Artificial gravity

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Artificial_gravity 

Gemini 11 tethered in 1966 the GATV-5006 Agena target vehicle performing various tests including a first artificial gravity test in a microgravity environment.

Proposed Nautilus-X International space station centrifuge demo concept, 2011

Artificial gravity is the creation of an inertial force that mimics the effects of a gravitational force, usually by rotation. Artificial gravity, or rotational gravity, is thus the appearance of a centrifugal force in a rotating frame of reference (the transmission of centripetal acceleration via normal force in the non-rotating frame of reference), as opposed to the force experienced in linear acceleration, which by the equivalence principle is indistinguishable from gravity. In a more general sense, "artificial gravity" may also refer to the effect of linear acceleration, e.g. by means of a rocket engine.

Rotational simulated gravity has been used in simulations to help astronauts train for extreme conditions. Rotational simulated gravity has been proposed as a solution in human spaceflight to the adverse health effects caused by prolonged weightlessness. However, there are no current practical outer space applications of artificial gravity for humans due to concerns about the size and cost of a spacecraft necessary to produce a useful centripetal force comparable to the gravitational field strength on Earth (g). Scientists are concerned about the effect of such a system on the inner ear of the occupants. The concern is that using centripetal force to create artificial gravity will cause disturbances in the inner ear leading to nausea and disorientation. The adverse effects may prove intolerable for the occupants.

Centrifugal force

Further information: Absolute rotation and Bucket argument

Artificial gravity space station. 1969 NASA concept. A drawback is that the astronauts would be moving between higher gravity near the ends and lower gravity near the center.

In the context of a rotating space station, it is the radial force provided by the spacecraft's hull that acts as centripetal force. Thus, the "gravity" force felt by an object is the centrifugal force perceived in the rotating frame of reference as pointing "downwards" towards the hull.

By Newton's third law, the value of little g (the perceived "downward" acceleration) is equal in magnitude and opposite in direction to the centripetal acceleration. It was tested with satellites like Bion 3 (1975) and Bion 4 (1977); they both had centrifuges on board to put some specimens in an artificial gravity environment.

Differences from normal gravity

Balls in a rotating spacecraft

From the perspective of people rotating with the habitat, artificial gravity by rotation behaves similarly to normal gravity but with the following differences, which can be mitigated by increasing the radius of a space station.

• Centrifugal force varies with distance: Unlike real gravity, the apparent force felt by observers in the habitat pushes radially outward from the axis, and the centrifugal force is directly proportional to the distance from the axis of the habitat. With a small radius of rotation, a standing person's head would feel significantly less gravity than their feet. Likewise, passengers who move in a space station experience changes in apparent weight in different parts of the body.
• The Coriolis effect gives an apparent force that acts on objects that are moving relative to a rotating reference frame. This apparent force acts at right angles to the motion and the rotation axis and tends to curve the motion in the opposite sense to the habitat's spin. If an astronaut inside a rotating artificial gravity environment moves towards or away from the axis of rotation, they will feel a force pushing them in or against the direction of spin. These forces act on the semicircular canals of the inner ear and can cause dizziness. Lengthening the period of rotation (lower spin rate) reduces the Coriolis force and its effects. It is generally believed that at 2 rpm or less, no adverse effects from the Coriolis forces will occur, although humans have been shown to adapt to rates as high as 23 rpm.
• Changes in the rotation axis or rate of a spin would cause a disturbance in the artificial gravity field and stimulate the semicircular canals (refer to above). Any movement of mass within the station, including a movement of people, would shift the axis and could potentially cause a dangerous wobble. Thus, the rotation of a space station would need to be adequately stabilized, and any operations to deliberately change the rotation would need to be done slowly enough to be imperceptible. One possible solution to prevent the station from wobbling would be to use its liquid water supply as ballast which could be pumped between different sections of the station as required.

Speed in rpm for a centrifuge of a given radius to achieve a given g-force

Human spaceflight

The Gemini 11 mission attempted in 1966 to produce artificial gravity by rotating the capsule around the Agena Target Vehicle to which it was attached by a 36-meter tether. They were able to generate a small amount of artificial gravity, about 0.00015 g, by firing their side thrusters to slowly rotate the combined craft like a slow-motion pair of bolas. The resultant force was too small to be felt by either astronaut, but objects were observed moving towards the "floor" of the capsule.

Health benefits

Artificial gravity has been suggested for interplanetary journeys to Mars

Artificial gravity has been suggested as a solution to various health risks associated with spaceflight. In 1964, the Soviet space program believed that a human could not survive more than 14 days in space for fear that the heart and blood vessels would be unable to adapt to the weightless conditions. This fear was eventually discovered to be unfounded as spaceflights have now lasted up to 437 consecutive days, with missions aboard the International Space Station commonly lasting 6 months. However, the question of human safety in space did launch an investigation into the physical effects of prolonged exposure to weightlessness. In June 1991, the Spacelab Life Sciences 1 on the Space Shuttle flight STS-40 flight performed 18 experiments on two men and two women over nine days. In an environment without gravity, it was concluded that the response of white blood cells and muscle mass decreased. Additionally, within the first 24 hours spent in a weightless environment, blood volume decreased by 10%. Long periods of weightlessness can cause brain swelling and eyesight problems. Upon return to Earth, the effects of prolonged weightlessness continue to affect the human body as fluids pool back to the lower body, the heart rate rises, a drop in blood pressure occurs, and there is a reduced tolerance for exercise.

Artificial gravity, for its ability to mimic the behavior of gravity on the human body, has been suggested as one of the most encompassing manners of combating the physical effects inherent in weightless environments. Other measures that have been suggested as symptomatic treatments include exercise, diet, and Pingvin suits. However, criticism of those methods lies in the fact that they do not fully eliminate health problems and require a variety of solutions to address all issues. Artificial gravity, in contrast, would remove the weightlessness inherent in space travel. By implementing artificial gravity, space travelers would never have to experience weightlessness or the associated side effects. Especially in a modern-day six-month journey to Mars, exposure to artificial gravity is suggested in either a continuous or intermittent form to prevent extreme debilitation to the astronauts during travel.

Proposals

Rotating Mars spacecraft – 1989 NASA concept

Several proposals have incorporated artificial gravity into their design:

• Discovery II: a 2005 vehicle proposal capable of delivering a 172-metric-ton crew to Jupiter's orbit in 118 days. A very small portion of the 1,690-metric-ton craft would incorporate a centrifugal crew station.
• Multi-Mission Space Exploration Vehicle (MMSEV): a 2011 NASA proposal for a long-duration crewed space transport vehicle; it included a rotational artificial gravity space habitat intended to promote crew health for a crew of up to six persons on missions of up to two years in duration. The torus-ring centrifuge would utilize both standard metal-frame and inflatable spacecraft structures and would provide 0.11 to 0.69 g if built with the 40 feet (12 m) diameter option.
• ISS Centrifuge Demo: a 2011 NASA proposal for a demonstration project preparatory to the final design of the larger torus centrifuge space habitat for the Multi-Mission Space Exploration Vehicle. The structure would have an outside diameter of 30 feet (9.1 m) with a ring interior cross-section diameter of 30 inches (760 mm). It would provide 0.08 to 0.51 g partial gravity. This test and evaluation centrifuge would have the capability to become a Sleep Module for the ISS crew.

Artist's rendering of TEMPO³ in orbit

• Mars Direct: A plan for a crewed Mars mission created by NASA engineers Robert Zubrin and David Baker in 1990, later expanded upon in Zubrin's 1996 book The Case for Mars. The "Mars Habitat Unit", which would carry astronauts to Mars to join the previously launched "Earth Return Vehicle", would have had artificial gravity generated during flight by tying the spent upper stage of the booster to the Habitat Unit, and setting them both rotating about a common axis.
• The proposed Tempo3 mission rotates two halves of a spacecraft connected by a tether to test the feasibility of simulating gravity on a crewed mission to Mars.
• The Mars Gravity Biosatellite was a proposed mission meant to study the effect of artificial gravity on mammals. An artificial gravity field of 0.38 g (equivalent to Mars's surface gravity) was to be produced by rotation (32 rpm, radius of ca. 30 cm). Fifteen mice would have orbited Earth (Low Earth orbit) for five weeks and then land alive. However, the program was canceled on 24 June 2009, due to a lack of funding and shifting priorities at NASA.
• Vast Space is a private company that proposes to build the world's first artificial gravity space station using the rotating spacecraft concept.

Issues with implementation

Some of the reasons that artificial gravity remains unused today in spaceflight trace back to the problems inherent in implementation. One of the realistic methods of creating artificial gravity is the centrifugal effect caused by the centripetal force of the floor of a rotating structure pushing up on the person. In that model, however, issues arise in the size of the spacecraft. As expressed by John Page and Matthew Francis, the smaller a spacecraft (the shorter the radius of rotation), the more rapid the rotation that is required. As such, to simulate gravity, it would be better to utilize a larger spacecraft that rotates slowly.

The requirements on size about rotation are due to the differing forces on parts of the body at different distances from the axis of rotation. If parts of the body closer to the rotational axis experience a force that is significantly different from parts farther from the axis, then this could have adverse effects. Additionally, questions remain as to what the best way is to initially set the rotating motion in place without disturbing the stability of the whole spacecraft's orbit. At the moment, there is not a ship massive enough to meet the rotation requirements, and the costs associated with building, maintaining, and launching such a craft are extensive.

In general, with the small number of negative health effects present in today's typically shorter spaceflights, as well as with the very large cost of research for a technology which is not yet really needed, the present day development of artificial gravity technology has necessarily been stunted and sporadic.

As the length of typical space flights increases, the need for artificial gravity for the passengers in such lengthy spaceflights will most certainly also increase, and so will the knowledge and resources available to create such artificial gravity, most likely also increase. In summary, it is probably only a question of time, as to how long it might take before the conditions are suitable for the completion of the development of artificial gravity technology, which will almost certainly be required at some point along with the eventual and inevitable development of an increase in the average length of a spaceflight.

In science fiction

Several science fiction novels, films, and series have featured artificial gravity production.

• In the movie 2001: A Space Odyssey, a rotating centrifuge in the Discovery spacecraft provides artificial gravity to the astronauts within it. The entirety of Space Station 5 rotates to provide artificial 1g downforce in the shirtsleeve environment of its outer rings; the central docking hub remains closer to zero gravity.
• The 1999 television series Cowboy Bebop, a rotating ring in the Bebop spacecraft creates artificial gravity throughout the spacecraft.
• In the novel The Martian, the Hermes spacecraft achieves artificial gravity by design; it employs a ringed structure, at whose periphery forces around 40% of Earth's gravity are experienced, similar to Mars' gravity.
  • In the novel Project Hail Mary by the same author, weight on the titular ship Hail Mary is provided initially by engine thrust, as the ship is capable of constant acceleration up to 2 ɡ and is also able to separate, turn the crew compartment inwards, and rotate to produce 1 ɡ while in orbit.
• The movie Interstellar features a spacecraft called the Endurance that can rotate on its central axis to create artificial gravity, controlled by retro thrusters on the ship.
• The 2021 film Stowaway features the upper stage of a launch vehicle connected by 450-meter long tethers to the ship's main hull, acting as a counterweight for inertia-based artificial gravity.
• The series The Expanse utilizes both rotational gravity and linear thrust gravity in various space stations and spaceships. Notably, Tycho Station and the Generation ship LDSS Nauvoo use rotational gravity. Linear gravity is provided by a fictitious 'Epstein Drive', which killed its creator Solomon Epstein during its maiden flight due to high gravity injuries.
• In the television series For All Mankind, the space hotel Polaris, later renamed Phoenix after being purchased and converted into a space vessel by Helios Aerospace for their own Mars mission, features a wheel-like structure controlled by thrusters to create artificial gravity, whilst a central axial hub operates in zero gravity as a docking station.

Linear acceleration

Further information: Equivalence principle

Linear acceleration is another method of generating artificial gravity, by using the thrust from a spacecraft's engines to create the illusion of being under a gravitational pull. A spacecraft under constant acceleration in a straight line would have the appearance of a gravitational pull in the direction opposite to that of the acceleration, as the thrust from the engines would cause the spacecraft to "push" itself up into the objects and persons inside of the vessel, thus creating the feeling of weight. This is because of Newton's third law: the weight that one would feel standing in a linearly accelerating spacecraft would not be a true gravitational pull, but simply the reaction of oneself pushing against the craft's hull as it pushes back. Similarly, objects that would otherwise be free-floating within the spacecraft if it were not accelerating would "fall" towards the engines when it started accelerating, as a consequence of Newton's first law: the floating object would remain at rest, while the spacecraft would accelerate towards it, and appear to an observer within that the object was "falling".

To emulate artificial gravity on Earth, spacecraft using linear acceleration gravity may be built similar to a skyscraper, with its engines as the bottom "floor". If the spacecraft were to accelerate at the rate of 1 g—Earth's gravitational pull—the individuals inside would be pressed into the hull at the same force, and thus be able to walk and behave as if they were on Earth.

This form of artificial gravity is desirable because it could functionally create the illusion of a gravity field that is uniform and unidirectional throughout a spacecraft, without the need for large, spinning rings, whose fields may not be uniform, not unidirectional with respect to the spacecraft, and require constant rotation. This would also have the advantage of relatively high speed: a spaceship accelerating at 1 g, 9.8 m/s², for the first half of the journey, and then decelerating for the other half, could reach Mars within a few days. Similarly, a hypothetical space travel using constant acceleration of 1 g for one year would reach relativistic speeds and allow for a round trip to the nearest star, Proxima Centauri. As such, low-impulse but long-term linear acceleration has been proposed for various interplanetary missions. For example, even heavy (100 ton) cargo payloads to Mars could be transported to Mars in 27 months and retain approximately 55 percent of the LEO vehicle mass upon arrival into a Mars orbit, providing a low-gravity gradient to the spacecraft during the entire journey.

This form of gravity is not without challenges, however. At present, the only practical engines that could propel a vessel fast enough to reach speeds comparable to Earth's gravitational pull require chemical reaction rockets, which expel reaction mass to achieve thrust, and thus the acceleration could only last for as long as a vessel had fuel. The vessel would also need to be constantly accelerating and at a constant speed to maintain the gravitational effect, and thus would not have gravity while stationary, and could experience significant swings in g-forces if the vessel were to accelerate above or below 1 g. Further, for point-to-point journeys, such as Earth-Mars transits, vessels would need to constantly accelerate for half the journey, turn off their engines, perform a 180° flip, reactivate their engines, and then begin decelerating towards the target destination, requiring everything inside the vessel to experience weightlessness and possibly be secured down for the duration of the flip.

A propulsion system with a very high specific impulse (that is, good efficiency in the use of reaction mass that must be carried along and used for propulsion on the journey) could accelerate more slowly producing useful levels of artificial gravity for long periods of time. A variety of electric propulsion systems provide examples. Two examples of this long-duration, low-thrust, high-impulse propulsion that have either been practically used on spacecraft or are planned in for near-term in-space use are Hall effect thrusters and Variable Specific Impulse Magnetoplasma Rockets (VASIMR). Both provide very high specific impulse but relatively low thrust, compared to the more typical chemical reaction rockets. They are thus ideally suited for long-duration firings which would provide limited amounts of, but long-term, milli-g levels of artificial gravity in spacecraft.

In a number of science fiction plots, acceleration is used to produce artificial gravity for interstellar spacecraft, propelled by as yet theoretical or hypothetical means.

This effect of linear acceleration is well understood, and is routinely used for 0 g cryogenic fluid management for post-launch (subsequent) in-space firings of upper stage rockets.

Roller coasters, especially launched roller coasters or those that rely on electromagnetic propulsion, can provide linear acceleration "gravity", and so can relatively high acceleration vehicles, such as sports cars. Linear acceleration can be used to provide air-time on roller coasters and other thrill rides.

Simulating lunar gravity

See also: Gravitation of the Moon § Simulating lunar gravity

In January 2022, China was reported by the South China Morning Post to have built a small (60 centimetres (24 in) diameter) research facility to simulate low lunar gravity with the help of magnets. The facility was reportedly partly inspired by the work of Andre Geim (who later shared the 2010 Nobel Prize in Physics for his research on graphene) and Michael Berry, who both shared the Ig Nobel Prize in Physics in 2000 for the magnetic levitation of a frog.

Graviton control or generator

Main article: Graviton

Speculative or fictional mechanisms

Main articles: Gravitational shielding and Anti-gravity

In science fiction, artificial gravity (or cancellation of gravity) or "paragravity" is sometimes present in spacecraft that are neither rotating nor accelerating. At present, there is no confirmed technique as such that can simulate gravity other than actual rotation or acceleration. There have been many claims over the years of such a device. Eugene Podkletnov, a Russian engineer, has claimed since the early 1990s to have made such a device consisting of a spinning superconductor producing a powerful "gravitomagnetic field." In 2006, a research group funded by ESA claimed to have created a similar device that demonstrated positive results for the production of gravitomagnetism, although it produced only 0.0001 g.

Anti-gravity

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Anti-gravity

Artistic depiction of a fictional anti-gravity vehicle

Anti-gravity (also known as non-gravitational field) is the phenomenon of creating a place or object that is free from the force of gravity. It does not refer to either the lack of weight under gravity experienced in free fall or orbit, or to balancing the force of gravity with some other force, such as electromagnetism or aerodynamic lift. Anti-gravity is a recurring concept in science fiction.

"Anti-gravity" is often used to refer to devices that look as if they reverse gravity even though they operate through other means, such as lifters, which fly in the air by moving air with electromagnetic fields.

Historical attempts at understanding gravity

Main article: History of gravitational theory

The possibility of creating anti-gravity depends upon a complete understanding and description of gravity and its interactions with other physical theories, such as general relativity and quantum mechanics; however, no quantum theory of gravity has yet been found.

During the summer of 1666, Isaac Newton observed an apple falling from the tree in his garden, thus realizing the principle of universal gravitation. Albert Einstein in 1915 considered the physical interaction between matter and space, where gravity occurs as a consequence of matter causing a geometric deformation of spacetime which is otherwise flat. Einstein, both independently and with Walther Mayer, attempted to unify his theory of gravity with electromagnetism using the work of Theodor Kaluza and James Clerk Maxwell to link gravity and quantum field theory.

Theoretical quantum physicists have postulated the existence of a quantum gravity particle, the graviton. Various theoretical explanations of quantum gravity have been created, including superstring theory, loop quantum gravity, E8 theory and asymptotic safety theory amongst many others.

Probable solutions

In Newton's law of universal gravitation, gravity was an external force transmitted by unknown means. In the 20th century, Newton's model was replaced by general relativity where gravity is not a force but the result of the geometry of spacetime. Under general relativity, anti-gravity is impossible except under contrived circumstances.

Gravity shields

A monument at Babson College dedicated to Roger Babson for research into anti-gravity and partial gravity insulators

Main article: Gravitational shielding

In 1948 businessman Roger Babson (founder of Babson College) formed the Gravity Research Foundation to study ways to reduce the effects of gravity. Their efforts were initially somewhat "crankish", but they held occasional conferences that drew such people as Clarence Birdseye, known for his frozen-food products, and helicopter pioneer Igor Sikorsky. Over time the Foundation turned its attention away from trying to control gravity, to simply better understanding it. The Foundation nearly disappeared after Babson's death in 1967. However, it continues to run an essay award, offering prizes of up to $4,000. As of 2017, it is still administered out of Wellesley, Massachusetts, by George Rideout Jr., son of the foundation's original director. Winners include California astrophysicist George F. Smoot (1993), who later won the 2006 Nobel Prize in Physics, and Gerard 't Hooft (2015) who previously won the 1999 Nobel Prize in Physics.

General relativity research in the 1950s

General relativity was introduced in the 1910s, but development of the theory was greatly slowed by a lack of suitable mathematical tools. It appeared that anti-gravity was outlawed under general relativity.

It is claimed the US Air Force also ran a study effort throughout the 1950s and into the 1960s. Former Lieutenant Colonel Ansel Talbert wrote two series of newspaper articles claiming that most of the major aviation firms had started gravity control propulsion research in the 1950s. However, there is no outside confirmation of these stories, and since they take place in the midst of the policy by press release era, it is not clear how much weight these stories should be given.

It is known that there were serious efforts underway at the Glenn L. Martin Company, who formed the Research Institute for Advanced Study. Major newspapers announced the contract that had been made between theoretical physicist Burkhard Heim and the Glenn L. Martin Company. Another effort in the private sector to master understanding of gravitation was the creation of the Institute for Field Physics, University of North Carolina at Chapel Hill in 1956, by Gravity Research Foundation trustee Agnew H. Bahnson.

Military support for anti-gravity projects was terminated by the Mansfield Amendment of 1973, which restricted Department of Defense spending to only the areas of scientific research with explicit military applications. The Mansfield Amendment was passed specifically to end long-running projects that had no results.

Under general relativity, gravity is the result of following spatial geometry (change in the normal shape of space) caused by local mass-energy. This theory holds that it is the altered shape of space, deformed by massive objects, that causes gravity, which is actually a property of deformed space rather than being a true force. Although the equations cannot normally produce a "negative geometry", it is possible to do so by using "negative mass". The same equations do not, of themselves, rule out the existence of negative mass.

Both general relativity and Newtonian gravity appear to predict that negative mass would produce a repulsive gravitational field. In particular, Sir Hermann Bondi proposed in 1957 that negative gravitational mass, combined with negative inertial mass, would comply with the strong equivalence principle of general relativity theory and the Newtonian laws of conservation of linear momentum and energy. Bondi's proof yielded singularity-free solutions for the relativity equations. In July 1988, Robert L. Forward presented a paper at the AIAA/ASME/SAE/ASEE 24th Joint Propulsion Conference that proposed a Bondi negative gravitational mass propulsion system.

Bondi pointed out that a negative mass will fall toward (and not away from) "normal" matter, since although the gravitational force is repulsive, the negative mass (according to Newton's law, F=ma) responds by accelerating in the opposite of the direction of the force. Normal mass, on the other hand, will fall away from the negative matter. He noted that two identical masses, one positive and one negative, placed near each other will therefore self-accelerate in the direction of the line between them, with the negative mass chasing after the positive mass. Notice that because the negative mass acquires negative kinetic energy, the total energy of the accelerating masses remains at zero. Forward pointed out that the self-acceleration effect is due to the negative inertial mass, and could be seen induced without the gravitational forces between the particles.

The Standard Model of particle physics, which describes all currently known forms of matter, does not include negative mass. Although cosmological dark matter may consist of particles outside the Standard Model whose nature is unknown, their mass is ostensibly known – since they were postulated from their gravitational effects on surrounding objects, which implies their mass is positive. The proposed cosmological dark energy, on the other hand, is more complicated, since according to general relativity the effects of both its energy density and its negative pressure contribute to its gravitational effect.

Unique force

Under general relativity any form of energy couples with spacetime to create the geometries that cause gravity. A longstanding question was whether or not these same equations applied to antimatter. The issue was considered solved in 1960 with the development of CPT symmetry, which demonstrated that antimatter follows the same laws of physics as "normal" matter, and therefore has positive energy content and also causes (and reacts to) gravity like normal matter (see gravitational interaction of antimatter).

For much of the last quarter of the 20th century, the physics community was involved in attempts to produce a unified field theory, a single physical theory that explains the four fundamental forces: gravity, electromagnetism, and the strong and weak nuclear forces. Scientists have made progress in unifying the three quantum forces, but gravity has remained "the problem" in every attempt. This has not stopped any number of such attempts from being made, however.

Generally these attempts tried to "quantize gravity" by positing a particle, the graviton, that carried gravity in the same way that photons (light) carry electromagnetism. Simple attempts along this direction all failed, however, leading to more complex examples that attempted to account for these problems. Two of these, supersymmetry and the relativity related supergravity, both required the existence of an extremely weak "fifth force" carried by a graviphoton, which coupled together several "loose ends" in quantum field theory, in an organized manner. As a side effect, both theories also all but required that antimatter be affected by this fifth force in a way similar to anti-gravity, dictating repulsion away from mass. Several experiments were carried out in the 1990s to measure this effect, but none yielded positive results.

In 2013 CERN looked for an antigravity effect in an experiment designed to study the energy levels within antihydrogen. The antigravity measurement was just an "interesting sideshow" and was inconclusive.

Breakthrough Propulsion Physics Program

During the close of the twentieth century NASA provided funding for the Breakthrough Propulsion Physics Program (BPP) from 1996 through 2002. This program studied a number of "far out" designs for space propulsion that were not receiving funding through normal university or commercial channels. Anti-gravity-like concepts were investigated under the name "diametric drive". The work of the BPP program continues in the independent, non-NASA affiliated Tau Zero Foundation.

Empirical claims and commercial efforts

There have been a number of attempts to build anti-gravity devices, and a small number of reports of anti-gravity-like effects in the scientific literature. None of the examples that follow are accepted as reproducible examples of anti-gravity.

Gyroscopic devices

A "kinemassic field" generator from U.S. patent 3,626,605: Method and apparatus for generating a secondary gravitational force field

Gyroscopes produce a force when twisted that operates "out of plane" and can appear to lift themselves against gravity. Although this force is well understood to be illusory, even under Newtonian models, it has nevertheless generated numerous claims of anti-gravity devices and any number of patented devices. None of these devices has ever been demonstrated to work under controlled conditions, and they have often become the subject of conspiracy theories as a result.

Another "rotating device" example is shown in a series of patents granted to Henry Wallace between 1968 and 1974. His devices consist of rapidly spinning disks of brass, a material made up largely of elements with a total half-integer nuclear spin. He claimed that by rapidly rotating a disk of such material, the nuclear spin became aligned, and as a result created a "gravitomagnetic" field in a fashion similar to the magnetic field created by the Barnett effect. No independent testing or public demonstration of these devices is known.

In 1989, it was reported that a weight decreases along the axis of a right spinning gyroscope. A test of this claim a year later yielded null results. A recommendation was made to conduct further tests at a 1999 AIP conference.

Thomas Townsend Brown's gravitator

Further information: Biefeld-Brown effect and Electrogravitics

In 1921, while still in high school, Thomas Townsend Brown found that a high-voltage Coolidge tube seemed to change mass depending on its orientation on a balance scale. Through the 1920s Brown developed this into devices that combined high voltages with materials with high dielectric constants (essentially large capacitors); he called such a device a "gravitator". Brown made the claim to observers and in the media that his experiments were showing anti-gravity effects. Brown would continue his work and produced a series of high-voltage devices in the following years in attempts to sell his ideas to aircraft companies and the military. He coined the names Biefeld–Brown effect and electrogravitics in conjunction with his devices. Brown tested his asymmetrical capacitor devices in a vacuum, supposedly showing it was not a more down-to-earth electrohydrodynamic effect generated by high voltage ion flow in air.

Electrogravitics is a popular topic in ufology, anti-gravity, free energy, with government conspiracy theorists and related websites, in books and publications with claims that the technology became highly classified in the early 1960s and that it is used to power UFOs and the B-2 bomber. There is also research and videos on the internet purported to show lifter-style capacitor devices working in a vacuum, therefore not receiving propulsion from ion drift or ion wind being generated in air.

Follow-up studies on Brown's work and other claims have been conducted by R. L. Talley in a 1990 US Air Force study, NASA scientist Jonathan Campbell in a 2003 experiment, and Martin Tajmar in a 2004 paper. They have found that no thrust could be observed in a vacuum and that Brown's and other ion lifter devices produce thrust along their axis regardless of the direction of gravity consistent with electrohydrodynamic effects.

Gravitoelectric coupling

In 1992, the Russian researcher Eugene Podkletnov claimed to have discovered, whilst experimenting with superconductors, that a fast rotating superconductor reduces the gravitational effect. Many studies have attempted to reproduce Podkletnov's experiment, always to negative results.

Douglas Torr, of the University of Alabama in Huntsville proposed how a time-dependent magnetic field could cause the spins of the lattice ions in a superconductor to generate detectable gravitomagnetic and gravitoelectric fields in a series of papers published between 1991 and 1993. In 1999, a Miss Li appeared in Popular Mechanics, claiming to have constructed a working prototype to generate what she described as "AC Gravity." No further evidence of this prototype has been offered.

Douglas Torr and Timir Datta were involved in the development of a "gravity generator" at the University of South Carolina. According to a leaked document from the Office of Technology Transfer at the University of South Carolina and confirmed to Wired reporter Charles Platt in 1998, the device would create a "force beam" in any desired direction and the university planned to patent and license this device. No further information about this university research project or the "Gravity Generator" device was ever made public.

Göde Award

The Institute for Gravity Research of the Göde Scientific Foundation has tried to reproduce many of the different experiments which claim any "anti-gravity" effects. All attempts by this group to observe an anti-gravity effect by reproducing past experiments have been unsuccessful thus far. The foundation has offered a reward of one million euros for a reproducible anti-gravity experiment.

In fiction

The existence of anti-gravity is a common theme in science fiction. The Encyclopedia of Science Fiction lists Francis Godwin's posthumously-published 1638 novel The Man in the Moone, where a "semi-magical" stone has the power to make gravity stronger or weaker, as the earliest variation of the theme. The first story to use anti-gravity for the purpose of space travel, as well as the first to treat the subject from a scientific rather than supernatural angle, was George Tucker's 1827 novel A Voyage to the Moon.

Apergy

Apergy is a term for a fictitious form of anti-gravitational energy first used by Percy Greg in his 1880 sword and planet novel Across the Zodiac. The term was later adopted by other fiction authors such as John Jacob Astor IV in his 1894 science fiction novel A Journey in Other Worlds, and it also appeared outside of explicit fiction writing.

Quantum electrodynamics

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Quantum_electrodynamics

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. It is the most precise and stringently tested theory in physics.

History

Main articles: History of quantum mechanics and History of quantum field theory

Paul Dirac

The first formulation of a quantum theory describing radiation and matter interaction is attributed to Paul Dirac, who during the 1920s computed the coefficient of spontaneous emission of an atom. He is credited with coining the term "quantum electrodynamics".

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 Enrico Fermi, physicists came to believe that, in principle, it was 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 doubt on the theory's internal consistency. This suggested that special relativity and quantum mechanics were fundamentally incompatible.

Hans Bethe

Difficulties 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, later known as the Lamb shift and magnetic moment of the electron. These experiments exposed discrepancies that the theory was unable to explain.

A first indication of a possible solution was given by Hans Bethe in 1947. He made the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured by Willis Lamb and Robert Retherford. Despite 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 with good experimental agreement. This procedure was named renormalization.

Feynman (center) and Oppenheimer (right) at Los Alamos.

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 produce fully covariant formulations that were finite at any order in a perturbation series of quantum electrodynamics. Tomonaga, Schwinger, and Feynman were jointly awarded the 1965 Nobel Prize in Physics for their work in this area. Their contributions, and Dyson's, 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 unlike the field-theoretic, operator-based approach of Schwinger and Tomonaga, but 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, became one of the fundamental aspects of quantum field theory and is seen as a criterion for a theory's general acceptability. Even though renormalization works well in practice, Feynman was never entirely comfortable with its mathematical validity, referring to renormalization as a "shell game" and "hocus pocus".

Neither Feynman nor Dirac were happy with that way to approach the observations made in theoretical physics, above all in quantum mechanics.

QED is 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, developed by H. David Politzer, Sidney Coleman, David Gross and Frank Wilczek. Building on Schwinger's pioneering work, 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 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.

Feynman diagram elements

These actions are represented in the form of visual shorthand by the three basic elements of 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 ${\displaystyle P(A\text{ to }B)}$, and it depends on only the momentum and polarization of the photon. The similar quantity for an electron moving from ${\displaystyle C}$ to ${\displaystyle D}$ is written ${\displaystyle E(C\text{ to }D)}$. It depends on the momentum and polarization of the electron, in addition to a constant Feynman calls n, sometimes called the "bare" mass of the electron: it is related to, but not the same as, the measured electron mass. Finally, the quantity that tells us about the probability amplitude for an electron to emit or absorb a photon Feynman calls j, and is sometimes called the "bare" charge of the electron: it is a constant, and 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:

1. If an event can occur via a number of indistinguishable alternative processes (a.k.a. "virtual" processes), then its probability amplitude is the sum of the probability amplitudes of the alternatives.
2. If a virtual process involves a number of independent or concomitant sub-processes, then the probability amplitude of the total (compound) process is the product of the probability amplitudes of the sub-processes.

The indistinguishability criterion in (a) is very important: it means that there is no observable feature present in the given system that in any way "reveals" which alternative is taken. In such a case, one cannot observe which alternative actually takes place without changing the experimental setup in some way (e.g. by introducing a new apparatus into the system). Whenever one is able to observe which alternative takes place, one always finds that the probability of the event is the sum of the probabilities of the alternatives. Indeed, if this were not the case, the very term "alternatives" to describe these processes would be inappropriate. What (a) says is that once the physical means for observing which alternative occurred is removed, one cannot still say that the event is occurring through "exactly one of the alternatives" in the sense of adding probabilities; one must add the amplitudes instead.

Similarly, the independence criterion in (b) is very important: it only applies to processes which are not "entangled".

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.

Compton scattering

But there are other ways in which the 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.

An infinite number of other intermediate "virtual" processes exist in which photons are absorbed or emitted. For each of these processes, a Feynman diagram could be drawn 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 nonzero probability amplitude 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

Feynman replaces complex numbers with spinning arrows, which start at emission and end at detection of a particle. The sum of all resulting arrows gives a final arrow whose length squared equals the probability of the event. In this diagram, light emitted by the source S can reach the detector at P by bouncing off the mirror (in blue) at various points. Each one of the paths has an arrow associated with it (whose direction changes uniformly with the time taken for the light to traverse the path). To correctly calculate the total probability for light to reach P starting at S, one needs to sum the arrows for all such paths. 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. 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 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:

${\displaystyle P(A\text{ to }B)\to D_F(x_B-x_A),E(C\text{ to }D)\to S_F(x_D-x_C),}$

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.

Mass renormalization

Main article: Self-energy

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", and Dirac also criticized this procedure, saying "in mathematics one does not get rid of infinities when it does not please you".

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."

Mathematical formulation

QED action

Mathematically, QED is an abelian gauge theory with the symmetry group U(1), defined on Minkowski space (flat spacetime). 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 in natural units gives rise to the action

QED Action

${\displaystyle S_{\text{QED}}=\int d^{4}x\left[-\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }+\overline{\psi }(i{\gamma }^{\mu }{D}_{\mu }-m)\psi \right]}$

where

• ${\displaystyle {\gamma }^{\mu }}$ are Dirac matrices.
• ${\displaystyle \psi }$ a bispinor field of spin-1/2 particles (e.g. electron–positron field).
• ${\displaystyle \overline{\psi }\equiv {\psi }^{†}{\gamma }^0}$, called "psi-bar", is sometimes referred to as the Dirac adjoint.
• ${\displaystyle D_{\mu }\equiv {\mathrm{\partial }}_{\mu }+ie{A}_{\mu }+ie{B}_{\mu }}$ is the gauge covariant derivative.
  • e is the coupling constant, equal to the electric charge of the bispinor field.
  • ${\displaystyle A_{\mu }}$ is the covariant four-potential of the electromagnetic field generated by the electron itself. It is also known as a gauge field or a ${\displaystyle \text{U}(1)}$ connection.
  • ${\displaystyle B_{\mu }}$ is the external field imposed by external source.
• m is the mass of the electron or positron.
• ${\displaystyle F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }}$ is the electromagnetic field tensor. This is also known as the curvature of the gauge field.

Expanding the covariant derivative reveals a second useful form of the Lagrangian (external field ${\displaystyle B_{\mu }}$ set to zero for simplicity)

${\displaystyle \mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\overline{\psi }(i{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }-m)\psi -e{j}^{\mu }{A}_{\mu }}$

where ${\displaystyle j^{\mu }}$ is the conserved ${\displaystyle \text{U}(1)}$ current arising from Noether's theorem. It is written

${\displaystyle j^{\mu }=\overline{\psi }{\gamma }^{\mu }\psi .}$

Equations of motion

Expanding the covariant derivative in the Lagrangian gives

${\displaystyle \mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+i\overline{\psi }{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }\psi -e\overline{\psi }{\gamma }^{\mu }{A}_{\mu }\psi -m\overline{\psi }\psi }$

${\displaystyle =-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+i\overline{\psi }{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }\psi -m\overline{\psi }\psi -e{j}^{\mu }{A}_{\mu }.}$

For simplicity, ${\displaystyle B_{\mu }}$ has been set to zero, with no loss of generality. Alternatively, we can absorb ${\displaystyle B_{\mu }}$ into a new gauge field ${\displaystyle A_{\mu }^{\prime }=A_{\mu }+B_{\mu }}$ and relabel the new field as ${\displaystyle A_{\mu }.}$

From this Lagrangian, the equations of motion for the ${\displaystyle \psi }$ and ${\displaystyle A_{\mu }}$ fields can be obtained.

Equation of motion for ψ

These arise most straightforwardly by considering the Euler-Lagrange equation for ${\displaystyle \overline{\psi }}$. Since the Lagrangian contains no ${\displaystyle {\mathrm{\partial }}_{\mu }\overline{\psi }}$ terms, we immediately get

${\displaystyle \frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\mu }\overline{\psi })}=0}$

so the equation of motion can be written ${\displaystyle (i{\gamma }^{\mu }{\mathrm{\partial }}_{\mu }-m)\psi =e{\gamma }^{\mu }{A}_{\mu }\psi .}$

Equation of motion for A_μ

• Using the Euler–Lagrange equation for the ${\displaystyle A_{\mu }}$ field,

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

3

the derivatives this time are $${\displaystyle {\mathrm{\partial }}_{\nu }\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\nu }{A}_{\mu })}\right)={\mathrm{\partial }}_{\nu }\left({\mathrm{\partial }}^{\mu }{A}^{\nu }-{\mathrm{\partial }}^{\nu }{A}^{\mu }\right),}$$ $${\displaystyle \frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }{A}_{\mu }}=-e\overline{\psi }{\gamma }^{\mu }\psi .}$$

Substituting back into (3) leads to

${\displaystyle {\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=e\overline{\psi }{\gamma }^{\nu }\psi }$

which can be written in terms of the ${\displaystyle \text{U}(1)}$ current ${\displaystyle j^{\mu }}$ as

${\displaystyle {\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=e{j}^{\nu }.}$

Now, if we impose the Lorenz gauge condition $${\displaystyle {\mathrm{\partial }}_{\mu }{A}^{\mu }=0,}$$ the equations reduce to $${\displaystyle ◻A^{\mu }=e{j}^{\mu },}$$ 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 wave operator, ${\displaystyle ◻={\mathrm{\partial }}_{\mu }{\mathrm{\partial }}^{\mu }}$.)

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 ${\displaystyle |i⟩}$ will give a final state ${\displaystyle ⟨f|}$ in such a way to have

$${\displaystyle M_{fi}=⟨f|U|i⟩.}$$

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:

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

Which can also be written in terms of an integral over the interaction Hamiltonian density ${\displaystyle {\mathcal{H}}_I=e\overline{\psi }{\gamma }^{\mu }\psi A_{\mu }}$. Thus, one has

$${\displaystyle U=T\exp \left[-\frac{i}{\hslash }{\int }_{t_0}^{t}d{t}^{\prime }V(t^{\prime })\right],}$$

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 expansion of the probability amplitude ${\displaystyle M_{fi}}$ is called the Dyson series, and is given by:

$${\displaystyle M_{fi}=⟨f|U|i⟩=⟨f\left|\sum _{n=0}^{\mathrm{\infty }}\frac{(-i{)}^n}{n!}\int d^4{x}_1\cdots \int d^4{x}_{n}T\{\mathcal{H}(x_1)\cdots \mathcal{H}(x_n)\}\right|i⟩}$$

Feynman diagrams

Despite the conceptual clarity of the 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

To these rules we must add a further one for closed loops that implies an integration on momenta $\int d^{4}p/(2\pi {)}^4$, since these internal ("virtual") particles are not constrained to any specific energy–momentum, even that usually required by special relativity (see Propagator for details). The signature of the metric ${\displaystyle {\eta }_{\mu \nu }}$ is ${\displaystyle \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(+---)}$.

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:

$${\displaystyle M_{fi}=(ie{)}^2\overline{u}({\overrightarrow{p}}^{\prime },s^{\prime })ϵ/{}^{\prime }({\overrightarrow{k}}^{\prime },{\lambda }^{\prime }{)}^{\ast }\frac{p/+k/+m_e}{(p+k{)}^2-m_e^2}ϵ/(\overrightarrow{k},\lambda )u(\overrightarrow{p},s)+(ie{)}^2\overline{u}({\overrightarrow{p}}^{\prime },s^{\prime })ϵ/(\overrightarrow{k},\lambda )\frac{p/-k{/}^{\prime }+m_e}{(p-k^{\prime }{)}^2-m_e^2}ϵ/{}^{\prime }({\overrightarrow{k}}^{\prime },{\lambda }^{\prime }{)}^{\ast }u(\overrightarrow{p},s),}$$

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

Nonperturbative phenomena

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, 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.

Renormalizability

Higher-order terms can be straightforwardly computed for the evolution operator, but these terms display diagrams containing the following simpler ones

• 
  One-loop contribution to the vacuum polarization function ${\displaystyle \mathrm{\Pi }}$
• 
  One-loop contribution to the electron self-energy function ${\displaystyle \mathrm{\Sigma }}$
• 
  One-loop contribution to the vertex function ${\displaystyle \mathrm{\Gamma }}$

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. 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 only conjectural and 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 is 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.

Electrodynamics in curved spacetime

See also: Maxwell's equations in curved spacetime and Dirac equation in curved spacetime

This theory can be extended, at least as a classical field theory, to curved spacetime. This arises similarly to the flat spacetime case, from coupling a free electromagnetic theory to a free fermion theory and including an interaction which promotes the partial derivative in the fermion theory to a gauge-covariant derivative.