Normal order

From Wikipedia, the free encyclopedia

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

In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operators in the product. The process of putting a product into normal order is called normal ordering (also called Wick ordering). The terms antinormal order and antinormal ordering are analogously defined, where the annihilation operators are placed to the left of the creation operators.

Normal ordering of a product quantum fields or creation and annihilation operators can also be defined in many other ways. Which definition is most appropriate depends on the expectation values needed for a given calculation. Most of this article uses the most common definition of normal ordering as given above, which is appropriate when taking expectation values using the vacuum state of the creation and annihilation operators.

The process of normal ordering is particularly important for a quantum mechanical Hamiltonian. When quantizing a classical Hamiltonian there is some freedom when choosing the operator order, and these choices lead to differences in the ground state energy.

Notation

If ${\displaystyle {\hat {O}}}$ denotes an arbitrary product of creation and/or annihilation operators (or equivalently, quantum fields), then the normal ordered form of ${\displaystyle {\hat {O}}}$ is denoted by ${\displaystyle {\mathopen {:}}{\hat {O}}{\mathclose {:}}}$.

An alternative notation is ${\displaystyle {\mathcal {N}}({\hat {O}})}$.

Note that normal ordering is a concept that only makes sense for products of operators. Attempting to apply normal ordering to a sum of operators is not useful as normal ordering is not a linear operation.

Bosons

Bosons are particles which satisfy Bose–Einstein statistics. We will now examine the normal ordering of bosonic creation and annihilation operator products.

Single bosons

If we start with only one type of boson there are two operators of interest:

• ${\displaystyle {\hat {b}}^{\dagger }}$: the boson's creation operator.
• ${\displaystyle {\hat {b}}}$: the boson's annihilation operator.

These satisfy the commutator relationship

${\displaystyle \left[{\hat {b}}^{\dagger },{\hat {b}}^{\dagger }\right]_{-}=0}$

${\displaystyle \left[{\hat {b}},{\hat {b}}\right]_{-}=0}$

${\displaystyle \left[{\hat {b}},{\hat {b}}^{\dagger }\right]_{-}=1}$

where ${\displaystyle \left[A,B\right]_{-}\equiv AB-BA}$ denotes the commutator. We may rewrite the last one as: ${\displaystyle {\hat {b}}\,{\hat {b}}^{\dagger }={\hat {b}}^{\dagger }\,{\hat {b}}+1.}$

Examples

1. We'll consider the simplest case first. This is the normal ordering of ${\displaystyle {\hat {b}}^{\dagger }{\hat {b}}}$:

${\displaystyle {:\,}{\hat {b}}^{\dagger }\,{\hat {b}}{\,:}={\hat {b}}^{\dagger }\,{\hat {b}}.}$

The expression ${\displaystyle {\hat {b}}^{\dagger }\,{\hat {b}}}$ has not been changed because it is already in normal order - the creation operator ${\displaystyle ({\hat {b}}^{\dagger })}$ is already to the left of the annihilation operator ${\displaystyle ({\hat {b}})}$.

2. A more interesting example is the normal ordering of ${\displaystyle {\hat {b}}\,{\hat {b}}^{\dagger }}$:

${\displaystyle {:\,}{\hat {b}}\,{\hat {b}}^{\dagger }{\,:}={\hat {b}}^{\dagger }\,{\hat {b}}.}$

Here the normal ordering operation has reordered the terms by placing ${\displaystyle {\hat {b}}^{\dagger }}$ to the left of ${\displaystyle {\hat {b}}}$.

These two results can be combined with the commutation relation obeyed by ${\displaystyle {\hat {b}}}$ and ${\displaystyle {\hat {b}}^{\dagger }}$ to get

${\displaystyle {\hat {b}}\,{\hat {b}}^{\dagger }={\hat {b}}^{\dagger }\,{\hat {b}}+1={:\,}{\hat {b}}\,{\hat {b}}^{\dagger }{\,:}\;+1.}$

or

${\displaystyle {\hat {b}}\,{\hat {b}}^{\dagger }-{:\,}{\hat {b}}\,{\hat {b}}^{\dagger }{\,:}=1.}$

This equation is used in defining the contractions used in Wick's theorem.

3. An example with multiple operators is:

${\displaystyle {:\,}{\hat {b}}^{\dagger }\,{\hat {b}}\,{\hat {b}}\,{\hat {b}}^{\dagger }\,{\hat {b}}\,{\hat {b}}^{\dagger }\,{\hat {b}}{\,:}={\hat {b}}^{\dagger }\,{\hat {b}}^{\dagger }\,{\hat {b}}^{\dagger }\,{\hat {b}}\,{\hat {b}}\,{\hat {b}}\,{\hat {b}}=({\hat {b}}^{\dagger })^{3}\,{\hat {b}}^{4}.}$

4. A simple example shows that normal ordering cannot be extended by linearity from the monomials to all operators in a self-consistent way:

${\displaystyle {:\,}{\hat {b}}{\hat {b}}^{\dagger }{\,:}={:\,}1+{\hat {b}}^{\dagger }{\hat {b}}{\,:}={:\,}1{\,:}+{:\,}{\hat {b}}^{\dagger }{\hat {b}}{\,:}=1+{\hat {b}}^{\dagger }{\hat {b}}\neq {\hat {b}}^{\dagger }{\hat {b}}={:\,}{\hat {b}}{\hat {b}}^{\dagger }{\,:}}$

The implication is that normal ordering is not a linear function on operators.

Multiple bosons

If we now consider ${\displaystyle N}$ different bosons there are ${\displaystyle 2N}$ operators:

• ${\displaystyle {\hat {b}}_{i}^{\dagger }}$: the ${\displaystyle i^{th}}$ boson's creation operator.
• ${\displaystyle {\hat {b}}_{i}}$: the ${\displaystyle i^{th}}$ boson's annihilation operator.

Here ${\displaystyle i=1,\ldots ,N}$.

These satisfy the commutation relations:

${\displaystyle \left[{\hat {b}}_{i}^{\dagger },{\hat {b}}_{j}^{\dagger }\right]_{-}=0}$

${\displaystyle \left[{\hat {b}}_{i},{\hat {b}}_{j}\right]_{-}=0}$

${\displaystyle \left[{\hat {b}}_{i},{\hat {b}}_{j}^{\dagger }\right]_{-}=\delta _{ij}}$

where ${\displaystyle i,j=1,\ldots ,N}$ and ${\displaystyle \delta _{ij}}$ denotes the Kronecker delta.

These may be rewritten as:

${\displaystyle {\hat {b}}_{i}^{\dagger }\,{\hat {b}}_{j}^{\dagger }={\hat {b}}_{j}^{\dagger }\,{\hat {b}}_{i}^{\dagger }}$

${\displaystyle {\hat {b}}_{i}\,{\hat {b}}_{j}={\hat {b}}_{j}\,{\hat {b}}_{i}}$

${\displaystyle {\hat {b}}_{i}\,{\hat {b}}_{j}^{\dagger }={\hat {b}}_{j}^{\dagger }\,{\hat {b}}_{i}+\delta _{ij}.}$

Examples

1. For two different bosons (${\displaystyle N=2}$) we have

${\displaystyle :{\hat {b}}_{1}^{\dagger }\,{\hat {b}}_{2}:\,={\hat {b}}_{1}^{\dagger }\,{\hat {b}}_{2}}$

${\displaystyle :{\hat {b}}_{2}\,{\hat {b}}_{1}^{\dagger }:\,={\hat {b}}_{1}^{\dagger }\,{\hat {b}}_{2}}$

2. For three different bosons (${\displaystyle N=3}$) we have

${\displaystyle :{\hat {b}}_{1}^{\dagger }\,{\hat {b}}_{2}\,{\hat {b}}_{3}:\,={\hat {b}}_{1}^{\dagger }\,{\hat {b}}_{2}\,{\hat {b}}_{3}}$

Notice that since (by the commutation relations) ${\displaystyle {\hat {b}}_{2}\,{\hat {b}}_{3}={\hat {b}}_{3}\,{\hat {b}}_{2}}$ the order in which we write the annihilation operators does not matter.

${\displaystyle :{\hat {b}}_{2}\,{\hat {b}}_{1}^{\dagger }\,{\hat {b}}_{3}:\,={\hat {b}}_{1}^{\dagger }\,{\hat {b}}_{2}\,{\hat {b}}_{3}}$

${\displaystyle :{\hat {b}}_{3}{\hat {b}}_{2}\,{\hat {b}}_{1}^{\dagger }:\,={\hat {b}}_{1}^{\dagger }\,{\hat {b}}_{2}\,{\hat {b}}_{3}}$

Bosonic operator functions

Normal ordering of bosonic operator functions ${\displaystyle f({\hat {n}})}$, with occupation number operator ${\displaystyle {\hat {n}}={\hat {b}}{\vphantom {\hat {n}}}^{\dagger }{\hat {b}}}$, can be accomplished using (falling) factorial powers ${\displaystyle {\hat {n}}^{\underline {k}}={\hat {n}}({\hat {n}}-1)\cdots ({\hat {n}}-k+1)}$ and Newton series instead of Taylor series: It is easy to show ^[1] that factorial powers ${\displaystyle {\hat {n}}^{\underline {k}}}$ are equal to normal-ordered (raw) powers ${\displaystyle {\hat {n}}^{k}}$ and are therefore normal ordered by construction,

${\displaystyle {\hat {n}}^{\underline {k}}={\hat {b}}{\vphantom {\hat {n}}}^{\dagger k}{\hat {b}}{\vphantom {\hat {n}}}^{k}={:\,}{\hat {n}}^{k}{\,:},}$

such that the Newton series expansion

${\displaystyle {\tilde {f}}({\hat {n}})=\sum _{k=0}^{\infty }\Delta _{n}^{k}{\tilde {f}}(0)\,{\frac {{\hat {n}}^{\underline {k}}}{k!}}}$

of an operator function ${\displaystyle {\tilde {f}}({\hat {n}})}$, with ${\displaystyle k}$-th forward difference ${\displaystyle \Delta _{n}^{k}{\tilde {f}}(0)}$ at ${\displaystyle n=0}$, is always normal ordered. Here, the eigenvalue equation ${\displaystyle {\hat {n}}|n\rangle =n|n\rangle }$ relates ${\displaystyle {\hat {n}}}$ and ${\displaystyle n}$.

As a consequence, the normal-ordered Taylor series of an arbitrary function ${\displaystyle f({\hat {n}})}$ is equal to the Newton series of an associated function ${\displaystyle {\tilde {f}}({\hat {n}})}$, fulfilling

${\displaystyle {\tilde {f}}({\hat {n}})={:\,}f({\hat {n}}){\,:},}$

if the series coefficients of the Taylor series of ${\displaystyle f(x)}$, with continuous ${\displaystyle x}$, match the coefficients of the Newton series of ${\displaystyle {\tilde {f}}(n)}$, with integer ${\displaystyle n}$,

${\displaystyle {\begin{aligned}f(x)&=\sum _{k=0}^{\infty }F_{k}\,{\frac {x^{k}}{k!}},\\{\tilde {f}}(n)&=\sum _{k=0}^{\infty }F_{k}\,{\frac {n^{\underline {k}}}{k!}},\\F_{k}&=\partial _{x}^{k}f(0)=\Delta _{n}^{k}{\tilde {f}}(0),\end{aligned}}}$

with ${\displaystyle k}$-th partial derivative ${\displaystyle \partial _{x}^{k}f(0)}$ at ${\displaystyle x=0}$. The functions ${\displaystyle f}$ and ${\displaystyle {\tilde {f}}}$ are related through the so-called normal-order transform ${\displaystyle {\mathcal {N}}[f]}$ according to

${\displaystyle {\begin{aligned}{\tilde {f}}(n)&={\mathcal {N}}_{x}[f(x)](n)\\&={\frac {1}{\Gamma (-n)}}\int _{-\infty }^{0}\mathrm {d} x\,e^{x}\,f(x)\,(-x)^{-(n+1)}\\&={\frac {1}{\Gamma (-n)}}{\mathcal {M}}_{-x}[e^{x}f(x)](-n),\end{aligned}}}$

which can be expressed in terms of the Mellin transform ${\displaystyle {\mathcal {M}}}$.

Fermions

Fermions are particles which satisfy Fermi–Dirac statistics. We will now examine the normal ordering of fermionic creation and annihilation operator products.

Single fermions

For a single fermion there are two operators of interest:

• ${\displaystyle {\hat {f}}^{\dagger }}$: the fermion's creation operator.
• ${\displaystyle {\hat {f}}}$: the fermion's annihilation operator.

These satisfy the anticommutator relationships

${\displaystyle \left[{\hat {f}}^{\dagger },{\hat {f}}^{\dagger }\right]_{+}=0}$

${\displaystyle \left[{\hat {f}},{\hat {f}}\right]_{+}=0}$

${\displaystyle \left[{\hat {f}},{\hat {f}}^{\dagger }\right]_{+}=1}$

where ${\displaystyle \left[A,B\right]_{+}\equiv AB+BA}$ denotes the anticommutator. These may be rewritten as

${\displaystyle {\hat {f}}^{\dagger }\,{\hat {f}}^{\dagger }=0}$

${\displaystyle {\hat {f}}\,{\hat {f}}=0}$

${\displaystyle {\hat {f}}\,{\hat {f}}^{\dagger }=1-{\hat {f}}^{\dagger }\,{\hat {f}}.}$

To define the normal ordering of a product of fermionic creation and annihilation operators we must take into account the number of interchanges between neighbouring operators. We get a minus sign for each such interchange.

Examples

1. We again start with the simplest cases:

${\displaystyle :{\hat {f}}^{\dagger }\,{\hat {f}}:\,={\hat {f}}^{\dagger }\,{\hat {f}}}$

This expression is already in normal order so nothing is changed. In the reverse case, we introduce a minus sign because we have to change the order of two operators:

${\displaystyle :{\hat {f}}\,{\hat {f}}^{\dagger }:\,=-{\hat {f}}^{\dagger }\,{\hat {f}}}$

These can be combined, along with the anticommutation relations, to show

${\displaystyle {\hat {f}}\,{\hat {f}}^{\dagger }\,=1-{\hat {f}}^{\dagger }\,{\hat {f}}=1+:{\hat {f}}\,{\hat {f}}^{\dagger }:}$

or

${\displaystyle {\hat {f}}\,{\hat {f}}^{\dagger }-:{\hat {f}}\,{\hat {f}}^{\dagger }:=1.}$

This equation, which is in the same form as the bosonic case above, is used in defining the contractions used in Wick's theorem.

2. The normal order of any more complicated cases gives zero because there will be at least one creation or annihilation operator appearing twice. For example:

${\displaystyle :{\hat {f}}\,{\hat {f}}^{\dagger }\,{\hat {f}}{\hat {f}}^{\dagger }:\,=-{\hat {f}}^{\dagger }\,{\hat {f}}^{\dagger }\,{\hat {f}}\,{\hat {f}}=0}$

Multiple fermions

For ${\displaystyle N}$ different fermions there are ${\displaystyle 2N}$ operators:

• ${\displaystyle {\hat {f}}_{i}^{\dagger }}$: the ${\displaystyle i^{th}}$ fermion's creation operator.
• ${\displaystyle {\hat {f}}_{i}}$: the ${\displaystyle i^{th}}$ fermion's annihilation operator.

Here ${\displaystyle i=1,\ldots ,N}$.

These satisfy the anti-commutation relations:

${\displaystyle \left[{\hat {f}}_{i}^{\dagger },{\hat {f}}_{j}^{\dagger }\right]_{+}=0}$

${\displaystyle \left[{\hat {f}}_{i},{\hat {f}}_{j}\right]_{+}=0}$

${\displaystyle \left[{\hat {f}}_{i},{\hat {f}}_{j}^{\dagger }\right]_{+}=\delta _{ij}}$

where ${\displaystyle i,j=1,\ldots ,N}$ and ${\displaystyle \delta _{ij}}$ denotes the Kronecker delta.

These may be rewritten as:

${\displaystyle {\hat {f}}_{i}^{\dagger }\,{\hat {f}}_{j}^{\dagger }=-{\hat {f}}_{j}^{\dagger }\,{\hat {f}}_{i}^{\dagger }}$

${\displaystyle {\hat {f}}_{i}\,{\hat {f}}_{j}=-{\hat {f}}_{j}\,{\hat {f}}_{i}}$

${\displaystyle {\hat {f}}_{i}\,{\hat {f}}_{j}^{\dagger }=\delta _{ij}-{\hat {f}}_{j}^{\dagger }\,{\hat {f}}_{i}.}$

When calculating the normal order of products of fermion operators we must take into account the number of interchanges of neighbouring operators required to rearrange the expression. It is as if we pretend the creation and annihilation operators anticommute and then we reorder the expression to ensure the creation operators are on the left and the annihilation operators are on the right - all the time taking account of the anticommutation relations.

Examples

1. For two different fermions (${\displaystyle N=2}$) we have

${\displaystyle :{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}:\,={\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}}$

Here the expression is already normal ordered so nothing changes.

${\displaystyle :{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger }:\,=-{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}}$

Here we introduce a minus sign because we have interchanged the order of two operators.

${\displaystyle :{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}^{\dagger }:\,={\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}^{\dagger }\,{\hat {f}}_{2}=-{\hat {f}}_{2}^{\dagger }\,{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}}$

Note that the order in which we write the operators here, unlike in the bosonic case, does matter.

2. For three different fermions (${\displaystyle N=3}$) we have

${\displaystyle :{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}\,{\hat {f}}_{3}:\,={\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}\,{\hat {f}}_{3}=-{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{3}\,{\hat {f}}_{2}}$

Notice that since (by the anticommutation relations) ${\displaystyle {\hat {f}}_{2}\,{\hat {f}}_{3}=-{\hat {f}}_{3}\,{\hat {f}}_{2}}$ the order in which we write the operators does matter in this case.

Similarly we have

${\displaystyle :{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{3}:\,=-{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}\,{\hat {f}}_{3}={\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{3}\,{\hat {f}}_{2}}$

${\displaystyle :{\hat {f}}_{3}{\hat {f}}_{2}\,{\hat {f}}_{1}^{\dagger }:\,={\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{3}\,{\hat {f}}_{2}=-{\hat {f}}_{1}^{\dagger }\,{\hat {f}}_{2}\,{\hat {f}}_{3}}$

Uses in quantum field theory

The vacuum expectation value of a normal ordered product of creation and annihilation operators is zero. This is because, denoting the vacuum state by ${\displaystyle |0\rangle }$, the creation and annihilation operators satisfy

${\displaystyle \langle 0|{\hat {a}}^{\dagger }=0\qquad {\textrm {and}}\qquad {\hat {a}}|0\rangle =0}$

(here ${\displaystyle {\hat {a}}^{\dagger }}$ and ${\displaystyle {\hat {a}}}$ are creation and annihilation operators (either bosonic or fermionic)).

Let ${\displaystyle {\hat {O}}}$ denote a non-empty product of creation and annihilation operators. Although this may satisfy

${\displaystyle \langle 0|{\hat {O}}|0\rangle \neq 0,}$

we have

${\displaystyle \langle 0|:{\hat {O}}:|0\rangle =0.}$

Normal ordered operators are particularly useful when defining a quantum mechanical Hamiltonian. If the Hamiltonian of a theory is in normal order then the ground state energy will be zero: ${\displaystyle \langle 0|{\hat {H}}|0\rangle =0}$.

Free fields

With two free fields φ and χ,

${\displaystyle :\phi (x)\chi (y):=\phi (x)\chi (y)-\langle 0|\phi (x)\chi (y)|0\rangle }$

where ${\displaystyle |0\rangle }$ is again the vacuum state. Each of the two terms on the right hand side typically blows up in the limit as y approaches x but the difference between them has a well-defined limit. This allows us to define :φ(x)χ(x):.

Wick's theorem

Main article: Wick's theorem

Wick's theorem states the relationship between the time ordered product of ${\displaystyle n}$ fields and a sum of normal ordered products. This may be expressed for ${\displaystyle n}$ even as

${\displaystyle {\begin{aligned}T\left[\phi (x_{1})\cdots \phi (x_{n})\right]=&:\phi (x_{1})\cdots \phi (x_{n}):+\sum _{\textrm {perm}}\langle 0|T\left[\phi (x_{1})\phi (x_{2})\right]|0\rangle :\phi (x_{3})\cdots \phi (x_{n}):\\&+\sum _{\textrm {perm}}\langle 0|T\left[\phi (x_{1})\phi (x_{2})\right]|0\rangle \langle 0|T\left[\phi (x_{3})\phi (x_{4})\right]|0\rangle :\phi (x_{5})\cdots \phi (x_{n}):\\\vdots \\&+\sum _{\textrm {perm}}\langle 0|T\left[\phi (x_{1})\phi (x_{2})\right]|0\rangle \cdots \langle 0|T\left[\phi (x_{n-1})\phi (x_{n})\right]|0\rangle \end{aligned}}}$

where the summation is over all the distinct ways in which one may pair up fields. The result for ${\displaystyle n}$ odd looks the same except for the last line which reads

${\displaystyle \sum _{\text{perm}}\langle 0|T\left[\phi (x_{1})\phi (x_{2})\right]|0\rangle \cdots \langle 0|T\left[\phi (x_{n-2})\phi (x_{n-1})\right]|0\rangle \phi (x_{n}).}$

This theorem provides a simple method for computing vacuum expectation values of time ordered products of operators and was the motivation behind the introduction of normal ordering.

Alternative definitions

The most general definition of normal ordering involves splitting all quantum fields into two parts (for example see Evans and Steer 1996) ${\displaystyle \phi _{i}(x)=\phi _{i}^{+}(x)+\phi _{i}^{-}(x)}$. In a product of fields, the fields are split into the two parts and the ${\displaystyle \phi ^{+}(x)}$ parts are moved so as to be always to the left of all the ${\displaystyle \phi ^{-}(x)}$ parts. In the usual case considered in the rest of the article, the ${\displaystyle \phi ^{+}(x)}$ contains only creation operators, while the ${\displaystyle \phi ^{-}(x)}$ contains only annihilation operators. As this is a mathematical identity, one can split fields in any way one likes. However, for this to be a useful procedure one demands that the normal ordered product of any combination of fields has zero expectation value

${\displaystyle \langle :\phi _{1}(x_{1})\phi _{2}(x_{2})\ldots \phi _{n}(x_{n}):\rangle =0}$

It is also important for practical calculations that all the commutators (anti-commutator for fermionic fields) of all ${\displaystyle \phi _{i}^{+}}$ and ${\displaystyle \phi _{j}^{-}}$ are all c-numbers. These two properties means that we can apply Wick's theorem in the usual way, turning expectation values of time-ordered products of fields into products of c-number pairs, the contractions. In this generalised setting, the contraction is defined to be the difference between the time-ordered product and the normal ordered product of a pair of fields.

The simplest example is found in the context of Thermal quantum field theory (Evans and Steer 1996). In this case the expectation values of interest are statistical ensembles, traces over all states weighted by ${\displaystyle \exp(-\beta {\hat {H}})}$. For instance, for a single bosonic quantum harmonic oscillator we have that the thermal expectation value of the number operator is simply the Bose–Einstein distribution

${\displaystyle \langle {\hat {b}}^{\dagger }{\hat {b}}\rangle ={\frac {\mathrm {Tr} (e^{-\beta \omega {\hat {b}}^{\dagger }{\hat {b}}}{\hat {b}}^{\dagger }{\hat {b}})}{\mathrm {Tr} (e^{-\beta \omega {\hat {b}}^{\dagger }{\hat {b}}})}}={\frac {1}{e^{\beta \omega }-1}}}$

So here the number operator ${\displaystyle {\hat {b}}^{\dagger }{\hat {b}}}$ is normal ordered in the usual sense used in the rest of the article yet its thermal expectation values are non-zero. Applying Wick's theorem and doing calculation with the usual normal ordering in this thermal context is possible but computationally impractical. The solution is to define a different ordering, such that the ${\displaystyle \phi _{i}^{+}}$ and ${\displaystyle \phi _{j}^{-}}$ are linear combinations of the original annihilation and creations operators. The combinations are chosen to ensure that the thermal expectation values of normal ordered products are always zero so the split chosen will depend on the temperature.

Species complex

From Wikipedia, the free encyclopedia

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

The butterfly genus Heliconius contains some species that are extremely difficult to tell apart.

In biology, a species complex is a group of closely related organisms that are so similar in appearance and other features that the boundaries between them are often unclear. The taxa in the complex may be able to hybridize readily with each other, further blurring any distinctions. Terms that are sometimes used synonymously but have more precise meanings are cryptic species for two or more species hidden under one species name, sibling species for two (or more) species that are each other's closest relative, and species flock for a group of closely related species that live in the same habitat. As informal taxonomic ranks, species group, species aggregate, macrospecies, and superspecies are also in use.

Two or more taxa that were once considered conspecific (of the same species) may later be subdivided into infraspecific taxa (taxa within a species, such as bacterial strains or plant varieties), that is complex but it is not a species complex.

A species complex is in most cases a monophyletic group with a common ancestor, but there are exceptions. It may represent an early stage after speciation but may also have been separated for a long time period without evolving morphological differences. Hybrid speciation can be a component in the evolution of a species complex.

Species complexes exist in all groups of organisms and are identified by the rigorous study of differences between individual species that uses minute morphological details, tests of reproductive isolation, or DNA-based methods, such as molecular phylogenetics and DNA barcoding. The existence of extremely similar species may cause local and global species diversity to be underestimated. The recognition of similar-but-distinct species is important for disease and pest control and in conservation biology although the drawing of dividing lines between species can be inherently difficult.

Definition

At least six treefrog species make up the Hypsiboas calcaratus–fasciatus species complex.

 

The fly agaric comprises several cryptic species, as is shown by genetic data.

 

The African forest elephant (shown) is the bush elephant's sibling species.

 

Mbuna cichlids form a species flock in Lake Malawi.

A species complex is typically considered as a group of close, but distinct species. Obviously, the concept is closely tied to the definition of a species. Modern biology understands a species as "separately evolving metapopulation lineage" but acknowledges that the criteria to delimit species may depend on the group studied. Thus, many traditionally defined species defined, based only on morphological similarity, have been found to be several distinct species when other criteria, such as genetic differentiation or reproductive isolation, are applied.

A more restricted use applies the term to close species between which hybridisation occurred or is occurring, which leads to intermediate forms and blurred species boundaries. The informal classification, superspecies, can be exemplified by the grizzled skipper butterfly, which is a superspecies that is further divided into three subspecies.

Some authors apply the term to a species with intraspecific variability, which might be a sign of ongoing or incipient speciation. Examples are ring species or species with subspecies, in which it is often unclear if they should be considered separate species.

Related concepts

Several terms are used synonymously for a species complex, but some of them may also have slightly different or narrower meanings. In the nomenclature codes of zoology and bacteriology, no taxonomic ranks are defined at the level between subgenera and species, but the botanical code defines four ranks below genera (section, subsections, series and subseries). Different informal taxonomic solutions have been used to indicate a species complex.

Cryptic species

Also called physiologic race (uncommon). This describes "distinct species that are erroneously classified (and hidden) under one species name". More generally, the term is often applied when species, even if they are known to be distinct, cannot be reliably distinguished by morphology. The usage physiologic race is not to be confused with physiological race.

Sibling species

Also called aphanic species. This term, introduced by Ernst Mayr in 1942, was initially used with the same meaning as cryptic species, but later authors emphasized the common phylogenetic origin. A recent article defines sibling species as "cryptic sister species", "two species that are the closest relative of each other and have not been distinguished from one another taxonomically".

Species flock

Also called species swarm. This refers to "a monophyletic group of closely related species all living in the same ecosystem". Conversely, the term has also been applied very broadly to a group of closely related species than can be variable and widespread. Not to be confused with a Mixed-species foraging flock, a behavior in which birds of different species feed together.

Superspecies

Sometimes used as an informal rank for a species complex around one "representative" species. Popularized by Bernhard Rensch and later Ernst Mayr, with the initial requirement that species forming a superspecies must have allopatric distributions. For the component species of a superspecies, allospecies was proposed.

Species aggregate

Used for a species complex, especially in plant taxa where polyploidy and apomixis are common. Historical synonyms are species collectiva, introduced by Adolf Engler, conspecies, and grex. Components of a species aggregate have been called segregates or microspecies. Used as abbreviation agg. after the binomial species name.

Sensu lato

A Latin phrase meaning "in the broad sense", it is often used after a binomial species name, often abbreviated as s.l., to indicate a species complex represented by that species.

Identification

Distinguishing close species within a complex requires the study of often very small differences. Morphological differences may be minute and visible only by the use of adapted methods, such as microscopy. However, distinct species sometimes have no morphological differences. In those cases, other characters, such as in the species' life history, behavior, physiology, and karyology, may be explored. For example, territorial songs are indicative of species in the treecreepers, a bird genus with few morphological differences. Mating tests are common in some groups such as fungi to confirm the reproductive isolation of two species.

Analysis of DNA sequences is becoming increasingly standard for species recognition and may, in many cases, be the only useful method. Different methods are used to analyse such genetic data, such as molecular phylogenetics or DNA barcoding. Such methods have greatly contributed to the discovery of cryptic species, including such emblematic species as the fly agaric or the African elephants.

Salamandra corsica

 

Salamandra atra

 

Salamandra salamandra

Similarity can be misleading: the Corsican fire salamander (top) was previously considered a subspecies of the fire salamander (bottom) but is in fact more closely related to the uniformly black Alpine salamander (center).

Evolution and ecology

Speciation process

A species complex typically forms a monophyletic group that has diversified rather recently, as is shown by the short branches between the species A–E (blue box) in this phylogenetic tree.

Species forming a complex have typically diverged very recently from each other, which sometimes allows the retracing of the process of speciation. Species with differentiated populations, such as ring species, are sometimes seen as an example of early, ongoing speciation: a species complex in formation. Nevertheless, similar but distinct species have sometimes been isolated for a long time without evolving differences, a phenomenon known as "morphological stasis". For example, the Amazonian frog Pristimantis ockendeni is actually at least three different species that diverged over 5 million years ago.

Stabilizing selection has been invoked as a force maintaining similarity in species complexes, especially when they adapted to special environments (such as a host in the case of symbionts or extreme environments). This may constrain possible directions of evolution; in such cases, strongly divergent selection is not to be expected. Also, asexual reproduction, such as through apomixis in plants, may separate lineages without producing a great degree of morphological differentiation.

Possible processes explaining similarity of species in a species complex:
a – morphological stasis
b – hybrid speciation

A species complex is usually a group that has one common ancestor (a monophyletic group), but closer examination can sometimes disprove that. For example, yellow-spotted "fire salamanders" in the genus Salamandra, formerly all classified as one species S. salamandra, are not monophyletic: the Corsican fire salamander's closest relative has been shown to be the entirely black Alpine salamander. In such cases, similarity has arisen from convergent evolution.

Hybrid speciation can lead to unclear species boundaries through a process of reticulate evolution, in which species have two parent species as their most recent common ancestors. In such cases, the hybrid species may have intermediate characters, such as in Heliconius butterflies. Hybrid speciation has been observed in various species complexes, such as insects, fungi and plants. In plants, hybridization often takes place through polyploidization, and hybrid plant species are called nothospecies.

Range and habitats

Sources differ on whether or not members of a species group share a range. A source from Iowa State University Department of Agronomy states that members of a species group usually have partially overlapping ranges but do not interbreed with one another. A Dictionary of Zoology (Oxford University Press 1999) describes a species group as complex of related species that exist allopatrically and explains that the "grouping can often be supported by experimental crosses in which only certain pairs of species will produce hybrids." The examples given below may support both uses of the term "species group."

Often, such complexes do not become evident until a new species is introduced into the system, which breaks down existing species barriers. An example is the introduction of the Spanish slug in Northern Europe, where interbreeding with the local black slug and red slug, which were traditionally considered clearly separate species that did not interbreed, shows that they may be actually just subspecies of the same species.

Where closely related species co-exist in sympatry, it is often a particular challenge to understand how the similar species persist without outcompeting each other. Niche partitioning is one mechanism invoked to explain that. Indeed, studies in some species complexes suggest that species divergence have gone in par with ecological differentiation, with species now preferring different microhabitats. Similar methods also found that the Amazonian frog Eleutherodactylus ockendeni is actually at least three different species that diverged over 5 million years ago.

A species flock may arise when a species penetrates a new geographical area and diversifies to occupy a variety of ecological niches, a process known as adaptive radiation. The first species flock to be recognized as such was the 13 species of Darwin's finches on the Galápagos Islands described by Charles Darwin.

Practical implications

Biodiversity estimates

It has been suggested that cryptic species complexes are very common in the marine environment. That suggestion came before the detailed analysis of many systems using DNA sequence data but has been proven to be correct. The increased use of DNA sequence in the investigation of organismal diversity (also called phylogeography and DNA barcoding) has led to the discovery of a great many cryptic species complexes in all habitats. In the marine bryozoan Celleporella hyalina, detailed morphological analyses and mating compatibility tests between the isolates identified by DNA sequence analysis were used to confirm that these groups consisted of more than 10 ecologically distinct species, which had been diverging for many millions of years.

Evidence from the identification of cryptic species has led some to conclude that current estimates of global species richness are too low.

Disease and pathogen control

The Anopheles gambiae mosquito complex contains both species that are a vector for malaria and species that are not.

Pests, species that cause diseases and their vectors, have direct importance for humans. When they are found to be cryptic species complexes, the ecology and the virulence of each of these species need to be re-evaluated to devise appropriate control strategies. Examples are cryptic species in the malaria vector genus of mosquito, Anopheles, the fungi causing cryptococcosis, and sister species of Bactrocera tryoni, or the Queensland fruit fly. That pest is indistinguishable from two sister species except that B. tryoni inflicts widespread, devastating damage to Australian fruit crops, but the sister species do not.

Conservation biology

When a species is found to be several phylogenetically distinct species, each typically has smaller distribution ranges and population sizes than had been reckoned. The different species can also differ in their ecology, such as by having different breeding strategies or habitat requirements, which must be taken into account for appropriate management. For example, giraffe populations and subspecies differ genetically to such an extent that they may be considered species. Although the giraffe, as a whole, is not considered to be threatened, if each cryptic species is considered separately, there is a much higher level of threat.