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.

Buddhism and sexual orientation

From Wikipedia, the free encyclopedia

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

The relationship between Buddhism and sexual orientation varies by tradition and teacher. According to some scholars, early Buddhism appears to have placed no special stigma on homosexual relations, since the subject was not mentioned.

Lay followers and monastic practices

In the early sutras of Buddhism, "accepted or unaccepted human sexual conduct" for laypersons "is not specifically mentioned." "Sexual misconduct" is a broad term, subject to interpretation according to followers' social norms. Early Buddhism appears to have been silent concerning homosexual relations.

According to the Pāli Canon and Āgama (the early Buddhist scriptures), it is not specified that same or opposite gender relations have inherently something to do with sexual misconduct, and some Theravada monks express that same-gender relations do not violate the rule to avoid sexual misconduct, which would mean not having sex with someone underage (thus protected by their parents or guardians), someone betrothed or married and who have taken vows of religious celibacy.

Some later traditions feature restrictions on non-vaginal sex (some Buddhist texts mention non-vaginal sex as sexual misconduct, including men having sex with men or paṇḍaka) though some Buddhist monks interpret such texts as being about forced sex. This non-vaginal sex view is not based on what Buddha said, but from some later scriptures such as the Abhidharma texts.

Regarding Buddhist monks, the Vinaya (code of monastic discipline) bans all sexual activity, but does so in purely physiological terms, making no moral distinctions among the many possible forms of intercourse it lists.

Among Buddhists, there is a wide diversity of opinion about homosexuality. Buddhism teaches that sensual enjoyment and desire in general are hindrances to enlightenment, and inferior to the kinds of pleasure (see, e.g. pīti, a Pāli word often translated as "rapture") that are integral to the practice of jhāna.

Buddhist texts

Early texts

Buddha is often portrayed as a male figure, such as in this painting from a monastery in Laos.

Within the earliest monastic texts such as the Vinaya (c. 4th century BCE), male monks are explicitly forbidden from having sexual relations with any of the four genders: male, female, ubhatovyañjanaka and paṇḍaka; various meanings of these words are given below. Later, the Buddha allowed the ordination of women, forbade ordination to these other types of people, with exceptions to a few particular types of paṇḍaka. The Buddha's proscriptions against certain types of people joining the monastic sangha (ordained community) are often understood to reflect his concern with upholding the public image of the sangha as virtuous; in some cases, this is explicitly stated. Social acceptability was vital for the sangha, as it could not survive without material support from lay society.

Ubhatovyañjanakas

The word ubhatovyañjanaka is usually thought to describe people who have both male and female sexual characteristics: hermaphrodites (intersex). In the Vinaya, it is said that ubhatovyañjanaka should not be ordained, on account of the possibility that they would entice a fellow monk or nun into having sex. Although it has been seen by some that the category of ubhatovyañjanaka is of later addition to the early buddhist texts, since it does not appear in the early suttas, the Pāṭimokkhas, nor in the early parts of the Vinaya.

Paṇḍakas

The paṇḍaka is a complex category that is variously defined in different Buddhist texts. In the earliest texts, the word seems to refer to a socially stigmatized class of trans-feminine and/or cross-dressing people, some of whom may have been sex workers. Paisarn Likhitpreechakul argue that these people are grouped together with groups who are excluded from ordination as well; those with physical disabilities such as deafness or dwarfism, or those who have committed crimes. "The Story of the Prohibition of the Ordination of Pandaka" from the Vinaya claims that the ban is a response to the example of a paṇḍaka monk with a desire to have sex. Being refused by other monks, he had sex with animal handlers, who then told the wider community and brought disgrace upon the sangha. Since the word paṇḍaka does not appear in either of the early Suttas nor in the early parts of the Vinaya, it has been seen by some as a possible indication that pandaka's inclusion in the Vinaya did not happen in the Buddha's lifetime but was added later.

In the Lotus Sutra, it said Bodhisattva should not go near Paṇḍaka, as like what monk rules said in Vinaya. The Theravadin text Milinda Panha, claims that Paṇḍakas let out secrets through their imperfection.

LGBTQ+ people in later traditions

Some modern commentators interpret the word ubhatovyanjañaka as including those who are not physically intersex, but display behavioral and psychological characteristics of both sexes, such as a woman who is attracted to other women. 5th-century Buddhist writer Buddhaghosa describes ubhatobyanjanaka as people with the body of one sex but the "power", or gender of the other. Leonard Zwilling argues that in this account Buddhaghosa does not in fact describe "hermaphroditism" but rather bisexuality or homosexuality. Janet Gyatso pointed out that Zwilling destroys his own argument that pandakas are homosexuals when he writes, "The Vinaya, in fact, goes so far as to distinguish sexual activity between normative males from sexual relations between a socially normative male and a pandaka."

In other texts, the term paṇḍaka can include those born sexually indeterminate or with no sex, eunuchs, those whose impotence changes every half month, males who gain sexual potency by absorbing other men's semen, or spying on other people having sex. It sometimes includes males or females with any sexual dysfunction, such as impotence or irregular menstrual cycles. The common element seems to be those whose sexuality is either limited physiologically, or those who are sexually impotent. Together these impotence types are almost always portrayed negatively as a pariah class, especially in the earliest texts. In modern contexts, paṇḍaka is sometimes alleged to include lesbians, gay men, and transgender and intersex people, although in ancient times, a man who sexually penetrated another man or a paṇḍaka was not himself considered a paṇḍaka.

Some texts of the Abhidharma state that a paṇḍaka cannot achieve enlightenment in their own lifetime, but must wait for reincarnation as a man or woman. Ananda — Buddha's cousin and disciple — was said to be a paṇḍaka in one of his many previous lives, as was the Buddhist nun Isidāsī (from the Therigatha). In both cases, birth as a paṇḍaka was a result of poor karma, and the idea that being a paṇḍaka stems from bad behaviour in a previous life is common in Buddhist literature. Asanga and Vasubandhu discussed if a pandaka was able to be enlightened or not.

In the Samantapasadika, a work of the 5th century CE Theravadin commentator and scholar Buddhaghosa's, paṇḍaka are described as being filled with defiling passions (ussanakilesa), unquenchable lusts (avapasantaparilaha) and are dominated by their libido (parilahavegabhibhuta). The 4th century Mahayana Buddhist writers Vasubandhu and Asanga contend that the paṇḍaka has no discipline for spiritual practice, due to their defiling passions of both male and female sexes. They lack the moral fortitude to counter these passions because they lack modesty and shame. Incapable of showing restraint, such a being is abandoned by their parents and lacking such ties are unable to hold strong views. Asanga, like Vasubandhu, refuses the pandaka recognition as a layman on the grounds that such persons are unfit to associate with or serve the Sangha. Asanga, however, considers them capable to practice the path of a layman individually if they so desire, but without receiving recognition as a layman or being introduced in the sangha. A position similar to Asanga view was also featured in the Lotus Sutra, where sangha members were advised to avoid the paṇḍaka.

Buddha's proscriptions against certain types of people joining the monastic sangha (ordained community) are often understood to reflect his concern with upholding the public image of the sangha as virtuous. Peter Jackson, the scholar of sexual politics and Buddhism in Thailand, speculates that the Buddha was initially reluctant to allow women to join the sangha for this reason. Jackson explains:

Buddhism, the middle path, has always been concerned with the maintenance of social order and since the Buddha's time the sangha has never claimed to provide a universal vehicle for the spiritual liberation of all individuals in society, explicitly excluding those who are considered to reflect badly on the monkhood in terms of prevailing social norms and attitudes.

Social acceptability was vital for the sangha, as it could not survive without material support from lay society.

Several Theravada Buddhist texts state that the members of the paṇḍaka are excluded from a variety of Buddhist practices (in addition to ordination):

• acting as preceptors in ordination ceremonies
• making donations to begging monks
• meditating and
• ability to understand the Dharma.

The Mahakala Ma Ning, a wrathful deity revered in Tibetan Buddhism, especially the Nyingma school, as a defender of the Dharma. The term ma ning has been translated as "genderless" or "eunuch", and equates to paṇḍaka. In this macabre 19th-century image the Ma Ning holds a human heart in their hand, and also a garland of hearts around their waist.

Classical Mahayana scholars like Shantideva and Aśvaghoṣa considered non-vagina sex (including men sex with men) to be sexual misconduct. Shantideva based his views on quotes from the Saddharma-smrtyupasthana Sutra. According to Mizuno Kōgen's study, Saddharma-smrtyupasthana Sutra is related to Ghoṣa's Abhidharmāmṛtarasaśāstra, and is compiled by Sarvastivada sect (possibly from someone related to Ghoṣa after the 2nd century). In the Great Treatise on the Perfection of Wisdom (Sanskrit: Mahā-prajñāpāramitā-śāstra), the Madhyamaka scholar Nagarjuna mentioned the non-vagina sex restriction as based on coerced action toward one's own spouse.

In contrast, later texts, particularly Tibetan Buddhist writings, occasionally value paṇḍaka positively for their "middleness" and balance. The paṇḍaka in these Tibetan works is translated with the term ma ning — "genderless" or "without genitals". The 13th-century Tibetan monk Gyalwa Yang Gönpa, who was one of the significant figures in the early Drukpa Kagyu sect, writes about ma ning as a balanced state between maleness and femaleness. Yang Gönpa describes ma ning as "the abiding breath between male exhalation and female inhalation" and "the balanced yogic channel, as opposed to the too tight male channel, and the too loose female one".

Most Mahayana teachings assert that all beings who correctly practice the dharma may reach enlightenment, since all possess innate Buddha nature. Enlightenment being achievable even in a single life.

Tibetan Buddhism

There are different views among Tibetan Buddhist teachers on acceptable expressions of sexuality.

Historically, Gampopa (12th century), one of the main early masters of the Kagyu school of Tibetan Buddhism, followed the Indian Buddhist tradition, starting with the 3rd-century Hinayana texts of Vasubandhu, and stated that oral and anal sex, whether with a man or a woman, are violations of the third precept regarding inappropriate sexual behavior. Longchenpa, the 13th-century founder of the Nyingma school, citing the 3rd-century Mahayana texts of the Indian master Asanga, elaborated that inappropriate sexual behavior also include the hands among inappropriate parts of the body for sexual activity. In the same way, Gelug predecessor Je Tsongkhapa also adheres to such rules in his studies.

Various contemporary teachers of Tibetan Buddhist lineages, including the 17th Gyalwang Karmapa, have offered understanding towards LGBTQ people while noting that same-sex relationships do not necessarily constitute misconduct for lay people. The Dalai Lama has maintained the views of Je Tsongkhapa

The 14th Dalai Lama has "voiced his support for the full recognition of human rights for all people, regardless of sexual orientation," while noting that from a Buddhist point of view, lesbian and gay sex is "generally considered sexual misconduct." In the most recent interview with the Dalai Lama on this topic (March 10, 2014), the Dalai Lama said gay marriage is OK provided it's not in contradiction with the values of one's chosen religion.

The Dalai Lama has also stated that any sex other than penis-vagina intercourse with one's own monogamous partner, including oral sex, anal sex, and masturbation is improper from the Buddhist perspective. In his 1996 book Beyond Dogma, he states, "A sexual act is deemed proper when the couples use the organs intended for sexual intercourse and nothing else... homosexuality, whether it is between men or between women, is not improper in itself. What is improper is the use of organs already defined as inappropriate for sexual contact."

The Dalai Lama cited the Indian Buddhist texts of Vasubandhu, Asanga, and Ashvaghosha as his sources concerning what constitutes inappropriate sexual behavior. In 1997 he stated: "Sexual organs were created for reproduction between the male element and the female element — and everything that deviates from that is not acceptable from a Buddhist point of view." The Dalai Lama has repeatedly said to LGBT groups that he can't rewrite the texts. He thinks that this is the type of issue that would need to be discussed by a council of Buddhist elders from all Buddhist traditions.

Theravada Buddhism

Peter Jackson, an Australian scholar of sexual politics and Buddhism in Thailand, writes that "Buddhism is a complex tradition and there is no single canonical or scripturally sanctioned position on homosexuality." Thailand is one of several countries with a large population of Theravada Buddhist.

In traditional Thai Buddhist accounts of sexuality, "[sexual] actions and desires have an involuntary cause [and] do not themselves accrue any future karmic consequences. They are the outworking of past karma, not sources for the accumulation of future karma. According to Bunmi, homosexual activity and the desire to engage in homosexual activity fall into this category and are not sinful and do not accrue karmic consequences." Jackson writes that this understanding of homosexuality "prevailed in Thailand until recent decades."

In the 1980s in Thailand, during the AIDS epidemic, there was "a shift in Buddhist attitudes from relative tolerance of homosexuality to condemnation." These views were "unprecedented in recent Thai history." During this time there were two ways Buddhists viewed homosexuality: in the sympathetic view, it was said that homosexuality arose from the karma of previous lifetimes; in the intolerant view, it was seen as arising from immoral conduct in one's present life.

In 1989, the supreme governing body of the Thai sangha affirmed that "gays" (here translated from Thai kathoey) are prohibited from being ordained. Their declaration has apparently gone unheeded in some quarters, as Phra Pisarn Thammapatee (AKA Phra Payom Kalayano), one of the most eminent monks in the country, demanded in 2003 that 1,000 gay monks be ousted from the sangha, and that better screening processes are put in place to keep out any gay postulants.

Recently, Phra Payom Kalayano, an eminent monk and abbot, affirmed the rights of gay monks to join the Sangha: "In the past, katoey had no hope of being ordained because the rules were stricter and society was less open minded. But they have just as much right as anyone else to join the monkhood." This view has been affirmed by other Thai Theravada monks. But it's not known if that statement given by Phra Payom was supported only by some monks of his sangha or by official structures of his Theravada school. There is no information if any other Thai branches of Theravada has changed their attitude towards LGBT ordination because other Theravada branches does not seem to have changed their attitude on this issue.

Japanese Buddhism

See also: Homosexuality in Japan § Monastic same-sex love

Several writers have noted the strong historical tradition of open bisexuality and homosexuality among male Buddhist institutions in Japan. When the Tendai priest Genshin harshly criticised homosexuality as immoral, others mistook his criticism as having been because the acolyte wasn't one's own. Chigo Monogatari (稚児物語), "acolyte stories" of love between monks and their chigo were popular, and such relationships appear to have been commonplace, alongside sex with women.

Western Christian travellers to Japan from the 16th century have noted (with distaste) the prevalence and acceptance of forms of homosexuality among Japanese Buddhists—Jesuit priest Francis Cabral wrote in 1596 that "abominations of the flesh" and "vicious habits" were "regarded in Japan as quite honourable; men of standing entrust their sons to the bonzes to be instructed in such things, and at the same time to serve their lust".

A 17th-century Japanese Buddhist scholar, Kitamura Kigin(北村季吟,きたむら きぎん), wrote that Buddha explained the pursuit of homosexuality over heterosexuality among priests:

It has been the nature of men's hearts to take pleasure in a beautiful woman since the age of male and female gods, but to become intoxicated by the blossom of a handsome youth... would seem to be both wrong and unusual. Nevertheless, the Buddha preached that [Mount] Imose was a place to be avoided and the priests of the law entered this Way as an outlet for their feelings, since their hearts were, after all, made of neither stone nor wood. Like water that plunges from the peak of Tsukubane to form the deep pools of the Minano River, this love has surpassed in depth the love between women and men in these latter days. It plagues the heart not only of courtier and aristocrat but also of brave warriors. Even the mountain dwellers who cut brush for fuel have learned to take pleasure in the shade of young saplings." — Wild Azaleas (1676)

A later Japanese legend attributed the introduction of monastic homosexuality to Japan to Shingon founder Kukai, although scholars now dismiss the veracity of this assertion, pointing out his strict adherence to the Vinaya. Nonetheless, the legend served to "affirm same-sex relation between men and boys in 17th century Japan."

Chinese Buddhism

About Buddhism and homosexuality in China, scholar A. L. De Silva writes, "Generally the attitude has been one of tolerance. Matteo Ricci, the Jesuit missionary who lived in China for 27 years from 1583, expressed horror at the open and tolerant attitude that the Chinese took to homosexuality and naturally enough saw this as proof of the degeneracy of Chinese society."

Venerable Hsing Yun, one of the premier figures in contemporary Chinese Buddhism, has stated that Buddhism should never teach intolerance toward homosexuality, and that people should expand their minds.

Marriage is an institution that reflects the values of the society that supports it. If the people of a society no longer believe that it is important to be married, then there is no reason why they cannot change the institution of marriage. Marriage is a custom. Customs can always be changed. We can find the same core point in this question as we have in others — the ultimate truth of the matter is that individuals can and should decide for themselves what is right. As long as they are not violating others or breaking the laws of the society in which they are living, then they are free to do what they believe is right. It is not for me or anyone else to tell them that they must get married if they want to live together. That is their choice and their choice alone.

The same analysis can be applied to homosexuality. People often ask me what I think about homosexuality. They wonder, is it right, is it wrong? The answer is, it is neither right nor wrong. It is just something that people do. If people are not harming each other, their private lives are their own business; we should be tolerant of them and not reject them.

However, it will still take some time for the world to fully accept homosexuality. All of us must learn to tolerate the behavior of others. Just as we hope to expand our minds to include all of the universe, so we should also seek to expand our minds to include all of the many forms of human behavior.

Tolerance is a form of generosity and it is a form of wisdom. There is nothing anywhere in the Dharma that should ever lead anyone to become intolerant. Our goal as Buddhists is to learn to accept all kinds of people and to help all kinds of people discover the wisdom of the teachings of Shakyamuni Buddha.

— Hsing Yun, Buddhism Pure and Simple, pp. 137–138

Hsing Yun is a best-selling author throughout the Sinosphere, as well as an advocate of Humanistic Buddhism, an approach to reform Chinese Buddhism to match the needs of contemporary laypeople. As such, his views may not be wholly reflective of the older Buddhist views in China. However, at an earlier point (1998), he is quoted as remarking that "according to Buddhism, any emotional involvement, whether homosexual or heterosexual, is a form of attachment ... and, therefore, is a source of suffering," and when the topic of homosexuality, in particular, came up "the master spoke more strongly, calling homosexuality a perverted view (xiejian)."

An example of an older view in opposition to homosexuality is given by the traditional master, Hsuan Hua, an important figure for Buddhism in both China and the United States. Master Hsuan Hua stated that homosexuality "plants the seeds that lead to rebirth in the lower realms of existence". In his commentary on the Sutra of Forty-two Chapters, he described homosexuality as behavior caused by confusion, as the product of sexual desires.

Some Mahayana Buddhist leaders were active in the movement for same-sex marriage rights in Taiwan which legalized same-sex marriages in 2019.

Buddhism in the West

Soeng Hyang, current headmaster of the Kwan Um School of Zen

In contrast to Buddhism in Asia, modern Buddhism in the Western world (European Union, United States of America, Canada, Australia and New Zealand) is typically associated with a concern for social equality—partly as a result of its largely middle-class intellectual membership base, and its philosophical roots in freethought and secular humanism. When applying Buddhist philosophy to the question of homosexuality, western Buddhists often emphasize the importance the Buddha placed on tolerance, compassion, and seeking answers within one's self. They stress these overarching values rather than examining specific passages or texts. As a result, western Buddhism is often relatively gay-friendly, especially since the 1990s. As an interpretation of what is sexual misconduct is an individual decision and not subject to judgment by any central authority, a view of accepting all peoples, but rejecting certain types of sexual acts is more predominant. LGBT people such as Issan Dorsey, Caitriona Reed, Pat Enkyo O'Hara and Soeng Hyang have been ordained as Buddhist monastics and clergy.

An interesting position comes from the western scholar Alexander Berzin,

The texts in Buddhist traditions have been written from the point of view of a heterosexual male. We need to explore the intention of the teachings on sexual misconduct, which ultimately is to eliminate attachment, obsessive desire, and dissatisfaction. If a heterosexual male finds no bounds to these disturbing emotions, he might have sex with someone else's partner, as well as other men. We can apply the same logic and explore what constitutes boundless attachment and dissatisfaction for homosexual and bisexual males or females. For example, having sex with somebody else's partner and so on could be destructive for these types of person as well".

The Dzogchen Ponlop Rinpoche, holder of the Karma Kagyu and Nyingma lineages, in a 2008 talk delivered to LGBT Dharma practitioners at the Shambhala Meditation Center of New York, stressed that for vajrayana lay practitioners, homosexual relationships are no better or worse than heterosexual relationships and that only unhealthy relationships, in general, are to be avoided. Both the Nalandabodhi sangha, which was founded by The Dzogchen Ponlop Rinpoche, and the Shambhala sanghas founded by Chögyam Trungpa Rinpoche have stated that they are welcoming of all sexual orientations. The Shambhala Meditation Center of New York hosts a weekly practice group, Queer Dharma, specifically catering to the needs of the LGBTQ Buddhist community. According to the Danish Karma Kagyu Lama Ole Nydahl, Buddha saw homosexuality as circumstances making life more difficult, but also explained the reason for homosexuality could be aversion against the opposite sex in a former life. Nydahl says however that sexual orientation is not really important in order to practice Buddhism. Well known Zen Buddhist, Thich Nhat Hanh, notes the spirit of Buddhism is inclusiveness and states "when you look at the ocean, you see different kinds of waves, many sizes and shapes, but all the waves have water as their foundation and substance. If you are born gay or lesbian, your ground of being in the same as mine. We are different, but we share the same ground of being."

The U.S. branch of Soka Gakkai International, a Japan-based new religious movement (Japanese new religion) influenced by Nichiren Buddhism, announced in 1995 that they would start holding wedding ceremonies for same-sex couples, and in 2001 established a conference for LGBT members and their supporters. A Buddhist temple in Salt Lake City connected with Jodo Shinshu, another Japanese school of Buddhism, also holds religious rites for same-sex couples.

Another Buddhist organization founded in the West, Juniper Foundation, wrote an article A Buddhist Vote for Same Sex Marriage demonstrating how Buddhist thinking embraces same-sex marriage:

The heart of Buddhist thought is its insight philosophy, which uses critical inquiry to challenge dogma and to reveal how seemingly fixed ideas are more arbitrary than we might think. Applying this philosophy, we see that social customs are not fixed laws but evolving conventions that serve a purpose in a particular culture and time. Marriage is one of these conventions. It is not a rigid law but a social custom that evolves.

In western traditions, there is widescale support for LGBT rights including the European Buddhist Union, the Buddhist Churches of America, many Shin Buddhist groups, and Zen leaders such as Thich Nhat Hanh. In a PEW research poll, 88% of American Buddhists stated that homosexuality should be accepted. This was a higher level of support than any other religious group studied. On 2012, the Australian branch of Buddhism voiced its support for same-sex marriage in a hearing of the Australian Parliament that sought to gather views on whether to legalize same-sex marriage.