Indistinguishable particles

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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.

Marsupial

From Wikipedia, the free encyclopedia

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

 

Marsupials Temporal range: Paleocene–Recent PreꞒ Ꞓ O S D C P T J K Pg N Possible Late Cretaceous records

Marsupials are a diverse group of mammals belonging to the infraclass Marsupialia. They are natively found in Australasia, Wallacea, and the Americas. One of marsupials' unique features is their reproductive strategy: the young are born in a relatively undeveloped state and then nurtured within a pouch on their mother's abdomen.

Extant marsupials encompass many species, including kangaroos, koalas, opossums, possums, Tasmanian devils, wombats, wallabies, and bandicoots.

Marsupials constitute a clade stemming from the last common ancestor of extant Metatheria, which encompasses all mammals more closely related to marsupials than to placentals. The evolutionary split between placentals and marsupials occurred 125–160 million years ago, in the Middle Jurassic–Early Cretaceous period.

Presently, close to 70% of the 334 extant marsupial species are concentrated on the Australian continent, including mainland Australia, Tasmania, New Guinea, and nearby islands. The remaining 30% are distributed across the Americas, primarily in South America, with thirteen species in Central America and a single species, the Virginia opossum, inhabiting North America north of Mexico.

Marsupial sizes range from a few grams in the long-tailed planigale, to several tonnes in the extinct Diprotodon.

The word marsupial comes from marsupium, the technical term for the abdominal pouch. It, in turn, is borrowed from the Latin marsupium and ultimately from the ancient Greek μάρσιππος mársippos, meaning "pouch".

Anatomy

Koala
(Phascolarctos cinereus)

Marsupials have typical mammalian characteristics—e.g., mammary glands, three middle ear bones (and ears that usually have tragi, varying in hearing thresholds), true hair and bone structure. However, striking differences including anatomical features separate them from eutherians.

Most female marsupials have a front pouch, which contains multiple nursing teats. Marsupials have other common structural features. Ossified patellae are absent in most modern marsupials (with exceptions) and epipubic bones are present. Marsupials (and monotremes) also lack a gross communication (corpus callosum) between the right and left brain hemispheres.

Skull and teeth

Marsupials exhibit distinct cranial features compared to placentals. Generally, their skulls are relatively small and compact. Notably, they possess frontal holes known as foramen lacrimale situated at the front of the orbit. Marsupials have enlarged cheekbones that extend further to the rear, and their lower jaw's angular extension (processus angularis) is bent toward the center. The hard palate of marsupials contains more openings than that of placentals.

Teeth differ significantly. Most Australian marsupials outside the order Diprotodontia have a varying number of incisors between their upper and lower jaws. Early marsupials had a dental formula of 5.1.3.4/4.1.3.4 per quadrant, consisting of five (maxillary) or four (mandibular) incisors, one canine, three premolars, and four molars, totaling 50 teeth. While some taxa, like the opossum, retain this original tooth count, others have reduced numbers.

For instance, members of the Macropodidae family, including kangaroos and wallabies, have a dental formula of 3/1 – (0 or 1)/0 – 2/2 – 4/4. Many marsupials typically have between 40 and 50 teeth, more than most placentals. In marsupials, the second set of teeth only grows in at the site of the third premolar and posteriorly; all teeth anterior to this erupt initially as permanent teeth.

Torso

Few general characteristics describe their skeleton. In addition to unique details in the construction of the ankle, epipubic bones (ossa epubica) are observed projecting forward from the pubic bone of the pelvis. Since these are present in males and pouchless species, it is believed that they originally had nothing to do with reproduction, but served in the muscular approach to the movement of the hind limbs. This could be explained by an original feature of mammals, as these epipubic bones are also found in monotremes. Marsupial reproductive organs differ from placentals. For them, the reproductive tract is doubled. Females have two uteri and two vaginas, and before birth, a birth canal forms between them, the median vagina. In most species, males have a split or double penis lying in front of the scrotum, which is not homologous to the placental scrota.

A pouch is present in most species. Many marsupials have a permanent bag, while in others such as the shrew opossum the pouch develops during gestation, where the young are hidden only by skin folds or in the maternal fur. The arrangement of the pouch is variable to allow the offspring to receive maximum protection. Locomotive kangaroos have a pouch opening at the front, while many others that walk or climb on all fours open in the back. Usually, only females have a pouch, but the male water opossum has a pouch that protects his genitalia while swimming or running.

General and convergences

The sugar glider, a marsupial, (left) and flying squirrel, a placental, (right) are examples of convergent evolution.

Marsupials have adapted to many habitats, reflected in the wide variety in their build. The largest living marsupial, the red kangaroo, grows up to 1.8 metres (5 ft 11 in) in height and 90 kilograms (200 lb) in weight. Extinct genera, such as Diprotodon, were significantly larger and heavier. The smallest marsupials are the marsupial mice, which reach only 5 centimetres (2.0 in) in body length.

Some species resemble placentals and are examples of convergent evolution. This convergence is evident in both brain evolution and behaviour. The extinct thylacine strongly resembled the placental wolf, hence one of its nicknames "Tasmanian wolf". The ability to glide evolved in both marsupials (as with sugar gliders) and some placentals (as with flying squirrels), which developed independently. Other groups such as the kangaroo, however, do not have clear placental counterparts, though they share similarities in lifestyle and ecological niches with ruminants.

Body temperature

Marsupials, along with monotremes (platypuses and echidnas), typically have lower body temperatures than similarly sized placentals (eutherians), with the averages being 35 °C (95 °F) for marsupials and 37 °C (99 °F) for placentals. Some species will bask to conserve energy.

Reproductive system

See also: Kangaroo § Reproduction and life cycle

Female eastern grey kangaroo with a joey in her pouch

Marsupials' reproductive systems differ markedly from those of placentals. During embryonic development, a choriovitelline placenta forms in all marsupials. In bandicoots, an additional chorioallantoic placenta forms, although it lacks the chorionic villi found in eutherian placentas.

Both sexes possess a cloaca, although modified by connecting to a urogenital sac and having a separate anal region in most species. The bladder of marsupials functions as a site to concentrate urine and empties into the common urogenital sinus in both females and males.

Males

Reproductive tract of a male macropod

Most male marsupials, except for macropods and marsupial moles, have a bifurcated penis, separated into two columns, so that the penis has two ends corresponding to the females' two vaginas. The penis is used only during copulation, and is separate from the urinary tract. It curves forward when erect, and when not erect, it is retracted into the body in an S-shaped curve. Neither marsupials nor monotremes possess a baculum. The shape of the glans penis varies among marsupial species.

The shape of the urethral grooves of the males' genitalia is used to distinguish between Monodelphis brevicaudata, M. domestica, and M. americana. The grooves form two channels that form the ventral and dorsal folds of the erectile tissue. Several species of dasyurid marsupials can also be distinguished by their penis morphology. Marsupials' only accessory sex glands are the prostate and bulbourethral glands. Male marsupials have one to three pairs of bulbourethral glands. Ampullae of vas deferens, seminal vesicles or coagulating glands are not present. The prostate is proportionally larger in marsupials than in placentals. During the breeding season, the male tammar wallaby's prostate and bulbourethral gland enlarge. However, the weight of the testes does not vary seasonally.

Females

See also: Birth § Marsupials

Female reproductive anatomy of several marsupial species

Female marsupials have two lateral vaginas, which lead to separate uteri, both accessed through the same orifice. A third canal, the median vagina, is used for birth. This canal can be transitory or permanent. Some marsupial species store sperm in the oviduct after mating.

Marsupials give birth very early in gestation; after birth, newborns crawl up their mothers' bodies and attach themselves to a teat, which is located on the underside of the mother, either inside a pouch called the marsupium, or externally. Mothers often lick their fur to leave a trail of scent for the newborn to follow to increase their chances of reaching the marsupium. There they remain for several weeks. Offspring eventually leave the marsupium for short periods, returning to it for warmth, protection, and nourishment.

Early development

A red-necked wallaby joey inside its mother's pouch

Gestation differs between marsupials and placentals. Key aspects of the first stages of placental embryo development, such as the inner cell mass and the process of compaction, are not found in marsupials. The cleavage stages of marsupial development vary among groups and aspects of marsupial early development are not yet fully understood.

Marsupials have a short gestation period—typically between 12 and 33 days, but as low as 10 days in the case of the stripe-faced dunnart and as long as 38 days for the long-nosed potoroo. The baby (joey) is born in a fetal state, equivalent to an 8–12 week human fetus, blind, furless, and small in comparison to placental newborns: sizes range from 4-800g+. A newborn can be categorized in one of three grades of development. The least developed are found in dasyurids, intermediates are found in didelphids and peramelids, and the most developed are macropods. The newborn crawls across its mother's fur to reach the pouch, where it latches onto a teat. It does not emerge for several months, during which time it relies on its mother's milk for essential nutrients, growth factors and immunological defence. Genes expressed in the eutherian placenta needed for the later stages of fetal development are expressed in females in their mammary glands during lactation. After this period, the joey spends increasing periods out of the pouch, feeding and learning survival skills. However, it returns to the pouch to sleep, and if danger threatens, it seeks refuge in its mother's pouch.

An early birth removes a developing marsupial from its mother's body much sooner than in placentals; thus marsupials lack a complex placenta to protect the embryo from its mother's immune system. Though early birth puts the newborn at greater environmental risk, it significantly reduces the dangers associated with long pregnancies, as the fetus cannot compromise the mother in bad seasons. Marsupials are altricial animals, needing intensive care following birth (cf. precocial). Newborns lack histologically mature immune tissues and are highly reliant on their mother's immune system for immunological protection.

Newborns front limbs and facial structures are much more developed than the rest of their bodies at birth. This requirement has been argued to have limited the range of locomotor adaptations in marsupials compared to placentals. Marsupials must develop grasping forepaws early, complicating the evolutive transition from these limbs into hooves, wings, or flippers. However, several marsupials do possess atypical forelimb morphologies, such as the hooved forelimbs of the pig-footed bandicoot, suggesting that the range of forelimb specialization is not as limited as assumed.

Joeys stay in the pouch for up to a year or until the next joey arrives. Joeys are unable to regulate their body temperature and rely upon an external heat source. Until the joey is well-furred and old enough to leave the pouch, a pouch temperature of 30–32 °C (86–90 °F) must be constantly maintained.

Joeys are born with "oral shields", soft tissue that reduces the mouth opening to a round hole just large enough to accept the teat. Once inside the mouth, a bulbous swelling on the end of the teat attaches it to the offspring till it has grown large enough to let go. In species without pouches or with rudimentary pouches these are more developed than in forms with well-developed pouches, implying an increased role in ensuring that the young remain attached to the teat.

Range

In Australasia, marsupials are found in Australia, Tasmania and New Guinea; throughout the Maluku Islands, Timor and Sulawesi to the west of New Guinea, and in the Bismarck Archipelago (including the Admiralty Islands) and Solomon Islands to the east of New Guinea.

In the Americas, marsupials are found throughout South America, excluding the central/southern Andes and parts of Patagonia; and through Central America and south-central Mexico, with a single species (the Virginia opossum Didelphis virginiana) widespread in the eastern United States and along the Pacific coast.

Interaction with Europeans

Europeans' first encounter with a marsupial was the common opossum. Vicente Yáñez Pinzón, commander of the Niña on Christopher Columbus' first voyage in the late fifteenth century, collected a female opossum with young in her pouch off the South American coast. He presented them to the Spanish monarchs, though by then the young were lost and the female had died. The animal was noted for its strange pouch or "second belly".

The Portuguese first described Australasian marsupials: António Galvão, a Portuguese administrator in Ternate (1536–1540), wrote a detailed account of the northern common cuscus (Phalanger orientalis):

Some animals resemble ferrets, only a little bigger. They are called Kusus. They have a long tail with which they hang from the trees in which they live continuously, winding it once or twice around a branch. On their belly they have a pocket like an intermediate balcony; as soon as they give birth to a young one, they grow it inside there at a teat until it does not need nursing anymore. As soon as she has borne and nourished it, the mother becomes pregnant again.

In the 17th century, more accounts of marsupials emerged. A 1606 record of an animal killed on the southern coast of New Guinea, described it as "in the shape of a dog, smaller than a greyhound", with a snakelike "bare scaly tail" and hanging testicles. The meat tasted like venison, and the stomach contained ginger leaves. This description appears to closely resemble the dusky pademelon (Thylogale brunii), the earliest European record of a member of the Macropodidae.

Taxonomy

Marsupials are taxonomically identified as members of mammalian infraclass Marsupialia, first described as a family under the order Pollicata by German zoologist Johann Karl Wilhelm Illiger in his 1811 work Prodromus Systematis Mammalium et Avium. However, James Rennie, author of The Natural History of Monkeys, Opossums and Lemurs (1838), pointed out that the placement of five different groups of mammals – monkeys, lemurs, tarsiers, aye-ayes and marsupials (with the exception of kangaroos, which were placed under the order Salientia) – under a single order (Pollicata) did not appear to have a strong justification. In 1816, French zoologist George Cuvier classified all marsupials under Marsupialia. In 1997, researcher J. A. W. Kirsch and others accorded infraclass rank to Marsupialia.

Classification

With seven living orders in total, Marsupialia is further divided as follows:† – Extinct

• Superorder Ameridelphia (American marsupials)
  • Order Didelphimorphia (93 species) – see list of didelphimorphs
    • Family Didelphidae: opossums
  • Order Paucituberculata (seven species)
    • Family Caenolestidae: shrew opossums
• Superorder Australidelphia (Australian marsupials)
  • Order Microbiotheria (one extant species)
    • Family Microbiotheriidae: monitos del monte
  • Order †Yalkaparidontia (incertae sedis)
  • Grandorder Agreodontia
    • Order Dasyuromorphia (73 species) – see list of dasyuromorphs
      • Family †Thylacinidae: thylacine
      • Family Dasyuridae: antechinuses, quolls, dunnarts, Tasmanian devil, and relatives
      • Family Myrmecobiidae: numbat
    • Order Notoryctemorphia (two species)
      • Family Notoryctidae: marsupial moles
    • Order Peramelemorphia (27 species)
      • Family Thylacomyidae: bilbies
      • Family †Chaeropodidae: pig-footed bandicoots
      • Family Peramelidae: bandicoots and allies
  • Order Diprotodontia (136 species) – see list of diprotodonts
    • Suborder Vombatiformes
      • Family Vombatidae: wombats
      • Family Phascolarctidae: koalas
      • Family † Diprotodontidae
      • Family † Palorchestidae: marsupial tapirs
      • Family † Thylacoleonidae: marsupial lions
    • Suborder Phalangeriformes – see list of phalangeriformes
      • Family Acrobatidae: feathertail glider and feather-tailed possum
      • Family Burramyidae: pygmy possums
      • Family †Ektopodontidae: sprite possums
      • Family Petauridae: striped possum, Leadbeater's possum, yellow-bellied glider, sugar glider, mahogany glider, squirrel glider
      • Family Phalangeridae: brushtail possums and cuscuses
      • Family Pseudocheiridae: ringtailed possums and relatives
      • Family Tarsipedidae: honey possum
    • Suborder Macropodiformes – see list of macropodiformes
      • Family Macropodidae: kangaroos, wallabies, and relatives
      • Family Potoroidae: potoroos, rat kangaroos, bettongs
      • Family Hypsiprymnodontidae: musky rat-kangaroo
      • Family † Balbaridae: basal quadrupedal kangaroos

Evolutionary history

See also: Metatheria, Evolution of Macropodidae, and Evolution of mammals

Comprising over 300 extant species, several attempts have been made to accurately interpret the phylogenetic relationships among the different marsupial orders. Studies differ on whether Didelphimorphia or Paucituberculata is the sister group to all other marsupials. Though the order Microbiotheria (which has only one species, the monito del monte) is found in South America, morphological similarities suggest it is closely related to Australian marsupials. Molecular analyses in 2010 and 2011 identified Microbiotheria as the sister group to all Australian marsupials. However, the relations among the four Australidelphid orders are not as well understood.

Isolated petrosals of Djarthia murgonensis, Australia's oldest marsupial fossils

Dentition of the herbivorous eastern grey kangaroo, as illustrated in Knight's Sketches in Natural History

Phylogenetic tree of marsupials derived from retroposon data

DNA evidence supports a South American origin for marsupials, with Australian marsupials arising from a single Gondwanan migration of marsupials from South America, across the Antarctic land bridge, to Australia. There are many small arboreal species in each group. The term "opossum" is used to refer to American species (though "possum" is a common abbreviation), while similar Australian species are properly called "possums".

The relationships among the three extant divisions of mammals (monotremes, marsupials, and placentals) were long a matter of debate among taxonomists. Most morphological evidence comparing traits such as number and arrangement of teeth and structure of the reproductive and waste elimination systems as well as most genetic and molecular evidence favors a closer evolutionary relationship between the marsupials and placentals than either has with the monotremes.

The ancestors of marsupials, part of a larger group called metatherians, probably split from those of placentals (eutherians) during the mid-Jurassic period, though no fossil evidence of metatherians themselves are known from this time. From DNA and protein analyses, the time of divergence of the two lineages has been estimated to be around 100 to 120 mya. Fossil metatherians are distinguished from eutherians by the form of their teeth; metatherians possess four pairs of molar teeth in each jaw, whereas eutherian mammals (including true placentals) never have more than three pairs. Using this criterion, the earliest known metatherian was thought to be Sinodelphys szalayi, which lived in China around 125 mya. However Sinodelphys was later reinterpreted as an early member of Eutheria. The unequivocal oldest known metatherians are now 110 million years old fossils from western North America. Metatherians were widespread in North America and Asia during the Late Cretaceous, but suffered a severe decline during the end-Cretaceous extinction event.

Metatheria

In 2022, a study provided strong evidence that the earliest known marsupial was Deltatheridium known from specimens from the Campanian age of the Late Cretaceous in Mongolia. This study placed both Deltatheridium and Pucadelphys as sister taxa to the modern large American opossums.

Marsupials spread to South America from North America during the Paleocene, possibly via the Aves Ridge. Northern Hemisphere metatherians, which were of low morphological and species diversity compared to contemporary placental mammals, eventually became extinct during the Miocene epoch.

In South America, the opossums evolved and developed a strong presence, and the Paleogene also saw the evolution of shrew opossums (Paucituberculata) alongside non-marsupial metatherian predators such as the borhyaenids and the saber-toothed Thylacosmilus. South American niches for mammalian carnivores were dominated by these marsupial and sparassodont metatherians, which seem to have competitively excluded South American placentals from evolving carnivory. While placental predators were absent, the metatherians did have to contend with avian (terror bird) and terrestrial crocodylomorph competition. Marsupials were excluded in turn from large herbivore niches in South America by the presence of native placental ungulates (now extinct) and xenarthrans (whose largest forms are also extinct). South America and Antarctica remained connected until 35 mya, as shown by the unique fossils found there. North and South America were disconnected until about three million years ago, when the Isthmus of Panama formed. This led to the Great American Interchange. Sparassodonts disappeared for unclear reasons – again, this has classically assumed as competition from carnivoran placentals, but the last sparassodonts co-existed with a few small carnivorans like procyonids and canines, and disappeared long before the arrival of macropredatory forms like felines, while didelphimorphs (opossums) invaded Central America, with the Virginia opossum reaching as far north as Canada.

Marsupials reached Australia via the Antarctic Land Bridge during the Early Eocene, around 50 mya, shortly after Australia had split off. This suggests a single dispersion event of just one species, most likely a relative to South America's monito del monte (a microbiothere, the only New World australidelphian). This progenitor may have rafted across the widening, but still narrow, gap between Australia and Antarctica. The journey must not have been easy; South American ungulate and xenarthran remains have been found in Antarctica, but these groups did not reach Australia.

In Australia, marsupials radiated into the wide variety seen today, including not only omnivorous and carnivorous forms such as were present in South America, but also into large herbivores. Modern marsupials appear to have reached the islands of New Guinea and Sulawesi relatively recently via Australia. A 2010 analysis of retroposon insertion sites in the nuclear DNA of a variety of marsupials has confirmed all living marsupials have South American ancestors. The branching sequence of marsupial orders indicated by the study puts Didelphimorphia in the most basal position, followed by Paucituberculata, then Microbiotheria, and ending with the radiation of Australian marsupials. This indicates that Australidelphia arose in South America, and reached Australia after Microbiotheria split off.

In Australia, terrestrial placentals disappeared early in the Cenozoic (their most recent known fossils being 55 million-year-old teeth resembling those of condylarths) for reasons that are not clear, allowing marsupials to dominate the Australian ecosystem. Extant native Australian terrestrial placentals (such as hopping mice) are relatively recent immigrants, arriving via island hopping from Southeast Asia.

Genetic analysis suggests a divergence date between the marsupials and the placentals at 160 million years ago. The ancestral number of chromosomes has been estimated to be 2n = 14.

A recent hypothesis suggests that South American microbiotheres resulted from a back-dispersal from eastern Gondwana. This interpretation is based on new cranial and post-cranial marsupial fossils of Djarthia murgonensis from the early Eocene Tingamarra Local Fauna in Australia that indicate this species is the most plesiomorphic ancestor, the oldest unequivocal australidelphian, and may be the ancestral morphotype of the Australian marsupial radiation.

In 2023, imaging of a partial skeleton found in Australia by paleontologists from Flinders University led to the identification of Ambulator keanei, the first long-distance walker in Australia.