values of the Fine-structure Constant Extracted from Different Experiments | |

Numbers in parentheses indicate ± uncertainties in the last three digits. | |

α^{-1}Value | Method |

credit: Mohr, P., and Taylor, B. "CODOTA Recommended Values of the Fundamental Physical Constants: 1998." Review of Modern Physics72 , 351(2000). | |

137.03600300(270) | Quantum Hall Effect |

137.03600840(330) | Rydberg + h /m_{neutron} Measurements |

137.03598710(430) | AC Josephson Effect |

137.03599520(790) | Muonium (μ^{+}e^{-}) Hyperfine Structure |

137.03600140(183) | Average |

of the fine-structure constant α. In Table 1, four precise values (actually its inverse) obtained in very different electromagnetic situations and their weighted average are given. Two involve quantized condensed matter effects and two come from atomic measurements.

The good agreement of such different methods to many significant figures confirms the validity and universality of QED. Perhaps even more impressive is the comparison of theory and experiment for the anomalous magnetic moment α_{e} of the electron and positron. That quantity describes the deviation of the gyromagnetic ratio *g*_{e} from its Dirac equation value of 2 (without quantum fluctuations): A deviation from zero is predicted as a result of QED quantum loop effects. Starting with the famous leading effect α_{e} = α/2π calculated by Schwinger, high-powered analytic and computer-aided calculations now give the prediction where the last term corresponds to very small calculable strong and weak interaction effects that lie outside the framework of QED. Comparing that prediction with the experimental average of electron and positron measurements (which are in perfect agreement with each other) leads to the current best determination of α: α^{-1} = 137.03599959(40). (11)Agreement with the average in Table 1 is impressive. It confirms QED to about a few parts in 10^{11}! An alternative method of comparison is to predict α_{e}using the average of Table 1 in (9). That procedure gives

**TABLE 2**

Elementary Particles | ||||||

Particle | Symbol | Spin | Charge | Colors | Mass (GeV) | |

credit: Courtest of William Marciano. | ||||||

Electron Neutrino | ν_{e} | 1/2 | 0 | 0 | <2 × 10^{-9} | |

Electron | e | 1/2 | -1 | 0 | 0.51 × 10^{-3} | 1st |

Up Quark | u | 1/2 | 2/3 | 3 | 5 × 10^{-3} | generation |

Down Quark | d | 1/2 | -1/3 | 3 | 9 × 10^{-3} | |

Muon Neutrino | ν_{μ} | 1/2 | 0 | 0 | <2 × 10^{-9} | |

Muon | μ | 1/2 | -1 | 0 | 0.106 | 2nd |

Charm Quark | c | 1/2 | 2/3 | 3 | 1.25 | generation |

Strange Quark | s | 1/2 | -1/3 | 3 | 0.175 | |

Tau Neutrino | ν_{τ} | 1/2 | 0 | 0 | <2 × 10^{-9} | |

Tau | τ | 1/2 | -1 | 0 | 1.777 | 3nd |

Top Quark | t | 1/2 | 2/3 | 3 | 174.3 | generation |

Bottom Quark | b | 1/2 | -1/3 | 3 | 4.5 | |

Photon | γ | 1 | 0 | 0 | 0 | |

W boson | W^{±} | 1 | ±1 | 0 | 80.43 | Gauge |

Z boson | Z | 1 | 0 | 0 | 91.187 | bosons |

Gluon | g | 1 | 0 | 8 | 0 | |

Higgs scalar | H | 0 | 0 | 0 | 114~200? |

which agrees with experiment. Future improvement in by a factor of 10 is expected. To utilize such a result will require a much better independent determination of and further improvements in the QED perturbative calculation of .

QED is now generally recognized to be part of a larger theory called the Standard Model that also describes strong and weak interactions. That theory is a generalization of the local gauge invariance principle used in QED. The Standard Model is based on the enlarged local gauge symmetry group SU(3)_{c} × SU(2)_{L} × U(1)_{Y} where *c* denotes color, and the SU(3)_{c} part of the theory is called quantum chromodynamics, *L* indicates that only left-handed chiral components of spin-½ (fermion) particles exhibit the SU(2) symmetry, and *Y* stands for weak hypercharge. SU(*N* ) is the symmetry group of *N* × *N* unitary matrices. That local gauge invariance requires the introduction of twelve gauge bosons: eight gluons, *W*^{±}*W*^{0}, and *B* spin-1 fields. Unlike QED, SU(2)_{L} × U(1)_{Y} is not an exact symmetry but is broken down to its U(1)_{QED} subgroup, which remains intact. The breaking mechanism (called the Higgs mechanism) endows three of the gauge bosons *W*^{±} and *Z* (a linear combination of *W*^{0} and *B* ) with masses, while the photon (the orthogonal combination of *W*^{0} and *B* ) and eight gluons remain massless. The particle content of the Standard Model is illustrated in Table 2. All the particles listed there have been discovered except the elusive spin-zero Higgs scalar left over as a remnant of the Higgs mechanism.

The Standard Model, like QED, is renormalizable. All its masses, couplings, and mixing parameters must undergo infinite renormalizations. Quantum loop effects in that framework have been computed and compared with a large body of precise experimental measurements. Those confrontations have confirmed the validity of the Standard Model and its quantum loop corrections at the 0.1 percent level. Although not as impressive as the 10^{-11} tests of QED, the electroweak tests of the Standard Model are much more powerful probes of "new physics" such as additional heavy mass particles or new (as yet undiscovered) interactions. Thus, for example, precise measurements of *W*^{±} and *Z* masses along with decay properties of muons, *Z* bosons, etc. have been used in conjunction with quantum loop calculations to give the bound

*m*_{H} < 200 GeV (13)

whereas direct (negative) searches yield

*m*_{H} > ≈ 114 GeV (14)

Therefore, with that predicted mass range rather narrow and relatively close at hand, it seems that the discovery of the Higgs particle should be possible either at Fermilab's existing Tevatron proton-antiproton collider facility or, if not, at the higher energy collider called the Large Hadron Collider (LHC) being constructed in Switzerland at the European Laboratory for Particle Physics (CERN). Discovery of the Higgs will complete the minimal content of the Standard Model but will leave unanswered many of the perplexing questions that plagued QED: What additional physics lies at shorter distances than are currently being explored? Does it finally tame the infinities of QED and the complete Standard Model? How do we unify gravity with local gauge interactions? Why is nature governed by local gauge invariance? Those fundamental questions and others drove scientists to explore beyond QED and establish the Standard Model. They are likely to continue guiding them in the quest to further unveil the simplicity and intricacy of nature.

*See also:*Boson, Gauge; Dirac, Paul; Feynman, Richard; Gauge Theory; Pauli, Wolfgang; Quantum Field Theory; Renormalization; Schwinger, Julian; Standard Model; Tomogana, Sinitiro

## Bibliography

Abers, E., and Lee, B. W. "Gauge Theories." *Physics Reports***9** , 1–143 (1973).

Kinoshita, T., ed. *Quantum Electrodynamics* (World Scientific, Singapore, 1990).

Marciano, W., and Pagels, H. "Quantum Chromodynamics." *Physics Reports***36C** , 137–276 (1978).

Mohr, P., and Taylor, B. "CODATA Recommended Values of the Fundamental Physical Constants: 1998." *Reviews of Modern Physics***72** (2), 351–495 (2000).

Peskin, M., and Schroeder, D. *An Introduction to Quantum Field Theory* (Addison-Wesley, New York, 1995).

Schwinger, J., ed. *Quantum Electrodynamics* (Dover, New York, 1958).

*William J. Marciano*

# Quantum Electrodynamics (QED)

# Quantum electrodynamics (QED)

Quantum electrodynamics (QED) is a scientific theory that is also known as the **quantum theory** of light. QED describes the quantum properties (properties that are conserved and that occur in discrete amounts called quanta) and mechanics associated with the interaction of light (i.e., electromagnetic radiation) with matter. The practical value of QED rests upon its ability, as a set of equations, to allow calculations related to the absorption and emission of light by atoms and to allow scientists to make very accurate predictions regarding the result of the interactions between photons and charged atomic particles such as electrons. QED is a fundamentally important scientific theory because it accounts for all observed physical phenomena except those associated with aspects of **relativity theory** and radioactive decay.

QED is a complex and highly mathematical theory that paints a picture of light that is counter-intuitive to everyday human experience. According to QED theory, light exists in a duality consisting of both particle and wave-like properties. More specifically, QED asserts that electromagnetism results from the quantum behavior of the photon, the fundamental "particle" responsible for the transmission electromagnetic radiation. According to QED theory, a seeming particle vacuum actually consists of electron-positron fields. An electron-positron pair (positrons are the positively charged antiparticle to electrons) comes into existence when photons interact with these fields. In turn, QED also accounts for the subsequent interactions of these electrons, positrons, and photons.

Photons, unlike other "solid" particles, are thought to be "virtual particles" constantly exchanged between charged particles such as electrons. Indeed, according to QED theory the forces of **electricity and magnetism** (i.e., the fundamental electromagnetic force) stem from the common exchange of virtual photons between particles and only under special circumstances do photons become observable as light.

According to QED theory, "virtual photons" are more like the wavelike disturbances on the surface of **water** after it is touched. The virtual photons are passed back and forth between the charged particles much like basketball players might pass a ball between them as they run down the court. As virtual particles, photons cannot be observed because they would violate the laws regarding the conservation of energy and momentum. Only in their veiled or hidden state do photons act as mediators of force between particles. The "force" caused by the exchange of virtual photons causes charged particles to change their velocity (speed and/or direction of travel) as they absorb or emit virtual photons.

Only under limited conditions do the photons escape the charged particles and thereby become observable as electro-magnetic radiation. Observable photons are created by perturbations (i.e., wave-like disruptions) of electrons and other charged particles. According to QED theory, the process also works in reverse as photons can create a particle and its antiparticle (e.g., an electron and its oppositely charged antiparticle, a positron).

In QED dynamics, the simplest interactions involve only two charged particles. The application of QED is, however, not limited to these simple systems; interactions involving an infinite number of photons are described by increasingly complex processes termed second-order (or higher) processes. Although QED can account for an infinite number of processes (i.e., an infinite number of interactions) the theory also dictates that more interactions also become increasingly rare as they become increasingly complex.

The genesis of QED was the need for physicists to reconcile theories initially advanced by British physicist **James Clerk Maxwell** regarding electromagnetism in the later half of the nineteenth century (i.e., that **electricity** and **magnetism** are two aspects of a single force) with quantum theory developed during the early decades of the twentieth century. Prior to WWII, British physicist Paul Dirac, German physicist Werner Heisenberg, and Austrian-born American physicist Wolfgang Pauli all made significant contributions to the mathematical foundations related to QED. Even for these experienced physicists, however, working with QED posed formidable obstacles because of the presence of "infinities" (infinite values) in the mathematical calculations (e.g., for emission rates or determinations of mass). It was often difficult to make predictions match observed phenomena and early attempts at using QED theory often gave physicists wrong or incomprehensible answers.

The calculations used to define QED were made more accessible and reliable by a process termed renormalization, independently developed by American physicist **Richard Feynman** (1918–1988), American physicist Julian Schwinger (1918–1994), and Japanese physicist Shin'ichiro Tomonaga (1906–1979). In essence the work of these three renowned scientists concentrated on making the needed corrections to Dirac's infinity problems and his advancement of QED theory, which helped reconcile quantum mechanics with Einstein's special theory of relativity. Their "renormalization" allowed positive infinities to cancel out negative infinities and thus, allowed measured values of mass and charge to be used in QED calculations.

The use of renormalization initially allowed QED predictions to accurately predict the observed interactions of electrons and photons. During the later half of the twentieth century, based principally on the work of Feynman, Schwinger, Tomonaga and another influential physicist Freeman Dyson, QED became an important model used to explain the structure, properties and reactions of quarks, gluons and other subatomic particles. Although Feynman, Schwinger, and Tomonaga each worked separately on the refinement of different aspects of QED theory, in 1965, these physicists jointly shared the Nobel Prize for their work.

Because QED is compatible with special relativity theory, and special relativity equations are part of QED equations, QED is termed a relativistic theory. QED is also termed a gauge-invariant theory, meaning that it makes accurate predictions regardless of where applied in **space** or time. Like **gravity** , QED mathematically describes a force that becomes weaker as the distance between charged particles increases, reducing in strength as the inverse square of the distance between particles. Although the photons themselves are electrically neutral, the predictions of interactions made possible by QED would not be possible between uncharged or electrically neutral particles. Accordingly, in QED theory there are two values for electric charge on particles, positive and negative.

QED theory was revolutionary in **physics** . In contrast to theories that strove to explain natural phenomena in terms of direct causes and effects, the development of QED stemmed from a growing awareness of the limitations on scientist's ability to make predictions regarding the subatomic realm. In fact, QED was unique precisely because QED did not always make specific predictions. QED relied instead on developing an understanding of the properties and behavior of subatomic particles characterized by probabilities rather than by traditional cause-and-effect certainties. Instead of allowing scientists to make specific predictions regarding the outcome of certain interactions—Tomonaga's predictions were often mystifyingly incompatible with human experience (e.g., that an electron could be in two places at once)—QED allowed the calculation of probabilities regarding outcomes (e.g., the probability that an electron would take one path as opposed to another).

In particular, Feynman's work, teaching, and contributions to QED theory reached near legendary status within the physics community. In 1986, Feynman published *QED: The Strange Theory of Light and Matter*. In his book, Feynman attempted to explain QED theory in much the same manner as Einstein's writing on relativity theory a half century earlier. In fact, although Feynman's profound contributions to QED theory were well beyond the understanding of the general public, no other physicist since Einstein and Oppenheimer had so captured the attention of the lay public. In addition, Feynman also became somewhat of a celebrity for chronicles relating to his life and studies.

Feynman's work redefined QED theory, quantum mechanics, and electrodynamics, and Feynman's writings remain the definitive explanation of QED theory. With regard to QED theory, Feynman is perhaps best remembered for his invention of simple diagrams, now widely known among physicists as "Feynman diagrams," to portray the complex interactions of atomic particles. The diagrams allow visual representation of the ways in which particles can interact by the exchange of virtual photons. In addition to providing a tangible picture of processes outside the human capacity for observation, Feynman's diagrams precisely portray the inter-actions of variables used in the complex QED mathematical calculations.

Schwinger and Tomonaga also refined the mathematical methodology of QED theory so that predictions became increasingly consistent with predictions of phenomena made by the special theory of relativity. Tomonaga also solved a perplexing inconsistency that vexed Dirac's work (e.g., that an electron could, inconceivably, and not in accord with observations, be calculated to have a seemingly infinite amount of energy). Tomonaga's mathematical improvements, along with refinements made by Schwinger and Feynman, resolved this incompatibility and allowed for the calculation of finite energies for electrons. In a master-stroke, Tomonaga renormalized and made more accurate the prediction of particle properties (e.g., magnetic properties) and the process of radiation.

QED went on to become, arguably, the best tested theory in science history. Most atomic interactions are electro-magnetic in nature and, no matter how accurate the equipment yet devised, the predictions made by renormalized QED theory hold true. Some tests of QED—for example, predictions of the mass of some subatomic particles—offer results accurate to six significant figures or more. Even with the improvements made by the renormalization of QED, however, the calculations often remain difficult. Although some predictions can be made using one Feynman diagram and a few pages of calculations, others may take hundreds of Feynman diagrams and the access to supercomputing facilities to complete the necessary calculations.

The development of QED theory allowed scientists to predict how subatomic particles are created or destroyed. Just as Feynman, Schwinger and Tomonaga's renormalization of QED allowed for calculation of finite properties relating to mass, energy, and charge-related properties of electrons, physicists hope that such improvements offer a model to improve other gauge theories (i.e., theories which explain how forces, such as the electroweak force, arise from underlying symmetries). The concept of forces such as electromagnetism arising from the exchange of virtual particles has intriguing ramifications for the advancement of theories regarding the working mechanisms underlying the strong, weak, and gravitational forces.

Many scientists assert that if a unified theory can be found, it will rest on the foundations established during the development of QED theory. Without speculation, however, is the fact that the development of QED theory was, and remains today, an essential element in the verification and development of quantum field theory.

** See also ** Atomic structure; Quantum theory and mechanics

# quantum electrodynamics

**quantum electrodynamics ( QED)** Use of quantum mechanics to study the properties of electromagnetic radiation and how it interacts with charged elementary particles. For example, the theory predicts that a collision between an electron and a proton should result in the production of a photon of electromagnetic radiation, which is exchanged between the colliding particles.

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