North American Man/Boy Love Association

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https://en.wikipedia.org/wiki/North_American_Man/Boy_Love_Association 

 

North American Man/Boy Love Association

A NAMBLA logo. The capital M and lowercase b symbolize a man and a boy.
Founded	December 2, 1978
Founder	David Thorstad
Type	Unincorporated association
Focus	Pedophilia and pederasty activism
Location	New York City and San Francisco
Area served	North America
Method	Removing age-of-consent laws

The North American Man/Boy Love Association (NAMBLA, stylized as NAMbLA) is a pedophilia and pederasty advocacy organization in the United States. It works to abolish age-of-consent laws criminalizing adult sexual involvement with minors and campaigns for the release of men who have been jailed for sexual contacts with minors that did not involve what it considers coercion.

The group no longer holds regular national meetings, and as of the late 1990s—to avoid local police infiltration—the organization discouraged the formation of local chapters. Around 1995, an undercover detective discovered there were 1,100 people on the organization's rolls. NAMBLA was the largest group in International Pedophile and Child Emancipation (IPCE), an international pro-pedophile activist organization. Since then, the organization has dwindled to only a handful of people, with many members joining online pedophile networks, according to Xavier Von Erck, director of operations at the anti-pedophile organization Perverted-Justice. As of 2005, a newspaper report stated that NAMBLA was based in New York and San Francisco.

History

Events such as Anita Bryant's 1977 "Save Our Children" campaign and a police raid of a Toronto-area newspaper, The Body Politic, for publishing an article by Gerald Hannon sympathetic to "boy-love" set the stage for the founding of NAMBLA.

In December 1977, police raided a house in the Boston suburb Revere. Twenty-four men were arrested and indicted on over 100 felony counts of the statutory rape of boys aged eight to fifteen. Suffolk County district attorney Garrett H. Byrne found the men had used drugs and video games to lure the boys into a house, where they photographed them as they engaged in sexual activity. The men were members of a "sex ring"; Byrne said the arrest was "the tip of the iceberg". Commenting on this issue, Boston magazine described NAMBLA as "the most despised group of men in America", which was "founded mostly by eccentric, boy-loving leftists". The "Boston-Boise Committee", a gay rights organization, was formed in response to these events (which they termed the "Boston witch-hunt"), allegedly in order to promote solidarity amongst gay men, saying in an official leaflet that: "The closet is weak. There is strength in unity and openness." NAMBLA's founding was inspired by this organization. It was co-founded by gay-rights activist and socialist David Thorstad.

In 1982, a NAMBLA member was falsely linked to the disappearance of Etan Patz. Although the accusation was groundless, the negative publicity was disastrous to the organization. NAMBLA published a book A Witchhunt Foiled: The FBI vs. NAMBLA, which documented these events. In testimony before the United States Senate, NAMBLA was exonerated from criminal activities; it said, "It is the pedophile with no organized affiliations who is the real threat to children".

Mike Echols, the author of I Know My First Name Is Steven, infiltrated NAMBLA and recorded his observations in his book, which was published in 1991. Echols published the names, addresses and telephone numbers of eighty suspected NAMBLA members on his website, which led to death threats being made to people who were not members of the organization.

Onell R. Soto, a San Diego Union-Tribune writer, wrote in February 2005, "Law enforcement officials and mental health professionals say that while NAMBLA's membership numbers are small, the group has a dangerous ripple effect through the Internet by sanctioning the behavior of those who would abuse children".

ILGA controversy

Main article: ILGA consultative status controversy

In 1993, the International Lesbian and Gay Association (ILGA) achieved United Nations consultative status. NAMBLA's membership in ILGA drew heavy criticism and caused the suspension of ILGA. Many gay organizations called for the ILGA to dissolve ties with NAMBLA. Republican Senator Jesse Helms proposed a bill to withhold US$119 million in UN contributions until U.S. President Bill Clinton could certify that no UN agency grants any official status to organizations that condoned pedophilia. The bill was unanimously approved by Congress and signed into law by Clinton in April 1994.

In 1994, ILGA expelled NAMBLA—the first U.S.-based organization to be a member—as well as Vereniging Martijn and Project Truth, because they were judged to be "organizations with a predominant aim of supporting or promoting pedophilia". Although ILGA removed NAMBLA, the UN reversed its decision to grant ILGA special consultative status. Repeated attempts by ILGA to regain special status with the UN succeeded in 2006.

Partially in response to the NAMBLA situation, Gregory King of the Human Rights Campaign later said, "NAMBLA is not a gay organization ... they are not part of our community and we thoroughly reject their efforts to insinuate that pedophilia is an issue related to gay and lesbian civil rights". NAMBLA said, "man/boy love is by definition homosexual", that "the Western homosexual tradition from Socrates to Wilde to Gide ... [and] many non Western homo sexualities from New Guinea and Persia to the Zulu and the Japanese" were formed by pederasty, that "man/boy lovers are part of the gay movement and central to gay history and culture", and that "homosexuals denying that it is 'not gay' to be attracted to adolescent boys are just as ludicrous as heterosexuals saying it's 'not heterosexual' to be attracted to adolescent girls".

Curley v. NAMBLA

Main article: Curley v. NAMBLA

In 2000, a Boston couple, Robert and Barbara Curley, sued NAMBLA for the wrongful death of their son. According to the suit, defendants Charles Jaynes and Salvatore Sicari, who were convicted of murdering the Curleys' son Jeffrey, "stalked ... tortured, murdered and mutilated [his] body on or about October 1, 1997. Upon information and belief immediately prior to said acts, Charles Jaynes accessed NAMBLA's website at the Boston Public Library." The lawsuit said, "NAMBLA serves as a conduit for an underground network of pedophiles in the United States who use their NAMBLA association and contacts therein and the Internet to obtain and promote pedophile activity". Jaynes wrote in his diary, "This was a turning point in discovery of myself ... NAMBLA's Bulletin helped me to become aware of my own sexuality and acceptance of it ... ".

Citing cases in which NAMBLA members were convicted of sexual offenses against children, Larry Frisoli, the attorney representing the Curleys, said the organization is a "training ground" for adults who wish to seduce children, in which men exchange strategies to find and groom child sex partners. Frisoli also said NAMBLA has sold on its website "The Rape and Escape Manual", which gave details about the avoidance of capture and prosecution. The American Civil Liberties Union (ACLU) stepped in to defend NAMBLA as a free speech matter; it won a dismissal because NAMBLA is organized as an unincorporated association rather than a corporation. John Reinstein, director of the ACLU Massachusetts, said although NAMBLA "may extol conduct which is currently illegal", there was nothing on its website that "advocated or incited the commission of any illegal acts, including murder or rape".

A NAMBLA founder said the case would "break our backs, even if we win, which we will". Media reports from 2006 said that for practical purposes the group no longer exists and that it consists only of a website maintained by a few enthusiasts. The Curleys continued the suit as a wrongful death action against individual NAMBLA members, some of whom were active in the group's leadership. Targets of the wrongful death suits included NAMBLA co-founder David Thorstad. The lawsuit was dropped in April 2008 after a judge ruled that a key witness was not competent to testify.

Support

Allen Ginsberg, poet and father of the Beat Generation, was an affiliated member of NAMBLA. Claiming to have joined the organization "in defense of free speech", Ginsberg said: "Attacks on NAMBLA stink of politics, witchhunting for profit, humorlessness, vanity, anger and ignorance ... I'm a member of NAMBLA because I love boys too—everybody does, who has a little humanity". He appeared in Chicken Hawk: Men Who Love Boys, produced and directed by Adi Sideman, a documentary in which members of NAMBLA gave interviews and presented defenses of the organization.

Pat Califia argued that politics played an important role in the gay community's rejection of NAMBLA. Califia has since withdrawn much of his earlier support for the association while still maintaining that discussing an issue does not constitute criminal activity.

Camille Paglia, feminist academic and social critic, signed a manifesto supporting the group in 1993. In 1994, Paglia supported lowering the legal age of consent to fourteen. She noted in a 1995 interview with pro-pedophile activist Bill Andriette "I fail to see what is wrong with erotic fondling with any age." In a 1997 Salon column, Paglia expressed the view that male pedophilia correlates with the heights of a civilization, stating "I have repeatedly protested the lynch-mob hysteria that dogs the issue of man-boy love. In Sexual Personae, I argued that male pedophilia is intricately intertwined with the cardinal moments of Western civilization." Paglia noted in several interviews, as well as Sexual Personae, that she supports the legalization of certain forms of child pornography. She later had a change of heart on the matter. In an interview for Radio New Zealand's Saturday Morning show, conducted on April 28, 2018, by Kim Hill, Paglia was asked, "Are you a libertarian on the issue of pedophilia?", to which she replied

In terms of the present day, I think it's absolutely impossible to think we could reproduce the Athenian code of pedophilia, of boy-love, that was central to culture at that time. ... We must protect children, and I feel that very very strongly. The age of consent for sexual interactions between a boy and an older man is obviously disputed, at what point that should be. I used to think that fourteen (the way it is in some places in the world) was adequate. I no longer think that. I think young people need greater protection than that. ... This is one of those areas that we must confine to the realm of imagination and the history of the arts.

Feigned support

See also: Anti-LGBTQ rhetoric § Conflation with pedophilia, and Societal attitudes toward homosexuality § Association with child abuse and pedophilia

In a 2017 protest at Columbia University against Mike Cernovich, an unidentified individual raised a pro-pedophilia banner showing logos from NAMBLA and some leftist organizations (all denying knowledge of any such cooperation). Fact-checking organizations consider this a false flag operation as alt-right personalities were quick to repost the photo without caveat and because NAMBLA had largely ceased operation by 2016. A similar 4chan hoax in 2018 connected NAMBLA with TED, following a controversial TEDx presentation—notably unvetted by the TED organization—referring to pedophilia as an "unchangeable sexual orientation".

Opposition

The first documented opposition to NAMBLA from LGBTQ organizations occurred at the conference that organized the first gay march on Washington in 1979.

In 1980, a group called the Lesbian Caucus distributed a flyer urging women to split from the annual New York City Gay Pride March, because according to the group, the organizing committee had been dominated by NAMBLA and its supporters. The next year, after some lesbians threatened to picket, the Cornell University group Gay People at Cornell (Gay PAC) rescinded its invitation to NAMBLA co-founder David Thorstad to be the keynote speaker at the annual May Gay Festival. In the following years, gay rights groups tried to block NAMBLA's participation in gay pride parades, prompting leading gay rights figure Harry Hay to wear a sign proclaiming "NAMBLA walks with me" as he participated in a 1986 gay pride march in Los Angeles.^[42]

By the mid-1980s, NAMBLA was virtually alone in its positions and found itself politically isolated.^[43] Support for "groups perceived as being on the fringe of the gay community," such as NAMBLA, vanished in the process.^[43]

In 1994, Stonewall 25, a New York LGBTQ rights group, voted to ban NAMBLA from its international march on the United Nations in June of that year.^[44] The same year, NAMBLA was again banned from the march commemorating Stonewall. Instead, members of NAMBLA and the Gay Liberation Front formed their own competing march called "The Spirit of Stonewall". The Gay & Lesbian Alliance Against Defamation (GLAAD) adopted a document called "Position Statement Regarding NAMBLA", which said GLAAD "deplores the North American Man Boy Love Association's (NAMBLA) goals, which include advocacy for sex between adult men and boys and the removal of legal protections for children. These goals constitute a form of child abuse and are repugnant to GLAAD."

That year, the Board of Directors of the National Gay and Lesbian Task Force (NGLTF) adopted a resolution on NAMBLA that said, "NGLTF condemns all abuse of minors, both sexual and any other kind, perpetrated by adults. Accordingly, NGLTF condemns the organizational goals of NAMBLA and any other such organization."

In 2000 in New York, a teacher was fired for his association with NAMBLA. There were no criminal charges or complaints about his conduct in class.

In April 2013, the hacktivist group Anonymous prevented NAMBLA's website from being accessed as part of an operation dubbed "Operation Alice Day". The timing of the attack coincided with Alice Day, a Pedophilia Pride Day celebrated by a small group of pedophiles and their supporters on April 25.

Associated individuals

Allen Ginsberg was a member of NAMBLA

• Bill Andriette, journalist. He joined NAMBLA at the age of 15 and edited the NAMBLA Bulletin for six years.
• Allen Ginsberg was a defender of NAMBLA and a member.
• Harry Hay, prominent LGBTQ rights activist. Hay supported NAMBLA's inclusion in gay pride parades and publicly addressed their meetings in support of the organization.
• Alan J. Horowitz, MD, convicted sex offender, ordained Orthodox rabbi, and psychiatrist. He specialized in working with adolescents, graduated magna cum laude from Harvard University, and earned a Ph.D. and medical degree from Duke University. Infamous for being the subject of a worldwide manhunt, Horowitz was known as "NAMBLA Rabbi".
• David Thorstad, founding member.
• Walter Breen, convicted sex offender. He wrote a book, Greek Love, and published a journal, The International Journal of Greek Love, both under the pseudonym "J.Z. Eglinton". As "Eglinton", he spoke at NAMBLA's founding convention.

In popular culture

• In the South Park episode "Cartman Joins NAMBLA", which first aired on June 21, 2000, Eric Cartman is convinced to become a poster boy for the organization after befriending older men online.
• In the Law and Order: Special Victims Unit episode "Angels", which aired on November 1, 2002, the body of a battered young boy found in a luggage compartment of an airport shuttle bus sends the detectives to his guardian who was discovered to be a pedophile only to find his corpse in bed with his genitals removed. The subsequent investigation leads them to a travel agency specializing in exotic trips for sexual predators, some of whom were NAMBLA members.

Indistinguishable particles

From Wikipedia, the free encyclopedia

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

 

Statistical mechanics
Thermodynamics Kinetic theory

In quantum mechanics, indistinguishable particles (also called identical or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (such as atomic nuclei), as well as atoms and molecules. Although all known indistinguishable particles only exist at the quantum scale, there is no exhaustive list of all possible sorts of particles nor a clear-cut limit of applicability, as explored in quantum statistics. They were first discussed by Werner Heisenberg and Paul Dirac in 1926.

There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which cannot (as described by the Pauli exclusion principle). Examples of bosons are photons, gluons, phonons, helium-4 nuclei and all mesons. Examples of fermions are electrons, neutrinos, quarks, protons, neutrons, and helium-3 nuclei.

The fact that particles can be identical has important consequences in statistical mechanics, where calculations rely on probabilistic arguments, which are sensitive to whether or not the objects being studied are identical. As a result, identical particles exhibit markedly different statistical behaviour from distinguishable particles. For example, the indistinguishability of particles has been proposed as a solution to Gibbs' mixing paradox.

Distinguishing between particles

There are two methods for distinguishing between particles. The first method relies on differences in the intrinsic physical properties of the particles, such as mass, electric charge, and spin. If differences exist, it is possible to distinguish between the particles by measuring the relevant properties. However, as far as can be determined, microscopic particles of the same species have completely equivalent physical properties. For instance, every electron has the same electric charge.

Even if the particles have equivalent physical properties, there remains a second method for distinguishing between particles, which is to track the trajectory of each particle. As long as the position of each particle can be measured with infinite precision (even when the particles collide), then there would be no ambiguity about which particle is which.

The problem with the second approach is that it contradicts the principles of quantum mechanics. According to quantum theory, the particles do not possess definite positions during the periods between measurements. Instead, they are governed by wavefunctions that give the probability of finding a particle at each position. As time passes, the wavefunctions tend to spread out and overlap. Once this happens, it becomes impossible to determine, in a subsequent measurement, which of the particle positions correspond to those measured earlier. The particles are then said to be indistinguishable.

Quantum mechanical description

Symmetrical and antisymmetrical states

Antisymmetric wavefunction for a (fermionic) 2-particle state in an infinite square well potential

Symmetric wavefunction for a (bosonic) 2-particle state in an infinite square well potential

What follows is an example to make the above discussion concrete, using the formalism developed in the article on the mathematical formulation of quantum mechanics.

Let n denote a complete set of (discrete) quantum numbers for specifying single-particle states (for example, for the particle in a box problem, take n to be the quantized wave vector of the wavefunction.) For simplicity, consider a system composed of two particles that are not interacting with each other. Suppose that one particle is in the state n₁, and the other is in the state n₂. The quantum state of the system is denoted by the expression

${\displaystyle |n_1⟩|n_2⟩}$

where the order of the tensor product matters ( if ${\displaystyle |n_2⟩|n_1⟩}$, then the particle 1 occupies the state n₂ while the particle 2 occupies the state n₁). This is the canonical way of constructing a basis for a tensor product space ${\displaystyle H\otimes H}$ of the combined system from the individual spaces. This expression is valid for distinguishable particles, however, it is not appropriate for indistinguishable particles since ${\displaystyle |n_1⟩|n_2⟩}$ and ${\displaystyle |n_2⟩|n_1⟩}$ as a result of exchanging the particles are generally different states.

• "the particle 1 occupies the n₁ state and the particle 2 occupies the n₂ state" ≠ "the particle 1 occupies the n₂ state and the particle 2 occupies the n₁ state".

Two states are physically equivalent only if they differ at most by a complex phase factor. For two indistinguishable particles, a state before the particle exchange must be physically equivalent to the state after the exchange, so these two states differ at most by a complex phase factor. This fact suggests that a state for two indistinguishable (and non-interacting) particles is given by following two possibilities:

${\displaystyle |n_1⟩|n_2⟩\pm |n_2⟩|n_1⟩}$

States where it is a sum are known as symmetric, while states involving the difference are called antisymmetric. More completely, symmetric states have the form

${\displaystyle |n_1,n_2;S⟩\equiv \text{constant}\times (|n_1⟩|n_2⟩+|n_2⟩|n_1⟩)}$

while antisymmetric states have the form

${\displaystyle |n_1,n_2;A⟩\equiv \text{constant}\times (|n_1⟩|n_2⟩-|n_2⟩|n_1⟩)}$

Note that if n₁ and n₂ are the same, the antisymmetric expression gives zero, which cannot be a state vector since it cannot be normalized. In other words, more than one identical particle cannot occupy an antisymmetric state (one antisymmetric state can be occupied only by one particle). This is known as the Pauli exclusion principle, and it is the fundamental reason behind the chemical properties of atoms and the stability of matter.

Exchange symmetry

The importance of symmetric and antisymmetric states is ultimately based on empirical evidence. It appears to be a fact of nature that identical particles do not occupy states of a mixed symmetry, such as

${\displaystyle |n_1,n_2;?⟩=\text{constant}\times (|n_1⟩|n_2⟩+i|n_2⟩|n_1⟩)}$

There is actually an exception to this rule, which will be discussed later. On the other hand, it can be shown that the symmetric and antisymmetric states are in a sense special, by examining a particular symmetry of the multiple-particle states known as exchange symmetry.

Define a linear operator P, called the exchange operator. When it acts on a tensor product of two state vectors, it exchanges the values of the state vectors:

${\displaystyle P(|\psi ⟩|\varphi ⟩)\equiv |\varphi ⟩|\psi ⟩}$

P is both Hermitian and unitary. Because it is unitary, it can be regarded as a symmetry operator. This symmetry may be described as the symmetry under the exchange of labels attached to the particles (i.e., to the single-particle Hilbert spaces).

Clearly, ${\displaystyle P^2=1}$ (the identity operator), so the eigenvalues of P are +1 and −1. The corresponding eigenvectors are the symmetric and antisymmetric states:

${\displaystyle P|n_1,n_2;S⟩=+|n_1,n_2;S⟩}$

${\displaystyle P|n_1,n_2;A⟩=-|n_1,n_2;A⟩}$

In other words, symmetric and antisymmetric states are essentially unchanged under the exchange of particle labels: they are only multiplied by a factor of +1 or −1, rather than being "rotated" somewhere else in the Hilbert space. This indicates that the particle labels have no physical meaning, in agreement with the earlier discussion on indistinguishability.

Since P is Hermitian, it can be regarded as an observable of the system: a measurement can be performed to find out if a state is symmetric or antisymmetric. Furthermore, the equivalence of the particles indicates that the Hamiltonian can be written in a symmetrical form, such as

${\displaystyle H=\frac{p_1^2}{2m}+\frac{p_2^2}{2m}+U(|x_1-x_2|)+V(x_1)+V(x_2)}$

It is possible to show that such Hamiltonians satisfy the commutation relation

${\displaystyle \left[P,H\right]=0}$

According to the Heisenberg equation, this means that the value of P is a constant of motion. If the quantum state is initially symmetric (antisymmetric), it will remain symmetric (antisymmetric) as the system evolves. Mathematically, this says that the state vector is confined to one of the two eigenspaces of P, and is not allowed to range over the entire Hilbert space. Thus, that eigenspace might as well be treated as the actual Hilbert space of the system. This is the idea behind the definition of Fock space.

Fermions and bosons

The choice of symmetry or antisymmetry is determined by the species of particle. For example, symmetric states must always be used when describing photons or helium-4 atoms, and antisymmetric states when describing electrons or protons.

Particles which exhibit symmetric states are called bosons. The nature of symmetric states has important consequences for the statistical properties of systems composed of many identical bosons. These statistical properties are described as Bose–Einstein statistics.

Particles which exhibit antisymmetric states are called fermions. Antisymmetry gives rise to the Pauli exclusion principle, which forbids identical fermions from sharing the same quantum state. Systems of many identical fermions are described by Fermi–Dirac statistics.

Parastatistics are mathematically possible, but no examples exist in nature.

In certain two-dimensional systems, mixed symmetry can occur. These exotic particles are known as anyons, and they obey fractional statistics. Experimental evidence for the existence of anyons exists in the fractional quantum Hall effect, a phenomenon observed in the two-dimensional electron gases that form the inversion layer of MOSFETs. There is another type of statistic, known as braid statistics, which are associated with particles known as plektons.

The spin-statistics theorem relates the exchange symmetry of identical particles to their spin. It states that bosons have integer spin, and fermions have half-integer spin. Anyons possess fractional spin.

N particles

The above discussion generalizes readily to the case of N particles. Suppose there are N particles with quantum numbers n₁, n₂, ..., n_N. If the particles are bosons, they occupy a totally symmetric state, which is symmetric under the exchange of any two particle labels:

${\displaystyle |n_1{n}_2\cdots n_N;S⟩=\sqrt{\frac{\prod _n{m}_n!}{N!}}\sum _p|n_{p(1)}⟩|n_{p(2)}⟩\cdots |n_{p(N)}⟩}$

Here, the sum is taken over all different states under permutations p acting on N elements. The square root left to the sum is a normalizing constant. The quantity m_n stands for the number of times each of the single-particle states n appears in the N-particle state. Note that Σ_n m_n = N.

In the same vein, fermions occupy totally antisymmetric states:

${\displaystyle |n_1{n}_2\cdots n_N;A⟩=\frac{1}{\sqrt{N!}}\sum _p\mathrm{sgn}(p)|n_{p(1)}⟩|n_{p(2)}⟩\cdots |n_{p(N)}⟩}$

Here, sgn(p) is the sign of each permutation (i.e. ${\displaystyle +1}$ if ${\displaystyle p}$ is composed of an even number of transpositions, and ${\displaystyle -1}$ if odd). Note that there is no ${\displaystyle {\mathrm{\Pi }}_n{m}_n}$ term, because each single-particle state can appear only once in a fermionic state. Otherwise the sum would again be zero due to the antisymmetry, thus representing a physically impossible state. This is the Pauli exclusion principle for many particles.

These states have been normalized so that

${\displaystyle ⟨n_1{n}_2\cdots n_N;S|n_1{n}_2\cdots n_N;S⟩=1,⟨n_1{n}_2\cdots n_N;A|n_1{n}_2\cdots n_N;A⟩=1.}$

Measurement

Suppose there is a system of N bosons (fermions) in the symmetric (antisymmetric) state

${\displaystyle |n_1{n}_2\cdots n_N;S/A⟩}$

and a measurement is performed on some other set of discrete observables, m. In general, this yields some result m₁ for one particle, m₂ for another particle, and so forth. If the particles are bosons (fermions), the state after the measurement must remain symmetric (antisymmetric), i.e.

${\displaystyle |m_1{m}_2\cdots m_N;S/A⟩}$

The probability of obtaining a particular result for the m measurement is

${\displaystyle P_{S/A}\left(n_1,\dots ,n_N\to m_1,\dots ,m_N\right)\equiv |⟨m_1\cdots m_N;S/A|n_1\cdots n_N;S/A⟩{|}^2}$

It can be shown that

${\displaystyle \sum _{m_1\le m_2\le \cdots \le m_N}{P}_{S/A}(n_1,\dots ,n_N\to m_1,\dots ,m_N)=1}$

which verifies that the total probability is 1. The sum has to be restricted to ordered values of m₁, ..., m_N to ensure that each multi-particle state is not counted more than once.

Wavefunction representation

So far, the discussion has included only discrete observables. It can be extended to continuous observables, such as the position x.

Recall that an eigenstate of a continuous observable represents an infinitesimal range of values of the observable, not a single value as with discrete observables. For instance, if a particle is in a state |ψ⟩, the probability of finding it in a region of volume d³x surrounding some position x is

${\displaystyle |⟨x|\psi ⟩{|}^2{d}^{3}x}$

As a result, the continuous eigenstates |x⟩ are normalized to the delta function instead of unity:

${\displaystyle ⟨x|x^{\prime }⟩={\delta }^3(x-x^{\prime })}$

Symmetric and antisymmetric multi-particle states can be constructed from continuous eigenstates in the same way as before. However, it is customary to use a different normalizing constant:

${\displaystyle \begin{array}{rl}|x_1{x}_2\cdots x_N;S⟩& =\sqrt{\frac{\prod _j{n}_j!}{N!}}\sum _p|x_{p(1)}⟩|x_{p(2)}⟩\cdots |x_{p(N)}⟩\\ |x_1{x}_2\cdots x_N;A⟩& =\frac{1}{\sqrt{N!}}\sum _p\mathrm{s}\mathrm{g}\mathrm{n}(p)|x_{p(1)}⟩|x_{p(2)}⟩\cdots |x_{p(N)}⟩\end{array}}$

A many-body wavefunction can be written,

${\displaystyle \begin{array}{rl}{\mathrm{\Psi }}_{n_1{n}_2\cdots n_N}^{(S)}(x_1,x_2,\dots ,x_N)& \equiv ⟨x_1{x}_2\cdots x_N;S|n_1{n}_2\cdots n_N;S⟩\\ & =\sqrt{\frac{\prod _j{n}_j!}{N!}}\sum _p{\psi }_{p(1)}(x_1){\psi }_{p(2)}(x_2)\cdots {\psi }_{p(N)}(x_N)\\ {\mathrm{\Psi }}_{n_1{n}_2\cdots n_N}^{(A)}(x_1,x_2,\dots ,x_N)& \equiv ⟨x_1{x}_2\cdots x_N;A|n_1{n}_2\cdots n_N;A⟩\\ & =\frac{1}{\sqrt{N!}}\sum _p\mathrm{s}\mathrm{g}\mathrm{n}(p){\psi }_{p(1)}(x_1){\psi }_{p(2)}(x_2)\cdots {\psi }_{p(N)}(x_N)\end{array}}$

where the single-particle wavefunctions are defined, as usual, by

${\displaystyle {\psi }_n(x)\equiv ⟨x|n⟩}$

The most important property of these wavefunctions is that exchanging any two of the coordinate variables changes the wavefunction by only a plus or minus sign. This is the manifestation of symmetry and antisymmetry in the wavefunction representation:

${\displaystyle \begin{array}{r}{\mathrm{\Psi }}_{n_1\cdots n_N}^{(S)}(\cdots x_i\cdots x_j\cdots )={\mathrm{\Psi }}_{n_1\cdots n_N}^{(S)}(\cdots x_j\cdots x_i\cdots )\\ {\mathrm{\Psi }}_{n_1\cdots n_N}^{(A)}(\cdots x_i\cdots x_j\cdots )=-{\mathrm{\Psi }}_{n_1\cdots n_N}^{(A)}(\cdots x_j\cdots x_i\cdots )\end{array}}$

The many-body wavefunction has the following significance: if the system is initially in a state with quantum numbers n₁, ..., n_N, and a position measurement is performed, the probability of finding particles in infinitesimal volumes near x₁, x₂, ..., x_N is

${\displaystyle N!{\left|{\mathrm{\Psi }}_{n_1{n}_2\cdots n_N}^{(S/A)}(x_1,x_2,\dots ,x_N)\right|}^2{d}^{3N}x}$

The factor of N! comes from our normalizing constant, which has been chosen so that, by analogy with single-particle wavefunctions,

${\displaystyle \int \int \cdots \int {\left|{\mathrm{\Psi }}_{n_1{n}_2\cdots n_N}^{(S/A)}(x_1,x_2,\dots ,x_N)\right|}^2{d}^3{x}_1{d}^3{x}_2\cdots d^3{x}_N=1}$

Because each integral runs over all possible values of x, each multi-particle state appears N! times in the integral. In other words, the probability associated with each event is evenly distributed across N! equivalent points in the integral space. Because it is usually more convenient to work with unrestricted integrals than restricted ones, the normalizing constant has been chosen to reflect this.

Finally, antisymmetric wavefunction can be written as the determinant of a matrix, known as a Slater determinant:

${\displaystyle {\mathrm{\Psi }}_{n_1\cdots n_N}^{(A)}(x_1,\dots ,x_N)=\frac{1}{\sqrt{N!}}\left|\begin{array}{cccc}{\psi }_{n_1}(x_1)& {\psi }_{n_1}(x_2)& \cdots & {\psi }_{n_1}(x_N)\\ {\psi }_{n_2}(x_1)& {\psi }_{n_2}(x_2)& \cdots & {\psi }_{n_2}(x_N)\\ ⋮& ⋮& \ddots & ⋮\\ {\psi }_{n_N}(x_1)& {\psi }_{n_N}(x_2)& \cdots & {\psi }_{n_N}(x_N)\end{array}\right|}$

Operator approach and parastatistics

The Hilbert space for ${\displaystyle n}$ particles is given by the tensor product $\underset{n}{⨂}H$. The permutation group of ${\displaystyle S_n}$ acts on this space by permuting the entries. By definition the expectation values for an observable ${\displaystyle a}$ of ${\displaystyle n}$ indistinguishable particles should be invariant under these permutations. This means that for all ${\displaystyle \psi \in H}$ and ${\displaystyle \sigma \in S_n}$

${\displaystyle (\sigma \mathrm{\Psi }{)}^{t}a(\sigma \mathrm{\Psi })={\mathrm{\Psi }}^{t}a\mathrm{\Psi },}$

or equivalently for each ${\displaystyle \sigma \in S_n}$

${\displaystyle {\sigma }^{t}a\sigma =a}$.

Two states are equivalent whenever their expectation values coincide for all observables. If we restrict to observables of ${\displaystyle n}$ identical particles, and hence observables satisfying the equation above, we find that the following states (after normalization) are equivalent

${\displaystyle \mathrm{\Psi }\sim \sum _{\sigma \in S_n}{\lambda }_{\sigma }\sigma \mathrm{\Psi }}$.

The equivalence classes are in bijective relation with irreducible subspaces of $\underset{n}{⨂}H$ under ${\displaystyle S_n}$.

Two obvious irreducible subspaces are the one dimensional symmetric/bosonic subspace and anti-symmetric/fermionic subspace. There are however more types of irreducible subspaces. States associated with these other irreducible subspaces are called parastatistic states. Young tableaux provide a way to classify all of these irreducible subspaces.

Statistical properties

Statistical effects of indistinguishability

The indistinguishability of particles has a profound effect on their statistical properties. To illustrate this, consider a system of N distinguishable, non-interacting particles. Once again, let n_j denote the state (i.e. quantum numbers) of particle j. If the particles have the same physical properties, the n_js run over the same range of values. Let ε(n) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies. The partition function of the system is

${\displaystyle Z=\sum _{n_1,n_2,\dots ,n_N}\exp \left\{-\frac{1}{kT}\left[\epsilon (n_1)+\epsilon (n_2)+\cdots +\epsilon (n_N)\right]\right\}}$

where k is the Boltzmann constant and T is the temperature. This expression can be factored to obtain

${\displaystyle Z={\xi }^N}$

where

${\displaystyle \xi =\sum _n\exp \left[-\frac{\epsilon (n)}{kT}\right].}$

If the particles are identical, this equation is incorrect. Consider a state of the system, described by the single particle states [n₁, ..., n_N]. In the equation for Z, every possible permutation of the ns occurs once in the sum, even though each of these permutations is describing the same multi-particle state. Thus, the number of states has been over-counted.

If the possibility of overlapping states is neglected, which is valid if the temperature is high, then the number of times each state is counted is approximately N!. The correct partition function is

${\displaystyle Z=\frac{{\xi }^N}{N!}.}$

Note that this "high temperature" approximation does not distinguish between fermions and bosons.

The discrepancy in the partition functions of distinguishable and indistinguishable particles was known as far back as the 19th century, before the advent of quantum mechanics. It leads to a difficulty known as the Gibbs paradox. Gibbs showed that in the equation Z = ξ^N, the entropy of a classical ideal gas is

${\displaystyle S=Nk\ln \left(V\right)+Nf(T)}$

where V is the volume of the gas and f is some function of T alone. The problem with this result is that S is not extensive – if N and V are doubled, S does not double accordingly. Such a system does not obey the postulates of thermodynamics.

Gibbs also showed that using Z = ξ^N/N! alters the result to

${\displaystyle S=Nk\ln \left(\frac{V}{N}\right)+Nf(T)}$

which is perfectly extensive.

Statistical properties of bosons and fermions

There are important differences between the statistical behavior of bosons and fermions, which are described by Bose–Einstein statistics and Fermi–Dirac statistics respectively. Roughly speaking, bosons have a tendency to clump into the same quantum state, which underlies phenomena such as the laser, Bose–Einstein condensation, and superfluidity. Fermions, on the other hand, are forbidden from sharing quantum states, giving rise to systems such as the Fermi gas. This is known as the Pauli Exclusion Principle, and is responsible for much of chemistry, since the electrons in an atom (fermions) successively fill the many states within shells rather than all lying in the same lowest energy state.

The differences between the statistical behavior of fermions, bosons, and distinguishable particles can be illustrated using a system of two particles. The particles are designated A and B. Each particle can exist in two possible states, labelled ${\displaystyle |0⟩}$ and ${\displaystyle |1⟩}$, which have the same energy.

The composite system can evolve in time, interacting with a noisy environment. Because the ${\displaystyle |0⟩}$ and ${\displaystyle |1⟩}$ states are energetically equivalent, neither state is favored, so this process has the effect of randomizing the states. (This is discussed in the article on quantum entanglement.) After some time, the composite system will have an equal probability of occupying each of the states available to it. The particle states are then measured.

If A and B are distinguishable particles, then the composite system has four distinct states: ${\displaystyle |0⟩|0⟩}$, ${\displaystyle |1⟩|1⟩}$, ${\displaystyle |0⟩|1⟩}$, and ${\displaystyle |1⟩|0⟩}$. The probability of obtaining two particles in the ${\displaystyle |0⟩}$ state is 0.25; the probability of obtaining two particles in the ${\displaystyle |1⟩}$ state is 0.25; and the probability of obtaining one particle in the ${\displaystyle |0⟩}$ state and the other in the ${\displaystyle |1⟩}$ state is 0.5.

If A and B are identical bosons, then the composite system has only three distinct states: ${\displaystyle |0⟩|0⟩}$, ${\displaystyle |1⟩|1⟩}$, and ${\displaystyle \frac{1}{\sqrt{2}}(|0⟩|1⟩+|1⟩|0⟩)}$. When the experiment is performed, the probability of obtaining two particles in the ${\displaystyle |0⟩}$ state is now 0.33; the probability of obtaining two particles in the ${\displaystyle |1⟩}$ state is 0.33; and the probability of obtaining one particle in the ${\displaystyle |0⟩}$ state and the other in the ${\displaystyle |1⟩}$ state is 0.33. Note that the probability of finding particles in the same state is relatively larger than in the distinguishable case. This demonstrates the tendency of bosons to "clump".

If A and B are identical fermions, there is only one state available to the composite system: the totally antisymmetric state ${\displaystyle \frac{1}{\sqrt{2}}(|0⟩|1⟩-|1⟩|0⟩)}$. When the experiment is performed, one particle is always in the ${\displaystyle |0⟩}$ state and the other is in the ${\displaystyle |1⟩}$ state.

The results are summarized in Table 1:

Table 1: Statistics of two particles

Particles	Both 0	Both 1	One 0 and one 1
Distinguishable	0.25	0.25	0.5
Bosons	0.33	0.33	0.33
Fermions	0	0	1

As can be seen, even a system of two particles exhibits different statistical behaviors between distinguishable particles, bosons, and fermions. In the articles on Fermi–Dirac statistics and Bose–Einstein statistics, these principles are extended to large number of particles, with qualitatively similar results.

Homotopy class

See also: Homotopy and Braid statistics

To understand why particle statistics work the way that they do, note first that particles are point-localized excitations and that particles that are spacelike separated do not interact. In a flat d-dimensional space M, at any given time, the configuration of two identical particles can be specified as an element of M × M. If there is no overlap between the particles, so that they do not interact directly, then their locations must belong to the space [M × M] \ {coincident points}, the subspace with coincident points removed. The element (x, y) describes the configuration with particle I at x and particle II at y, while (y, x) describes the interchanged configuration. With identical particles, the state described by (x, y) ought to be indistinguishable from the state described by (y, x). Now consider the homotopy class of continuous paths from (x, y) to (y, x), within the space [M × M] \ {coincident points} . If M is ⁠${\displaystyle {\mathbb{R}}^d}$⁠ where d ≥ 3, then this homotopy class only has one element. If M is ⁠${\displaystyle {\mathbb{R}}^2}$⁠, then this homotopy class has countably many elements (i.e. a counterclockwise interchange by half a turn, a counterclockwise interchange by one and a half turns, two and a half turns, etc., a clockwise interchange by half a turn, etc.). In particular, a counterclockwise interchange by half a turn is not homotopic to a clockwise interchange by half a turn. Lastly, if M is ⁠${\displaystyle \mathbb{R}}$⁠, then this homotopy class is empty.

Suppose first that d ≥ 3. The universal covering space of [M × M] ∖ {coincident points}, which is none other than [M × M] ∖ {coincident points} itself, only has two points which are physically indistinguishable from (x, y), namely (x, y) itself and (y, x). So, the only permissible interchange is to swap both particles. This interchange is an involution, so its only effect is to multiply the phase by a square root of 1. If the root is +1, then the points have Bose statistics, and if the root is –1, the points have Fermi statistics.

In the case ${\displaystyle M={\mathbb{R}}^2,}$ the universal covering space of [M × M] ∖ {coincident points} has infinitely many points that are physically indistinguishable from (x, y). This is described by the infinite cyclic group generated by making a counterclockwise half-turn interchange. Unlike the previous case, performing this interchange twice in a row does not recover the original state; so such an interchange can generically result in a multiplication by exp(iθ) for any real θ (by unitarity, the absolute value of the multiplication must be 1). This is called anyonic statistics. In fact, even with two distinguishable particles, even though (x, y) is now physically distinguishable from (y, x), the universal covering space still contains infinitely many points which are physically indistinguishable from the original point, now generated by a counterclockwise rotation by one full turn. This generator, then, results in a multiplication by exp(iφ). This phase factor here is called the mutual statistics.

Finally, in the case ${\displaystyle M=\mathbb{R},}$ the space [M × M] ∖ {coincident points} is not connected, so even if particle I and particle II are identical, they can still be distinguished via labels such as "the particle on the left" and "the particle on the right". There is no interchange symmetry here.