Cult

From Wikipedia, the free encyclopedia

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

Cults are social groups which have unusual, and often extreme, religious, spiritual, or philosophical beliefs and rituals. Extreme devotion to a particular person, object, or goal is another characteristic often ascribed to cults. The term has different, divergent and often pejorative, definitions both in popular culture and academia and has been an ongoing source of contention among scholars across several fields of study.

Beginning in the 1930s, new religious movements became an object of sociological study within the context of the study of religious behavior. Since the 1940s, the Christian countercult movement has opposed some sects and new religious movements, labeling them cults because of their unorthodox beliefs. Since the 1970s, the secular anti-cult movement has opposed certain groups, which they call cults, accusing them of practicing brainwashing.

Groups labelled cults are found around the world and range in size from small localized groups to some international organizations with up to millions of members.

Definition and usage

The word cult is derived from the Latin term cultus, which means 'worship'. In modern English the term cult is generally a pejorative, carrying derogatory connotations. The term is variously applied to abusive or coercive groups of many categories, including gangs, organized crime, and terrorist organizations.

An older sense of the word cult, which is not pejorative, indicates a set of religious devotional practices that is conventional within its culture, is related to a particular figure, and is frequently associated with a particular place, or generally the collective participation in rites of religion. References to the imperial cult of ancient Rome, for example, use the word in this sense. A derived sense of "excessive devotion" arose in the 19th century, and usage is not always strictly religious.

Sociological classifications of religious movements may identify a cult as a social group with socially deviant or novel beliefs and practices, although this is often unclear. Other researchers present a less-organized picture of cults, saying that they arise spontaneously around novel beliefs and practices. Cults have been compared to miniature totalitarian political systems. Such groups are typically described as being led by a charismatic leader who tightly controls its members.

In its pejorative sense, the term is often used for new religious movements and other social groups defined by their unusual religious, spiritual, or philosophical beliefs and rituals, or their group belief in a particular person, object, or goal. This sense of the term is weakly defined, having divergent definitions both in popular culture and in academia, where it has been an ongoing source of contention among scholars across several fields of study. According to Susannah Crockford, "[t]he word 'cult' is a shapeshifter, semantically morphing with the intentions of whoever uses it. As an analytical term, it resists rigorous definition." She argues that the least subjective definition of cult refers to a religion or religion-like group "self-consciously building a new form of society", but that the rest of society rejects as unacceptable.

The term cult has been criticized as lacking "scholarly rigour"; Benjamin E. Zeller stated "[l]abelling any group with which one disagrees and considers deviant as a cult may be a common occurrence, but it is not scholarship". Religious scholar Catherine Wessinger argued the term was dehumanizing of the people within the group, as well as their children; following the Waco siege, it was argued by some scholars that the defining of the Branch Davidians as a cult by the media, government and former members is a significant factor as to what led to the deaths. However, it has also been viewed as empowering for ex-members of groups who have had traumatic experiences. The term was noted to carry "considerable cultural legitimacy".

In the 1970s, with the rise of secular anti-cult movements, scholars (though not the general public) began to abandon the use of the term cult, regarding it as pejorative. By the end of the 1970s, the term cult was largely replaced in academia with the term "new religion" or "new religious movement". Other proposed alternative terms that have been used were "emergent religion", "alternative religious movement", or "marginal religious movement", though new religious movement is the most popular term. The anti-cult movement mostly regards the term "new religious movement" as a euphemism for "cult" that loses the implication that they are harmful.

Scholarly studies

Further information: Academic study of new religious movements

Howard P. Becker's church–sect typology, based on Ernst Troeltsch's original theory and providing the basis for a modern classification of cults, sects, and new religious movements

Beginning in the 1930s, new religious movements perceived as cults became an object of sociological study within the context of the study of religious behavior. The term "cult" in this context saw its origins in the work of sociologist Max Weber (1864–1920). Weber is an important theorist in the academic study of cults, which often draws on his theorizations of charismatic authority, and of the distinction he drew between churches and sects. This concept of a church–sect division was further elaborated upon by German theologian Ernst Troeltsch, who added a "mystical" categorization to accommodate more personal or individual religious experiences.

American sociologist Howard P. Becker further bisected Troeltsch's first two categories, splitting church into ecclesia and denomination; and sect into sect and cult. Like Troeltsch's "mystical religion", Becker's cult refers to small religious groups that lack organization and that emphasize the private nature of personal beliefs.

Max Weber (1864–1920), an important theorist in the study of cults

Later sociological formulations built on such characteristics, placing an additional emphasis on cults as deviant religious groups, "deriving their inspiration from outside of the predominant religious culture." This is often thought to lead to a high degree of tension between the group and the more mainstream culture surrounding it, a characteristic shared with religious sects. According to this sociological terminology, sects are products of religious schism and therefore maintain a continuity with traditional beliefs and practices, whereas cults arise spontaneously around novel beliefs and practices.

Scholars William Sims Bainbridge and Rodney Stark have argued for a further distinction between three kinds of cults: cult movements, client cults, and audience cults, all of which share a "compensator" or rewards for the things invested into the group. In the Bainbridge–Stark typology, a "cult movement" is an actual complete organization, differing from a "sect" in that it is not a splinter of a bigger religion, while "audience cults" are loosely organized, and propagated through media, and "client cults" offer services (i.e. psychic readings or meditation sessions). One type can turn into another, for example the Church of Scientology changing from audience to client cult. Sociologists who follow the Bainbridge–Stark classification tend to continue using the word "cult", unlike most other academics; however Bainbridge later stated he regretted having used the word at all. Stark and Bainbridge, in discussing the process by which individuals join new religious groups, have even questioned the utility of the concept of conversion, suggesting affiliation as a more useful concept.

In the early 1960s, sociologist John Lofland studied the activities of Unification Church members in California in trying to promote their beliefs and win new members. Lofland noted that most of their efforts were ineffective and that most of the people who joined did so because of personal relationships with other members (often family relationships). Lofland published his findings in 1964 as a doctoral thesis entitled "The World Savers: A Field Study of Cult Processes", and in 1966 in book form by as Doomsday Cult: A Study of Conversion, Proselytization, and Maintenance of Faith. This work is considered to be one of the most important and widely cited studies of the process of religious conversion.

J. Gordon Melton stated that, in 1970, "one could count the number of active researchers on new religions on one's hands". However, James R. Lewis in 2004 wrote that the "meteoric growth" in this field of study can be attributed to the cult controversy of the early 1970s. Because of "a wave of nontraditional religiosity" in the late 1960s and early 1970s, academics perceived new religious movements as different phenomena from previous religious innovations.

Some Stanford University research published in 1994 has documented and discussed "cultism" and "cult-like" characteristics in "visionary companies".

Types

Destructive cults

Destructive cult is a term frequently used by the anti-cult movement. Members of the anti-cult movement typically define a destructive cult as a group that is unethical, deceptive, and one that uses "strong influence" or mind control techniques to affect critical thinking skills. This term is sometimes presented in contrast to a "benign cult", which implies that not all "cults" would be harmful, though others apply it to all cults. Psychologist Michael Langone, executive director of the anti-cult group International Cultic Studies Association, defines a destructive cult as "a highly manipulative group which exploits and sometimes physically and/or psychologically damages members and recruits."

In Cults and the Family, the authors cite Eli Shapiro, who defines a destructive cultism as a sociopathic syndrome, whose distinctive qualities include: "behavioral and personality changes, loss of personal identity, cessation of scholastic activities, estrangement from family, disinterest in society and pronounced mental control and enslavement by cult leaders." Writing about Bruderhof communities in the book Misunderstanding Cults, Julius H. Rubin said that American religious innovation created an unending diversity of sects. These "new religious movements…gathered new converts and issued challenges to the wider society. Not infrequently, public controversy, contested narratives and litigation result." In his work Cults in Context author Lorne L. Dawson writes that although the Unification Church "has not been shown to be violent or volatile," it has been described as a destructive cult by "anticult crusaders." In 2002, the German government was held by the Federal Constitutional Court to have defamed the Osho movement by referring to it, among other things, as a "destructive cult" with no factual basis.

Some researchers have criticized the term destructive cult, writing that it is used to describe groups which are not necessarily harmful in nature to themselves or others. In his book Understanding New Religious Movements, John A. Saliba writes that the term is overgeneralized. Saliba sees the Peoples Temple as the "paradigm of a destructive cult", where those that use the term are implying that other groups will also commit mass suicide.

Doomsday cults

Main article: Doomsday cult

Doomsday cult is a term which is used to describe groups that believe in apocalypticism and millenarianism, and it can also be used to refer both to groups that predict disaster, and groups that attempt to bring it about. In the 1950s, American social psychologist Leon Festinger and his colleagues observed members of a small UFO religion called the Seekers for several months, and recorded their conversations both prior to and after a failed prophecy from their charismatic leader. Their work was later published in the book When Prophecy Fails: A Social and Psychological Study of a Modern Group that Predicted the Destruction of the World.

In the late 1980s, doomsday cults were a major topic of news reports, with some reporters and commentators considering them a serious threat to society. A 1997 psychological study by Festinger, Riecken, and Schachter found that people turned to a cataclysmic world view after they had repeatedly failed to find meaning in mainstream movements.

Political cults

A political cult is a cult with a primary interest in political action and ideology. Groups that some have described as "political cults", mostly advocating far-left or far-right agendas, have received some attention from journalists and scholars. In their 2000 book On the Edge: Political Cults Right and Left, Dennis Tourish and Tim Wohlforth discuss about a dozen organizations in the United States and Great Britain that they characterize as cults.

Anti-cult movements

Christian countercult movement

Main article: Christian countercult movement

In the 1940s, the long-held opposition by some established Christian denominations to non-Christian religions and heretical or counterfeit Christian sects crystallized into a more organized Christian countercult movement in the United States. For those belonging to the movement, all religious groups claiming to be Christian, but deemed outside of Christian orthodoxy, were considered cults. The countercult movement is mostly evangelical Protestants. The Christian countercult movement asserts that Christian groups whose teachings deviate from the belief that the bible is inerrant, but also focuses on non-Christian religions like Hinduism. Christian countercult activist writers also emphasize the need for Christians to evangelize to followers of cults.

Secular anti-cult movement

Main article: Anti-cult movement

An anti-Aum Shinrikyo protest in Japan, 2009

Starting in the late 1960s, a different strand of anti-cult groups arose, with the formation of the secular anti-cult movement (ACM). This was in response to the rise of new religions in the 1960s and 1970s, particularly the events at Jonestown and the deaths of nearly 1000 people. The organizations that formed the secular anti-cult movement (ACM) often acted on behalf of relatives of "cult" converts who did not believe their loved ones could have altered their lives so drastically by their own free will. A few psychologists and sociologists working in this field suggested that brainwashing techniques were used to maintain the loyalty of cult members.

The belief that cults brainwashed their members became a unifying theme among cult critics and in the more extreme corners of the anti-cult movement techniques like the sometimes forceful "deprogramming" of cult members was practised. In the mass media, and among average citizens, "cult" gained an increasingly negative connotation, becoming associated with things like kidnapping, brainwashing, psychological abuse, sexual abuse, and other criminal activity, and mass suicide. While most of these negative qualities usually have real documented precedents in the activities of a very small minority of new religious groups, mass culture often extends them to any religious group viewed as culturally deviant, however peaceful or law abiding it may be.

While some psychologists were receptive to these theories, sociologists were for the most part sceptical of their ability to explain conversion to NRMs. In the late 1980s, psychologists and sociologists started to abandon theories like brainwashing and mind control. While scholars may believe that various less dramatic coercive psychological mechanisms could influence group members, they came to see conversion to new religious movements principally as an act of a rational choice.

Governmental policies and actions

Main article: Governmental lists of cults and sects

The application of the labels cult or sect to religious movements in government documents signifies the popular and negative use of the term cult in English and a functionally similar use of words translated as 'sect' in several European languages. Sociologists critical to this negative politicized use of the word cult argue that it may adversely impact the religious freedoms of group members. At the height of the counter-cult movement and ritual abuse scare of the 1990s, some governments published lists of cults. Groups labelled "cults" are found around the world and range in size from local groups with a few members to international organizations with millions.

While these documents utilize similar terminology, they do not necessarily include the same groups nor is their assessment of these groups based on agreed criteria. Other governments and world bodies also report on new religious movements but do not use these terms to describe the groups. Since the 2000s, some governments have again distanced themselves from such classifications of religious movements. While the official response to new religious groups has been mixed across the globe, some governments aligned more with the critics of these groups to the extent of distinguishing between "legitimate" religion and "dangerous", "unwanted" cults in public policy.

China

Main article: Heterodox teachings (Chinese law)

Falun Gong books being symbolically destroyed by the Chinese government

For centuries, governments in China have categorized certain religions as xiéjiào (邪教), translated as "evil cults" or "heterodox teachings". In imperial China, the classification of a religion as xiejiao did not necessarily mean that a religion's teachings were believed to be false or inauthentic; rather, the label was applied to religious groups that were not authorized by the state, or it was applied to religious groups that were believed to challenge the legitimacy of the state. Groups branded xiejiao face suppression and punishment by authorities. The most notable xiejiao is Falun Gong, a ultra-conservative new religious movement, headquartered in Deerpark, New York, USA.

Russia

In 2008 the Russian Interior Ministry prepared a list of "extremist groups". At the top of the list were Islamic groups outside of "traditional Islam", which is supervised by the Russian government. Next listed were "Pagan cults". In 2009 the Russian Ministry of Justice created a council which it named the "Council of Experts Conducting State Religious Studies Expert Analysis." The new council listed 80 large sects which it considered potentially dangerous to Russian society, and it also mentioned that there were thousands of smaller ones. The large sects which were listed included: The Church of Jesus Christ of Latter-day Saints, the Jehovah's Witnesses, and other sects which were loosely referred to as "neo-Pentecostals".

United States

In the 1970s, the scientific status of the "brainwashing theory" became a central topic in U.S. court cases where the theory was used to try to justify the use of the forceful deprogramming of cult members. Meanwhile, sociologists who were critical of these theories assisted advocates of religious freedom in defending the legitimacy of new religious movements in court. In the United States the religious activities of cults are protected under the First Amendment of the United States Constitution, which prohibits governmental establishment of religion and protects freedom of religion, freedom of speech, freedom of the press, and freedom of assembly; however, no members of religious groups or cults are granted any special immunity from criminal prosecution.

In 1990, the court case of United States v. Fishman (1990) ended the usage of brainwashing theories by expert witnesses such as Margaret Singer and Richard Ofshe. In the case's ruling, the court cited the Frye standard, which states that the scientific theory which is utilized by expert witnesses must be generally accepted in their respective fields. The court deemed brainwashing to be inadmissible in expert testimonies, using supporting documents which were published by the APA Task Force on Deceptive and Indirect Methods of Persuasion and Control, literature from previous court cases in which brainwashing theories were used, and expert testimonies which were delivered by scholars such as Dick Anthony.

Western Europe

See also: MIVILUDES, Union nationale des associations de défense des familles et de l'individu, and Parliamentary Commission on Cults in France

The governments of France and Belgium have taken policy positions which accept "brainwashing" theories uncritically, while the governments of other European nations, such as those of Sweden and Italy, are cautious with regard to brainwashing and as a result, they have responded more neutrally with regard to new religions. Scholars have suggested that the outrage which followed the mass murder/suicides perpetuated by the Solar Temple, have significantly contributed to European anti-cult positions. In the 1980s, clergymen and officials of the French government expressed concern that some orders and other groups within the Roman Catholic Church would be adversely affected by anti-cult laws which were then being considered.

Perturbation theory

From Wikipedia, the free encyclopedia

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

In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In regular perturbation theory, the solution is expressed as a power series in a small parameter ${\displaystyle \epsilon }$. The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of ${\displaystyle \epsilon }$ usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction.

Perturbation theory is used in a wide range of fields and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. The field in general remains actively and heavily researched across multiple disciplines.

Description

Perturbation theory develops an expression for the desired solution in terms of a formal power series known as a perturbation series in some "small" parameter, that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, while further terms describe the deviation in the solution, due to the deviation from the initial problem. Formally, we have for the approximation to the full solution ${\displaystyle A,}$ a series in the small parameter (here called ε), like the following:

${\displaystyle A\equiv A_0+{\epsilon }^1{A}_1+{\epsilon }^2{A}_2+{\epsilon }^3{A}_3+\cdots }$

In this example, ${\displaystyle A_0}$ would be the known solution to the exactly solvable initial problem, and the terms ${\displaystyle A_1,A_2,A_3,\dots }$ represent the first-order, second-order, third-order, and higher-order terms, which may be found iteratively by a mechanistic but increasingly difficult procedure. For small ${\displaystyle \epsilon }$ these higher-order terms in the series generally (but not always) become successively smaller. An approximate "perturbative solution" is obtained by truncating the series, often by keeping only the first two terms, expressing the final solution as a sum of the initial (exact) solution and the "first-order" perturbative correction

${\displaystyle A\to A_0+\epsilon A_1\mathsf{f}\mathsf{o}\mathsf{r}\epsilon \to 0}$

Some authors use big O notation to indicate the order of the error in the approximate solution: ${\displaystyle A=A_0+\epsilon A_1+\mathcal{O}({\epsilon }^2).}$

If the power series in ${\displaystyle \epsilon }$ converges with a nonzero radius of convergence, the perturbation problem is called a regular perturbation problem. In regular perturbation problems, the asymptotic solution smoothly approaches the exact solution. However, the perturbation series can also diverge, and the truncated series can still be a good approximation to the true solution if it is truncated at a point at which its elements are minimum. This is called an asymptotic series. If the perturbation series is divergent or not a power series (for example, if the asymptotic expansion must include non-integer powers ${\displaystyle {\epsilon }^{\left(1/2\right)}}$ or negative powers ${\displaystyle {\epsilon }^{-2}}$) then the perturbation problem is called a singular perturbation problem. Many special techniques in perturbation theory have been developed to analyze singular perturbation problems.

Shell-crossing

A shell-crossing (sc) occurs in perturbation theory when matter trajectories intersect, forming a singularity. This limits the predictive power of physical simulations at small scales.

Prototypical example

The earliest use of what would now be called perturbation theory was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: for example the orbit of the Moon, which moves noticeably differently from a simple Keplerian ellipse because of the competing gravitation of the Earth and the Sun.

Perturbation methods start with a simplified form of the original problem, which is simple enough to be solved exactly. In celestial mechanics, this is usually a Keplerian ellipse. Under Newtonian gravity, an ellipse is exactly correct when there are only two gravitating bodies (say, the Earth and the Moon) but not quite correct when there are three or more objects (say, the Earth, Moon, Sun, and the rest of the Solar System) and not quite correct when the gravitational interaction is stated using formulations from general relativity.

Perturbative expansion

Keeping the above example in mind, one follows a general recipe to obtain the perturbation series. The perturbative expansion is created by adding successive corrections to the simplified problem. The corrections are obtained by forcing consistency between the unperturbed solution, and the equations describing the system in full. Write ${\displaystyle D}$ for this collection of equations; that is, let the symbol ${\displaystyle D}$ stand in for the problem to be solved. Quite often, these are differential equations, thus, the letter "D".

The process is generally mechanical, if laborious. One begins by writing the equations ${\displaystyle D}$ so that they split into two parts: some collection of equations ${\displaystyle D_0}$ which can be solved exactly, and some additional remaining part ${\displaystyle \epsilon D_1}$ for some small ${\displaystyle \epsilon \ll 1.}$ The solution ${\displaystyle A_0}$ (to ${\displaystyle D_0}$) is known, and one seeks the general solution ${\displaystyle A}$ to ${\displaystyle D=D_0+\epsilon D_1.}$

Next the approximation ${\displaystyle A\approx A_0+\epsilon A_1}$ is inserted into ${\displaystyle \epsilon D_1}$. This results in an equation for ${\displaystyle A_1,}$ which, in the general case, can be written in closed form as a sum over integrals over ${\displaystyle A_0.}$ Thus, one has obtained the first-order correction ${\displaystyle A_1}$ and thus ${\displaystyle A\approx A_0+\epsilon A_1}$ is a good approximation to ${\displaystyle A.}$ It is a good approximation, precisely because the parts that were ignored were of size ${\displaystyle {\epsilon }^2.}$ The process can then be repeated, to obtain corrections ${\displaystyle A_2,}$ and so on.

In practice, this process rapidly explodes into a profusion of terms, which become extremely hard to manage by hand. Isaac Newton is reported to have said, regarding the problem of the Moon's orbit, that "It causeth my head to ache." This unmanageability has forced perturbation theory to develop into a high art of managing and writing out these higher order terms. One of the fundamental breakthroughs in quantum mechanics for controlling the expansion are the Feynman diagrams, which allow quantum mechanical perturbation series to be represented by a sketch.

Examples

Perturbation theory has been used in a large number of different settings in physics and applied mathematics. Examples of the "collection of equations" ${\displaystyle D}$ include algebraic equations, differential equations (e.g., the equations of motion and commonly wave equations), thermodynamic free energy in statistical mechanics, radiative transfer, and Hamiltonian operators in quantum mechanics.

Examples of the kinds of solutions that are found perturbatively include the solution of the equation of motion (e.g., the trajectory of a particle), the statistical average of some physical quantity (e.g., average magnetization), and the ground state energy of a quantum mechanical problem.

Examples of exactly solvable problems that can be used as starting points include linear equations, including linear equations of motion (harmonic oscillator, linear wave equation), statistical or quantum-mechanical systems of non-interacting particles (or in general, Hamiltonians or free energies containing only terms quadratic in all degrees of freedom).

Examples of systems that can be solved with perturbations include systems with nonlinear contributions to the equations of motion, interactions between particles, terms of higher powers in the Hamiltonian/free energy.

For physical problems involving interactions between particles, the terms of the perturbation series may be displayed (and manipulated) using Feynman diagrams.

In chemistry

Many of the ab initio quantum chemistry methods use perturbation theory directly or are closely related methods. Implicit perturbation theory works with the complete Hamiltonian from the very beginning and never specifies a perturbation operator as such. Møller–Plesset perturbation theory uses the difference between the Hartree–Fock Hamiltonian and the exact non-relativistic Hamiltonian as the perturbation. The zero-order energy is the sum of orbital energies. The first-order energy is the Hartree–Fock energy and electron correlation is included at second-order or higher. Calculations to second, third or fourth order are very common and the code is included in most ab initio quantum chemistry programs. A related but more accurate method is the coupled cluster method.

History

Perturbation theory was first devised to solve otherwise intractable problems in the calculation of the motions of planets in the solar system. For instance, Newton's law of universal gravitation explained the gravitation between two astronomical bodies, but when a third body is added, the problem was, "How does each body pull on each?" Kepler's orbital equations only solve Newton's gravitational equations when the latter are limited to just two bodies interacting. The gradually increasing accuracy of astronomical observations led to incremental demands in the accuracy of solutions to Newton's gravitational equations, which led many eminent 18th and 19th century mathematicians, notably Joseph-Louis Lagrange and Pierre-Simon Laplace, to extend and generalize the methods of perturbation theory.

These well-developed perturbation methods were adopted and adapted to solve new problems arising during the development of quantum mechanics in 20th century atomic and subatomic physics. Paul Dirac developed quantum perturbation theory in 1927 to evaluate when a particle would be emitted in radioactive elements. This was later named Fermi's golden rule. Perturbation theory in quantum mechanics is fairly accessible, mainly because quantum mechanics is limited to linear wave equations, but also since the quantum mechanical notation allows expressions to be written in fairly compact form, thus making them easier to comprehend. This resulted in an explosion of applications, ranging from the Zeeman effect to the hyperfine splitting in the hydrogen atom.

Despite the simpler notation, perturbation theory applied to quantum field theory still easily gets out of hand. Richard Feynman developed the celebrated Feynman diagrams by observing that many terms repeat in a regular fashion. These terms can be replaced by dots, lines, squiggles and similar marks, each standing for a term, a denominator, an integral, and so on; thus complex integrals can be written as simple diagrams, with absolutely no ambiguity as to what they mean. The one-to-one correspondence between the diagrams, and specific integrals is what gives them their power. Although originally developed for quantum field theory, it turns out the diagrammatic technique is broadly applicable to many other perturbative series (although not always worthwhile).

In the second half of the 20th century, as chaos theory developed, it became clear that unperturbed systems were in general completely integrable systems, while the perturbed systems were not. This promptly lead to the study of "nearly integrable systems", of which the KAM torus is the canonical example. At the same time, it was also discovered that many (rather special) non-linear systems, which were previously approachable only through perturbation theory, are in fact completely integrable. This discovery was quite dramatic, as it allowed exact solutions to be given. This, in turn, helped clarify the meaning of the perturbative series, as one could now compare the results of the series to the exact solutions.

The improved understanding of dynamical systems coming from chaos theory helped shed light on what was termed the small denominator problem or small divisor problem. In the 19th century Poincaré observed (as perhaps had earlier mathematicians) that sometimes 2nd and higher order terms in the perturbative series have "small denominators": That is, they have the general form ${\displaystyle \frac{{\psi }_{n}V{\varphi }_m}{({\omega }_n-{\omega }_m)}}$ where ${\displaystyle {\psi }_n,}$ ${\displaystyle V,}$ and ${\displaystyle {\varphi }_m}$ are some complicated expressions pertinent to the problem to be solved, and ${\displaystyle {\omega }_n}$ and ${\displaystyle {\omega }_m}$ are real numbers; very often they are the energy of normal modes. The small divisor problem arises when the difference ${\displaystyle {\omega }_n-{\omega }_m}$ is small, causing the perturbative correction to "blow up", becoming as large or maybe larger than the zeroth order term. This situation signals a breakdown of perturbation theory: It stops working at this point, and cannot be expanded or summed any further. In formal terms, the perturbative series is an asymptotic series: A useful approximation for a few terms, but at some point becomes less accurate if even more terms are added. The breakthrough from chaos theory was an explanation of why this happened: The small divisors occur whenever perturbation theory is applied to a chaotic system. The one signals the presence of the other.

Beginnings in the study of planetary motion

Since the planets are very remote from each other, and since their mass is small as compared to the mass of the Sun, the gravitational forces between the planets can be neglected, and the planetary motion is considered, to a first approximation, as taking place along Kepler's orbits, which are defined by the equations of the two-body problem, the two bodies being the planet and the Sun.

Since astronomic data came to be known with much greater accuracy, it became necessary to consider how the motion of a planet around the Sun is affected by other planets. This was the origin of the three-body problem; thus, in studying the system Moon-Earth-Sun, the mass ratio between the Moon and the Earth was chosen as the "small parameter". Lagrange and Laplace were the first to advance the view that the so-called "constants" which describe the motion of a planet around the Sun gradually change: They are "perturbed", as it were, by the motion of other planets and vary as a function of time; hence the name "perturbation theory".

Perturbation theory was investigated by the classical scholars – Laplace, Siméon Denis Poisson, Carl Friedrich Gauss – as a result of which the computations could be performed with a very high accuracy. The discovery of the planet Neptune in 1848 by Urbain Le Verrier, based on the deviations in motion of the planet Uranus. He sent the coordinates to J.G. Galle who successfully observed Neptune through his telescope – a triumph of perturbation theory.

Perturbation orders

The standard exposition of perturbation theory is given in terms of the order to which the perturbation is carried out: first-order perturbation theory or second-order perturbation theory, and whether the perturbed states are degenerate, which requires singular perturbation. In the singular case extra care must be taken, and the theory is slightly more elaborate.