title: USS Scorpion (SSN-589)

text: Both "All Hands Down" and "Scorpion Down" point toward involvement by the KGB spy ring (the so-called Walker spy ring) led by John Anthony Walker, Jr., in the heart of the U.S. Navy's communications, stating that it could have known that "Scorpion" was coming to investigate the Soviet flotilla. According to this theory, both navies agreed to hide the truth about both USS "Scorpion" and K-129 incidents. Several USN and RN submarines collided with Soviet "Echo"-class subs in Russian and British waters in this period, showing greatly enhanced aggression in Soviet Navy sub operations in 1968. The navy Minister in the British Labour government, noted 11 such deliberate collisions. Commander Roger Lane Nott, Royal Navy commander of during the 1982 Falklands War, stated that in 1972, during his service as a junior navigation officer on , a Soviet submarine entered the Firth of Clyde channel in Scotland and "Conqueror" was given the order to "chase it out". Having realized it was being pursued, "a very aggressive Soviet captain turned his submarine and drove her straight at HMS "Conqueror". It had been an extremely close call."


title: USS Scorpion (SSN-589)

text: According to a translated article from "Pravda", Moscow never issued a "fire" command during the Cold War. This is disputed by Royal Navy officers, "there had been other occasions when harassed Russians had fired torpedoes to scare off trails". The Navy court of inquiry official statement confirms that the Soviet naval group, including an Echo-II submarine, was conducting a "hydro-acoustic operation" in the area, but they were about 200 miles to the west of "Scorpion"'s position at the time of the sinking. Adding to the body of evidence against a Soviet torpedo-attack theory, U.S. Navy submarine Captain Robert LaGassa has flatly stated, "no Soviet submarine in 1968 could detect, track, approach, and attack any "Skipjack-" or later-class U.S. submarine".


title: USS Scorpion (SSN-589)

text: During 1967, though, the large Soviet nuclear submarine-building program, and the view of naval officers and in particular Admiral Rickover, that Defense Secretary R. McNamara, Naval Intelligence and CIA assessments underrated the speed of even existing Soviet subs and their threat led to 2 major tests on the request of Rickover. On 3–5 January 1968, the CVN USS "Enterprise" proved unable to outrun a Soviet November SSN at flank speed of 30/31 knots. This showed the Soviet November SSN, 5 knots faster than 'shatteringly' wrong intelligence estimates and underwater tracking a US CVN on sonar at 30 knots. Sec. McNamara remained opposed to a new fast SSN 688 class leader capable of 30 knots. His deputy and the chief sea warfare advisor to head of USN scientific engineering research James Nunan advised against SSN 688 or the need or reason for an enlarged, fast, 30-knot "Sturgeon" on 18 December 1967 in favor of long-term research into a new small 'conform' nuclear design. McNamara apparently attempted to have Rickover court-martialed and removed from office at this point.


title: USS Scorpion (SSN-589)

text: However, the USN case for SSN 688 after the hijacking of USS "Pueblo" in January 1968 and the Tet offensive in February 1968, which showed new communist aggression, was resubmitted. The plan and requirement for new fast SSNs was accepted by USN Chief of Staff Admiral Thomas Moorer after further inquiry in March 1968, but was not accepted by the US government. Events in May 1968 led to Admiral Rickover and Chief of USN Scientific Research and Engineering, John S Foster, appearing before the US Senate and House armed forces committees in the first week of June 1968, and it was decided to order an immediate test to illustrate to Foster that the tactical advantage of speed in a SSN could outweigh stealth and quietness. The radical test was conducted with a top USN "Permit"-class SSN crew aboard USS" Dace" captained by Cdr K. McKee and a crew with experience running in Russian waters engaging in a hunt and attempt to simulate a torpedo attack on a fast "Skipjack"-class, the USS "Shark", with a declared speed of 29 knots. While the trial was successful, it showed just how difficult a faster, noisier submarine like "Shark" was to engage, and by implication that an even faster" November"-class Soviet sub, while noisy, might well have been able to engage a 29-knot "Skipjack".


title: USS Scorpion (SSN-589)

text: The results of the U.S. Navy's various investigations into the loss of "Scorpion" are inconclusive. While the court of inquiry never endorsed Dr. Craven's torpedo theory regarding the loss of "Scorpion", its "findings of facts" released in 1993 carried Craven's torpedo theory at the head of a list of possible causes of "Scorpion"s loss.


title: USS Scorpion (SSN-589)

text: Released in 2006, Stephen Johnson's "Silent Steel: The Mysterious Death of the Nuclear Attack Sub USS Scorpion" provides a detailed listing of every mechanical problem on the submarine cited by the Navy or mentioned in crewmen's letters, but does not solve the "Scorpion"s sinking. Johnson, a critic of Dr. Craven, agrees with Navy scientists who, in 1970, gave their opinion that the sub's hull was smashed by implosion damage and not a torpedo blast, a finding they support with their interpretation of certain evidence about the condition of the hull and hydroacoustic recordings of the disaster. "Silent Steel" portrays an overworked submarine denied needed maintenance and manned by a demoralized crew, a depiction contradicted by many former "Scorpion" enlisted men and officers, and based in part on the testimony of sailors who had applied for transfer from the boat. Johnson also enumerates many of the Navy-wide submarine maintenance issues that denied "Scorpion" an overhaul and overdue safety improvements, though the Navy would maintain that virtually all necessary and vital improvements and repairs were made on the submarine before her final deployment. The Submarine Safety Program (SUBSAFE), initiated following the 1963 loss of "Thresher", delayed new submarine construction and sub overhauls by monopolizing skilled workers and critical spare parts. Fearing that a normal overhaul and safety work during 1967 might sideline "Scorpion" for three years, it was selected for a brief experimental overhaul, but this was canceled due to a shortage of workers. "Scorpion" sank eight months after leaving Norfolk Naval Shipyard.


title: USS Scorpion (SSN-589)

text: In 1998, two "New York Times" reporters published "", a book providing a rare look into the world of nuclear submarines and espionage during the Cold War. One lengthy chapter deals extensively with "Scorpion" and her loss. The book reports that concerns about the Mk 37 conventional torpedo carried aboard "Scorpion" were raised in 1967 and 1968, before "Scorpion" left Norfolk for her last mission. The concerns focused on the battery that powered the torpedoes. The battery had a thin metal-foil barrier separating two types of volatile chemicals. When mixed slowly and in a controlled fashion, the chemicals generated heat and electricity, powering the motor that pushed the torpedo through the water, but vibrations normally experienced on a nuclear submarine were found to cause the thin foil barrier to break down, allowing the chemicals to interact intensely. This interaction generated excessive heat, which in tests, could readily have caused an inadvertent torpedo explosion. The authors of "Blind Man's Bluff" do not directly contradict the official findings, but highlight information discovered during the investigation, which contradicts the investigation's findings, but are not addressed in the report. Notably, the book cites a hot-running torpedo incident on the USS "Sargo" (SSN-583) prior to the loss of "Scorpion", although "Sargo" was moored at the time and not lost.


title: USS Scorpion (SSN-589)

text: In 2005, the book "Red Star Rogue: The Untold Story of a Soviet Submarine's Nuclear Strike Attempt on the U.S.", by former American submariner Kenneth Sewell in collaboration with journalist Clint Richmond, claimed that Soviet submarine was sunk northwest of Oahu on 7 March 1968 while attempting to launch her three ballistic missiles, in a rogue attempt to destroy Pearl Harbor.


title: USS Scorpion (SSN-589)

text: Sewell claims that the sinking of "Scorpion" was caused by a retaliatory strike for the sinking of "K-129", which the Soviets had attributed to a collision with .


title: USS Scorpion (SSN-589)

text: In 1995, when Peter Huchthausen began work on a book about the Soviet underwater fleet, he interviewed former Soviet Admiral Victor Dygalo, who stated that the true history of "K-129" has not been revealed because of the informal agreement between the two countries' senior naval commands. The purpose of that secrecy, he alleged, is to stop any further research into the losses of either "Scorpion" or "K-129". Huchthausen states that Dygalo told him to "forget about ever resolving these sad issues for the surviving families."


title: USS Scorpion (SSN-589)

text: "All Hands Down" was written by Kenneth R. Sewell, a nuclear engineer and a U.S. Navy veteran who spent five years aboard , a fast attack submarine. It attempts to link the sinking of "Scorpion" with the "Pueblo" incident, the John Anthony Walker spy ring, and Cold War Soviet aggression. The thesis of this book is that action off the Canary Islands was the direct cause of the sinking. The author purports that this is supported by motives in the Soviet Navy following the sinking of "K-129", which caused the Russian Navy to trap a U.S. submarine. The bait for this trap would be strange military operations and furtive naval manoeuvres in the Atlantic, accompanied by countermeasures that would seemingly be defeated only by the deployment of a nuclear submarine. With information from spying by Walker, the position and arrival time of "Scorpion" was known by the Russians, and its sinking followed the springing of the trap. The book claims "Scorpion" was sunk by a Ka-25 helicopter equipped with antisubmarine torpedoes, which took off from one ship and landed on a different one. This was so that no one, other than the aircrew of the helicopter, would notice one torpedo missing.


title: USS Scorpion (SSN-589)

text: The book then purports a cover-up by American and Soviet officials, to avoid public outrage and an increase in Cold War tension.


title: USS Scorpion (SSN-589)

text: Ed Offley, a reporter on military affairs, has closely followed developments in information concerning the sinking of the "Scorpion". His most recent article on the subject is "Buried at Sea" published in the Winter 2008 issue of the "Quarterly Journal of Military History". This article summarizes the facts in the case as presented in his 2007 book "Scorpion Down: Sunk by the Soviets, Buried by the Pentagon: The Untold Story of the USS Scorpion". In the book, Offley, gathering decades of his own research, hypothesizes that "Scorpion" was sunk by the Soviets, possibly in retaliation for the loss of "K-129" earlier that year. The book paints a picture of increasing Soviet anger at U.S. Navy provocations — specifically, close-in monitoring of Soviet naval operations by almost every U.S. nuclear submarine. Around the same time, the Soviet intelligence community scored a huge boon in receiving the mechanical cryptologic devices (TSAC/KW- 7) from . These machines, combined with daily crypto keys from the Walker spy ring, likely allowed the Soviets to monitor U.S. Navy ship dispositions and communications.


title: USS Scorpion (SSN-589)

text: Offley contends that the "Scorpion" was tracked by several Soviet Navy assets from the Mediterranean to its final operational area south of the Azores, where it was then sunk by a Soviet torpedo. He claims the U.S. Navy was aware of the loss of the "Scorpion" on 21 May 1968, and by the night of 22 May 68, deep concern had arisen over the "Scorpion" after not receiving the required 24-hr, four-word communication check required on operational duty, in the SUBCOMLANT communication center at Norfolk on the evidence of two 2nd-class radiomen on the deck that night, among junior USN officers, and the unit supervisor, Warrant Officer J. Walker, confirmed to interested staff that no check transmission from "Scorpion" was received that night, and by the morning of 23 May 1968, the SUBCOMLANT center was full of admirals and a Marine general, and no doubt existed about the loss of "Scorpion", and the US government and Navy were engaged in a massive cover-up, within days destroying much of the sound and communication data at SOSUS ground stations in the U.S. and Europe, and delaying any public indication of the loss until the ship's scheduled arrival at Norfolk, Virginia, five days later, partly to disguise the fact that U.S. nuclear subs were in constant or frequent communication with U.S. naval communication bases and that the subsequent search for the "Scorpion" was a five-month-long deception to pretend they had no idea of the hull's location.


title: USS Scorpion (SSN-589)

text: The oral testimony Offley relied upon are recollections of surviving SOSUS recordings documenting torpedo sounds, evasion sounds, an explosion, and eventually the sounds of implosions as "Scorpion" plunged past crush depth.


title: USS Scorpion (SSN-589)

text: In a section from this 2014 book titled "The Danger of Culture", retired U.S. Navy Rear Admiral Dave Oliver offers the theory based on his own experiences that hydrogen explosion, either during or immediately following a battery charge, possibly destroyed USS "Scorpion" and killed her crew. The proximate cause in that scenario would have been the procedural carryover from diesel-boat days wherein the boat was effectively rigged for collision—with subsequent changes in ventilation flow and watertight condition—before proceeding to periscope depth by way of setting "Condition Baker". Oliver had personally witnessed dangerously high hydrogen level spikes under such conditions aboard a nuclear submarine, specifically while going to periscope depth and setting Condition Baker during a battery charge. Diesel boats, in contrast, were not capable of doing a battery charge while deeply submerged, but were instead dealing with the risk of collision while on antisurface-ship operations when proceeding to periscope depth while in or near shipping lanes. In regard to NAVSEA responsibility, he further states: "I always felt that the investigators closed their eyes to the most likely cause because they did not want to acknowledge their own involvement in this tragedy. I had forwarded my letter about Condition Baker via some of the same people responsible for the "Scorpion" investigation."


title: USS Scorpion (SSN-589)

text: Phil Ochs released a song on his album "Rehearsals for Retirement" (1969) titled "The Scorpion Departs but Never Returns".


title: African elephant

text: African elephants (Loxodonta) are a genus comprising two living elephant species, the African bush elephant ("L. africana") and the smaller African forest elephant ("L. cyclotis"). Both are social herbivores with grey skin, but differ in the size and colour of their tusks and in the shape and size of their ears and skulls.


title: African elephant

text: Both species are considered at heavy risk of extinction on the IUCN Red List; as of 2021, the bush elephant is considered endangered and the forest elephant is considered critically endangered. They are threatened by habitat loss and fragmentation, and poaching for the illegal ivory trade is a threat in several range countries as well.


title: African elephant

text: "Loxodonta" is one of two extant genera of the family Elephantidae. The name refers to the lozenge-shaped enamel of their molar teeth. Fossil remains of "Loxodonta" species have been excavated in Africa, dating to the Middle Pliocene.


title: African elephant

text: The first scientific description of the African elephant was written in 1797 by Johann Friedrich Blumenbach, who proposed the scientific name "Elephas africanus".


title: African elephant

text: "Loxodonte" was proposed as a generic name for the African elephant by Frédéric Cuvier in 1825. This name refers to the lozenge-shaped enamel of the molar teeth, which differs significantly from the rounded shape of the Asian elephant's molar enamel.


title: African elephant

text: An anonymous author used the Latinized spelling "Loxodonta" in 1827. Anonymous was recognized as authority by the International Code of Zoological Nomenclature in 1999.


title: African elephant

text: "Elephas" ("Loxodonta") "cyclotis" was proposed by Paul Matschie in 1900, who described three African elephant zoological specimens from Cameroon whose skulls differed in shape from elephant skulls collected elsewhere in Africa. In 1936, Glover Morrill Allen considered this elephant to be a distinct species and called it 'forest elephant'; later authors considered it to be a subspecies. Morphological and genetic analyses provided evidence for species-level differences between the African bush elephant and the African forest elephant.


title: African elephant

text: In 1907, Richard Lydekker proposed six African elephant subspecies based on the different sizes and shapes of their ears. They are all considered synonymous with the African bush elephant.


title: African elephant

text: A third species, the West African elephant, has also been proposed but it needs confirmation. It is thought that this lineage has been isolated from the others for 2.4 million years.


title: African elephant

text: Between the late 18th and 20th centuries, the following extinct African elephants were described on the basis of fossil remains:


title: African elephant

text: Analysis of nuclear DNA sequences indicates that the genetic divergence between African bush and forest elephants dates 2.6 – 5.6 million years ago. The divergence between the Asian elephant and the woolly mammoths is estimated 2.5 – 5.4 million years ago, which strongly supports their status as distinct species. The African forest elephant was found to have a high degree of genetic diversity, perhaps reflecting periodic fragmentation of their habitat during the climatic changes in the Pleistocene.


title: African elephant

text: Gene flow between the two African elephant species was examined at 21 locations. The analysis revealed that several African bush elephants carried mitochondrial DNA of African forest elephants, indicating they hybridised in the savanna-forest transition zone in ancient times.


title: African elephant

text: Sequence analysis of DNA from fossils of the extinct Eurasian "Palaeoloxodon antiquus" shows it to be much closer related to the African forest elephant than to the African bush elephant. The validity of "Palaeoloxodon" has therefore been questioned.


title: African elephant

text: African elephants have grey folded skin up to thick that is covered with sparse, bristled dark-brown to black hair. Short tactile hair grows on the trunk, which has two finger-like processes at the tip, whereas Asian elephants only have one. Their large ears help to reduce body heat; flapping them creates air currents and exposes the ears' inner sides where large blood vessels increase heat loss during hot weather. The trunk is a prehensile elongation of its upper lip and nose. This highly sensitive organ is innervated primarily by the trigeminal nerve, and thought to be manipulated by about 40,000–60,000 muscles. Because of this muscular structure, the trunk is so strong that elephants can use it for lifting about 3% of their own body weight. They use it for smelling, touching, feeding, drinking, dusting, producing sounds, loading, defending and attacking. Elephants sometimes swim underwater and use their trunks as snorkels.


title: African elephant

text: Both male and female African elephants have tusks that grow from deciduous teeth called tushes, which are replaced by tusks when calves are about one year old. Tusks are composed of dentin, which forms small diamond-shaped structures in the tusk's center that become larger at its periphery. Tusks are primarily used to dig for roots and strip the bark from trees for food, for fighting each other during mating season, and for defending themselves against predators. The tusks weigh from and can be from long. They are curved forward and continue to grow throughout the elephant's lifetime.


title: African elephant

text: The dental formula of elephants is . Elephants have four molars; each weighs about and measures about long. As the front pair wears down and drops out in pieces, the back pair moves forward, and two new molars emerge in the back of the mouth. Elephants replace their teeth four to six times in their lifetimes. At around 40 to 60 years of age, the elephant loses the last of its molars and will likely die of starvation which is a common cause of death. African elephants have 24 teeth in total, six on each quadrant of the jaw. The enamel plates of the molars are fewer in number than in Asian elephants.


title: African elephant

text: The African bush elephant is the largest terrestrial animal. Cows are tall at the shoulder and weigh , while bulls are tall and weigh . Its back is concave-shaped, while the back of the African forest elephant is nearly straight.


title: African elephant

text: The largest recorded individual stood at the shoulder, and is estimated to have weighed . The tallest recorded individual stood at the shoulder and weighed .


title: African elephant

text: The African forest elephant is smaller with a weight of up and a shoulder height of in females and in males. It is the third largest terrestrial animal.


title: African elephant

text: African elephants are distributed in Sub-Saharan Africa, where they inhabit Sahelian scrubland and arid regions, tropical rainforests, mopane and miombo woodlands. African forest elephant populations occur only in Central Africa.


title: African elephant

text: Elephants are the animals with the lowest sleep times, especially African elephants. Their average sleep was found to be only 2 hours in 24-hour cycles.


title: African elephant

text: Both African elephant species live in family units comprising several adult cows, their daughters and their subadult sons. Each family unit is led by an older cow known as the matriarch. African forest elephant groups are less cohesive than African bush elephant groups, probably because of the lack of predators.


title: African elephant

text: When separate family units bond, they form kinship or bond groups. After puberty, male elephants tend to form close alliances with other males. While females are the most active members of African elephant groups, both male and female elephants are capable of distinguishing between hundreds of different low frequency infrasonic calls to communicate with and identify each other.


title: African elephant

text: Elephants use some vocalisations that are beyond the hearing range of humans, to communicate across large distances. Elephant mating rituals include the gentle entwining of trunks.


title: African elephant

text: The bulls were believed to be solitary animals, becoming independent once reaching maturity. New research suggests that bulls maintain ecological knowledge for the herd, facilitating survival when searching for food and water, which also benefits the young bulls who associate with them. Bulls only return to the herd to breed or to socialize; they do not provide prenatal care to their offspring, but rather play a fatherly role to younger bulls to show dominance.


title: African elephant

text: While feeding, the African elephant uses its trunk to pluck leaves and its tusk to tear at branches, which can cause enormous damage to foliage. Fermentation of the food takes place in the hindgut, thus enabling large food intakes. The large size and hindgut of the African elephant also allows for digestion of various plant parts, including fibrous stems, bark and roots.


title: African elephant

text: African elephants are highly intelligent. They have a very large and highly convoluted neocortex, a trait they share with humans, apes and some dolphin species. They are amongst the world's most intelligent species. With a mass of just over , the elephant brain is larger than that of any other terrestrial animal. The elephant's brain is similar to a human brain in terms of structure and complexity; the elephant's cortex has as many neurons as that of a human brain, suggesting convergent evolution.


title: African elephant

text: Elephants exhibit a wide variety of behaviours, including those associated with grief, learning, mimicry, art, play, a sense of humor, altruism, use of tools, compassion, cooperation, self-awareness, memory and possibly language. All of these behaviors point to a highly intelligent species that is thought to be equal with cetaceans and primates.


title: African elephant

text: African elephants are at their most fertile between the ages of 25 and 45. Calves are born after a gestation period of up to nearly two years. The calves are cared for by their mother and other young females in the group, known as allomothering.


title: African elephant

text: African elephants show sexual dimorphism in weight and shoulder height by age 20, due to the rapid early growth of males. By age 25, males are double the weight of females; however, both sexes continue to grow throughout their lives.


title: African elephant

text: Female African elephants are able to start reproducing at around 10 to 12 years of age, and are in estrus for about 2 to 7 days. They do not mate at a specific time; however, they are less likely to reproduce in times of drought than when water is plentiful. The gestation period of an elephant is 22 months and fertile females usually give birth every 3–6 years, so if they live to around 50 years of age, they may produce 7 offspring. Females are a scarce and mobile resource for the males so there is intense competition to gain access to estrous females.


title: African elephant

text: Post sexual maturity, males begin to experience musth, a physical and behavioral condition that is characterized by elevated testosterone, aggression and more sexual activity. Musth also serves a purpose of calling attention to the females that they are of good quality, and it cannot be mimicked as certain calls or noises may be. Males sire few offspring in periods when they are not in musth. During the middle of estrus, female elephants look for males in musth to guard them. The females will yell, in a loud, low way to attract males from far away. Male elephants can also smell the hormones of a female ready for breeding. This leads males to compete with each other to mate, which results in the females mating with older, healthier males. Females choose to a point who they mate with, since they are the ones who try to get males to compete to guard them. However, females are not guarded in the early and late stages of estrus, which may permit mating by younger males not in musth.


title: African elephant

text: Males over the age of 25 compete strongly for females in estrus, and are more successful the larger and more aggressive they are. Bigger males tend to sire bigger offspring. Wild males begin breeding in their thirties when they are at a size and weight that is competitive with other adult males. Male reproductive success is maximal in mid-adulthood and then begins to decline. However, this can depend on the ranking of the male within their group, as higher-ranking males maintain a higher rate of reproduction. Most observed matings are by males in musth over 35 years of age. Twenty-two long observations showed that age and musth are extremely important factors; "… older males had markedly elevated paternity success compared with younger males, suggesting the possibility of sexual selection for longevity in this species."


title: African elephant

text: Males usually stay with a female and her herd for about a month before moving on in search of another mate. Less than a third of the population of female elephants will be in estrus at any given time and the gestation period of an elephant is long, so it makes more evolutionary sense for a male to search for as many females as possible rather than stay with one group.


title: African elephant

text: Both species are threatened by habitat loss and fragmentation, and poaching for the illegal ivory trade is a threat in several range countries as well. The African bush elephant is listed as Endangered and the African forest elephant as Critically Endangered on the respective IUCN Red Lists.


title: African elephant

text: Based on vegetation types that provide suitable habitat for African elephants, it was estimated that in the early 19th century a maximum of 26,913,000 African elephants might have been present from the Sahel in the north to the Highveld in the south. Decrease of suitable habitat was the major cause for the decline of elephant populations until the 1950s. Hunting African elephants for the ivory trade accelerated the decline from the 1970s onwards. The carrying capacity of remaining suitable habitats was estimated at 8,985,000 elephants at most by 1987.


title: African elephant

text: In the 1970s and 1980s, the price for ivory rose, and poaching for ivory increased in particular in Central African range countries where access to elephant habitats was facilitated by logging and petroleum mining industries.


title: African elephant

text: Between 1976 and 1980, about raw ivory was exported from Africa to Hong Kong and Japan, equivalent to tusks of about 222,000 African elephants.


title: African elephant

text: The first continental elephant census was carried out in 1976. At the time, 1.34 million elephants were estimated to range over .


title: African elephant

text: In the 1980s, it was difficult to carry out systematic surveys in several East African range countries due to civil wars.


title: African elephant

text: In 1987, it was estimated that the African elephant population had declined to 760,000 individuals. In 1989, only 608,000 African elephants were estimated to have survived.


title: African elephant

text: When the international ivory trade reopened in 2006, the demand and price for ivory increased in Asia. In Chad's Zakouma National Park, more than 3,200 elephants were killed between 2005 and 2010. The park did not have sufficient guards to combat poaching and their weapons were outdated. Well organized networks facilitated smuggling the ivory through Sudan.


title: African elephant

text: The government of Tanzania estimated that more than 85,000 elephants were lost to poaching in Tanzania between 2009 and 2014, representing a 60% loss.


title: African elephant

text: In 2012, a large upsurge in ivory poaching was reported, with about 70% of the product flowing to China.


title: African elephant

text: China was the biggest market for poached ivory but announced that it would phase out the legal domestic manufacture and sale of ivory products in May 2015.


title: African elephant

text: Conflicts between elephants and a growing human population are a major issue in elephant conservation. Human encroachment into natural areas where bush elephants occur or their increasing presence in adjacent areas has spurred research into methods of safely driving groups of elephants away from humans. Playback of the recorded sounds of angry Western honey bees has been found to be remarkably effective at prompting elephants to flee an area. Farmers have tried scaring elephants away by more aggressive means such as fire or the use of chili peppers along fences to protect their crops.


title: African elephant

text: In 1986, the African Elephant Database was initiated with the aim to monitor the status of African elephant populations. This database includes results from aerial surveys, dung counts, interviews with local people and data on poaching.


title: African elephant

text: In 1989, the Convention on International Trade in Endangered Species of Wild Fauna and Flora listed the African elephant on CITES Appendix I. This listing banned international trade of African elephants and their body parts by countries that signed the CITES agreement. Hunting elephants is banned in the Central African Republic, Democratic Republic of Congo, Gabon, Côte d'Ivoire, and Senegal. After the ban came into force in 1990, retail sales of ivory carvings in South Africa have plummeted by more than 95% within 10 years.


title: African elephant

text: As a result of the trade ban, African elephant populations recovered in Southern African range countries.


title: African elephant

text: The African Elephant Specialist Group has set up a "Human-Elephant Conflict Task Force" with the aim to develop conflict mitigation strategies.


title: African elephant

text: In 2005, the West African Elephant Memorandum of Understanding was signed by 12 West African countries. The Convention on the Conservation of Migratory Species of Wild Animals provided financial support for four years to implement the West African Elephant Conservation Strategy, which forms the central component of this intergovernmental treaty.


title: African elephant

text: In 2019, the export of wild African elephants to zoos around the world was banned, with an exception added by the EU to allow export in "exceptional cases where … it is considered that a transfer to ex-situ locations will provide demonstrable in-situ conservation benefits for African elephants". Previously, export had been allowed in Southern Africa with Zimbabwe capturing and exporting more than 100 baby elephants to Chinese zoos since 2012.


title: African elephant

text: It was found that elephant conservation does not pose a trade-off with climate change mitigation. Although animals typically cause a reduction of woody biomass and therewith above-ground carbon, they foster soil carbon sequestration.


title: African elephant

text: Many African cultures revere the African elephant as a symbol of strength and power. It is also praised for its size, longevity, stamina, mental faculties, cooperative spirit, and loyalty. Its religious importance is mostly totemic. Many societies believed that their chiefs would be reincarnated as elephants. In the 10th century, the people of Igbo-Ukwu in Nigeria buried their leaders with elephant tusks.


title: African elephant

text: South Africa uses elephant tusks in their coat of arms to represent wisdom, strength, moderation and eternity.


title: African elephant

text: In the western African Kingdom of Dahomey, the elephant was associated with the 19th century rulers of the Fon people, Guezo and his son Glele. The animal is believed to evoke strength, royal legacy, and enduring memory as related by the proverbs: ""There where the elephant passes in the forest, one knows"" and ""The animal steps on the ground, but the elephant steps down with strength"." Their flag depicted an elephant wearing a royal crown.


title: African elephant

text: The coat of arms of the Central African Republic features the head of an elephant in the upper left quadrant of the shield. The version of the coat of arms of Guinea used from 1958 to 1984 featured a golden elephant in the centre of the shield. The coat of arms of Ivory Coast features the head of an elephant as the focal point of the emblem. The coat of arms of the Republic of the Congo has two elephants supporting the shield. The coat of arms of Eswatini has an elephant and a lion supporting the shield.


title: Bob Balaban

text: Robert Elmer Balaban (born August 16, 1945) is an American actor, author, comedian, director and producer. He was one of the producers nominated for the Academy Award for Best Picture for "Gosford Park" (2001), in which he also appeared.


title: Bob Balaban

text: Balaban is most known for his appearances in the Christopher Guest comedies "Waiting for Guffman" (1996), "Best in Show" (2000), "A Mighty Wind" (2003), and "For Your Consideration" (2006) and in the Wes Anderson films "Moonrise Kingdom" (2012), "The Grand Budapest Hotel" (2014), "Isle of Dogs" (2018) and "The French Dispatch" (2021). Balaban's other film roles include the drama "Midnight Cowboy" (1969); the science fiction films "Close Encounters of the Third Kind" (1977), "Altered States" (1980), "" (1984), the comedy "Deconstructing Harry" (1997), and the historical drama "Capote" (2005).


title: Bob Balaban

text: Balaban has directed three feature films, in addition to numerous television episodes and films. He is also an author of children's novels.


title: Bob Balaban

text: Balaban was born to a Jewish family on August 16, 1945, in Chicago, Illinois, the son of Eleanor (née Pottasch) and Elmer Balaban, who owned several movie theatres and later was a pioneer in cable television. His mother acted under the name Eleanor Barry. His paternal grandparents emigrated from Russia to Chicago, while his mother's family was from Germany, Russia, and Romania.


title: Bob Balaban

text: His uncles were dominant forces in the theatre business; they founded the Balaban and Katz Theatre circuit in Chicago, a chain which included the Chicago and Uptown Theatres. Balaban's father, Elmer, and uncle, Harry, founded the H & E Balaban Corporation in Chicago, which operated its own movie palaces, including the Esquire Theatre in Chicago. They later owned a powerful group of television stations and cable television franchises. His uncle Barney Balaban was president of Paramount Pictures for nearly 30 years from 1936 to 1964. His maternal grandmother's second husband, Sam Katz, was a vice president at Metro-Goldwyn-Mayer beginning in 1936. Sam had been an early partner of Bob's uncles Abe, Barney, John, and Max in forming Balaban and Katz. Sam served as president of the Publix theatre division of Paramount Pictures.


title: Bob Balaban

text: Balaban began his college career at Colgate University where he joined Phi Kappa Tau fraternity and then transferred to New York University. He studied acting at HB Studio under Uta Hagen.


title: Bob Balaban

text: Balaban's first notable role was on stage; he created the role of Linus in the original off-Broadway production of "You're A Good Man, Charlie Brown" in 1967.


title: Bob Balaban

text: One of his earliest appearances in film was in "Midnight Cowboy" (1969). In the 1970s, he appeared as Grady Garrett on an episode of "Room 222", Orr in "Catch-22", Elliot the Organizer in "The Strawberry Statement", and the interpreter David Laughlin in the 1977 Steven Spielberg science fiction film "Close Encounters of the Third Kind". In 1979, he received a Tony Award nomination for his role in "The Inspector General". During the 1980s he appeared in films including Ken Russell's "Altered States" (1980) and the 1984 "" sequel "2010" (as Dr Chandra, the creator of HAL 9000). He also directed the Randy Quaid horror comedy film "Parents", and the Armin Mueller-Stahl drama film "The Last Good Time" (1994).


title: Bob Balaban

text: Balaban had supporting roles in films such as "Absence of Malice", "Bob Roberts", "Deconstructing Harry", "Ghost World", "The Majestic", "Lady in the Water", and Christopher Guest's "Waiting for Guffman", "Best in Show", "A Mighty Wind", and "For Your Consideration".


title: Bob Balaban

text: Balaban appeared in "Miami Vice" as reporter Ira Stone. In the 1990s, Balaban had a recurring role on the fourth season of "Seinfeld" as Russell Dalrymple, the fictional president of NBC. He also played Warren Littlefield, a real-world NBC executive, in "The Late Shift", about the battle between Jay Leno and David Letterman for NBC's "The Tonight Show". His tie to Littlefield continued in 2012 when he read the audiobook of Littlefield's autobiography, "Top of the Rock: Inside the Rise and Fall of Must See TV". In 1999, Balaban made a guest appearance in the sitcom "Friends" as Phoebe Buffay's father Frank in "The One With Joey's Bag". In 2010, Balaban appeared as Judge Clayton Horn, the real-life judge who presided over the obscenity trial of Lawrence Ferlinghetti and City Lights Bookstore in the movie "Howl".


title: Bob Balaban

text: In 2001, Balaban produced "Gosford Park", for which he received an Academy Award nomination for Best Picture. He also appeared in the movie as Morris Weissman, a Hollywood producer. He appeared in an episode of "Entourage" as a doctor known for writing prescriptions for medical marijuana. He directed the film "Bernard and Doris" (2006), starring Susan Sarandon; and also the biopic "Georgia O'Keeffe" (2009), starring Joan Allen and Jeremy Irons. He has directed several episodes of the Showtime series "Nurse Jackie".


title: Bob Balaban

text: In September 2011, he was featured with Morgan Freeman and John Lithgow in the Broadway debut of the play, "8"—a staged reenactment of the federal trial that overturned California's Prop 8 ban on same-sex marriage—as Judge Vaughn Walker. The production was held at the Eugene O'Neill Theatre to raise money for the American Foundation for Equal Rights.


title: Bob Balaban

text: In January 2016, Balaban appeared in the short play "Milton Bradley" by Peter Sagal, for Playing On Air, a non-profit organization that "records short plays [for public radio and podcast] written by top playwrights and performed by outstanding actors."


title: Bob Balaban

text: In early 2021, Balaban provided the voice of the narrator in "The Simpsons" episode “The Dad-Feelings Limited”.


title: Bob Balaban

text: Balaban wrote a series of six children's novels featuring a bionic dog named McGrowl. He also co-authored "Spielberg, Truffaut & Me: An Actor's Diary with Steven Spielberg" and "The Creature from the Seventh Grade: Sink or Swim (Creature from the Seventh Grade, #2)" which Andy Rash illustrated.


title: Bob Balaban

text: Balaban is married to Lynn Grossman; they have two daughters. He resides on the Upper West Side of Manhattan, New York City, New York.


title: David Hogg

text: David Miles Hogg (born April 12, 2000) is an American gun control activist. He rose to prominence during the 2018 United States gun violence protests as a student survivor of the Stoneman Douglas High School shooting, helping lead several high-profile protests, marches, and boycotts, including the boycott of "The Ingraham Angle". He has also been a target and scapegoat of several conspiracy theories.


title: David Hogg

text: With his sister Lauren Hogg, he wrote "#NeverAgain: A New Generation Draws the Line", a book that made "The New York Times" bestseller list. They pledged to donate to charity all income from the book. , Hogg is a student at Harvard University.


title: David Hogg

text: Hogg was included in "Time" magazine's "100 Most Influential People of 2018." He is the co-founder of Good Pillow, a pillow manufacturing company.


title: David Hogg

text: Before attending Marjory Stoneman Douglas High School, Hogg lived in Los Angeles, California. He is the son of Kevin Hogg, a former agent of the Federal Bureau of Investigation. His mother is Rebecca Boldrick, born in San Diego County, California and a teacher for Broward County Public Schools in Broward County, Florida.


title: David Hogg

text: Hogg chose to attend Stoneman Douglas because it offered television production classes. He was a Teenlink reporter for the "Sun Sentinel". He graduated on June 3, 2018. Hogg has dyslexia, attention deficit hyperactivity disorder, and post-traumatic stress disorder.


title: David Hogg

text: Hogg had been accepted to several universities, but decided to take a year off before starting college to work on the 2018 mid-term elections. In December 2018, he announced that he had been accepted by Harvard University and started in the fall of 2019. Hogg reportedly had a 1270 SAT score and 4.2 weighted GPA.


title: David Hogg

text: After Hogg's sister Lauren Hogg graduated from Stoneman Douglas in 2021, he and his family relocated to Washington, D.C.


title: David Hogg

text: On February 14, 2018, Hogg was a senior at Stoneman Douglas and on campus when a 19-year-old former student of the school entered Building 12 and started shooting with a semi-automatic rifle. Hogg, who was in his AP environmental science class, told the teacher that the repeated "pop" sounds the class heard sounded like gunshots. When the fire alarm went off, Hogg and other students made an attempt to exit the building, but a janitor instructed the students to go back into the classroom. Hogg credits the janitor for saving them, as the group of students were inadvertently heading towards the shooter. A culinary arts teacher pulled Hogg and others inside her classroom and they hid in a closet.


title: David Hogg

text: Hogg checked social media and discovered that the shooting was occurring at his high school. He used his cell phone to record the scene in real time, to interview the other students hiding in the closet, and to leave a record in the event that they did not survive the shooting. Hogg's sister, Lauren Hogg, who was a freshman student at the time, corresponded with her brother via text message while the shooting was taking place. After about an hour, SWAT team police officers came into the classroom and escorted them out. Hogg reunited with his sister and father later that day.


title: David Hogg

text: After the school shooting, Hogg emerged as a leader in the 2018 United States gun violence protests. Along with Alfonso Calderon, Sarah Chadwick, X González, Cameron Kasky and other students, he turned to the media to talk about their role as survivors in the shooting and voice his opinion on gun control and gun violence. He called on elected officials to pass gun control measures and has been a vocal critic of officials who take donations from the NRA, and he has been urging them to compromise on legislation to save lives.


title: David Hogg

text: Hogg joined the social media movement and student-led gun control advocacy group Never Again MSD shortly after its formation. Hogg flew to Los Angeles on February 21, 2018, to be on "The Dr. Phil Show", along with his sister, to discuss the shooting. There, they met with survivors of the Columbine High School massacre. Hogg, along with González, blamed the National Rifle Association and the politicians to which they donate as being complicit in school shootings. He declined to go to the White House on February 21 to meet with President Donald Trump, saying that he had to be in Tallahassee, and that Trump could come to Parkland if he wanted to talk.


title: David Hogg

text: When Republican candidate Leslie Gibson, who was running unopposed for the Maine House of Representatives, described González as a "skinhead lesbian", and also insulted Hogg as a "bald-faced liar", Hogg called for somebody to challenge the Republican; Eryn Gilchrist, who was "horrified and embarrassed" by Gibson's comment, decided to run as a Democrat to challenge Gibson for the position, as did Republican former State Senator Thomas Martin, Jr., who said Gibson's remarks did not represent the Maine Republican Party and that he planned to contact the survivors to commend their courage. Gibson dropped out of the race in response to public reaction critical of his comments.


title: David Hogg

text: Following the 2021 storming of the United States Capitol, David Hogg tweeted, "They can put up all the fencing around the capitol the real threats of @mtgreenee and @laurenboebert will still be inside until @GOPLeader takes a stand." Congresswoman Lauren Boebert (R-CO) retorted, "Give your keyboard a rest, child." A video of Congresswoman Marjorie Taylor Greene (R-GA) harassing Hogg in 2019 went viral in January 2021.


title: David Hogg

text: Hogg was featured on the cover of an April 2018 edition of "Time", along with fellow activists Alex Wind, Jaclyn Corin, González, and Kasky.


title: David Hogg

text: Hogg states that he is a supporter of the Second Amendment and supports NRA members' right to own guns legally, saying, "We're calling out the NRA a lot and 99.9 percent of the people that are in the NRA are responsible, safe gun owners and I respect them for that, joining an organization that wants to support safe gun ownership is excellent."


title: David Hogg

text: In an interview with Fox News, Hogg said he was for reasonable gun control such as regulations that prohibit those with mental illnesses from acquiring guns.


title: David Hogg

text: Hogg criticized the media coverage of the Parkland shooting as well as its aftermath in that black students were not given a voice by the media; he said that his school was 25% black but "the way we're covered doesn't reflect that".


title: David Hogg

text: In April 2018, Hogg initiated an effort to urge Speaker of the House Paul Ryan to bring a bill to the House of Representatives that required mandatory background checks for gun buyers; on Twitter, Hogg urged people to contact speaker Ryan and demand a vote on universal background checks.


title: David Hogg

text: Hogg worked to develop an anti-NRA advocacy group to encourage young people to register and vote in the 2018 midterm elections and elect candidates who promise better gun control legislation.


title: David Hogg

text: In May 2018, Hogg and other Never Again MSD students led a "die-in" protest at a Publix supermarket, with a mass of students lying down on the store's floor, as a rebuke of the supermarket's financial support of pro-NRA gubernatorial candidate Adam Putnam; the supermarket had contributed more than $670,000 to Putnam's campaign over three years. In addition, Hogg called on people to boycott Publix until the chain's support of Putnam stopped. As a result of the protest, Publix made a statement suspending support for Putnam.


title: David Hogg

text: In August 2018, Hogg announced he was planning on running to become a member of the United States House of Representatives when he turns 25 years old (due to ).


title: David Hogg

text: On February 10, 2021, March For Our Lives announced that Hogg would take a leave of absence "to take some time for himself to reflect and recommit to the mission."


title: David Hogg

text: On July 20, 2022, Hogg interrupted a House Judiciary Committee hearing to mark up the 2021 Assault Weapons Ban and the Equal Access for Victims of Gun Violence Act. During Republican member Andy Biggs' time, Biggs said that Americans should be armed should an invasion of the southern border happen. Hogg then shouted: "The shooter at my high school: antisemitic, anti-black and racist. The shooter in El Paso described it as an invasion", in reference to the 2019 El Paso shooting. He added, as he was escorted out of the hearing room: "Those guns are coming from the United States of America. They aren’t coming from Mexico. You are reiterating the points of a mass shooter sir".


title: David Hogg

text: Hogg called for students to boycott spring break in Florida and instead travel to Puerto Rico if gun control legislation was not passed by the Florida state government. Having finished high school in May 2018, Hogg took a gap year to campaign for politicians in favor of gun reform in the midterm elections.


title: David Hogg

text: Hogg initiated a boycott of companies who advertise during "The Ingraham Angle". Hogg called for the boycott after television host Laura Ingraham attacked him in a tweet about his lack of college acceptances, which Hogg characterized as cyberbullying. In response to the boycott, 24 advertisers left the show. Following the loss of advertisers, Ingraham apologized. Hogg dismissed the apology as insincere. The boycott drew mixed reactions. Ingraham was supported by Ted Nugent, Bill Maher, and by Russian bots on Twitter. Fox News continued to support Ingraham. Public polling showed that public perception of Fox News declined more than that of any advertiser. Simultaneously, Ingraham's viewership increased in the weeks following the boycott. Before, her viewership averaged 2.5 million. It jumped to 3 million when she returned after the boycott.


title: David Hogg

text: Shortly after the shooting, false claims appeared on social media claiming that the event never happened, and others accused Hogg and other students of being "crisis actors". After a series of televised interviews following the shooting, far-right figures and conspiracy theorists attacked Hogg in online media. Hogg's family received death threats from various conspiracy theorists, according to David's mother. Facebook, YouTube, and Instagram reported removing posts that attack the students or accuse them of being actors. The conspiracy theories about Hogg and other Parkland activists were named "PolitiFact"'s 2018 Lie of the Year.


title: David Hogg

text: In April 2018, Hogg was threatened by Sinclair Broadcast Group TV host Jamie Allman from station KDNL-TV in St. Louis, who wrote a tweet threatening to insert a hot poker in his anus. Following a boycott of advertisers initiated by Hogg on "The Allman Report", Allman resigned and his show was canceled.


title: David Hogg

text: On the morning of June 5, 2018, Broward County Sheriff's Office received a false report from an anonymous caller claiming that there was a hostage situation in Hogg's family home. The harassment tactic known as swatting was described by Hogg, several media organizations, and the sheriff's department as a prank.


title: David Hogg

text: During a June 2019 interview with the "Washington Post Magazine", Hogg said there have been seven attempts on his life.


title: David Hogg

text: In March 2019, now congresswoman Marjorie Taylor Greene followed Hogg down the street in Washington D.C. for over a minute demanding to know how he was able to get media coverage and meetings with over 30 U.S. lawmakers. She called him a "coward" and then, on video, accused him of being paid by George Soros and others. Hogg later responded to Greene on Twitter, saying, "how embarrassing a sitting congresswoman argues that an 18 year old is both a more affective "[sic]" lobbyist and communicator than her."


title: David Hogg

text: In February 2021, Hogg announced that he and progressive tech entrepreneur William LeGate would start a company to compete with MyPillow, whose CEO Mike Lindell has spread unsubstantiated claims of voter fraud as a staunch supporter of former President Trump. Hogg announced the new venture in a series of tweets. The company name was later announced as Good Pillow. In early April 2021, Hogg announced in another series of tweets that he had resigned, released all his interest in the company to LeGate, and was leaving the venture.


title: Zero-point energy

text: Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty principle. Therefore, even at absolute zero, atoms and molecules retain some vibrational motion. Apart from atoms and molecules, the empty space of the vacuum also has these properties. According to quantum field theory, the universe can be thought of not as isolated particles but continuous fluctuating fields: matter fields, whose quanta are fermions (i.e., leptons and quarks), and force fields, whose quanta are bosons (e.g., photons and gluons). All these fields have zero-point energy. These fluctuating zero-point fields lead to a kind of reintroduction of an aether in physics since some systems can detect the existence of this energy. However, this aether cannot be thought of as a physical medium if it is to be Lorentz invariant such that there is no contradiction with Einstein's theory of special relativity.


title: Zero-point energy

text: The notion of a zero-point energy is also important for cosmology, and physics currently lacks a full theoretical model for understanding zero-point energy in this context; in particular, the discrepancy between theorized and observed vacuum energy in the universe is a source of major contention. Physicists Richard Feynman and John Wheeler calculated the zero-point radiation of the vacuum to be an order of magnitude greater than nuclear energy, with a single light bulb containing enough energy to boil all the world's oceans. Yet according to Einstein's theory of general relativity, any such energy would gravitate, and the experimental evidence from the expansion of the universe, dark energy and the Casimir effect shows any such energy to be exceptionally weak. A popular proposal that attempts to address this issue is to say that the fermion field has a negative zero-point energy, while the boson field has positive zero-point energy and thus these energies somehow cancel each other out. This idea would be true if supersymmetry were an exact symmetry of nature; however, the LHC at CERN has so far found no evidence to support it. Moreover, it is known that if supersymmetry is valid at all, it is at most a broken symmetry, only true at very high energies, and no one has been able to show a theory where zero-point cancellations occur in the low-energy universe we observe today. This discrepancy is known as the cosmological constant problem and it is one of the greatest unsolved mysteries in physics. Many physicists believe that "the vacuum holds the key to a full understanding of nature".


title: Zero-point energy

text: Sometimes used interchangeably with it are the terms zero-point radiation and ground state energy. The term zero-point field (ZPF) can be used when referring to a specific vacuum field, for instance the QED vacuum which specifically deals with quantum electrodynamics (e.g., electromagnetic interactions between photons, electrons and the vacuum) or the QCD vacuum which deals with quantum chromodynamics (e.g., color charge interactions between quarks, gluons and the vacuum). A vacuum can be viewed not as empty space but as the combination of all zero-point fields. In quantum field theory this combination of fields is called the vacuum state, its associated zero-point energy is called the vacuum energy and the average energy value is called the vacuum expectation value (VEV) also called its condensate.


title: Zero-point energy

text: In classical mechanics all particles can be thought of as having some energy made up of their potential energy and kinetic energy. Temperature, for example, arises from the intensity of random particle motion caused by kinetic energy (known as Brownian motion). As temperature is reduced to absolute zero, it might be thought that all motion ceases and particles come completely to rest. In fact, however, kinetic energy is retained by particles even at the lowest possible temperature. The random motion corresponding to this zero-point energy never vanishes; it is a consequence of the uncertainty principle of quantum mechanics.


title: Zero-point energy

text: The uncertainty principle states that no object can ever have precise values of position and velocity simultaneously. The total energy of a quantum mechanical object (potential and kinetic) is described by its Hamiltonian which also describes the system as a harmonic oscillator, or wave function, that fluctuates between various energy states (see wave-particle duality). All quantum mechanical systems undergo fluctuations even in their ground state, a consequence of their wave-like nature. The uncertainty principle requires every quantum mechanical system to have a fluctuating zero-point energy greater than the minimum of its classical potential well. This results in motion even at absolute zero. For example, liquid helium does not freeze under atmospheric pressure regardless of temperature due to its zero-point energy.


title: Zero-point energy

text: Given the equivalence of mass and energy expressed by Albert Einstein's , any point in space that contains energy can be thought of as having mass to create particles. Virtual particles spontaneously flash into existence at every point in space due to the energy of quantum fluctuations caused by the uncertainty principle. Modern physics has developed quantum field theory (QFT) to understand the fundamental interactions between matter and forces, it treats every single point of space as a quantum harmonic oscillator. According to QFT the universe is made up of matter fields, whose quanta are fermions (i.e. leptons and quarks), and force fields, whose quanta are bosons (e.g. photons and gluons). All these fields have zero-point energy. Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying quantum vacuum, and that all properties of matter are merely vacuum fluctuations arising from interactions of the zero-point field.


title: Zero-point energy

text: The idea that "empty" space can have an intrinsic energy associated to it, and that there is no such thing as a "true vacuum" is seemingly unintuitive. It is often argued that the entire universe is completely bathed in the zero-point radiation, and as such it can add only some constant amount to calculations. Physical measurements will therefore reveal only deviations from this value. For many practical calculations zero-point energy is dismissed by fiat in the mathematical model as a term that has no physical effect. Such treatment causes problems however, as in Einstein's theory of general relativity the absolute energy value of space is not an arbitrary constant and gives rise to the cosmological constant. For decades most physicists assumed that there was some undiscovered fundamental principle that will remove the infinite zero-point energy and make it completely vanish. If the vacuum has no intrinsic, absolute value of energy it will not gravitate. It was believed that as the universe expands from the aftermath of the Big Bang, the energy contained in any unit of empty space will decrease as the total energy spreads out to fill the volume of the universe; galaxies and all matter in the universe should begin to decelerate. This possibility was ruled out in 1998 by the discovery that the expansion of the universe is not slowing down but is in fact accelerating, meaning empty space does indeed have some intrinsic energy. The discovery of dark energy is best explained by zero-point energy, though it still remains a mystery as to why the value appears to be so small compared to the huge value obtained through theory - the cosmological constant problem.


title: Zero-point energy

text: Many physical effects attributed to zero-point energy have been experimentally verified, such as spontaneous emission, Casimir force, Lamb shift, magnetic moment of the electron and Delbrück scattering. These effects are usually called "radiative corrections". In more complex nonlinear theories (e.g. QCD) zero-point energy can give rise to a variety of complex phenomena such as multiple stable states, symmetry breaking, chaos and emergence. Many physicists believe that "the vacuum holds the key to a full understanding of nature" and that studying it is critical in the search for the theory of everything. Active areas of research include the effects of virtual particles, quantum entanglement, the difference (if any) between inertial and gravitational mass, variation in the speed of light, a reason for the observed value of the cosmological constant and the nature of dark energy.


title: Zero-point energy

text: Zero-point energy evolved from historical ideas about the vacuum. To Aristotle the vacuum was , "the empty"; i.e., space independent of body. He believed this concept violated basic physical principles and asserted that the elements of fire, air, earth, and water were not made of atoms, but were continuous. To the atomists the concept of emptiness had absolute character: it was the distinction between existence and nonexistence. Debate about the characteristics of the vacuum were largely confined to the realm of philosophy, it was not until much later on with the beginning of the renaissance, that Otto von Guericke invented the first vacuum pump and the first testable scientific ideas began to emerge. It was thought that a totally empty volume of space could be created by simply removing all gases. This was the first generally accepted concept of the vacuum.


title: Zero-point energy

text: Late in the 19th century, however, it became apparent that the evacuated region still contained thermal radiation. The existence of the aether as a substitute for a true void was the most prevalent theory of the time. According to the successful electromagnetic aether theory based upon Maxwell's electrodynamics, this all-encompassing aether was endowed with energy and hence very different from nothingness. The fact that electromagnetic and gravitational phenomena were easily transmitted in empty space indicated that their associated aethers were part of the fabric of space itself. Maxwell himself noted that:


title: Zero-point energy

text: However, the results of the Michelson–Morley experiment in 1887 were the first strong evidence that the then-prevalent aether theories were seriously flawed, and initiated a line of research that eventually led to special relativity, which ruled out the idea of a stationary aether altogether. To scientists of the period, it seemed that a true vacuum in space might be created by cooling and thus eliminating all radiation or energy. From this idea evolved the second concept of achieving a real vacuum: cool a region of space down to absolute zero temperature after evacuation. Absolute zero was technically impossible to achieve in the 19th century, so the debate remained unsolved.


title: Zero-point energy

text: In 1900, Max Planck derived the average energy of a single "energy radiator", e.g., a vibrating atomic unit, as a function of absolute temperature:


title: Zero-point energy

text: where is Planck's constant, is the frequency, is the Boltzmann constant, and is the absolute temperature. The zero-point energy makes no contribution to Planck's original law, as its existence was unknown to Planck in 1900.


title: Zero-point energy

text: The concept of zero-point energy was developed by Max Planck in Germany in 1911 as a corrective term added to a zero-grounded formula developed in his original quantum theory in 1900.


title: Zero-point energy

text: In 1912, Max Planck published the first journal article to describe the discontinuous emission of radiation, based on the discrete quanta of energy. In Planck's "second quantum theory" resonators absorbed energy continuously, but emitted energy in discrete energy quanta only when they reached the boundaries of finite cells in phase space, where their energies became integer multiples of . This theory led Planck to his new radiation law, but in this version energy resonators possessed a zero-point energy, the smallest average energy a resonator could take on. Planck's radiation equation contained a residual energy factor, one , as an additional term dependent on the frequency , which was greater than zero (where is Planck's constant). It is therefore widely agreed that "Planck's equation marked the birth of the concept of zero-point energy." In a series of papers from 1911 to 1913, Planck found the average energy of an oscillator to be:


title: Zero-point energy

text: Soon, the idea of zero-point energy attracted the attention of Albert Einstein and his assistant Otto Stern. In 1913 they published a paper that attempted to prove the existence of zero-point energy by calculating the specific heat of hydrogen gas and compared it with the experimental data. However, after assuming they had succeeded, they retracted support for the idea shortly after publication because they found Planck's second theory may not apply to their example. In a letter to Paul Ehrenfest of the same year Einstein declared zero-point energy "dead as a doornail" Zero-point energy was also invoked by Peter Debye, who noted that zero-point energy of the atoms of a crystal lattice would cause a reduction in the intensity of the diffracted radiation in X-ray diffraction even as the temperature approached absolute zero. In 1916 Walther Nernst proposed that empty space was filled with zero-point electromagnetic radiation. With the development of general relativity Einstein found the energy density of the vacuum to contribute towards a cosmological constant in order to obtain static solutions to his field equations; the idea that empty space, or the vacuum, could have some intrinsic energy associated to it had returned, with Einstein stating in 1920:


title: Zero-point energy

text: In 1913 Niels Bohr had proposed what is now called the Bohr model of the atom, but despite this it remained a mystery as to why electrons do not fall into their nuclei. According to classical ideas, the fact that an accelerating charge loses energy by radiating implied that an electron should spiral into the nucleus and that atoms should not be stable. This problem of classical mechanics was nicely summarized by James Hopwood Jeans in 1915: "There would be a very real difficulty in supposing that the (force) law held down to the zero values of . For the forces between two charges at zero distance would be infinite; we should have charges of opposite sign continually rushing together and, when once together, no force would tend to shrink into nothing or to diminish indefinitely in size." The resolution to this puzzle came in 1926 with Schrödinger's famous equation. This equation explained the new, non-classical fact that an electron confined to be close to a nucleus would necessarily have a large kinetic energy so that the minimum total energy (kinetic plus potential) actually occurs at some positive separation rather than at zero separation; in other words, zero-point energy is essential for atomic stability.


title: Zero-point energy

text: In 1926 Pascual Jordan published the first attempt to quantize the electromagnetic field. In a joint paper with Max Born and Werner Heisenberg he considered the field inside a cavity as a superposition of quantum harmonic oscillators. In his calculation he found that in addition to the "thermal energy" of the oscillators there also had to exist an infinite zero-point energy term. He was able to obtain the same fluctuation formula that Einstein had obtained in 1909. However, Jordan did not think that his infinite zero-point energy term was "real", writing to Einstein that "it is just a quantity of the calculation having no direct physical meaning". Jordan found a way to get rid of the infinite term, publishing a joint work with Pauli in 1928, performing what has been called "the first infinite subtraction, or renormalisation, in quantum field theory".


title: Zero-point energy

text: Building on the work of Heisenberg and others Paul Dirac's theory of emission and absorption (1927) was the first application of the quantum theory of radiation. Dirac's work was seen as crucially important to the emerging field of quantum mechanics; it dealt directly with the process in which "particles" are actually created: spontaneous emission. 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. The theory showed that spontaneous emission depends upon the zero-point energy fluctuations of the electromagnetic field in order to get started. In a process in which a photon is annihilated (absorbed), the photon can be thought of as making a transition into the vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state. In the words of Dirac:


title: Zero-point energy

text: Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the zero-point energy of the electromagnetic field. This view was popularized by Victor Weisskopf who in 1935 wrote:


title: Zero-point energy

text: This view was also later supported by (1948), who argued that spontaneous emission "can be thought of as forced emission taking place under the action of the fluctuating field." This new theory, which Dirac coined quantum electrodynamics (QED) predicted a fluctuating zero-point or "vacuum" field existing even in the absence of sources.


title: Zero-point energy

text: Throughout the 1940s improvements in microwave technology made it possible to take more precise measurements of the shift of the levels of a hydrogen atom, now known as the Lamb shift, and measurement of the magnetic moment of the electron. Discrepancies between these experiments and Dirac's theory led to the idea of incorporating renormalisation into QED to deal with zero-point infinities. Renormalization was originally developed by Hans Kramers and also Victor Weisskopf (1936), and first successfully applied to calculate a finite value for the Lamb shift by Hans Bethe (1947). As per spontaneous emission, these effects can in part be understood with interactions with the zero-point field. But in light of renormalisation being able to remove some zero-point infinities from calculations, not all physicists were comfortable attributing zero-point energy any physical meaning, viewing it instead as a mathematical artifact that might one day be fully eliminated. In Wolfgang Pauli's 1945 Nobel lecture he made clear his opposition to the idea of zero-point energy stating "It is clear that this zero-point energy has no physical reality".


title: Zero-point energy

text: In 1948 Hendrik Casimir showed that one consequence of the zero-point field is an attractive force between two uncharged, perfectly conducting parallel plates, the so-called Casimir effect. At the time, Casimir was studying the properties of "colloidal solutions". These are viscous materials, such as paint and mayonnaise, that contain micron-sized particles in a liquid matrix. The properties of such solutions are determined by Van der Waals forces – short-range, attractive forces that exist between neutral atoms and molecules. One of Casimir's colleagues, Theo Overbeek, realized that the theory that was used at the time to explain Van der Waals forces, which had been developed by Fritz London in 1930, did not properly explain the experimental measurements on colloids. Overbeek therefore asked Casimir to investigate the problem. Working with Dirk Polder, Casimir discovered that the interaction between two neutral molecules could be correctly described only if the fact that light travels at a finite speed was taken into account. Soon afterwards after a conversation with Bohr about zero-point energy, Casimir noticed that this result could be interpreted in terms of vacuum fluctuations. He then asked himself what would happen if there were two mirrors – rather than two molecules – facing each other in a vacuum. It was this work that led to his famous prediction of an attractive force between reflecting plates. The work by Casimir and Polder opened up the way to a unified theory of Van der Waals and Casimir forces and a smooth continuum between the two phenomena. This was done by Lifshitz (1956) in the case of plane parallel dielectric plates. The generic name for both Van der Waals and Casimir forces is dispersion forces, because both of them are caused by dispersions of the operator of the dipole moment. The role of relativistic forces becomes dominant at orders of a hundred nanometers.


title: Zero-point energy

text: In 1951 Herbert Callen and Theodore Welton proved the quantum fluctuation-dissipation theorem (FDT) which was originally formulated in classical form by Nyquist (1928) as an explanation for observed Johnson noise in electric circuits. The fluctuation-dissipation theorem showed that when something dissipates energy, in an effectively irreversible way, a connected heat bath must also fluctuate. The fluctuations and the dissipation go hand in hand; it is impossible to have one without the other. The implication of FDT being that the vacuum could be treated as a heat bath coupled to a dissipative force and as such energy could, in part, be extracted from the vacuum for potentially useful work. FDT has been shown to be true experimentally under certain quantum, non-classical, conditions.


title: Zero-point energy

text: In 1963 the Jaynes–Cummings model was developed describing the system of a two-level atom interacting with a quantized field mode (i.e. the vacuum) within an optical cavity. It gave nonintuitive predictions such as that an atom's spontaneous emission could be driven by field of effectively constant frequency (Rabi frequency). In the 1970s experiments were being performed to test aspects of quantum optics and showed that the rate of spontaneous emission of an atom could be controlled using reflecting surfaces. These results were at first regarded with suspicion in some quarters: it was argued that no modification of a spontaneous emission rate would be possible, after all, how can the emission of a photon be affected by an atom's environment when the atom can only "see" its environment by emitting a photon in the first place? These experiments gave rise to cavity quantum electrodynamics (CQED), the study of effects of mirrors and cavities on radiative corrections. Spontaneous emission can be suppressed (or "inhibited") or amplified. Amplification was first predicted by Purcell in 1946 (the Purcell effect) and has been experimentally verified. This phenomenon can be understood, partly, in terms of the action of the vacuum field on the atom.


title: Zero-point energy

text: Zero-point energy is fundamentally related to the Heisenberg uncertainty principle. Roughly speaking, the uncertainty principle states that complementary variables (such as a particle's position and momentum, or a field's value and derivative at a point in space) cannot simultaneously be specified precisely by any given quantum state. In particular, there cannot exist a state in which the system simply sits motionless at the bottom of its potential well, for then its position and momentum would both be completely determined to arbitrarily great precision. Therefore, the lowest-energy state (the ground state) of the system must have a distribution in position and momentum that satisfies the uncertainty principle, which implies its energy must be greater than the minimum of the potential well.


title: Zero-point energy

text: Near the bottom of a potential well, the Hamiltonian of a general system (the quantum-mechanical operator giving its energy) can be approximated as a quantum harmonic oscillator,


title: Zero-point energy

text: A more thorough treatment, showing that the energy of the ground state actually saturates this bound and is exactly , requires solving for the ground state of the system.


title: Zero-point energy

text: The idea of a quantum harmonic oscillator and its associated energy can apply to either an atom or a subatomic particle. In ordinary atomic physics, the zero-point energy is the energy associated with the ground state of the system. The professional physics literature tends to measure frequency, as denoted by above, using angular frequency, denoted with and defined by . This leads to a convention of writing Planck's constant with a bar through its top () to denote the quantity . In these terms, the most famous such example of zero-point energy is the above associated with the ground state of the quantum harmonic oscillator. In quantum mechanical terms, the zero-point energy is the expectation value of the Hamiltonian of the system in the ground state.


title: Zero-point energy

text: If more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator which acts non-trivially on a ground state and commutes with the Hamiltonian of the system.


title: Zero-point energy

text: According to the third law of thermodynamics, a system at absolute zero temperature exists in its ground state; thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a perfect crystal lattice, have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit negative temperature.


title: Zero-point energy

text: The wave function of the ground state of a particle in a one-dimensional well is a half-period sine wave which goes to zero at the two edges of the well. The energy of the particle is given by:


title: Zero-point energy

text: where is the Planck constant, is the mass of the particle, is the energy state ( corresponds to the ground-state energy), and is the width of the well.


title: Zero-point energy

text: In quantum field theory (QFT), the fabric of "empty" space is visualized as consisting of fields, with the field at every point in space and time being a quantum harmonic oscillator, with neighboring oscillators interacting with each other. According to QFT the universe is made up of matter fields whose quanta are fermions (e.g. electrons and quarks), force fields whose quanta are bosons (i.e. photons and gluons) and a Higgs field whose quantum is the Higgs boson. The matter and force fields have zero-point energy. A related term is "zero-point field" (ZPF), which is the lowest energy state of a particular field. The vacuum can be viewed not as empty space, but as the combination of all zero-point fields.


title: Zero-point energy

text: In QFT the zero-point energy of the vacuum state is called the vacuum energy and the average expectation value of the Hamiltonian is called the vacuum expectation value (also called condensate or simply VEV). The QED vacuum is a part of the vacuum state which specifically deals with quantum electrodynamics (e.g. electromagnetic interactions between photons, electrons and the vacuum) and the QCD vacuum deals with quantum chromodynamics (e.g. color charge interactions between quarks, gluons and the vacuum). Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying quantum vacuum, and that all properties of matter are merely vacuum fluctuations arising from interactions with the zero-point field.


title: Zero-point energy

text: Each point in space makes a contribution of , resulting in a calculation of infinite zero-point energy in any finite volume; this is one reason renormalization is needed to make sense of quantum field theories. In cosmology, the vacuum energy is one possible explanation for the cosmological constant and the source of dark energy.


title: Zero-point energy

text: Scientists are not in agreement about how much energy is contained in the vacuum. Quantum mechanics requires the energy to be large as Paul Dirac claimed it is, like a sea of energy. Other scientists specializing in General Relativity require the energy to be small enough for curvature of space to agree with observed astronomy. The Heisenberg uncertainty principle allows the energy to be as large as needed to promote quantum actions for a brief moment of time, even if the average energy is small enough to satisfy relativity and flat space. To cope with disagreements, the vacuum energy is described as a virtual energy potential of positive and negative energy.


title: Zero-point energy

text: In quantum perturbation theory, it is sometimes said that the contribution of one-loop and multi-loop Feynman diagrams to elementary particle propagators are the contribution of vacuum fluctuations, or the zero-point energy to the particle masses.


title: Zero-point energy

text: The oldest and best known quantized force field is the electromagnetic field. Maxwell's equations have been superseded by quantum electrodynamics (QED). By considering the zero-point energy that arises from QED it is possible to gain a characteristic understanding of zero-point energy that arises not just through electromagnetic interactions but in all quantum field theories.


title: Zero-point energy

text: In the quantum theory of the electromagnetic field, classical wave amplitudes and are replaced by operators and that satisfy:


title: Zero-point energy

text: The classical quantity appearing in the classical expression for the energy of a field mode is replaced in quantum theory by the photon number operator . The fact that:


title: Zero-point energy

text: implies that quantum theory does not allow states of the radiation field for which the photon number and a field amplitude can be precisely defined, i.e., we cannot have simultaneous eigenstates for and . The reconciliation of wave and particle attributes of the field is accomplished via the association of a probability amplitude with a classical mode pattern. The calculation of field modes is entirely classical problem, while the quantum properties of the field are carried by the mode "amplitudes" and associated with these classical modes.


title: Zero-point energy

text: The zero-point energy of the field arises formally from the non-commutativity of and . This is true for any harmonic oscillator: the zero-point energy appears when we write the Hamiltonian:


title: Zero-point energy

text: It is often argued that the entire universe is completely bathed in the zero-point electromagnetic field, and as such it can add only some constant amount to expectation values. Physical measurements will therefore reveal only deviations from the vacuum state. Thus the zero-point energy can be dropped from the Hamiltonian by redefining the zero of energy, or by arguing that it is a constant and therefore has no effect on Heisenberg equations of motion. Thus we can choose to declare by fiat that the ground state has zero energy and a field Hamiltonian, for example, can be replaced by:


title: Zero-point energy

text: without affecting any physical predictions of the theory. The new Hamiltonian is said to be normally ordered (or Wick ordered) and is denoted by a double-dot symbol. The normally ordered Hamiltonian is denoted , i.e.:


title: Zero-point energy

text: In other words, within the normal ordering symbol we can commute and . Since zero-point energy is intimately connected to the non-commutativity of and , the normal ordering procedure eliminates any contribution from the zero-point field. This is especially reasonable in the case of the field Hamiltonian, since the zero-point term merely adds a constant energy which can be eliminated by a simple redefinition for the zero of energy. Moreover, this constant energy in the Hamiltonian obviously commutes with and and so cannot have any effect on the quantum dynamics described by the Heisenberg equations of motion.


title: Zero-point energy

text: However, things are not quite that simple. The zero-point energy cannot be eliminated by dropping its energy from the Hamiltonian: When we do this and solve the Heisenberg equation for a field operator, we must include the vacuum field, which is the homogeneous part of the solution for the field operator. In fact we can show that the vacuum field is essential for the preservation of the commutators and the formal consistency of QED. When we calculate the field energy we obtain not only a contribution from particles and forces that may be present but also a contribution from the vacuum field itself i.e. the zero-point field energy. In other words, the zero-point energy reappears even though we may have deleted it from the Hamiltonian.


title: Zero-point energy

text: From Maxwell's equations, the electromagnetic energy of a "free" field i.e. one with no sources, is described by:


title: Zero-point energy

text: We wish to "quantize" the electromagnetic energy of free space for a multimode field. The field intensity of free space should be independent of position such that should be independent of for each mode of the field. The mode function satisfying these conditions is:


title: Zero-point energy

text: where in order to have the transversality condition satisfied for the Coulomb gauge in which we are working.


title: Zero-point energy

text: To achieve the desired normalization we pretend space is divided into cubes of volume and impose on the field the periodic boundary condition:


title: Zero-point energy

text: where can assume any integer value. This allows us to consider the field in any one of the imaginary cubes and to define the mode function:


title: Zero-point energy

text: where is chosen to be a unit vector which specifies the polarization of the field mode. The condition means that there are two independent choices of , which we call and where and . Thus we define the mode functions:


title: Zero-point energy

text: where and , are photon annihilation and creation operators for the mode with wave vector and polarization . This gives the vector potential for a plane wave mode of the field. The condition for shows that there are infinitely many such modes. The linearity of Maxwell's equations allows us to write:


title: Zero-point energy

text: This is the Hamiltonian for an infinite number of uncoupled harmonic oscillators. Thus different modes of the field are independent and satisfy the commutation relations:


title: Zero-point energy

text: This state describes the zero-point energy of the vacuum. It appears that this sum is divergent – in fact highly divergent, as putting in the density factor


title: Zero-point energy

text: There are two separate questions to consider. First, is the divergence a real one such that the zero-point energy really is infinite? If we consider the volume is contained by perfectly conducting walls, very high frequencies can only be contained by taking more and more perfect conduction. No actual method of containing the high frequencies is possible. Such modes will not be stationary in our box and thus not countable in the stationary energy content. So from this physical point of view the above sum should only extend to those frequencies which are countable; a cut-off energy is thus eminently reasonable. However, on the scale of a "universe" questions of general relativity must be included. Suppose even the boxes could be reproduced, fit together and closed nicely by curving spacetime. Then exact conditions for running waves may be possible. However the very high frequency quanta will still not be contained. As per John Wheeler's "geons" these will leak out of the system. So again a cut-off is permissible, almost necessary. The question here becomes one of consistency since the very high energy quanta will act as a mass source and start curving the geometry.


title: Zero-point energy

text: This leads to the second question. Divergent or not, finite or infinite, is the zero-point energy of any physical significance? The ignoring of the whole zero-point energy is often encouraged for all practical calculations. The reason for this is that energies are not typically defined by an arbitrary data point, but rather changes in data points, so adding or subtracting a constant (even if infinite) should be allowed. However this is not the whole story, in reality energy is not so arbitrarily defined: in general relativity the seat of the curvature of spacetime is the energy content and there the absolute amount of energy has real physical meaning. There is no such thing as an arbitrary additive constant with density of field energy. Energy density curves space, and an increase in energy density produces an increase of curvature. Furthermore, the zero-point energy density has other physical consequences e.g. the Casimir effect, contribution to the Lamb shift, or anomalous magnetic moment of the electron, it is clear it is not just a mathematical constant or artifact that can be cancelled out.


title: Zero-point energy

text: The vacuum state of the "free" electromagnetic field (that with no sources) is defined as the ground state in which for all modes . The vacuum state, like all stationary states of the field, is an eigenstate of the Hamiltonian but not the electric and magnetic field operators. In the vacuum state, therefore, the electric and magnetic fields do not have definite values. We can imagine them to be fluctuating about their mean value of zero.


title: Zero-point energy

text: In a process in which a photon is annihilated (absorbed), we can think of the photon as making a transition into the vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state. An atom, for instance, can be considered to be "dressed" by emission and reabsorption of "virtual photons" from the vacuum. The vacuum state energy described by is infinite. We can make the replacement:


title: Zero-point energy

text: This can be large even in relatively narrow "low frequency" regions of the spectrum. In the optical region from 400 to 700 nm, for instance, the above equation yields around 220 erg/cm.


title: Zero-point energy

text: We showed in the above section that the zero-point energy can be eliminated from the Hamiltonian by the normal ordering prescription. However, this elimination does not mean that the vacuum field has been rendered unimportant or without physical consequences. To illustrate this point we consider a linear dipole oscillator in the vacuum. The Hamiltonian for the oscillator plus the field with which it interacts is:


title: Zero-point energy

text: This has the same form as the corresponding classical Hamiltonian and the Heisenberg equations of motion for the oscillator and the field are formally the same as their classical counterparts. For instance the Heisenberg equations for the coordinate and the canonical momentum of the oscillator are:


title: Zero-point energy

text: since the rate of change of the vector potential in the frame of the moving charge is given by the convective derivative


title: Zero-point energy

text: Above we have made the electric dipole approximation in which the spatial dependence of the field is neglected. The Heisenberg equation for is found similarly from the Hamiltonian to be:


title: Zero-point energy

text: In deriving these equations for , , and we have used the fact that equal-time particle and field operators commute. This follows from the assumption that particle and field operators commute at some time (say, ) when the matter-field interpretation is presumed to begin, together with the fact that a Heisenberg-picture operator evolves in time as , where is the time evolution operator satisfying


title: Zero-point energy

text: Alternatively, we can argue that these operators must commute if we are to obtain the correct equations of motion from the Hamiltonian, just as the corresponding Poisson brackets in classical theory must vanish in order to generate the correct Hamilton equations. The formal solution of the field equation is:


title: Zero-point energy

text: It can be shown that in the radiation reaction field, if the mass is regarded as the "observed" mass then we can take:


title: Zero-point energy

text: The total field acting on the dipole has two parts, and . is the free or zero-point field acting on the dipole. It is the homogeneous solution of the Maxwell equation for the field acting on the dipole, i.e., the solution, at the position of the dipole, of the wave equation


title: Zero-point energy

text: satisfied by the field in the (source free) vacuum. For this reason is often referred to as the "vacuum field", although it is of course a Heisenberg-picture operator acting on whatever state of the field happens to be appropriate at . is the source field, the field generated by the dipole and acting on the dipole.


title: Zero-point energy

text: Using the above equation for we obtain an equation for the Heisenberg-picture operator formula_48 that is formally the same as the classical equation for a linear dipole oscillator:


title: Zero-point energy

text: where . in this instance we have considered a dipole in the vacuum, without any "external" field acting on it. the role of the external field in the above equation is played by the vacuum electric field acting on the dipole.


title: Zero-point energy

text: Classically, a dipole in the vacuum is not acted upon by any "external" field: if there are no sources other than the dipole itself, then the only field acting on the dipole is its own radiation reaction field. In quantum theory however there is always an "external" field, namely the source-free or vacuum field .


title: Zero-point energy

text: According to our earlier equation for the free field is the only field in existence at as the time at which the interaction between the dipole and the field is "switched on". The state vector of the dipole-field system at is therefore of the form


title: Zero-point energy

text: where is the vacuum state of the field and is the initial state of the dipole oscillator. The expectation value of the free field is therefore at all times equal to zero:


title: Zero-point energy

text: The important point of this is that the zero-point field energy does not affect the Heisenberg equation for since it is a c-number or constant (i.e. an ordinary number rather than an operator) and commutes with . We can therefore drop the zero-point field energy from the Hamiltonian, as is usually done. But the zero-point field re-emerges as the homogeneous solution for the field equation. A charged particle in the vacuum will therefore always see a zero-point field of infinite density. This is the origin of one of the infinities of quantum electrodynamics, and it cannot be eliminated by the trivial expedient dropping of the term in the field Hamiltonian.


title: Zero-point energy

text: The free field is in fact necessary for the formal consistency of the theory. In particular, it is necessary for the preservation of the commutation relations, which is required by the unitary of time evolution in quantum theory:


title: Zero-point energy

text: For the dipole oscillator under consideration it can be assumed that the radiative damping rate is small compared with the natural oscillation frequency, i.e., . Then the integrand above is sharply peaked at and:


title: Zero-point energy

text: the necessity of the vacuum field can also be appreciated by making the small damping approximation in


title: Zero-point energy

text: Without the free field in this equation the operator would be exponentially dampened, and commutators like would approach zero for . With the vacuum field included, however, the commutator is at all times, as required by unitarity, and as we have just shown. A similar result is easily worked out for the case of a free particle instead of a dipole oscillator.