Geochemistry of carbon

From Wikipedia, the free encyclopedia

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

The geochemistry of carbon is the study of the transformations involving the element carbon within the systems of the Earth. To a large extent this study is organic geochemistry, but it also includes the very important carbon dioxide. Carbon is transformed by life, and moves between the major phases of the Earth, including the water bodies, atmosphere, and the rocky parts. Carbon is important in the formation of organic mineral deposits, such as coal, petroleum or natural gas. Most carbon is cycled through the atmosphere into living organisms and then respirated back into the atmosphere. However an important part of the carbon cycle involves the trapping of living matter into sediments. The carbon then becomes part of a sedimentary rock when lithification happens. Human technology or natural processes such as weathering, or underground life or water can return the carbon from sedimentary rocks to the atmosphere. From that point it can be transformed in the rock cycle into metamorphic rocks, or melted into igneous rocks. Carbon can return to the surface of the Earth by volcanoes or via uplift in tectonic processes. Carbon is returned to the atmosphere via volcanic gases. Carbon undergoes transformation in the mantle under pressure to diamond and other minerals, and also exists in the Earth's outer core in solution with iron, and may also be present in the inner core.

Carbon can form a huge variety stable compounds. It is an essential component of living matter. Living organisms can live in a limited range of conditions on the Earth that are limited by temperature and the existence of liquid water. The potential habitability of other planets or moons can also be assessed by the existence of liquid water.

Carbon makes up only 0.08% of the combination of the lithosphere, hydrosphere, and atmosphere. Yet it is the twelfth most common element there. In the rock of the lithosphere, carbon commonly occurs as carbonate minerals containing calcium or magnesium. It is also found as fossil fuels in coal and petroleum and gas. Native forms of carbon are much rarer, requiring pressure to form. Pure carbon exists as graphite or diamond.

The deeper parts of Earth such as the mantle are very hard to discover. Few samples are known, in the form of uplifted rocks, or xenoliths. Even fewer remain in the same state they were in where the pressure and temperature is much higher. Some diamonds retain inclusions held at pressures they were formed at, but the temperature is much lower at the surface. Iron meteorites may represent samples of the core of an asteroid, but it would have formed under different conditions to the Earth's core. Therefore, experimental studies are conducted in which minerals or substances are compressed and heated to determine what happens in similar conditions to the planetary interior.

The two common isotopes of carbon are stable. On Earth, carbon 12, ¹²C is by far the most common at 98.894%. Carbon 13 is much rarer averaging 1.106%. This percentage can vary slightly and its value is important in isotope geochemistry whereby the origin of the carbon is suggested.

Origins

Formation

Carbon can be produced in stars at least as massive as the Sun by fusion of three helium-4 nuclei: ⁴He + ⁴He + ⁴He --> ¹²C. This is the triple alpha process. In stars as massive as the Sun, carbon-12 is also converted to carbon-13 and then onto nitrogen-14 by fusion with protons. ¹²C + ¹H --> ¹³C + e⁺. ¹³C + ¹H --> ¹⁴N. In more massive stars, two carbon nuclei can fuse to magnesium, or a carbon and an oxygen to sulfur.

Astrochemistry

In molecular clouds, simple carbon molecules are formed, including carbon monoxide and dicarbon. Reactions with the trihydrogen cation of the simple carbon molecules yield carbon containing ions that readily react to form larger organic molecules. Carbon compounds that exist as ions, or isolated gas molecules in the interstellar medium, can condense onto dust grains. Carbonaceous dust grains consist mostly of carbon. Grains can stick together to form larger aggregates.

Earth formation

Meteorites and interplanetary dust shows the composition of solid material at the start of the Solar System, as they have not been modified since its formation. Carbonaceous chondrites are meteorites with around 5% carbon compounds. Their composition resembles the Sun's minus the very volatile elements like hydrogen and noble gases. The Earth is believed to have formed by the gravitational collapse of material like meteorites.

Important effects on Earth in the first Hadian Era include strong solar winds during the T-Tauri stage of the Sun. The Moon forming impact caused major changes to the surface. Juvenile volatiles outgased from the early molten surface of the Earth. These included carbon dioxide and carbon monoxide. The emissions probably did not include methane, but the Earth was probably free of molecular oxygen. The Late Heavy Bombardment was between 4.0 and 3.8 billion years ago (Ga). To start with, the Earth did not have a crust as it does today. Plate tectonics in its present form commenced about 2.5 Ga.

Early sedimentary rocks formed under water date to 3.8 Ga. Pillow lavas dating from 3.5 Ga prove the existence of oceans. Evidence of early life is given by fossils of stromatolites, and later by chemical tracers.

Organic matter continues to be added to the Earth from space via interplanetary dust, which also includes some interstellar particles. The amounts added to the Earth were around 60,000 tonnes per year about 4 Ga.

Isotope

Biological sequestration of carbon causes enrichment of carbon-12, so that substances that originate from living organisms have a higher carbon-12 content. Due to the kinetic isotope effect, chemical reactions can happen faster with lighter isotopes, so that photosynthesis fixes lighter carbon-12 faster than carbon-13. Also lighter isotopes diffuse across a biological membrane faster. Enrichment in carbon 13 is measured by delta ¹³C(o/oo) = [(¹³C/¹²C)sample/(¹³C/¹²C)standard - 1] * 1000. The common standard for carbon is Cretaceous Peedee formation belemnite.

Stereoisomers

Complex molecules, in particular those containing carbon can be in the form of stereoisomers. With abiotic processes they would be expected to be equally likely, but in carbonaceous chondrites this is not the case. The reasons for this are unknown.

Crust

The outer layer of the Earth, the crust along with its outer layers contain about 10²⁰ kg of carbon. This is enough for each square meter of the surface to have 200 tons of carbon.

Sedimentation

Carbon added to sedimentary rocks can take the form of carbonates, or organic carbon compounds. In order of source quantity the organic carbon comes from phytoplankton, plants, bacteria and zooplankton. However terrestrial sediments may be mostly from higher plants, and some oxygen deficient sediments from water may be mostly bacteria. Fungi and other animals make insignificant contributions. On the oceans the main contributor of organic matter to sediments is plankton, either dead fragments or faecal pellets termed marine snow. Bacteria degrade this matter in the water column, and the amount surviving to the ocean floor is inversely proportional to the depth. This is accompanied by biominerals consisting of silicates and carbonates. The particulate organic matter in sediments is about 20% of known molecules 80% of material that cannot be analysed. Detritivores consume some of the fallen organic materials. Aerobic bacteria and fungi also consume organic matter in the oxic surface parts of the sediment. Coarse-grained sediments are oxygenated to about half a meter, but fine grained clays may only have a couple of millimetres exposed to oxygen. The organic matter in the oxygenated zone will become completely mineralized if it stays there long enough.

Deeper in sediments where oxygen is exhausted, anaerobic biological processes continue at a slower rate. These include anaerobic mineralization making ammonium, phosphate and sulfide ions; fermentation making short chain alcohols, acids or methyl amines; acetogenesis making acetic acid; methanogenesis making methane, and sulfate, nitrite and nitrate reduction. Carbon dioxide and hydrogen are also outputs. Under freshwater, sulfate is usually very low, so methanogenesis is more important. Yet other bacteria can convert methane, back into living matter, by oxidising with other substrates. Bacteria can reside at great depths in sediments. However sedimentary organic matter accumulates the indigestible components.

Deep bacteria may be lithotrophes, using hydrogen, and carbon dioxide as a carbon source.

In the oceans and other waters there is much dissolved organic materials. These are several thousand years old on average, and are called gelbstoff (yellow substance) particularly in fresh waters. Much of this is tannins. The nitrogen containing materials here appear to be amides, perhaps from peptidoglycans from bacteria. Microorganisms have trouble consuming the high molecular weight dissolved substances, but quickly consume small molecules.

From terrestrial sources black carbon produced by charring is an important component. Fungi are important decomposers in soil.

Macromolecules

Proteins are normally hydrolysed slowly even withoutenzymes or bacteria, with a half-life of 460 years, but can be preserved if they are desiccated, pickled or frozen. Being enclosed in bone also helps preservation. Over time the amino acids tend to racemize, and those with more functional groups are lost earlier. Protein still will degrade on the timescale of a million years. DNA degrades rapidly, lasting only about four years in water. Cellulose and chitin have a half-life in water at 25° of about 4.7 million years. Enzymes can accelerate this by a factor of 10¹⁷. About 10¹¹ tons of chitin are produced each year, but it is almost all degraded.

Lignin is only efficiently degraded by fungi, white rot, or brown rot. These require oxygen.

Lipids are hydrolysed to fatty acids over long time periods. Plant cuticle waxes are very difficult to degrade, and may survive over geological time periods.

Preservation

More organic matter is preserved in sediments if there is high primary production, or the sediment is fine-grained. The lack of oxygen helps preservation greatly, and that also is caused by a large supply of organic matter. Soil does not usually preserve organic matter, it would need to be acidified or water logged, as in the bog. Rapid burial ensures the material gets to an oxygen free depth, but also dilutes the organic matter. A low energy environment ensures the sediment is not stirred up and oxygenated. Salt marshes and mangroves meet some of these requirements, but unless the sea level is rising will not have a chance to accumulate much. Coral reefs are very productive, but are well oxygenated, and recycle everything before it is buried.

Sphagnum bog

In dead Sphagnum, sphagnan a polysaccharide with D-lyxo-5-hexosulouronic acid is a major remaining substance. It makes the bog very acidic, so that bacteria cannot grow. Not only that, the plant ensures there is no available nitrogen. Holocellulose also absorbs any digestive enzymes around. Together this leads to major accumulation of peat under sphagnum bogs.

Mantle

Earth's mantle is a significant reservoir of carbon. The mantle contains more carbon than the crust, oceans, biosphere, and atmosphere put together. The figure is estimated to be very roughly 10²² kg. Carbon concentration in the mantle is very variable, varying by more than a factor of 100 between different parts.

The form carbon takes depends on its oxidation state, which depends on the oxygen fugacity of the environment. Carbon dioxide and carbonate are found where the oxygen fugacity is high. Lower oxygen fugacity results in diamond formation, first in eclogite, then peridotite, and lastly in fluid water mixtures. At even lower oxygen fugacity, methane is stable in contact with water, and even lower, metallic iron and nickel form along with carbides. Iron carbides include Fe₃C and Fe₇C₃.

Minerals that contain carbon include calcite and its higher density polymorphs. Other significant carbon minerals include magnesium and iron carbonates. Dolomite is stable above 100 km depth. Below 100 km, dolomite reacts with orthopyroxine (found in peridotite) to yield magnesite (an iron magnesium carbonate). Below 200 km deep, carbon dioxide is reduced by ferrous iron (Fe²⁺), forming diamond, and ferric iron (Fe³⁺). Even deeper pressure induced disproportionation of iron minerals produces more ferric iron, and metallic iron. The metallic iron combines with carbon to form the mineral cohenite with formula Fe₃C. Cohenite also contains some nickel substituting for iron. This form or carbon is called "carbide". Diamond forms in the mantle below 150 km deep, but because it is so durable, it can survive in eruptions to the surface in kimberlites, lamproites, or ultramafic lamprophyres.

Xenoliths can come from the mantle, and different compositions come from different depths. Above 90 km (3.2 GPa) spinel peridotite occurs, below this garnet peridotite is found.

Inclusions trapped in diamond can reveal the material and conditions much deeper in the mantle. Large gem diamonds are usually formed in the transition zone part of the mantle, (410 to 660 km deep) and crystallise from a molten iron-nickel-carbon solution, that also contains sulfur and trace amounts of hydrogen, chromium, phosphorus and oxygen. Carbon atoms constitute about 12% of the melt (about 3% by mass). Inclusions of the crystallised metallic melt are sometimes included in diamonds. Diamond can be caused to precipitate from the liquid metal, by increasing pressure, or by adding sulfur.

Fluid inclusions in crystals from the mantle have contents that most often are liquid carbon dioxide, but which also include carbon oxysulfide, methane and carbon monoxide.

Material is added by subduction from the crust. This includes the major carbon containing sediments such as limestone, or coal. Each year 2×10¹¹ kg of CO₂ is transferred from the crust to the mantle by subduction. (1700 tons of carbon per second).

Upwelling mantle material can add to the crust at mid oceanic ridges. Fluids can extract carbon from the mantle and erupt in volcanoes. At 330 km deep a liquid consisting of carbon dioxide and water can form. It is highly corrosive, and dissolves incompatible elements from the solid mantle. These elements include uranium, thorium, potassium, helium and argon. The fluids can then go on to cause metasomatism or extend to the surface in carbonatite eruptions. The total mid oceanic ridge, and hot spot volcanic emissions of carbon dioxide match the loss due to subduction: 2×10¹¹ kg of CO₂ per year.

In slowly convecting mantle rocks, diamond that slowly rises above 150 km will slowly turn into graphite or be oxidised to carbon dioxide or carbonate minerals.

Core

Earth's core is believed to be mostly an alloy of iron and nickel. The density indicates that it also contains a significant amount of lighter elements. Elements such as hydrogen would be stable in the Earth's core, however the conditions at the formation of the core would not be suitable for its inclusion. Carbon is a very likely constituent of the core. Preferential partitioning of the carbon isotope¹²C into the metallic core, during its formation, may explain why there seems to be more ¹³C on the surface and mantle of the Earth compared to other solar system bodies (−5‰ compared to -20‰). The difference can also help to predict the value of the carbon proportion of the core.

The outer core has a density around 11 cm⁻³, and a mass of 1.3×10²⁴kg. It contains roughly 10²² kg of carbon. Carbon dissolved in liquid iron affect the solution of other elements. Dissolved carbon changes lead from a siderophile to a lithophile. It has the opposite effect on tungsten and molybdenum, causing more tungsten or molybdenum to dissolve in the metallic phase. The measured amounts of these elements in the rocks compared to the Solar System can be explained by a 0.6% carbon composition of the core.

The inner core is about 1221 km in radius. It has a density of 13 g cm⁻³, and a total mass of 9×10²² kg and a surface area of 18,000,000 square kilometers. Experiments with mixtures under pressure and temperature attempt to reproduce the known properties of the inner and outer core. Carbides are among the first to precipitate from a molten metal mix, and so the inner core may be mostly iron carbides, Fe₇C₃ or Fe₃C. At atmospheric pressure (100 kPa) the iron-Fe₃C eutectic point is at 4.1% carbon. This percentage decreases as pressure increases to around 50 GPa. Above that pressure the percentage of carbon at the eutectic increases. The pressure on the inner core ranges from 330 GPa to 360 GPa at the centre of the Earth. The temperature at the inner core surface is about 6000 K. The material of the inner core must be stable at the pressure and temperature found there, and more dense than that of the outer core liquid. Extrapolations show that either Fe₃C or Fe₇C₃ match the requirements. Fe₇C₃ is 8.4% carbon, and Fe₃C is 6.7% carbon. The inner core is growing by about 1 mm per year, or adding about 18 cubic kilometres per year. This is about 18×10¹² kg of carbon added to the inner core every year. It contains about 8×10²¹ kg of carbon.

High pressure experimentation

In order to determine the fate of natural carbon containing substances deep in the Earth, experiments have been conducted to see what happens when high pressure, and or temperatures are applied. Such substances include carbon dioxide, carbon monoxide, graphite, methane, and other hydrocarbons such as benzene, carbon dioxide water mixtures and carbonate minerals such as calcite, magnesium carbonate, or ferrous carbonate. Under super high pressures carbon may take on a higher coordination number than the four found in sp³ compounds like diamond, or the three found in carbonates. Perhaps carbon can substitute into silicates, or form a silicon oxycarbide. Carbides may be possible.

Carbon

At 15 GPa graphite changes to a hard transparent form, that is not diamond. Diamond is very resistant to pressure, but at about 1 TPa (1000 GPa) transforms to a BC-8 form. But these conditions are not found in the Earth.

Carbides

Carbides are predicted to be more likely lower in the mantle as experiments have shown a much lower oxygen fugacity for high pressure iron silicates. Cohenite remains stable to over 187 GPa, but is predicted to have a denser orthorhombic Cmcm form in the inner core.

Carbon dioxide

Under 0.3 GPa pressure, carbon dioxide is stable at room temperature in the same form as dry ice. Over 0.5 GPa carbon dioxide forms a number of different solid forms containing molecules. At pressures over 40 GPa and high temperatures, carbon dioxide forms a covalent solid that contains CO₄ tetrahedra, and has the same structure as β-cristobalite. This is called phase V or CO₂-V. When CO₂-V is subjected to high temperatures, or higher pressures, experiments show it breaks down to form diamond and oxygen. In the mantle the geotherm would mean that carbon dioxide would be a liquid till a pressure of 33 GPa, then it would adopt the solid CO₂-V form till 43 GPa, and deeper than that would make diamond and fluid oxygen.

Carbonyls

High pressure carbon monoxide forms the high energy polycarbonyl covalent solid, however it is not expected to be present inside the Earth.

Hydrocarbons

Under 1.59 GPa pressure at 25 °C, methane converts to a cubic solid. The molecules are rotationally disordered. But over 5.25 GPa the molecules become locked into position and cannot spin. Other hydrocarbons under high pressure have hardly been studied.

Carbonates

Calcite changes to calcite-II and calcite-III at pressures of 1.5, and 2.2 GPa. Siderite undergoes a chemical change at 10 GPa at 1800K to form Fe₄O₅. Dolomite decomposes 7GPa and below 1000 °C to yield aragonite and magnesite. However, there are forms of iron containing dolomite stable at higher pressures and temperatures. Over 130 GPa aragonite undergoes a transformation to a SP³ tetrahedrally connected carbon, in a covalent network in a C222₁ structure. Magnesite can survive 80 GPa, but with more than 100 GPa (as at a depth of 1800 km it changes to forms with three-member rings of CO₄ tetrahedra (C₃O₉⁶⁻). If iron is present in this mineral, at these pressures it will convert to magnetite and diamond. Melted carbonates with SP³ carbon are predicted to be very viscous.

Some minerals that contain both silicate and carbonate exist, spurrite and tilleyite. But high-pressure forms have not been studied. There have been attempts to make silicon carbonate. Six coordinated silicates mixed with carbonate should not exist on Earth, but may exist on more massive planets.

Neutrino astronomy

From Wikipedia, the free encyclopedia

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

An optical module from a neutrino telescope. Neutrino telescopes consist of hundreds to thousands of optical modules distributed over a large volume.

Neutrino astronomy is a branch of astronomy that gathers information about astronomical objects by observing and studying neutrinos emitted by them with the help of neutrino detectors in special Earth observatories. It is an emerging field in astroparticle physics providing insights into the high-energy and non-thermal processes in the universe.

Neutrinos are nearly massless and electrically neutral or chargeless elementary particles. They are created as a result of certain types of radioactive decay, nuclear reactions such as those that take place in the Sun or high energy astrophysical phenomena, in nuclear reactors, or when cosmic rays hit atoms in the atmosphere. Neutrinos rarely interact with matter (only via the weak nuclear force), travel at nearly the speed of light in straight lines, pass through large amounts of matter without any notable absorption or without being deflected by magnetic fields. Unlike photons, neutrinos rarely scatter along their trajectory. But like photons, neutrinos are some of the most common particles in the universe. Because of this, neutrinos offer a unique opportunity to observe processes that are inaccessible to optical telescopes, such as reactions in the Sun's core. Neutrinos that are created in the Sun's core are barely absorbed, so a large quantity of them escape from the Sun and reach the Earth. Neutrinos can also offer a very strong pointing direction compared to charged particle cosmic rays.

Neutrinos are very hard to detect due to their non-interactive nature. In order to detect neutrinos, scientists have to shield the detectors from cosmic rays, which can penetrate hundreds of meters of rock. Neutrinos, on the other hand, can go through the entire planet without being absorbed, like "ghost particles". That's why neutrino detectors are placed many hundreds of meter underground, usually at the bottom of mines. There a neutrino detection liquid such as a Chlorine-rich solution is placed; the neutrinos react with a Chlorine isotope and can create radioactive Argon. Gallium to Germanium conversion has also been used. The IceCube Neutrino Observatory built in 2010 in the south pole is the biggest neutrino detector, consisting of thousands of optical sensors buried 500 meters underneath a cubic kilometer of deep, ultra-transparent ice, detects light emitted by charged particles that are produced when a single neutrino collides with a proton or neutron inside an atom. The resulting nuclear reaction produces secondary particles traveling at high speeds that give off a blue light called Cherenkov radiation. Super-Kamiokande in Japan and ANTARES and KM3NeT in the Mediterranean are some other important neutrino detectors.

Since neutrinos interact weakly, neutrino detectors must have large target masses (often thousands of tons). The detectors also must use shielding and effective software to remove background signal. Since neutrinos are very difficult to detect, the only bodies that have been studied in this way are the sun and the supernova SN1987A, which exploded in 1987. Scientist predicted that supernova explosions would produce bursts of neutrinos, and a similar burst was actually detected from Supernova 1987A.

In the future neutrino astronomy promises to discover other aspects of the universe, including coincidental gravitational waves, gamma ray bursts, the cosmic neutrino background, origins of ultra-high-energy neutrinos, neutrino properties (such as neutrino mass hierarchy), dark matter properties, etc. It will become an integral part of multi-messenger astronomy, complementing gravitational astronomy and traditional telescopic astronomy.

History

Neutrinos were first recorded in 1956 by Clyde Cowan and Frederick Reines in an experiment employing a nearby nuclear reactor as a neutrino source. Their discovery was acknowledged with a Nobel Prize in Physics in 1995.

This was followed by the first atmospheric neutrino detection in 1965 by two groups almost simultaneously. One was led by Frederick Reines who operated a liquid scintillator - the Case-Witwatersrand-Irvine or CWI detector - in the East Rand gold mine in South Africa at an 8.8 km water depth equivalent. The other was a Bombay-Osaka-Durham collaboration that operated in the Indian Kolar Gold Field mine at an equivalent water depth of 7.5 km. Although the KGF group detected neutrino candidates two months later than Reines CWI, they were given formal priority due to publishing their findings two weeks earlier.

In 1968, Raymond Davis, Jr. and John N. Bahcall successfully detected the first solar neutrinos in the Homestake experiment. Davis, along with Japanese physicist Masatoshi Koshiba were jointly awarded half of the 2002 Nobel Prize in Physics "for pioneering contributions to astrophysics, in particular for the detection of cosmic neutrinos (the other half went to Riccardo Giacconi for corresponding pioneering contributions which have led to the discovery of cosmic X-ray sources)."

The first generation of undersea neutrino telescope projects began with the proposal by Moisey Markov in 1960 "...to install detectors deep in a lake or a sea and to determine the location of charged particles with the help of Cherenkov radiation."

The first underwater neutrino telescope began as the DUMAND project. DUMAND stands for Deep Underwater Muon and Neutrino Detector. The project began in 1976 and although it was eventually cancelled in 1995, it acted as a precursor to many of the following telescopes in the following decades.

The Baikal Neutrino Telescope is installed in the southern part of Lake Baikal in Russia. The detector is located at a depth of 1.1 km and began surveys in 1980. In 1993, it was the first to deploy three strings to reconstruct the muon trajectories as well as the first to record atmospheric neutrinos underwater.

AMANDA (Antarctic Muon And Neutrino Detector Array) used the 3 km thick ice layer at the South Pole and was located several hundred meters from the Amundsen-Scott station. Holes 60 cm in diameter were drilled with pressurized hot water in which strings with optical modules were deployed before the water refroze. The depth proved to be insufficient to be able to reconstruct the trajectory due to the scattering of light on air bubbles. A second group of 4 strings were added in 1995/96 to a depth of about 2000 m that was sufficient for track reconstruction. The AMANDA array was subsequently upgraded until January 2000 when it consisted of 19 strings with a total of 667 optical modules at a depth range between 1500 m and 2000 m. AMANDA would eventually be the predecessor to IceCube in 2005.

An example of an early neutrino detector is the Artyomovsk Scintillation Detector [ru] (ASD), located in the Soledar Salt Mine in Ukraine at a depth of more than 100 m. It was created in the Department of High Energy Leptons and Neutrino Astrophysics of the Institute of Nuclear Research of the USSR Academy of Sciences in 1969 to study antineutrino fluxes from collapsing stars in the Galaxy, as well as the spectrum and interactions of muons of cosmic rays with energies up to 10 ^ 13 eV. A feature of the detector is a 100-ton scintillation tank with dimensions on the order of the length of an electromagnetic shower with an initial energy of 100 GeV.

21st century

After the decline of DUMAND the participating groups split into three branches to explore deep sea options in the Mediterranean Sea. ANTARES was anchored to the sea floor in the region off Toulon at the French Mediterranean coast. It consists of 12 strings, each carrying 25 "storeys" equipped with three optical modules, an electronic container, and calibration devices down to a maximum depth of 2475 m.

NEMO (NEutrino Mediterranean Observatory) was pursued by Italian groups to investigate the feasibility of a cubic-kilometer scale deep-sea detector. A suitable site at a depth of 3.5 km about 100 km off Capo Passero at the South-Eastern coast of Sicily has been identified. From 2007 to 2011 the first prototyping phase tested a "mini-tower" with 4 bars deployed for several weeks near Catania at a depth of 2 km. The second phase as well as plans to deploy the full-size prototype tower will be pursued in the KM3NeT framework.

The NESTOR Project was installed in 2004 to a depth of 4 km and operated for one month until a failure of the cable to shore forced it to be terminated. The data taken still successfully demonstrated the detector's functionality and provided a measurement of the atmospheric muon flux. The proof of concept will be implemented in the KM3Net framework.

The second generation of deep-sea neutrino telescope projects reach or even exceed the size originally conceived by the DUMAND pioneers. IceCube, located at the South Pole and incorporating its predecessor AMANDA, was completed in December 2010. Consisting of 5160 digital optical modules installed on 86 strings at depths of 1450 to 2550 m in the Antarctic ice, in 2013 it became the first experiment to detect astrophysical (cosmic) neutrinos. The KM3NeT in the Mediterranean Sea and the GVD are in their preparatory/prototyping phase. IceCube instruments 1 km³ of ice. GVD is also planned to cover 1 km³ but at a much higher energy threshold. KM3NeT is planned to cover several km³ and have two components; ARCA (Astroparticle Research with Cosmics in the Abyss) and ORCA (Oscillations Research with Cosmics in the Abyss). Both KM3NeT and GVD have completed at least part of their construction and it is expected that these two along with IceCube will form a global neutrino observatory.

In July 2018, the IceCube Neutrino Observatory announced that they have traced an extremely-high-energy neutrino that hit their Antarctica-based research station in September 2017 back to its point of origin in the blazar TXS 0506+056 located 3.7 billion light-years away in the direction of the constellation Orion. This was the first time that a neutrino detector has been used to locate an object in space and that a source of cosmic rays has been identified. In November 2022, another significant progress towards identifying the origin of cosmic rays came when IceCube reported the observation of 79 neutrinos with an energy over 1 TeV originated from the nearby galaxy M77. These findings in a well-known object are expected to help study the active nucleus of this galaxy, as well as serving as a baseline for future observations. And in June 2023, IceCube reported the first detection of neutrinos from the galactic plane of the Milky Way.

Detection methods

Main article: Neutrino detector

Neutrinos interact incredibly rarely with matter, so the vast majority of neutrinos will pass through a detector without interacting. If a neutrino does interact, it will only do so once. Therefore, to perform neutrino astronomy, large detectors must be used to obtain enough statistics.

The IceCube Neutrino Detector at the South Pole. The PMTs are under more than a kilometer of ice, and will detect the photons from neutrino interactions within a cubic kilometer of ice

The method of neutrino detection depends on the energy and type of the neutrino. A famous example is that anti-electron neutrinos can interact with a nucleus in the detector by inverse beta decay and produce a positron and a neutron. The positron immediately will annihilate with an electron, producing two 511keV photons. The neutron will attach to another nucleus and give off a gamma with an energy of a few MeV. In general, neutrinos can interact through neutral-current and charged-current interactions. In neutral-current interactions, the neutrino interacts with a nucleus or electron and the neutrino retains its original flavor. In charged-current interactions, the neutrino is absorbed by the nucleus and produces a lepton corresponding to the neutrino's flavor (${\displaystyle {\nu }_{\text{e}}^{}⟶{\text{e}}^{-}}$,${\displaystyle {\nu }_{\mu }^{}⟶{\mu }^{-}}$, etc.). If the charged resultants are moving fast enough, they can create Cherenkov light.

To observe neutrino interactions, detectors use photomultiplier tubes (PMTs) to detect individual photons. From the timing of the photons, it is possible to determine the time and place of the neutrino interaction. If the neutrino creates a muon during its interaction, then the muon will travel in a line, creating a "track" of Cherenkov photons. The data from this track can be used to reconstruct the directionality of the muon. For high-energy interactions, the neutrino and muon directions are the same, so it's possible to tell where the neutrino came from. This is pointing direction is important in extra-solar system neutrino astronomy. Along with time, position, and possibly direction, it's possible to infer the energy of the neutrino from the interactions. The number of photons emitted is related to the neutrino energy, and neutrino energy is important for measuring the fluxes from solar and geo-neutrinos.

Due to the rareness of neutrino interactions, it is important to maintain a low background signal. For this reason, most neutrino detectors are constructed under a rock or water overburden. This overburden shields against most cosmic rays in the atmosphere; only some of the highest-energy muons are able to penetrate to the depths of our detectors. Detectors must include ways of dealing with data from muons so as to not confuse them with neutrinos. Along with more complicated measures, if a muon track is first detected outside of the desired "fiducial" volume, the event is treated as a muon and not considered. Ignoring events outside the fiducial volume also decreases the signal from radiation outside the detector.

Despite shielding efforts, it is inevitable that some background will make it into the detector, many times in the form of radioactive impurities within the detector itself. At this point, if it is impossible to differentiate between the background and true signal, a Monte Carlo simulation must be used to model the background. While it may be unknown if an individual event is background or signal, it is possible to detect an excess about the background, signifying existence of the desired signal.

Applications

When astronomical bodies, such as the Sun, are studied using light, only the surface of the object can be directly observed. Any light produced in the core of a star will interact with gas particles in the outer layers of the star, taking hundreds of thousands of years to make it to the surface, making it impossible to observe the core directly. Since neutrinos are also created in the cores of stars (as a result of stellar fusion), the core can be observed using neutrino astronomy. Other sources of neutrinos- such as neutrinos released by supernovae- have been detected. Several neutrino experiments have formed the Supernova Early Warning System (SNEWS), where they search for an increase of neutrino flux that could signal a supernova event. There are currently goals to detect neutrinos from other sources, such as active galactic nuclei (AGN), as well as gamma-ray bursts and starburst galaxies. Neutrino astronomy may also indirectly detect dark matter.

Supernova warning

Seven neutrino experiments (Super-K, LVD, IceCube, KamLAND, Borexino, Daya Bay, and HALO) work together as the Supernova Early Warning System (SNEWS). In a core collapse supernova, ninety-nine percent of the energy released will be in neutrinos. While photons can be trapped in the dense supernova for hours, neutrinos are able to escape on the order of seconds. Since neutrinos travel at roughly the speed of light, they can reach Earth before photons do. If two or more of SNEWS detectors observe a coincidence of an increased flux of neutrinos, an alert is sent to professional and amateur astronomers to be on the lookout for supernova light. By using the distance between detectors and the time difference between detections, the alert can also include directionality as to the supernova's location in the sky.

Stellar processes

The proton-proton fusion chain that occurs within the Sun. This process is responsible for the majority of the Sun's energy.

The Sun, like other stars, is powered by nuclear fusion in its core. The core is incredibly large, meaning that photons produced in the core will take a long time to diffuse outward. Therefore, neutrinos are the only way that we can obtain real-time data about the nuclear processes in the Sun.

There are two main processes for stellar nuclear fusion. The first is the Proton-Proton (PP) chain, in which protons are fused together into helium, sometimes temporarily creating the heavier elements of lithium, beryllium, and boron along the way. The second is the CNO cycle, in which carbon, nitrogen, and oxygen are fused with protons, and then undergo alpha decay (helium nucleus emission) to begin the cycle again. The PP chain is the primary process in the Sun, while the CNO cycle is more dominant in stars more massive than the Sun.

Each step in the process has an allowed spectra of energy for the neutrino (or a discrete energy for electron capture processes). The relative rates of the Sun's nuclear processes can be determined by observations in its flux at different energies. This would shed insight into the Sun's properties, such as metallicity, which is the composition of heavier elements.

Borexino is one of the detectors studying solar neutrinos. In 2018, they found 5σ significance for the existence of neutrinos from the fusing of two protons with an electron (pep neutrinos). In 2020, they found for the first time evidence of CNO neutrinos in the Sun. Improvements on the CNO measurement will be especially helpful in determining the Sun's metallicity.

Composition and structure of Earth

The interior of Earth contains radioactive elements such as ${\displaystyle {}^{40}\text{K}}$ and the decay chains of ${\displaystyle {}^{238}\text{U}}$ and ${\displaystyle {}^{232}\text{Th}}$. These elements decay via Beta decay, which emits an anti-neutrino. The energies of these anti-neutrinos are dependent on the parent nucleus. Therefore, by detecting the anti-neutrino flux as a function of energy, we can obtain the relative compositions of these elements and set a limit on the total power output of Earth's geo-reactor. Most of our current data about the core and mantle of Earth comes from seismic data, which does not provide any information as to the nuclear composition of these layers.

Borexino has detected these geo-neutrinos through the process ${\displaystyle \overline{\nu }+{\text{p}}^{+}⟶{\text{e}}^{+}+\text{n}}$. The resulting positron will immediately annihilate with an electron and produce two gamma-rays each with an energy of 511keV (the rest mass of an electron). The neutron will later be captured by another nucleus, which will lead to a 2.22MeV gamma-ray as the nucleus de-excites. This process on average takes on the order of 256 microseconds. By searching for time and spatial coincidence of these gamma rays, the experimenters can be certain there was an event.

Using over 3,200 days of data, Borexino used geoneutrinos to place constraints on the composition and power output of the mantle. They found that the ratio of ${\displaystyle {}^{238}\text{U}}$ to ${\displaystyle {}^{232}\text{Th}}$ is the same as chondritic meteorites. The power output from uranium and thorium in Earth's mantle was found to be 14.2-35.7 TW with a 68% confidence interval.

Neutrino tomography also provides insight into the interior of Earth. For neutrinos with energies of a few TeV, the interaction probability becomes non-negligible when passing through Earth. The interaction probability will depend on the number of nucleons the neutrino passed along its path, which is directly related to density. If the initial flux is known (as it is in the case of atmospheric neutrinos), then detecting the final flux provides information about the interactions that occurred. The density can then be extrapolated from knowledge of these interactions. This can provide an independent check on the information obtained from seismic data.

The interior of the Earth as we know it. Currently, our information comes only from seismic data. Neutrinos would be an independent check on this data

In 2018, one year worth of IceCube data was evaluated to perform neutrino tomography. The analysis studied upward going muons, which provide both the energy and directionality of the neutrinos after passing through the Earth. A model of Earth with five layers of constant density was fit to the data, and the resulting density agreed with seismic data. The values determined for the total mass of Earth, the mass of the core, and the moment of inertia all agree with the data obtained from seismic and gravitational data. With the current data, the uncertainties on these values are still large, but future data from IceCube and KM3NeT will place tighter restrictions on this data.

High-energy astrophysical events

Neutrinos can either be primary cosmic rays (astrophysical neutrinos), or be produced from cosmic ray interactions. In the latter case, the primary cosmic ray will produce pions and kaons in the atmosphere. As these hadrons decay, they produce neutrinos (called atmospheric neutrinos). At low energies, the flux of atmospheric neutrinos is many times greater than astrophysical neutrinos. At high energies, the pions and kaons have a longer lifetime (due to relativistic time dilation). The hadrons are now more likely to interact before they decay. Because of this, the astrophysical neutrino flux will dominate at high energies (~100TeV). To perform neutrino astronomy of high-energy objects, experiments rely on the highest energy neutrinos.

To perform astronomy of distant objects, a strong angular resolution is required. Neutrinos are electrically neutral and interact weakly, so they travel mostly unperturbed in straight lines. If the neutrino interacts within a detector and produces a muon, the muon will produce an observable track. At high energies, the neutrino direction and muon direction are closely correlated, so it is possible to trace back the direction of the incoming neutrino.

These high-energy neutrinos are either the primary or secondary cosmic rays produced by energetic astrophysical processes. Observing neutrinos could provide insights into these processes beyond what is observable with electromagnetic radiation. In the case of the neutrino detected from a distant blazar, multi-wavelength astronomy was used to show spatial coincidence, confirming the blazar as the source. In the future, neutrinos could be used to supplement electromagnetic and gravitational observations, leading to multi-messenger astronomy.

Introduction to gauge theory

From Wikipedia, the free encyclopedia

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

 

Quantum field theory
Feynman diagram

A gauge theory is a type of theory in physics. The word gauge means a measurement, a thickness, an in-between distance (as in railroad tracks), or a resulting number of units per certain parameter (a number of loops in an inch of fabric or a number of lead balls in a pound of ammunition). Modern theories describe physical forces in terms of fields, e.g., the electromagnetic field, the gravitational field, and fields that describe forces between the elementary particles. A general feature of these field theories is that the fundamental fields cannot be directly measured; however, some associated quantities can be measured, such as charges, energies, and velocities. For example, say you cannot measure the diameter of a lead ball, but you can determine how many lead balls, which are equal in every way, are required to make a pound. Using the number of balls, the density of lead, and the formula for calculating the volume of a sphere from its diameter, one could indirectly determine the diameter of a single lead ball.

In field theories, different configurations of the unobservable fields can result in identical observable quantities. A transformation from one such field configuration to another is called a gauge transformation; the lack of change in the measurable quantities, despite the field being transformed, is a property called gauge invariance. For example, if you could measure the color of lead balls and discover that when you change the color, you still fit the same number of balls in a pound, the property of "color" would show gauge invariance. Since any kind of invariance under a field transformation is considered a symmetry, gauge invariance is sometimes called gauge symmetry. Generally, any theory that has the property of gauge invariance is considered a gauge theory.

For example, in electromagnetism the electric field E and the magnetic field B are observable, while the potentials V ("voltage") and A (the vector potential) are not. Under a gauge transformation in which a constant is added to V, no observable change occurs in E or B.

With the advent of quantum mechanics in the 1920s, and with successive advances in quantum field theory, the importance of gauge transformations has steadily grown. Gauge theories constrain the laws of physics, because all the changes induced by a gauge transformation have to cancel each other out when written in terms of observable quantities. Over the course of the 20th century, physicists gradually realized that all forces (fundamental interactions) arise from the constraints imposed by local gauge symmetries, in which case the transformations vary from point to point in space and time. Perturbative quantum field theory (usually employed for scattering theory) describes forces in terms of force-mediating particles called gauge bosons. The nature of these particles is determined by the nature of the gauge transformations. The culmination of these efforts is the Standard Model, a quantum field theory that accurately predicts all of the fundamental interactions except gravity.

History and importance

The earliest field theory having a gauge symmetry was James Clerk Maxwell's formulation, in 1864–65, of electrodynamics in "A Dynamical Theory of the Electromagnetic Field". The importance of this symmetry remained unnoticed in the earliest formulations. Similarly unnoticed, David Hilbert had derived Einstein's equations of general relativity by postulating a symmetry under any change of coordinates, just as Einstein was completing his work. Later Hermann Weyl, inspired by success in Einstein's general relativity, conjectured (incorrectly, as it turned out) in 1919 that invariance under the change of scale or "gauge" (a term inspired by the various track gauges of railroads) might also be a local symmetry of electromagnetism. Although Weyl's choice of the gauge was incorrect, the name "gauge" stuck to the approach. After the development of quantum mechanics, Weyl, Vladimir Fock and Fritz London modified their gauge choice by replacing the scale factor with a change of wave phase, and applying it successfully to electromagnetism. Gauge symmetry was generalized mathematically in 1954 by Chen Ning Yang and Robert Mills in an attempt to describe the strong nuclear forces. This idea, dubbed Yang–Mills theory, later found application in the quantum field theory of the weak force, and its unification with electromagnetism in the electroweak theory.

The importance of gauge theories for physics stems from their tremendous success in providing a unified framework to describe the quantum-mechanical behavior of electromagnetism, the weak force and the strong force. This gauge theory, known as the Standard Model, accurately describes experimental predictions regarding three of the four fundamental forces of nature.

In classical physics

Electromagnetism

Main article: Gauge fixing

Historically, the first example of gauge symmetry to be discovered was classical electromagnetism. A static electric field can be described in terms of an electric potential (voltage, V) that is defined at every point in space, and in practical work it is conventional to take the Earth as a physical reference that defines the zero level of the potential, or ground. But only differences in potential are physically measurable, which is the reason that a voltmeter must have two probes, and can only report the voltage difference between them. Thus one could choose to define all voltage differences relative to some other standard, rather than the Earth, resulting in the addition of a constant offset. If the potential V is a solution to Maxwell's equations then, after this gauge transformation, the new potential V → V + C is also a solution to Maxwell's equations and no experiment can distinguish between these two solutions. In other words, the laws of physics governing electricity and magnetism (that is, Maxwell equations) are invariant under gauge transformation. Maxwell's equations have a gauge symmetry.

Generalizing from static electricity to electromagnetism, we have a second potential, the magnetic vector potential A, which can also undergo gauge transformations. These transformations may be local. That is, rather than adding a constant onto V, one can add a function that takes on different values at different points in space and time. If A is also changed in certain corresponding ways, then the same E (electric) and B (magnetic) fields result. The detailed mathematical relationship between the fields E and B and the potentials V and A is given in the article Gauge fixing, along with the precise statement of the nature of the gauge transformation. The relevant point here is that the fields remain the same under the gauge transformation, and therefore Maxwell's equations are still satisfied.

Gauge symmetry is closely related to charge conservation. Suppose that there existed some process by which one could briefly violate conservation of charge by creating a charge q at a certain point in space, 1, moving it to some other point 2, and then destroying it. We might imagine that this process was consistent with conservation of energy. We could posit a rule stating that creating the charge required an input of energy E₁ = qV₁ and destroying it released E₂ = qV₂, which would seem natural since qV measures the extra energy stored in the electric field because of the existence of a charge at a certain point. Outside of the interval during which the particle exists, conservation of energy would be satisfied, because the net energy released by creation and destruction of the particle, qV₂ − qV₁, would be equal to the work done in moving the particle from 1 to 2, qV₂ − qV₁. But although this scenario salvages conservation of energy, it violates gauge symmetry. Gauge symmetry requires that the laws of physics be invariant under the transformation V → V + C, which implies that no experiment should be able to measure the absolute potential, without reference to some external standard such as an electrical ground. But the proposed rules E₁ = qV₁ and E₂ = qV₂ for the energies of creation and destruction would allow an experimenter to determine the absolute potential, simply by comparing the energy input required to create the charge q at a particular point in space in the case where the potential is V and V + C respectively. The conclusion is that if gauge symmetry holds, and energy is conserved, then charge must be conserved.

The Cartesian coordinate grid on this square has been distorted by a coordinate transformation, so that there is a nonlinear relationship between the old (x, y) coordinates and the new ones. Einstein's equations of general relativity are still valid in the new coordinate system. Such changes of coordinate system are the gauge transformations of general relativity.

General relativity

As discussed above, the gauge transformations for classical (i.e., non-quantum mechanical) general relativity are arbitrary coordinate transformations. Technically, the transformations must be invertible, and both the transformation and its inverse must be smooth, in the sense of being differentiable an arbitrary number of times.

An example of a symmetry in a physical theory: translation invariance

Some global symmetries under changes of coordinate predate both general relativity and the concept of a gauge. For example, Galileo and Newton introduced the notion of translation invariance, an advancement from the Aristotelian concept that different places in space, such as the earth versus the heavens, obeyed different physical rules.

Suppose, for example, that one observer examines the properties of a hydrogen atom on Earth, the other—on the Moon (or any other place in the universe), the observer will find that their hydrogen atoms exhibit completely identical properties. Again, if one observer had examined a hydrogen atom today and the other—100 years ago (or any other time in the past or in the future), the two experiments would again produce completely identical results. The invariance of the properties of a hydrogen atom with respect to the time and place where these properties were investigated is called translation invariance.

Recalling our two observers from different ages: the time in their experiments is shifted by 100 years. If the time when the older observer did the experiment was t, the time of the modern experiment is t + 100 years. Both observers discover the same laws of physics. Because light from hydrogen atoms in distant galaxies may reach the earth after having traveled across space for billions of years, in effect one can do such observations covering periods of time almost all the way back to the Big Bang, and they show that the laws of physics have always been the same.

In other words, if in the theory we change the time t to t + 100 years (or indeed any other time shift) the theoretical predictions do not change.

Another example of a symmetry: invariance of Einstein's field equation under arbitrary coordinate transformations

In Einstein's general relativity, coordinates like x, y, z, and t are not only "relative" in the global sense of translations like t → t + C, rotations, etc., but become completely arbitrary, so that, for example, one can define an entirely new time-like coordinate according to some arbitrary rule such as t → t + t³/t₀², where t₀ has dimensions of time, and yet the Einstein equations will have the same form.

Invariance of the form of an equation under an arbitrary coordinate transformation is customarily referred to as general covariance, and equations with this property are referred to as written in the covariant form. General covariance is a special case of gauge invariance.

Maxwell's equations can also be expressed in a generally covariant form, which is as invariant under general coordinate transformation as the Einstein field equation.

In quantum mechanics

Quantum electrodynamics

Until the advent of quantum mechanics, the only well known example of gauge symmetry was in electromagnetism, and the general significance of the concept was not fully understood. For example, it was not clear whether it was the fields E and B or the potentials V and A that were the fundamental quantities; if E and B were the fundamental quantities, then the gauge transformations could be considered as nothing more than a mathematical trick to use on V and A.

Aharonov–Bohm experiment

Double-slit diffraction and interference pattern

Main article: Aharonov–Bohm effect

In quantum mechanics, a particle such as an electron is also described as a wave. For example, if the double-slit experiment is performed with electrons, then a wave-like interference pattern is observed. The electron has the highest probability of being detected at locations where the parts of the wave passing through the two slits are in phase with one another, resulting in constructive interference. The frequency, f, of the electron wave is related to the kinetic energy of an individual electron particle via the quantum-mechanical relation E = hf. If there are no electric or magnetic fields present in this experiment, then the electron's energy is constant, and, for example, there will be a high probability of detecting the electron along the central axis of the experiment, where by symmetry the two parts of the wave are in phase.

But now suppose that the electrons in the experiment are subject to electric or magnetic fields. For example, if an electric field were imposed on one side of the axis but not on the other, the results of the experiment would be affected. The part of the electron wave passing through that side oscillates at a different rate, since its energy has had −eV added to it, where −e is the charge of the electron and V the electrical potential. The results of the experiment will be different, because phase relationships between the two parts of the electron wave have changed, and therefore the locations of constructive and destructive interference will be shifted to one side or the other. It is the electric potential that occurs here, not the electric field, and this is a manifestation of the fact that it is the potentials and not the fields that are of fundamental significance in quantum mechanics.

Schematic of double-slit experiment in which Aharonov–Bohm effect can be observed: electrons pass through two slits, interfering at an observation screen, with the interference pattern shifted when a magnetic field B is turned on in the cylindrical solenoid, marked in blue on the diagram.

Explanation with potentials

It is even possible to have cases in which an experiment's results differ when the potentials are changed, even if no charged particle is ever exposed to a different field. One such example is the Aharonov–Bohm effect, shown in the figure. In this example, turning on the solenoid only causes a magnetic field B to exist within the solenoid. But the solenoid has been positioned so that the electron cannot possibly pass through its interior. If one believed that the fields were the fundamental quantities, then one would expect that the results of the experiment would be unchanged. In reality, the results are different, because turning on the solenoid changed the vector potential A in the region that the electrons do pass through. Now that it has been established that it is the potentials V and A that are fundamental, and not the fields E and B, we can see that the gauge transformations, which change V and A, have real physical significance, rather than being merely mathematical artifacts.

Gauge invariance: independence of results of gauge for the potentials

Note that in these experiments, the only quantity that affects the result is the difference in phase between the two parts of the electron wave. Suppose we imagine the two parts of the electron wave as tiny clocks, each with a single hand that sweeps around in a circle, keeping track of its own phase. Although this cartoon ignores some technical details, it retains the physical phenomena that are important here. If both clocks are sped up by the same amount, the phase relationship between them is unchanged, and the results of experiments are the same. Not only that, but it is not even necessary to change the speed of each clock by a fixed amount. We could change the angle of the hand on each clock by a varying amount θ, where θ could depend on both the position in space and on time. This would have no effect on the result of the experiment, since the final observation of the location of the electron occurs at a single place and time, so that the phase shift in each electron's "clock" would be the same, and the two effects would cancel out. This is another example of a gauge transformation: it is local, and it does not change the results of experiments.

Summary

In summary, gauge symmetry attains its full importance in the context of quantum mechanics. In the application of quantum mechanics to electromagnetism, i.e., quantum electrodynamics, gauge symmetry applies to both electromagnetic waves and electron waves. These two gauge symmetries are in fact intimately related. If a gauge transformation θ is applied to the electron waves, for example, then one must also apply a corresponding transformation to the potentials that describe the electromagnetic waves. Gauge symmetry is required in order to make quantum electrodynamics a renormalizable theory, i.e., one in which the calculated predictions of all physically measurable quantities are finite.

Types of gauge symmetries

The description of the electrons in the subsection above as little clocks is in effect a statement of the mathematical rules according to which the phases of electrons are to be added and subtracted: they are to be treated as ordinary numbers, except that in the case where the result of the calculation falls outside the range of 0≤θ<360°, we force it to "wrap around" into the allowed range, which covers a circle. Another way of putting this is that a phase angle of, say, 5° is considered to be completely equivalent to an angle of 365°. Experiments have verified this testable statement about the interference patterns formed by electron waves. Except for the "wrap-around" property, the algebraic properties of this mathematical structure are exactly the same as those of the ordinary real numbers.

In mathematical terminology, electron phases form an Abelian group under addition, called the circle group or U(1). "Abelian" means that addition is commutative, so that θ + φ = φ + θ. "Group" means that addition is associative, has an identity element, namely "0", and for every phase there exists an inverse such that the sum of a phase and its inverse is 0. Other examples of abelian groups are the integers under addition, 0, and negation, and the nonzero fractions under product, 1, and reciprocal.

Gauge fixing of a twisted cylinder

As a way of visualizing the choice of a gauge, consider whether it is possible to tell if a cylinder has been twisted. If the cylinder has no bumps, marks, or scratches on it, we cannot tell. We could, however, draw an arbitrary curve along the cylinder, defined by some function θ(x), where x measures distance along the axis of the cylinder. Once this arbitrary choice (the choice of gauge) has been made, it becomes possible to detect it if someone later twists the cylinder.

In 1954, Chen Ning Yang and Robert Mills proposed to generalize these ideas to noncommutative groups. A noncommutative gauge group can describe a field that, unlike the electromagnetic field, interacts with itself. For example, general relativity states that gravitational fields have energy, and special relativity concludes that energy is equivalent to mass. Hence a gravitational field induces a further gravitational field. The nuclear forces also have this self-interacting property.

Gauge bosons

Surprisingly, gauge symmetry can give a deeper explanation for the existence of interactions, such as the electric and nuclear interactions. This arises from a type of gauge symmetry relating to the fact that all particles of a given type are experimentally indistinguishable from one another. Imagine that Alice and Betty are identical twins, labeled at birth by bracelets reading A and B. Because the girls are identical, nobody would be able to tell if they had been switched at birth; the labels A and B are arbitrary, and can be interchanged. Such a permanent interchanging of their identities is like a global gauge symmetry. There is also a corresponding local gauge symmetry, which describes the fact that from one moment to the next, Alice and Betty could swap roles while nobody was looking, and nobody would be able to tell. If we observe that Mom's favorite vase is broken, we can only infer that the blame belongs to one twin or the other, but we cannot tell whether the blame is 100% Alice's and 0% Betty's, or vice versa. If Alice and Betty are in fact quantum-mechanical particles rather than people, then they also have wave properties, including the property of superposition, which allows waves to be added, subtracted, and mixed arbitrarily. It follows that we are not even restricted to complete swaps of identity. For example, if we observe that a certain amount of energy exists in a certain location in space, there is no experiment that can tell us whether that energy is 100% A's and 0% B's, 0% A's and 100% B's, or 20% A's and 80% B's, or some other mixture. The fact that the symmetry is local means that we cannot even count on these proportions to remain fixed as the particles propagate through space. The details of how this is represented mathematically depend on technical issues relating to the spins of the particles, but for our present purposes we consider a spinless particle, for which it turns out that the mixing can be specified by some arbitrary choice of gauge θ(x), where an angle θ = 0° represents 100% A and 0% B, θ = 90° means 0% A and 100% B, and intermediate angles represent mixtures.

According to the principles of quantum mechanics, particles do not actually have trajectories through space. Motion can only be described in terms of waves, and the momentum p of an individual particle is related to its wavelength λ by p = h/λ. In terms of empirical measurements, the wavelength can only be determined by observing a change in the wave between one point in space and another nearby point (mathematically, by differentiation). A wave with a shorter wavelength oscillates more rapidly, and therefore changes more rapidly between nearby points. Now suppose that we arbitrarily fix a gauge at one point in space, by saying that the energy at that location is 20% A's and 80% B's. We then measure the two waves at some other, nearby point, in order to determine their wavelengths. But there are two entirely different reasons that the waves could have changed. They could have changed because they were oscillating with a certain wavelength, or they could have changed because the gauge function changed from a 20–80 mixture to, say, 21–79. If we ignore the second possibility, the resulting theory does not work; strange discrepancies in momentum will show up, violating the principle of conservation of momentum. Something in the theory must be changed.

Again there are technical issues relating to spin, but in several important cases, including electrically charged particles and particles interacting via nuclear forces, the solution to the problem is to impute physical reality to the gauge function θ(x). We say that if the function θ oscillates, it represents a new type of quantum-mechanical wave, and this new wave has its own momentum p = h/λ, which turns out to patch up the discrepancies that otherwise would have broken conservation of momentum. In the context of electromagnetism, the particles A and B would be charged particles such as electrons, and the quantum mechanical wave represented by θ would be the electromagnetic field. (Here we ignore the technical issues raised by the fact that electrons actually have spin 1/2, not spin zero. This oversimplification is the reason that the gauge field θ comes out to be a scalar, whereas the electromagnetic field is actually represented by a vector consisting of V and A.) The result is that we have an explanation for the presence of electromagnetic interactions: if we try to construct a gauge-symmetric theory of identical, non-interacting particles, the result is not self-consistent, and can only be repaired by adding electric and magnetic fields that cause the particles to interact.

Although the function θ(x) describes a wave, the laws of quantum mechanics require that it also have particle properties. In the case of electromagnetism, the particle corresponding to electromagnetic waves is the photon. In general, such particles are called gauge bosons, where the term "boson" refers to a particle with integer spin. In the simplest versions of the theory gauge bosons are massless, but it is also possible to construct versions in which they have mass. This is the case for the gauge bosons that carry the weak interaction: the force responsible for nuclear decay.