Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, as well as composite microscopic particles such as atoms and molecules. Particle statistics refers to the particular description of particles in statistical mechanics. ... It has been suggested that the section Physical applications of the Maxwell-Boltzmann distribution from the article Maxwell-Boltzmann distribution be merged into this article or section. ... In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... Fermi-Dirac distribution as a function of ε/μ plotted for 4 different temperatures. ... In quantum mechanics, despite what many textbooks and articles erronously claim, the Bose-Einstein and Fermi-Dirac statistics (and Maxwell-Boltzmann statistics) are NOT the only alternatives. ... In mathematics and physics, an anyon is a type of projective representation of a Lie group. ... In mathematics and theoretical physics, braid statistics is a generalization of the statistics of bosons and fermions based on the concept of braid group. ... In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. ... Properties The electron is a lightweight fundamental subatomic particle that carries a negative electric charge. ... Properties For other uses, see Atom (disambiguation). ... In chemistry, a molecule is an aggregate of at least two atoms in a definite arrangement held together by special forces. ...

There are two main categories of identical particles: bosons, which can share quantum states, and fermions, which are forbidden from sharing quantum states (this property of fermions is known as the Pauli exclusion principle.) Examples of bosons are photons, gluons, phonons, and helium-4 atoms. Examples of fermions are electrons, neutrinos, quarks, protons and neutrons, and helium-3 atoms. In physics, bosons, named after Satyendra Nath Bose, are particles with integer spin. ... A quantum state is any possible state in which a quantum mechanical system can be. ... In particle physics, fermions, (named after Enrico Fermi), are particles with semi-integer spin. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925, which states that no two identical fermions may occupy the same quantum state simultaneously. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves and radio waves are all forms of light. ... In particle physics, gluons are vector gauge bosons that mediate strong color charge interactions of quarks in quantum chromodynamics (QCD). ... In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ... General Name, Symbol, Number helium, He, 2 Chemical series noble gases Group, Period, Block 18, 1, s Appearance colorless Atomic mass 4. ... Properties The electron is a lightweight fundamental subatomic particle that carries a negative electric charge. ... The neutrino is an elementary particle. ... Quarks are one of the two basic constituents of matter in the Standard Model of particle physics. ... Properties [1][2] In physics, the proton (Greek proton = first) is a subatomic particle with an electric charge of one positive fundamental unit (1. ... This article or section does not cite its references or sources. ... Helium-3 is a non-radioactive and light isotope of helium. ...

The fact that particles can be identical has important consequences in statistical mechanics. Calculations in statistical mechanics 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 behavior from distinguishable particles. For example, the indistinguishability of particles has been proposed as a solution to Gibb's mixing paradox. Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In thermodynamics and statistical mechanics, the mixing paradox involves the calculation of the entropy of mixing of two thermodynamic systems before and after their contents are mixed. ...

Distinguishing between particles

There are two ways in which one might distinguish between particles. The first method relies on differences in the particles' intrinsic physical properties, such as mass, electric charge, and spin. If differences exist, we can distinguish between the particles by measuring the relevant properties. However, it is an empirical fact that microscopic particles of the same species have completely equivalent physical properties. For instance, every electron in the universe has exactly the same electric charge; this is why we can speak of such a thing as "the charge of the electron". Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interactions. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. ... The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ...

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 we can measure the position of each particle with infinite precision (even when the particles collide), there would be no ambiguity about which particle is which.

The problem with this 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. Fig. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

Quantum mechanical description of identical particles

Symmetrical and antisymmetrical states

We will now make the above discussion concrete, using the formalism developed in the article on the mathematical formulation of quantum mechanics. One of the remarkable characteristics of the mathematical formulation of quantum mechanics, which distinguishes it from mathematical formulations of theories developed prior to the early 1900s, is its use of abstract mathematical structures, such as Hilbert spaces and operators on these spaces. ...

For simplicity, consider a system composed of two identical particles. As the particles possess equivalent physical properties, their state vectors occupy mathematically identical Hilbert spaces. If we denote the Hilbert space of a single particle as H, then the Hilbert space of the combined system is formed by the tensor product . In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

Let n denote a complete set of (discrete) quantum numbers for specifying single-particle states (for example, for the particle in a box problem we can take n to be the quantized wave vector of the wavefunction.) Suppose that one particle is in the state n₁, and another is in the state n₂. What is the quantum state of the system? We might guess that it is In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a very simple problem consisting of a single particle bouncing around inside of an immovable box, from which it cannot escape, and which loses no energy when it collides with... A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ...

which is simply the canonical way of constructing a basis for a tensor product space from the individual spaces. However, this expression implies that we can identify the particle with n₁ as "particle 1" and the particle with n₂ as "particle 2", which conflicts with the ideas about indistinguishability discussed earlier.

Actually, it is an empirical fact that identical particles occupy special types of multi-particle states, called symmetric states and antisymmetric states. Symmetric states have the form

Antisymmetric states have the form

Note that if n₁ and n₂ are the same, our equation for the antisymmetric state gives the zero set, which cannot be a state vector as it cannot be normalized. In other words, in an antisymmetric state the particles cannot occupy the same single-particle states. 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. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925, which states that no two identical fermions may occupy the same quantum state simultaneously. ... Chemistry (from the Greek word χημεία (chemeia) meaning cast together or pour together) is the science of matter at the atomic to molecular scale, dealing primarily with collections of atoms (such as molecules, crystals, and metals). ... Matter is commonly defined as the substance of which physical objects are composed. ...

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

There is actually an exception to this rule, which we will discuss later. On the other hand, we can show 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.

Let us 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:

P is both Hermitian and unitary. Because it is unitary, we can regard it as a symmetry operator. We can describe this symmetry as the symmetry under the exchange of labels attached to the particles (i.e., to the single-particle Hilbert spaces). A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse...

Clearly, P² = 1 (the identity operator), so the eigenvalues of P are +1 and −1. The corresponding eigenvectors are the symmetric and antisymmetric states: In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

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 our earlier discussion on indistinguishability.

We have mentioned that P is Hermitian. As a result, it can be regarded as an observable of the system, which means that we can, in principle, perform a measurement 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 The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ...

It is possible to show that such Hamiltonians satisfy the commutation relation In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ...

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, we might as well treat that eigenspace as the actual Hilbert space of the system. This is the idea behind the definition of Fock space. The Heisenberg Picture of quantum mechanics is also known as Matrix mechanics. ... The Fock space is an algebraic system (Hilbert space) used in quantum mechanics to describe quantum states with a variable or unknown number of identical particles. ...

Fermions and bosons

The choice of symmetry or antisymmetry is determined by the species of particle. For example, we must always use symmetric states when describing photons or helium-4 atoms, and antisymmetric states when describing electrons or protons. The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves and radio waves are all forms of light. ... General Name, Symbol, Number helium, He, 2 Chemical series noble gases Group, Period, Block 18, 1, s Appearance colorless Atomic mass 4. ... Properties The electron is a lightweight fundamental subatomic particle that carries a negative electric charge. ... Properties [1][2] In physics, the proton (Greek proton = first) is a subatomic particle with an electric charge of one positive fundamental unit (1. ...

Particles which exhibit symmetric states are called bosons. As we will see, 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. In physics, bosons, named after Satyendra Nath Bose, are particles with integer spin. ... In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ...

Particles which exhibit antisymmetric states are called fermions. As we have seen, 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. In particle physics, fermions, (named after Enrico Fermi), are particles with semi-integer spin. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925, which states that no two identical fermions may occupy the same quantum state simultaneously. ... Fermi-Dirac distribution as a function of ε/μ plotted for 4 different temperatures. ...

Parastatistics are also possible. In quantum mechanics, despite what many textbooks and articles erronously claim, the Bose-Einstein and Fermi-Dirac statistics (and Maxwell-Boltzmann statistics) are NOT the only alternatives. ...

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. In mathematics and physics, an anyon is a type of projective representation of a Lie group. ... In mathematics and physics, an anyon is a type of projective representation of a Lie group. ... The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional systems of electrons subjected to low temperatures and strong magnetic fields, in which the Hall conductance σ takes on the quantized values where e is the elementary charge and h is Plancks constant. ... Photomicrograph of two MOSFETs in a test pattern The metal-oxide-semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET), is by far the most common field-effect transistor in both digital and analog circuits. ... In mathematics and theoretical physics, braid statistics is a generalization of the statistics of bosons and fermions based on the concept of braid group. ... In physics, a plekton is a hypothetical kind of elementary particle, which would obey a different style of statistics with respect to the interchange of identical particles. ...

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. The spin-statistics theorem in quantum mechanics relates the spin of a particle to the statistics obeyed by that particle. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. ...

N particles

The above discussion generalizes readily to the case of N particles. Suppose we have 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:

Here, the sum is taken over all possible permutations p acting on N elements. The square root on the right hand side is a normalizing constant. The quantity N_j stands for the number of times each of the single-particle states appears in the N-particle state. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ... The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ...

In the same vein, fermions occupy totally antisymmetric states:

Here, sgn(p) is the signature of each permutation (i.e. +1 if p is composed of an even number of transpositions, and −1 if odd.) Note that we have omitted the Π_jN_j term, because each single-particle state can appear only once in a fermionic state. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

These states have been normalized so that

Measurements of identical particles

Suppose we have a system of N bosons (fermions) in the symmetric (antisymmetric) state

and we perform a measurement of some other set of discrete observables, m. In general, this would yield 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.

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

We can show that

which verifies that the total probability is 1. Note that we have to restrict the sum to ordered values of m₁, ..., m_N to ensure that we do not count each multi-particle state more than once.

Wavefunction representation

So far, we have worked with discrete observables. We will now extend the discussion to continuous observables, such as the position x. Look up position in Wiktionary, the free dictionary. ...

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

As a result, the continuous eigenstates |x> are normalized to the delta function instead of unity: The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...

We can construct symmetric and antisymmetric multi-particle states out of continuous eigenstates in the same way as before. However, it is customary to use a different normalizing constant:

We can then write a many-body wavefunction, This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

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

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:

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

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

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, we have chosen our normalizing constant to reflect this.

Finally, it is interesting to note that that antisymmetric wavefunction can be written as the determinant of a matrix, known as a Slater determinant: In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... A Slater determinant (named after the physicist John C. Slater) is an expression in quantum mechanics for the wavefunction of a many-fermion system, which by construction satisfies the Pauli principle. ...

Statistical properties

Statistical effects of indistinguishability

The indistinguishability of particles has a profound effect on their statistical properties. To illustrate this, let us 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_j's 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 In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ...

where k is Boltzmann's constant and T is the temperature. We can factorize this expression to obtain The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... In thermodynamics, temperature is the physical property of a system that underlies the common notions of hot and cold —something that is hotter has the greater temperature. ... In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. ...

Z = ξ^N

where

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 n's occurs once in the sum, even though each of these permutations is describing the same multi-particle state. We have thus over-counted the actual number of states.

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

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 if we use the equation Z = ξ^N, the entropy of a classical ideal gas is Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ... For other uses of the term entropy, see Entropy (disambiguation) The thermodynamic entropy S, often simply called the entropy in the context of thermodynamics, is a measure of the amount of energy in a physical system that cannot be used to do work. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ...

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 we double N and V, S does not double accordingly. Such a system does not obey the postulates of thermodynamics. Volume, also called capacity, is a quantification of how much space an object occupies. ... In physics and chemistry, an extensive quantity (also referred to as an extensive variable) is a physical quantity whose value is proportional to the size of the system it describes. ... ‹ The template below has been proposed for deletion. ...

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

which is perfectly extensive. However, the reason for this correction to the partition function remained obscure until the discovery of quantum mechanics.

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 by the Pauli exclusion principle from sharing quantum states, giving rise to systems such as the Fermi gas. In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... Fermi-Dirac distribution as a function of ε/μ plotted for 4 different temperatures. ... Experiment using a (likely argon) laser. ... A Bose-Einstein condensate is a phase of matter formed by bosons cooled to temperatures very near to absolute zero. ... Superfluidity is a phase of matter characterised by the complete absence of viscosity. ... A Fermi gas is a collection of non-interacting fermions. ...

We can illustrate the differences between the statistical behavior of fermions, bosons, and distinguishable particles using a system of two particles. Let us call the particles A and B. Each particle can exist in two possible states, labelled |0> and |1>, which have the same energy.

We let the composite system evolve in time, interacting with a noisy environment. Because the |0> and |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. We then measure the particle states. Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. ...

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

If A and B are identical bosons, then the composite system has only three distinct states: |0>|0>, |1>|1>, and 2^−1/2(|0>|1> + |1>|0>). When we perform the experiment, the probability of obtaining two particles in the |0> state is now 0.33; the probability of obtaining two particles in the |1> state is 0.33; and the probability of obtaining one particle in the |0> state and the other in the |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 2^−1/2(|0>|1> - |1>|0>). When we perform the experiment, we inevitably find that one particle is in the |0> state and the other is in the |1> state.

The results are summarized in Table 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. Fermi-Dirac distribution as a function of ε/μ plotted for 4 different temperatures. ... In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ...

The homotopy class

To understand why we have the statistics that we do for particles, we first have to note 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 (at the same time, we are not referring to time delayed interactions here, which are mediated at the speed of light or slower), then we are dealing with the space [M × M]/{coincident points}, the subspace with coincident points removed. (x,y) describes the configuration with particle I at x and particle II at y. (y,x) describes the interchanged configuration. With identical particles, the state described by (x,y) ought to be indistinguishable (which ISN'T the same thing as identical!) from the state described by (y,x). Let's look at the homotopy class of continuous paths from (x,y) to (y,x). If M is R^d where , then this homotopy class only has one element. If M is R², 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 R, then this homotopy class is empty. Obviously, if M is not isomorphic to R^d, we can have more complicated homotopy classes... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

What does this all mean?

Let's first look at the case . 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 two swap both particles. Performing this interchange twice gives us (x,y) back again. If this interchange results in a multiplication by +1, then we have Bose statistics and if this interchange results in a multiplication by -1, we have Fermi statistics. In mathematics, specifically topology, a covering map on a topological space X is a continuous surjective map p : C → X, with C another topological space, with the property that for every x in X there exists an open neighborhood U such that the inverse image of U under p is...

Now how about R²? The universal covering space of [M × M]/{coincident points} has infinitely many points which 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 lead us back to the original state. So, such an interchange can generically result in a multiplication by exp(iθ) (its absolute value is 1 because of unitarity...). This is called anyonic statistics. In fact, even with two DISTINGUISHABLE particles, even though (x,y) is now physically distinguishable from (y,x), if we go over to the universal covering space, we still end up with infinitely many points which are physically indistinguishable from the original point and the interchanges are generated by a counterclockwise rotation by one full turn which results in a multiplication by exp(iφ). This phase factor here is called the mutual statistics. In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In mathematics and physics, unitarity is the property of an operator (or a matrix) that is unitary. ... In mathematics and physics, an anyon is a type of projective representation of a Lie group. ...

As for R, even if particle I and particle II are identical, we can always distinguish between them by the labels "the particle on the left" and "the particle on the right". There is no interchange symmetry here and such particles are called plektons.

The generalization to n identical particles doesn't give us anything qualitatively new because they are generated from the exchanges of two identical particles.