Lorentz group

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

Hendrik Antoon Lorentz (1853–1928), after whom the Lorentz group is named.

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

For example, the following laws, equations, and theories respect Lorentz symmetry:

• The kinematical laws of special relativity
• Maxwell's field equations in the theory of electromagnetism
• The Dirac equation in the theory of the electron
• The Standard Model of particle physics

The Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In general relativity physics, in cases involving small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as that of special relativity physics.

Basic properties

The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is an isotropy subgroup of the isometry group of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the homogeneous Lorentz group while the Poincaré group is sometimes called the inhomogeneous Lorentz group. Lorentz transformations are examples of linear transformations; general isometries of Minkowski spacetime are affine transformations. Mathematically, the Lorentz group may be described as the indefinite orthogonal group O(1,3), the matrix Lie group that preserves the quadratic form

${\displaystyle (t,x,y,z)\mapsto t^{2}-x^{2}-y^{2}-z^{2}}$

on ${\displaystyle \mathbb {R} ^{4}}$ (The vector space equipped with this quadratic form is sometimes written ${\displaystyle \mathbb {R} ^{1,3}}$). This quadratic form is, when put on matrix form (see classical orthogonal group), interpreted in physics as the metric tensor of Minkowski spacetime.

The Lorentz group is a six-dimensional noncompact non-abelian real Lie group that is not connected. The four connected components are not simply connected. The identity component (i.e., the component containing the identity element) of the Lorentz group is itself a group, and is often called the restricted Lorentz group, and is denoted SO⁺(1,3). The restricted Lorentz group consists of those Lorentz transformations that preserve the orientation of space and direction of time. Its fundamental group has order 2, and its universal cover, the indefinite spin group Spin(1,3), is isomorphic to both the special linear group SL(2, C) and to the symplectic group Sp(2, C). These isomorphisms allow the Lorentz group to act on a large number of mathematical structures important to physics, most notably the spinors. Thus, in relativistic quantum mechanics and in quantum field theory, it is very common to call SL(2, C) the Lorentz group, with the understanding that SO⁺(1,3) is a specific representation (the vector representation) of it. The biquaternions, popular in geometric algebra, are also isomorphic to SL(2, C).

The restricted Lorentz group also arises as the point symmetry group of a certain ordinary differential equation.

Connected components

Light cone in 2D space plus a time dimension.

Because it is a Lie group, the Lorentz group O(1,3) is both a group and admits a topological description as a smooth manifold. As a manifold, it has four connected components. Intuitively, this means that it consists of four topologically separated pieces.

The four connected components can be categorized by two transformation properties its elements have:

• Some elements are reversed under time-inverting Lorentz transformations, for example, a future-pointing timelike vector would be inverted to a past-pointing vector
• Some elements have orientation reversed by improper Lorentz transformations, for example, certain vierbein (tetrads)

Lorentz transformations that preserve the direction of time are called orthochronous. The subgroup of orthochronous transformations is often denoted O⁺(1, 3). Those that preserve orientation are called proper, and as linear transformations they have determinant +1. (The improper Lorentz transformations have determinant −1.) The subgroup of proper Lorentz transformations is denoted SO(1, 3).

The subgroup of all Lorentz transformations preserving both orientation and direction of time is called the proper, orthochronous Lorentz group or restricted Lorentz group, and is denoted by SO⁺(1, 3). (Note that some authors refer to SO(1,3) or even O(1,3) when they actually mean SO⁺(1, 3).)

The set of the four connected components can be given a group structure as the quotient group O(1, 3)/SO⁺(1, 3), which is isomorphic to the Klein four-group. Every element in O(1,3) can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group

{1, P, T, PT}

where P and T are the parity and time reversal operators:

P = diag(1, −1, −1, −1)

T = diag(−1, 1, 1, 1).

Thus an arbitrary Lorentz transformation can be specified as a proper, orthochronous Lorentz transformation along with a further two bits of information, which pick out one of the four connected components. This pattern is typical of finite-dimensional Lie groups.

Restricted Lorentz group

The restricted Lorentz group ${\displaystyle {\text{SO}}^{+}(1,3)}$ is the identity component of the Lorentz group, which means that it consists of all Lorentz transformations that can be connected to the identity by a continuous curve lying in the group. The restricted Lorentz group is a connected normal subgroup of the full Lorentz group with the same dimension, in this case with dimension six.

The restricted Lorentz group is generated by ordinary spatial rotations and Lorentz boosts (which are rotations in a hyperbolic space that includes a time-like direction). Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation (specified by 3 real parameters) and a boost (also specified by 3 real parameters), it takes 6 real parameters to specify an arbitrary proper orthochronous Lorentz transformation. This is one way to understand why the restricted Lorentz group is six-dimensional. (See also the Lie algebra of the Lorentz group.)

The set of all rotations forms a Lie subgroup isomorphic to the ordinary rotation group SO(3). The set of all boosts, however, does not form a subgroup, since composing two boosts does not, in general, result in another boost. (Rather, a pair of non-colinear boosts is equivalent to a boost and a rotation, and this relates to Thomas rotation.) A boost in some direction, or a rotation about some axis, generates a one-parameter subgroup.

Surfaces of transitivity

Hyperboloid of one sheet

 

Common conical surface

 

Hyperboloid of two sheets

If a group G acts on a space V, then a surface S ⊂ V is a surface of transitivity if S is invariant under G (i.e., ∀g ∈ G, ∀s ∈ S: gs ∈ S) and for any two points s₁, s₂ ∈ S there is a g ∈ G such that gs₁ = s₂. By definition of the Lorentz group, it preserves the quadratic form

${\displaystyle Q(x)=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}.}$

The surfaces of transitivity of the orthochronous Lorentz group O⁺(1, 3), Q(x) = const. acting on flat spacetime ${\displaystyle \mathbb {R} ^{1,3}}$ are the following:

• Q(x) > 0, x₀ > 0 is the upper branch of a hyperboloid of two sheets. Points on this sheet are separated from the origin by a future time-like vector.
• Q(x) > 0, x₀ < 0 is the lower branch of this hyperboloid. Points on this sheet are the past time-like vectors.
• Q(x) = 0, x₀ > 0 is the upper branch of the light cone, the future light cone.
• Q(x) = 0, x₀ < 0 is the lower branch of the light cone, the past light cone.
• Q(x) < 0 is a hyperboloid of one sheet. Points on this sheet are space-like separated from the origin.
• The origin x₀ = x₁ = x₂ = x₃ = 0.

These surfaces are 3-dimensional, so the images are not faithful, but they are faithful for the corresponding facts about O⁺(1, 2). For the full Lorentz group, the surfaces of transitivity are only four since the transformation T takes an upper branch of a hyperboloid (cone) to a lower one and vice versa.

As symmetric spaces

An equivalent way to formulate the above surfaces of transitivity is as a symmetric space in the sense of Lie theory. For example, the upper sheet of the hyperboloid can be written as the quotient space ${\displaystyle {\text{SO}}^{+}(1,3)/{\text{SO}}(3)}$, due to the orbit-stabilizer theorem. Furthermore, this upper sheet also provides a model for three-dimensional hyperbolic space.

Representations of the Lorentz group

These observations constitute a good starting point for finding all infinite-dimensional unitary representations of the Lorentz group, in fact, of the Poincaré group, using the method of induced representations.^[4] One begins with a "standard vector", one for each surface of transitivity, and then ask which subgroup preserves these vectors. These subgroups are called little groups by physicists. The problem is then essentially reduced to the easier problem of finding representations of the little groups. For example, a standard vector in one of the hyperbolas of two sheets could be suitably chosen as (m, 0, 0, 0). For each m ≠ 0, the vector pierces exactly one sheet. In this case the little group is SO(3), the rotation group, all of whose representations are known. The precise infinite-dimensional unitary representation under which a particle transforms is part of its classification. Not all representations can correspond to physical particles (as far as is known). Standard vectors on the one-sheeted hyperbolas would correspond to tachyons. Particles on the light cone are photons, and more hypothetically, gravitons. The "particle" corresponding to the origin is the vacuum.

Homomorphisms and isomorphisms

See also: Algebra of physical space

Several other groups are either homomorphic or isomorphic to the restricted Lorentz group SO⁺(1, 3). These homomorphisms play a key role in explaining various phenomena in physics.

• The special linear group SL(2, C) is a double covering of the restricted Lorentz group. This relationship is widely used to express the Lorentz invariance of the Dirac equation and the covariance of spinors. In other words, the (restricted) Lorentz group is isomorphic to SL(2, C)/${\displaystyle \mathbb {Z_{2}} }$
• The symplectic group Sp(2, C) is isomorphic to SL(2, C); it is used to construct Weyl spinors, as well as to explain how spinors can have a mass.
• The spin group Spin(1, 3) is isomorphic to SL(2, C); it is used to explain spin and spinors in terms of the Clifford algebra, thus making it clear how to generalize the Lorentz group to general settings in Riemannian geometry, including theories of supergravity and string theory.
• The restricted Lorentz group is isomorphic to the projective special linear group PSL(2, C) which is, in turn, isomorphic to the Möbius group, the symmetry group of conformal geometry on the Riemann sphere. This relationship is central to the classification of the subgroups of the Lorentz group according to an earlier classification scheme developed for the Möbius group.

The Weyl representation

The Weyl representation or spinor map is a pair of surjective homomorphisms from SL(2,C) to SO⁺(1, 3). They form a matched pair under parity transformations, corresponding to left and right chiral spinors.

One may define an action of SL(2,C) on Minkowski spacetime by writing a point of spacetime as a two-by-two Hermitian matrix in the form

${\displaystyle {\overline {X}}={\begin{bmatrix}ct+z&x-iy\\x+iy&ct-z\end{bmatrix}}=ct1\!\!1+x\sigma _{x}+y\sigma _{y}+z\sigma _{z}=ct1\!\!1+{\vec {x}}\cdot {\vec {\sigma }}}$

in terms of Pauli matrices.

This presentation, the Weyl presentation, satisfies

${\displaystyle \det \,{\overline {X}}=(ct)^{2}-x^{2}-y^{2}-z^{2}.}$

Therefore, one has identified the space of Hermitian matrices (which is four-dimensional, as a real vector space) with Minkowski spacetime, in such a way that the determinant of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime. An element ${\displaystyle S\in SL(2,\mathbb {C} )}$ acts on the space of Hermitian matrices via

${\displaystyle {\overline {X}}\mapsto S{\overline {X}}S^{\dagger }~,}$

where ${\displaystyle S^{\dagger }}$ is the Hermitian transpose of ${\displaystyle S}$. This action preserves the determinant and so SL(2,C) acts on Minkowski spacetime by (linear) isometries. The parity-inverted form of the above is

${\displaystyle X=ct1\!\!1-{\vec {x}}\cdot {\vec {\sigma }}}$

which transforms as

${\displaystyle X\mapsto \left(S^{-1}\right)^{\dagger }XS^{-1}}$

That this is the correct transformation follows by noting that

${\displaystyle {\overline {X}}X=\left(c^{2}t^{2}-{\vec {x}}\cdot {\vec {x}}\right)1\!\!1=\left(c^{2}t^{2}-x^{2}-y^{2}-z^{2}\right)1\!\!1}$

remains invariant under the above pair of transformations.

These maps are surjective, and kernel of either map is the two element subgroup ±I. By the first isomorphism theorem, the quotient group PSL(2, C) = SL(2, C) / {±I} is isomorphic to SO⁺(1, 3).

The parity map swaps these two coverings. It corresponds to Hermitian conjugation being an automorphism of ${\displaystyle SL(2,\mathbb {C} ).}$ These two distinct coverings corresponds to the two distinct chiral actions of the Lorentz group on spinors. The non-overlined form corresponds to right-handed spinors transforming as ${\displaystyle \psi _{R}\mapsto S\psi _{R}~,}$ while the overline form corresponds to left-handed spinors transforming as ${\displaystyle \psi _{L}\mapsto \left(S^{\dagger }\right)^{-1}\psi _{L}~.}$

It is important to observe that this pair of coverings does not survive quantization; when quantized, this leads to the peculiar phenomenon of the chiral anomaly. The classical (i.e., non-quantized) symmetries of the Lorentz group are broken by quantization; this is the content of the Atiyah–Singer index theorem.

Notational conventions

In physics, it is conventional to denote a Lorentz transformation ${\displaystyle \Lambda \in SO^{+}(1,3)}$ as ${\displaystyle {\Lambda ^{\mu }}_{\nu }~,}$ thus showing the matrix with spacetime indexes ${\displaystyle \mu ,\nu =0,1,2,3.}$ A four-vector can be created from the Pauli matrices in two different ways: as ${\displaystyle \sigma ^{\mu }=(I,{\vec {\sigma }})}$ and as ${\displaystyle {\overline {\sigma }}^{\mu }=\left(I,-{\vec {\sigma }}\right)~.}$ The two forms are related by a parity transformation. Note that ${\displaystyle {\overline {\sigma }}_{\mu }=\sigma ^{\mu }~.}$

Given a Lorentz transformation ${\displaystyle x^{\mu }\mapsto x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu }~,}$ the double-covering of the orthochronous Lorentz group by ${\displaystyle S\in SL(2,\mathbb {C} )}$ given above can be written as

${\displaystyle x^{\prime \mu }{\overline {\sigma }}_{\mu }={\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }x^{\nu }=Sx^{\nu }{\overline {\sigma }}_{\nu }S^{\dagger }}$

Dropping the ${\displaystyle x^{\mu }}$ this takes the form

${\displaystyle {\overline {\sigma }}_{\mu }{\Lambda ^{\mu }}_{\nu }=S{\overline {\sigma }}_{\nu }S^{\dagger }}$

The parity conjugate form is

${\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}$

Proof

That the above is the correct form for indexed notation is not immediately obvious, partly because, when working in indexed notation, it is quite easy to accidentally confuse a Lorentz transform with its inverse, or its transpose. This confusion arises due to the identity ${\displaystyle \eta \Lambda ^{\textsf {T}}\eta =\Lambda ^{-1}}$ being difficult to recognize when written in indexed form. Lorentz transforms are not tensors under Lorentz transformations! Thus a direct proof of this identity is useful, for establishing its correctness. It can be demonstrated by starting with the identity

${\displaystyle \omega \sigma ^{k}\omega ^{-1}=-\left(\sigma ^{k}\right)^{\textsf {T}}=-\left(\sigma ^{k}\right)^{*}}$

where ${\displaystyle k=1,2,3}$ so that the above are just the usual Pauli matrices, and ${\displaystyle (\cdot )^{\textsf {T}}}$ is the matrix transpose, and ${\displaystyle (\cdot )^{*}}$ is complex conjugation. The matrix ${\displaystyle \omega }$ is

${\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}$

Written as the four-vector, the relationship is

${\displaystyle \sigma _{\mu }^{\textsf {T}}=\sigma _{\mu }^{*}=\omega {\overline {\sigma }}_{\mu }\omega ^{-1}}$

This transforms as

${\displaystyle {\begin{aligned}\sigma _{\mu }^{\textsf {T}}{\Lambda ^{\mu }}_{\nu }&=\omega {\overline {\sigma }}_{\mu }\omega ^{-1}{\Lambda ^{\mu }}_{\nu }\\&=\omega S\;{\overline {\sigma }}_{\nu }\,S^{\dagger }\omega ^{-1}\\&=\left(\omega S\omega ^{-1}\right)\,\left(\omega {\overline {\sigma }}_{\nu }\omega ^{-1}\right)\,\left(\omega S^{\dagger }\omega ^{-1}\right)\\&=\left(S^{-1}\right)^{\textsf {T}}\,\sigma _{\nu }^{\textsf {T}}\,\left(S^{-1}\right)^{*}\end{aligned}}}$

Taking one more transpose, one gets

${\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}$

The symplectic group

The symplectic group Sp(2, C) is isomorphic to SL(2, C). This isomorphism is constructed so as to preserve a symplectic bilinear form on ${\displaystyle \mathbb {C} ^{2},}$ that is, to leave the form invariant under Lorentz transformations. This may be articulated as follows. The symplectic group is defined as

${\displaystyle Sp(2,\mathbb {C} )=\left\{S\in GL(2,\mathbb {C} ):S^{\textsf {T}}\omega S=\omega \right\}}$

where

${\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}$

Other common notations are ${\displaystyle \omega =\epsilon }$ for this element; sometimes ${\displaystyle J}$ is used, but this invites confusion with the idea of almost complex structures, which are not the same, as they transform differently.

Given a pair of Weyl spinors (two-component spinors)

${\displaystyle u={\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}~,\quad v={\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}}$

the invariant bilinear form is conventionally written as

${\displaystyle \langle u,v\rangle =-\langle v,u\rangle =u_{1}v_{2}-u_{2}v_{1}=u^{\textsf {T}}\omega v}$

This form is invariant under the Lorentz group, so that for ${\displaystyle S\in SL(2,\mathbb {C} )}$ one has

${\displaystyle \langle Su,Sv\rangle =\langle u,v\rangle }$

This defines a kind of "scalar product" of spinors, and is commonly used to defined a Lorentz-invariant mass term in Lagrangians. There are several notable properties to be called out that are important to physics. One is that ${\displaystyle \omega ^{2}=-1}$ and so ${\displaystyle \omega ^{-1}=\omega ^{\textsf {T}}=\omega ^{\dagger }=-\omega }$

The defining relation can be written as

${\displaystyle \omega S^{\textsf {T}}\omega ^{-1}=S^{-1}}$

which closely resembles the defining relation for the Lorentz group

${\displaystyle \eta \Lambda ^{\textsf {T}}\eta ^{-1}=\Lambda ^{-1}}$

where ${\displaystyle \eta =\operatorname {diag} (+1,-1,-1,-1)}$ is the metric tensor for Minkowski space and of course, ${\displaystyle \Lambda \in SO(1,3)}$ as before.

Covering groups

Since SL(2, C) is simply connected, it is the universal covering group of the restricted Lorentz group SO⁺(1, 3). By restriction, there is a homomorphism SU(2) → SO(3). Here, the special unitary group SU(2), which is isomorphic to the group of unit norm quaternions, is also simply connected, so it is the covering group of the rotation group SO(3). Each of these covering maps are twofold covers in the sense that precisely two elements of the covering group map to each element of the quotient. One often says that the restricted Lorentz group and the rotation group are doubly connected. This means that the fundamental group of the each group is isomorphic to the two-element cyclic group Z₂.

Twofold coverings are characteristic of spin groups. Indeed, in addition to the double coverings

Spin⁺(1, 3) = SL(2, C) → SO⁺(1, 3)

Spin(3) = SU(2) → SO(3)

we have the double coverings

Pin(1, 3) → O(1, 3)

Spin(1, 3) → SO(1, 3)

Spin⁺(1, 2) = SU(1, 1) → SO(1, 2)

These spinorial double coverings are constructed from Clifford algebras.

Topology

The left and right groups in the double covering

SU(2) → SO(3)

are deformation retracts of the left and right groups, respectively, in the double covering

SL(2, C) → SO⁺(1, 3).

But the homogeneous space SO⁺(1, 3)/SO(3) is homeomorphic to hyperbolic 3-space H³, so we have exhibited the restricted Lorentz group as a principal fiber bundle with fibers SO(3) and base H³. Since the latter is homeomorphic to R³, while SO(3) is homeomorphic to three-dimensional real projective space RP³, we see that the restricted Lorentz group is locally homeomorphic to the product of RP³ with R³. Since the base space is contractible, this can be extended to a global homeomorphism.

Conjugacy classes

Because the restricted Lorentz group SO⁺(1, 3) is isomorphic to the Möbius group PSL(2, C), its conjugacy classes also fall into five classes:

• Elliptic transformations
• Hyperbolic transformations
• Loxodromic transformations
• Parabolic transformations
• The trivial identity transformation

In the article on Möbius transformations, it is explained how this classification arises by considering the fixed points of Möbius transformations in their action on the Riemann sphere, which corresponds here to null eigenspaces of restricted Lorentz transformations in their action on Minkowski spacetime.

An example of each type is given in the subsections below, along with the effect of the one-parameter subgroup it generates (e.g., on the appearance of the night sky).

The Möbius transformations are the conformal transformations of the Riemann sphere (or celestial sphere). Then conjugating with an arbitrary element of SL(2,C) obtains the following examples of arbitrary elliptic, hyperbolic, loxodromic, and parabolic (restricted) Lorentz transformations, respectively. The effect on the flow lines of the corresponding one-parameter subgroups is to transform the pattern seen in the examples by some conformal transformation. For example, an elliptic Lorentz transformation can have any two distinct fixed points on the celestial sphere, but points still flow along circular arcs from one fixed point toward the other. The other cases are similar.

Elliptic

An elliptic element of SL(2, C) is

${\displaystyle P_{1}={\begin{bmatrix}\exp \left({\frac {i}{2}}\theta \right)&0\\0&\exp \left(-{\frac {i}{2}}\theta \right)\end{bmatrix}}}$

and has fixed points ξ = 0, ∞. Writing the action as X ↦ P₁ X P₁^† and collecting terms, the spinor map converts this to the (restricted) Lorentz transformation

${\displaystyle Q_{1}={\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&-\sin(\theta )&0\\0&\sin(\theta )&\cos(\theta )&0\\0&0&0&1\end{bmatrix}}=\exp \left(\theta {\begin{bmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{bmatrix}}\right)~.}$

This transformation then represents a rotation about the z axis, exp(iθJ_z). The one-parameter subgroup it generates is obtained by taking θ to be a real variable, the rotation angle, instead of a constant.

The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same two fixed points, the North and South poles. The transformations move all other points around latitude circles so that this group yields a continuous counter-clockwise rotation about the z axis as θ increases. The angle doubling evident in the spinor map is a characteristic feature of spinorial double coverings.

Hyperbolic

A hyperbolic element of SL(2,C) is

${\displaystyle P_{2}={\begin{bmatrix}\exp \left({\frac {\eta }{2}}\right)&0\\0&\exp \left(-{\frac {\eta }{2}}\right)\end{bmatrix}}}$

and has fixed points ξ = 0, ∞. Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this Möbius transformation is a dilation from the origin.

The spinor map converts this to the Lorentz transformation

${\displaystyle Q_{2}={\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&1&0&0\\0&0&1&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}=\exp \left(\eta {\begin{bmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{bmatrix}}\right)~.}$

This transformation represents a boost along the z axis with rapidity η. The one-parameter subgroup it generates is obtained by taking η to be a real variable, instead of a constant. The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same fixed points (the North and South poles), and they move all other points along longitudes away from the South pole and toward the North pole.

Loxodromic

A loxodromic element of SL(2, C) is

${\displaystyle P_{3}=P_{2}P_{1}=P_{1}P_{2}={\begin{bmatrix}\exp \left({\frac {1}{2}}(\eta +i\theta )\right)&0\\0&\exp \left(-{\frac {1}{2}}(\eta +i\theta )\right)\end{bmatrix}}}$

and has fixed points ξ = 0, ∞. The spinor map converts this to the Lorentz transformation

${\displaystyle Q_{3}=Q_{2}Q_{1}=Q_{1}Q_{2}.}$

The one-parameter subgroup this generates is obtained by replacing η + iθ with any real multiple of this complex constant. (If η, θ vary independently, then a two-dimensional abelian subgroup is obtained, consisting of simultaneous rotations about the z axis and boosts along the z-axis; in contrast, the one-dimensional subgroup discussed here consists of those elements of this two-dimensional subgroup such that the rapidity of the boost and angle of the rotation have a fixed ratio.)

The corresponding continuous transformations of the celestial sphere (excepting the identity) all share the same two fixed points (the North and South poles). They move all other points away from the South pole and toward the North pole (or vice versa), along a family of curves called loxodromes. Each loxodrome spirals infinitely often around each pole.

Parabolic

A parabolic element of SL(2, C) is

${\displaystyle P_{4}={\begin{bmatrix}1&\alpha \\0&1\end{bmatrix}}}$

and has the single fixed point ξ = ∞ on the Riemann sphere. Under stereographic projection, it appears as an ordinary translation along the real axis.

The spinor map converts this to the matrix (representing a Lorentz transformation)

${\displaystyle {\begin{aligned}Q_{4}&={\begin{bmatrix}1+{\frac {1}{2}}\vert \alpha \vert ^{2}&\operatorname {Re} (\alpha )&\operatorname {Im} (\alpha )&-{\frac {1}{2}}\vert \alpha \vert ^{2}\\\operatorname {Re} (\alpha )&1&0&-\operatorname {Re} (\alpha )\\-\operatorname {Im} (\alpha )&0&1&\operatorname {Im} (\alpha )\\{\frac {1}{2}}\vert \alpha \vert ^{2}&\operatorname {Re} (\alpha )&\operatorname {Im} (\alpha )&1-{\frac {1}{2}}\vert \alpha \vert ^{2}\end{bmatrix}}\\[6pt]&=\exp {\begin{bmatrix}0&\operatorname {Re} (\alpha )&\operatorname {Im} (\alpha )&0\\\operatorname {Re} (\alpha )&0&0&-\operatorname {Re} (\alpha )\\-\operatorname {Im} (\alpha )&0&0&\operatorname {Im} (\alpha )\\0&\operatorname {Re} (\alpha )&\operatorname {Im} (\alpha )&0\end{bmatrix}}~.\end{aligned}}}$

This generates a two-parameter abelian subgroup, which is obtained by considering α a complex variable rather than a constant. The corresponding continuous transformations of the celestial sphere (except for the identity transformation) move points along a family of circles that are all tangent at the North pole to a certain great circle. All points other than the North pole itself move along these circles.

Parabolic Lorentz transformations are often called null rotations. Since these are likely to be the least familiar of the four types of nonidentity Lorentz transformations (elliptic, hyperbolic, loxodromic, parabolic), it is illustrated here how to determine the effect of an example of a parabolic Lorentz transformation on Minkowski spacetime.

The matrix given above yields the transformation

${\displaystyle {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}\rightarrow {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}+\operatorname {Re} (\alpha )\;{\begin{bmatrix}x\\t-z\\0\\x\end{bmatrix}}+\operatorname {Im} (\alpha )\;{\begin{bmatrix}y\\0\\z-t\\y\end{bmatrix}}+{\frac {\vert \alpha \vert ^{2}}{2}}\;{\begin{bmatrix}t-z\\0\\0\\t-z\end{bmatrix}}.}$

Now, without loss of generality, pick Im(α) = 0. Differentiating this transformation with respect to the now real group parameter α and evaluating at α = 0 produces the corresponding vector field (first order linear partial differential operator),

${\displaystyle x\,\left(\partial _{t}+\partial _{z}\right)+(t-z)\,\partial _{x}.}$

Apply this to a function f(t, x, y, z), and demand that it stays invariant; i.e., it is annihilated by this transformation. The solution of the resulting first order linear partial differential equation can be expressed in the form

${\displaystyle f(t,x,y,z)=F\left(y,\,t-z,\,t^{2}-x^{2}-z^{2}\right),}$

where F is an arbitrary smooth function. The arguments of F give three rational invariants describing how points (events) move under this parabolic transformation, as they themselves do not move,

${\displaystyle y=c_{1},~~~~t-z=c_{2},~~~~t^{2}-x^{2}-z^{2}=c_{3}.}$

Choosing real values for the constants on the right hand sides yields three conditions, and thus specifies a curve in Minkowski spacetime. This curve is an orbit of the transformation.

The form of the rational invariants shows that these flowlines (orbits) have a simple description: suppressing the inessential coordinate y, each orbit is the intersection of a null plane, t = z + c₂, with a hyperboloid, t² − x² − z² = c₃. The case c₃ = 0 has the hyperboloid degenerate to a light cone with the orbits becoming parabolas lying in corresponding null planes.

A particular null line lying on the light cone is left invariant; this corresponds to the unique (double) fixed point on the Riemann sphere mentioned above. The other null lines through the origin are "swung around the cone" by the transformation. Following the motion of one such null line as α increases corresponds to following the motion of a point along one of the circular flow lines on the celestial sphere, as described above.

A choice Re(α) = 0 instead, produces similar orbits, now with the roles of x and y interchanged.

Parabolic transformations lead to the gauge symmetry of massless particles (like photons) with helicity |h| ≥ 1. In the above explicit example, a massless particle moving in the z direction, so with 4-momentum P = (p, 0, 0, p), is not affected at all by the x-boost and y-rotation combination K_x − J_y defined below, in the "little group" of its motion. This is evident from the explicit transformation law discussed: like any light-like vector, P itself is now invariant; i.e., all traces or effects of α have disappeared. c₁ = c₂ = c₃ = 0, in the special case discussed. (The other similar generator, K_y+J_x as well as it and J_z comprise altogether the little group of the light-like vector, isomorphic to E(2).)

Appearance of the night sky

This isomorphism has the consequence that Möbius transformations of the Riemann sphere represent the way that Lorentz transformations change the appearance of the night sky, as seen by an observer who is maneuvering at relativistic velocities relative to the "fixed stars".

Suppose the "fixed stars" live in Minkowski spacetime and are modeled by points on the celestial sphere. Then a given point on the celestial sphere can be associated with ξ = u + iv, a complex number that corresponds to the point on the Riemann sphere, and can be identified with a null vector (a light-like vector) in Minkowski space

${\displaystyle {\begin{bmatrix}u^{2}+v^{2}+1\\2u\\-2v\\u^{2}+v^{2}-1\end{bmatrix}}}$

or, in the Weyl representation (the spinor map), the Hermitian matrix

${\displaystyle N=2{\begin{bmatrix}u^{2}+v^{2}&u+iv\\u-iv&1\end{bmatrix}}.}$

The set of real scalar multiples of this null vector, called a null line through the origin, represents a line of sight from an observer at a particular place and time (an arbitrary event we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars. Then the points of the celestial sphere (equivalently, lines of sight) are identified with certain Hermitian matrices.

Projective geometry and different views of the 2-sphere

This picture emerges cleanly in the language of projective geometry. The (restricted) Lorentz group acts on the projective celestial sphere. This is the space of non-zero null vectors with ${\displaystyle t>0}$ under the given quotient for projective spaces: ${\displaystyle (t,x,y,z)\sim (t',x',y',z')}$ if ${\displaystyle (t',x',y',z')=(\lambda t,\lambda x,\lambda y,\lambda z)}$ for ${\displaystyle \lambda >0}$. This is referred to as the celestial sphere as this allows us to rescale the time coordinate ${\displaystyle t}$ to 1 after acting using a Lorentz transformation, ensuring the space-like part sits on the unit sphere.

From the Möbius side, ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ acts on complex projective space ${\displaystyle \mathbb {CP} ^{2}}$, which can be shown to be diffeomorphic to the 2-sphere - this is sometimes referred to as the Riemann sphere. The quotient on projective space leads to a quotient on the group ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$.

Finally, these two can be linked together by using the complex projective vector to construct a null-vector. If ${\displaystyle \xi }$ is a ${\displaystyle \mathbb {CP} ^{2}}$ projective vector, it can be tensored with its Hermitian conjugate to produce a ${\displaystyle 2\times 2}$ Hermitian matrix. From elsewhere in this article we know this space of matrices can be viewed as 4-vectors. The space of matrices coming from turning each projective vector in the Riemann sphere into a matrix is known as the Bloch sphere.

Lie algebra

As with any Lie group, a useful way to study many aspects of the Lorentz group is via its Lie algebra. Since the Lorentz group ${\displaystyle {\text{SO}}(1,3)}$ is a matrix Lie group, its corresponding Lie algebra ${\displaystyle {\mathfrak {so}}(1,3)}$ is a matrix Lie algebra, which may be computed as

${\displaystyle {\mathfrak {so}}(1,3)=\left\{4\times 4\,\,\,\mathbb {R} {\text{-valued matrices}}\,X\mid e^{tX}\in \mathrm {SO} (1,3)\,\mathrm {for} \,\mathrm {all} \,t\right\}}$.

If ${\displaystyle \eta }$ is the diagonal matrix with diagonal entries ${\displaystyle (1,-1,-1,-1)}$, then the Lie algebra ${\displaystyle {\mathfrak {o}}(1,3)}$ consists of ${\displaystyle 4\times 4}$ matrices ${\displaystyle X}$ such that

${\displaystyle \eta X\eta =-X^{\textsf {T}}}$.

Explicitly, ${\displaystyle {\mathfrak {so}}(1,3)}$ consists of ${\displaystyle 4\times 4}$ matrices of the form

${\displaystyle {\begin{pmatrix}0&a&b&c\\a&0&d&e\\b&-d&0&f\\c&-e&-f&0\end{pmatrix}}}$,

where ${\displaystyle a,b,c,d,e,f}$ are arbitrary real numbers. This Lie algebra is six dimensional. The subalgebra of ${\displaystyle {\mathfrak {so}}(1,3)}$ consisting of elements in which ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ equal zero is isomorphic to ${\displaystyle {\mathfrak {so}}(3)}$.

The full Lorentz group ${\displaystyle {\text{O}}(1,3)}$, the proper Lorentz group ${\displaystyle {\text{SO}}(1,3)}$ and the proper orthochronous Lorentz group ${\displaystyle \mathrm {SO} ^{+}(1,3)}$ (the component connected to the identity) all have the same Lie algebra, which is typically denoted ${\displaystyle {\mathfrak {so}}(1,3)}$.

Since the identity component of the Lorentz group is isomorphic to a finite quotient of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ (see the section above on the connection of the Lorentz group to the Möbius group), the Lie algebra of the Lorentz group is isomorphic to the Lie algebra ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$. As a complex Lie algebra ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ is three dimensional, but is six dimensional when viewed as a real Lie algebra.

Commutation relations of the Lorentz algebra

The standard basis matrices can be indexed as ${\displaystyle M^{\mu \nu }}$ where ${\displaystyle \mu ,\nu }$ take values in ${\displaystyle \{0,1,2,3\}}$. These arise from taking only one of ${\displaystyle a,b,\cdots ,f}$ to be one, and others zero, in turn. The components can be written as

${\displaystyle (M^{\mu \nu })_{\rho \sigma }=\delta ^{\mu }{}_{\rho }\delta ^{\nu }{}_{\sigma }-\delta ^{\nu }{}_{\rho }\delta ^{\mu }{}_{\sigma }}$.

The commutation relations are

${\displaystyle [M^{\mu \nu },M^{\rho \sigma }]=M^{\mu \sigma }\eta ^{\nu \rho }-M^{\nu \sigma }\eta ^{\mu \rho }+M^{\nu \rho }\eta ^{\mu \sigma }-M^{\mu \rho }\eta ^{\nu \sigma }.}$

There are different possible choices of convention in use. In physics, it is common to include a factor of ${\displaystyle i}$ with the basis elements, which gives a factor of ${\displaystyle i}$ in the commutation relations.

Then ${\displaystyle M^{0i}}$ generate boosts and ${\displaystyle M^{ij}}$ generate rotations.

The structure constants for the Lorentz algebra can be read off from the commutation relations. Any set of basis elements which satisfy these relations form a representation of the Lorentz algebra.

Generators of boosts and rotations

The Lorentz group can be thought of as a subgroup of the diffeomorphism group of R⁴ and therefore its Lie algebra can be identified with vector fields on R⁴. In particular, the vectors that generate isometries on a space are its Killing vectors, which provides a convenient alternative to the left-invariant vector field for calculating the Lie algebra. We can write down a set of six generators:

• Vector fields on R⁴ generating three rotations i J,

  ${\displaystyle -y\partial _{x}+x\partial _{y}\equiv iJ_{z}~,\qquad -z\partial _{y}+y\partial _{z}\equiv iJ_{x}~,\qquad -x\partial _{z}+z\partial _{x}\equiv iJ_{y}~;}$

• Vector fields on R⁴ generating three boosts i K,

  ${\displaystyle x\partial _{t}+t\partial _{x}\equiv iK_{x}~,\qquad y\partial _{t}+t\partial _{y}\equiv iK_{y}~,\qquad z\partial _{t}+t\partial _{z}\equiv iK_{z}.}$

The factor of ${\displaystyle i}$ appears to ensure that the generators of rotations are Hermitian.

It may be helpful to briefly recall here how to obtain a one-parameter group from a vector field, written in the form of a first order linear partial differential operator such as

${\displaystyle {\mathcal {L}}=-y\partial _{x}+x\partial _{y}.}$

The corresponding initial value problem (consider ${\displaystyle r=(x,y)}$ a function of a scalar ${\displaystyle \lambda }$ and solve ${\displaystyle \partial _{\lambda }r={\mathcal {L}}r}$ with some initial conditions) is

${\displaystyle {\frac {\partial x}{\partial \lambda }}=-y,\;{\frac {\partial y}{\partial \lambda }}=x,\;x(0)=x_{0},\;y(0)=y_{0}.}$

The solution can be written

${\displaystyle x(\lambda )=x_{0}\cos(\lambda )-y_{0}\sin(\lambda ),\;y(\lambda )=x_{0}\sin(\lambda )+y_{0}\cos(\lambda )}$

or

${\displaystyle {\begin{bmatrix}t\\x\\y\\z\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&\cos(\lambda )&-\sin(\lambda )&0\\0&\sin(\lambda )&\cos(\lambda )&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}t_{0}\\x_{0}\\y_{0}\\z_{0}\end{bmatrix}}}$

where we easily recognize the one-parameter matrix group of rotations exp(i λ J_z) about the z axis.

Differentiating with respect to the group parameter λ and setting it λ=0 in that result, we recover the standard matrix,

${\displaystyle iJ_{z}={\begin{bmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{bmatrix}}~,}$

which corresponds to the vector field we started with. This illustrates how to pass between matrix and vector field representations of elements of the Lie algebra. The exponential map plays this special role not only for the Lorentz group but for Lie groups in general.

Reversing the procedure in the previous section, we see that the Möbius transformations that correspond to our six generators arise from exponentiating respectively η/2 (for the three boosts) or iθ/2 (for the three rotations) times the three Pauli matrices

${\displaystyle \sigma _{1}={\begin{bmatrix}0&1\\1&0\end{bmatrix}},\;\;\sigma _{2}={\begin{bmatrix}0&-i\\i&0\end{bmatrix}},\;\;\sigma _{3}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}.}$

Generators of the Möbius group

Another generating set arises via the isomorphism to the Möbius group. The following table lists the six generators, in which

• The first column gives a generator of the flow under the Möbius action (after stereographic projection from the Riemann sphere) as a real vector field on the Euclidean plane.
• The second column gives the corresponding one-parameter subgroup of Möbius transformations.
• The third column gives the corresponding one-parameter subgroup of Lorentz transformations (the image under our homomorphism of preceding one-parameter subgroup).
• The fourth column gives the corresponding generator of the flow under the Lorentz action as a real vector field on Minkowski spacetime.

Notice that the generators consist of

• Two parabolics (null rotations)
• One hyperbolic (boost in the ${\displaystyle \partial _{z}}$ direction)
• Three elliptics (rotations about the x, y, z axes, respectively)

Vector field on ${\displaystyle \mathbb {R} ^{2}}$	One-parameter subgroup of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$, representing Möbius transformations	One-parameter subgroup of ${\displaystyle {\text{SO}}^{+}(1,3)}$, representing Lorentz transformations	Vector field on ${\displaystyle \mathbb {R} ^{1,3}}$
Parabolic
${\displaystyle \partial _{u}\,\!}$	${\displaystyle {\begin{bmatrix}1&\alpha \\0&1\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}1+{\frac {1}{2}}\alpha ^{2}&\alpha &0&-{\frac {1}{2}}\alpha ^{2}\\\alpha &1&0&-\alpha \\0&0&1&0\\{\frac {1}{2}}\alpha ^{2}&\alpha &0&1-{\frac {1}{2}}\alpha ^{2}\end{bmatrix}}}$	${\displaystyle {\begin{aligned}X_{1}=x&(\partial _{t}+\partial _{z})+{}\\&(t-z)\partial _{x}\end{aligned}}}$
${\displaystyle \partial _{v}\,\!}$	${\displaystyle {\begin{bmatrix}1&i\alpha \\0&1\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}1+{\frac {1}{2}}\alpha ^{2}&0&\alpha &-{\frac {1}{2}}\alpha ^{2}\\0&1&0&0\\\alpha &0&1&-\alpha \\{\frac {1}{2}}\alpha ^{2}&0&\alpha &1-{\frac {1}{2}}\alpha ^{2}\end{bmatrix}}}$	${\displaystyle {\begin{aligned}X_{2}=y&(\partial _{t}+\partial _{z})+{}\\&(t-z)\partial _{y}\end{aligned}}}$
Hyperbolic
${\displaystyle {\frac {1}{2}}\left(u\partial _{u}+v\partial _{v}\right)}$	${\displaystyle {\begin{bmatrix}\exp \left({\frac {\eta }{2}}\right)&0\\0&\exp \left(-{\frac {\eta }{2}}\right)\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}\cosh(\eta )&0&0&\sinh(\eta )\\0&1&0&0\\0&0&1&0\\\sinh(\eta )&0&0&\cosh(\eta )\end{bmatrix}}}$	${\displaystyle X_{3}=z\partial _{t}+t\partial _{z}\,\!}$
Elliptic
${\displaystyle {\frac {1}{2}}\left(-v\partial _{u}+u\partial _{v}\right)}$	${\displaystyle {\begin{bmatrix}\exp \left({\frac {i\theta }{2}}\right)&0\\0&\exp \left({\frac {-i\theta }{2}}\right)\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&-\sin(\theta )&0\\0&\sin(\theta )&\cos(\theta )&0\\0&0&0&1\end{bmatrix}}}$	${\displaystyle X_{4}=-y\partial _{x}+x\partial _{y}}$
${\displaystyle {\frac {v^{2}-u^{2}-1}{2}}\partial _{u}-uv\,\partial _{v}}$	${\displaystyle {\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&-\sin \left({\frac {\theta }{2}}\right)\\\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&0&\sin(\theta )\\0&0&1&0\\0&-\sin(\theta )&0&\cos(\theta )\end{bmatrix}}}$	${\displaystyle X_{5}=-x\partial _{z}+z\partial _{x}}$
${\displaystyle uv\,\partial _{u}+{\frac {1-u^{2}+v^{2}}{2}}\partial _{v}}$	${\displaystyle {\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&i\sin \left({\frac {\theta }{2}}\right)\\i\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}}$	${\displaystyle {\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos(\theta )&-\sin(\theta )\\0&0&\sin(\theta )&\cos(\theta )\end{bmatrix}}}$	${\displaystyle X_{6}=-z\partial _{y}+y\partial _{z}}$

Worked example: rotation about the y-axis

Start with

${\displaystyle \sigma _{2}={\begin{bmatrix}0&i\\-i&0\end{bmatrix}}.}$

Exponentiate:

${\displaystyle \exp \left({\frac {i\theta }{2}}\,\sigma _{2}\right)={\begin{bmatrix}\cos \left({\frac {\theta }{2}}\right)&-\sin \left({\frac {\theta }{2}}\right)\\\sin \left({\frac {\theta }{2}}\right)&\cos \left({\frac {\theta }{2}}\right)\end{bmatrix}}.}$

This element of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ represents the one-parameter subgroup of (elliptic) Möbius transformations:

${\displaystyle \xi \mapsto \xi '={\frac {\cos \left({\frac {\theta }{2}}\right)\,\xi -\sin \left({\frac {\theta }{2}}\right)}{\sin \left({\frac {\theta }{2}}\right)\,\xi +\cos \left({\frac {\theta }{2}}\right)}}.}$

Next,

${\displaystyle \left.{\frac {d\xi '}{d\theta }}\right|_{\theta =0}=-{\frac {1+\xi ^{2}}{2}}.}$

The corresponding vector field on ${\displaystyle \mathbb {C} }$ (thought of as the image of ${\displaystyle S^{2}}$ under stereographic projection) is

${\displaystyle -{\frac {1+\xi ^{2}}{2}}\,\partial _{\xi }.}$

Writing ${\displaystyle \xi =u+iv}$, this becomes the vector field on ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle -{\frac {1+u^{2}-v^{2}}{2}}\,\partial _{u}-uv\,\partial _{v}.}$

Returning to our element of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$, writing out the action ${\displaystyle X\mapsto PXP^{\dagger }}$ and collecting terms, we find that the image under the spinor map is the element of ${\displaystyle {\text{SO}}^{+}(1,3)}$

${\displaystyle {\begin{bmatrix}1&0&0&0\\0&\cos(\theta )&0&\sin(\theta )\\0&0&1&0\\0&-\sin(\theta )&0&\cos(\theta )\end{bmatrix}}.}$

Differentiating with respect to ${\displaystyle \theta }$ at ${\displaystyle \theta =0}$, yields the corresponding vector field on ${\displaystyle \mathbb {R} ^{1,3}}$,

${\displaystyle z\partial _{x}-x\partial _{z}.\,\!}$

This is evidently the generator of counterclockwise rotation about the ${\displaystyle y}$-axis.

Subgroups of the Lorentz group

The subalgebras of the Lie algebra of the Lorentz group can be enumerated, up to conjugacy, from which the closed subgroups of the restricted Lorentz group can be listed, up to conjugacy. (See the book by Hall cited below for the details.) These can be readily expressed in terms of the generators ${\displaystyle X_{n}}$ given in the table above.

The one-dimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group:

• ${\displaystyle X_{1}}$ generates a one-parameter subalgebra of parabolics SO(0, 1),
• ${\displaystyle X_{3}}$ generates a one-parameter subalgebra of boosts SO(1, 1),
• ${\displaystyle X_{4}}$ generates a one-parameter of rotations SO(2),
• ${\displaystyle X_{3}+aX_{4}}$ (for any ${\displaystyle a\neq 0}$) generates a one-parameter subalgebra of loxodromic transformations.

(Strictly speaking the last corresponds to infinitely many classes, since distinct ${\displaystyle a}$ give different classes.) The two-dimensional subalgebras are:

• ${\displaystyle X_{1},X_{2}}$ generate an abelian subalgebra consisting entirely of parabolics,
• ${\displaystyle X_{1},X_{3}}$ generate a nonabelian subalgebra isomorphic to the Lie algebra of the affine group Aff(1),
• ${\displaystyle X_{3},X_{4}}$ generate an abelian subalgebra consisting of boosts, rotations, and loxodromics all sharing the same pair of fixed points.

The three-dimensional subalgebras use the Bianchi classification scheme:

• ${\displaystyle X_{1},X_{2},X_{3}}$ generate a Bianchi V subalgebra, isomorphic to the Lie algebra of Hom(2), the group of euclidean homotheties,
• ${\displaystyle X_{1},X_{2},X_{4}}$ generate a Bianchi VII₀ subalgebra, isomorphic to the Lie algebra of E(2), the euclidean group,
• ${\displaystyle X_{1},X_{2},X_{3}+aX_{4}}$, where ${\displaystyle a\neq 0}$, generate a Bianchi VII_a subalgebra,
• ${\displaystyle X_{1},X_{3},X_{5}}$ generate a Bianchi VIII subalgebra, isomorphic to the Lie algebra of SL(2, R), the group of isometries of the hyperbolic plane,
• ${\displaystyle X_{4},X_{5},X_{6}}$ generate a Bianchi IX subalgebra, isomorphic to the Lie algebra of SO(3), the rotation group.

The Bianchi types refer to the classification of three-dimensional Lie algebras by the Italian mathematician Luigi Bianchi.

The four-dimensional subalgebras are all conjugate to

• ${\displaystyle X_{1},X_{2},X_{3},X_{4}}$ generate a subalgebra isomorphic to the Lie algebra of Sim(2), the group of Euclidean similitudes.

The subalgebras form a lattice (see the figure), and each subalgebra generates by exponentiation a closed subgroup of the restricted Lie group. From these, all subgroups of the Lorentz group can be constructed, up to conjugation, by multiplying by one of the elements of the Klein four-group.

The lattice of subalgebras of the Lie algebra SO(1, 3), up to conjugacy.

As with any connected Lie group, the coset spaces of the closed subgroups of the restricted Lorentz group, or homogeneous spaces, have considerable mathematical interest. A few, brief descriptions:

• The group Sim(2) is the stabilizer of a null line; i.e., of a point on the Riemann sphere—so the homogeneous space SO⁺(1, 3)/Sim(2) is the Kleinian geometry that represents conformal geometry on the sphere S².
• The (identity component of the) Euclidean group SE(2) is the stabilizer of a null vector, so the homogeneous space SO⁺(1, 3)/SE(2) is the momentum space of a massless particle; geometrically, this Kleinian geometry represents the degenerate geometry of the light cone in Minkowski spacetime.
• The rotation group SO(3) is the stabilizer of a timelike vector, so the homogeneous space SO⁺(1, 3)/SO(3) is the momentum space of a massive particle; geometrically, this space is none other than three-dimensional hyperbolic space H³.

Generalization to higher dimensions

Main article: Indefinite orthogonal group

The concept of the Lorentz group has a natural generalization to spacetime of any number of dimensions. Mathematically, the Lorentz group of (n + 1)-dimensional Minkowski space is the indefinite orthogonal group O(n, 1) of linear transformations of Rⁿ⁺¹ that preserves the quadratic form

${\displaystyle (x_{1},x_{2},\ldots ,x_{n},x_{n+1})\mapsto x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}-x_{n+1}^{2}.}$

The group O(1, n) preserves the quadratic form

${\displaystyle (x_{1},x_{2},\ldots ,x_{n},x_{n+1})\mapsto x_{1}^{2}-x_{2}^{2}-\cdots -x_{n+1}^{2}}$

It is isomorphic to O(n, 1) but enjoys greater popularity in mathematical physics, primarily because the algebra of the Dirac equation and, more generally, spinor and Clifford algebras, are "more natural" with this signature.

A common notation for the vector space ${\displaystyle \mathbb {R} ^{n+1}}$, equipped with this choice of quadratic form, is ${\displaystyle \mathbb {R} ^{1,n}}$.

Many of the properties of the Lorentz group in four dimensions (where n = 3) generalize straightforwardly to arbitrary n. For instance, the Lorentz group O(n, 1) has four connected components, and it acts by conformal transformations on the celestial (n−1)-sphere in (n+1)-dimensional Minkowski space. The identity component SO⁺(n, 1) is an SO(n)-bundle over hyperbolic n-space Hⁿ.

The low-dimensional cases n = 1 and n = 2 are often useful as "toy models" for the physical case n = 3, while higher-dimensional Lorentz groups are used in physical theories such as string theory that posit the existence of hidden dimensions. The Lorentz group O(n, 1) is also the isometry group of n-dimensional de Sitter space dS_n, which may be realized as the homogeneous space O(n, 1)/O(n − 1, 1). In particular O(4, 1) is the isometry group of the de Sitter universe dS₄, a cosmological model.

Drug

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Drug 

Uncoated aspirin tablets, consisting of about 90% acetylsalicylic acid, along with a minor amount of inert fillers and binders. Aspirin is a pharmaceutical drug often used to treat pain, fever, and inflammation.

A drug is any chemical substance that causes a change in an organism's physiology or psychology when consumed. Drugs are typically distinguished from food and substances that provide nutritional support. Consumption of drugs can be via inhalation, injection, smoking, ingestion, absorption via a patch on the skin, suppository, or dissolution under the tongue.

In pharmacology, a drug is a chemical substance, typically of known structure, which, when administered to a living organism, produces a biological effect. A pharmaceutical drug, also called a medication or medicine, is a chemical substance used to treat, cure, prevent, or diagnose a disease or to promote well-being. Traditionally drugs were obtained through extraction from medicinal plants, but more recently also by organic synthesis. Pharmaceutical drugs may be used for a limited duration, or on a regular basis for chronic disorders.

Pharmaceutical drugs are often classified into drug classes—groups of related drugs that have similar chemical structures, the same mechanism of action (binding to the same biological target), a related mode of action, and that are used to treat the same disease. The Anatomical Therapeutic Chemical Classification System (ATC), the most widely used drug classification system, assigns drugs a unique ATC code, which is an alphanumeric code that assigns it to specific drug classes within the ATC system. Another major classification system is the Biopharmaceutics Classification System. This classifies drugs according to their solubility and permeability or absorption properties.

Psychoactive drugs are chemical substances that affect the function of the central nervous system, altering perception, mood or consciousness. These drugs are divided into different groups like: stimulants, depressants, antidepressants, anxiolytics, antipsychotics, and hallucinogens. These psychoactive drugs have been proven useful in treating wide range of medical conditions including mental disorders around the world. The most widely used drugs in the world include caffeine, nicotine and alcohol, which are also considered recreational drugs, since they are used for pleasure rather than medicinal purposes. All drugs can have potential side effects. Abuse of several psychoactive drugs can cause addiction and/or physical dependence. Excessive use of stimulants can promote stimulant psychosis. Many recreational drugs are illicit and international treaties such as the Single Convention on Narcotic Drugs exist for the purpose of their prohibition.

Etymology

In English, the noun "drug" is thought to originate from Old French "drogue", possibly deriving from "droge (vate)" from Middle Dutch meaning "dry (barrels)", referring to medicinal plants preserved as dry matter in barrels.

Medication

Nexium (Esomeprazole) is a proton-pump inhibitor. It is used to reduce the production of stomach acid.

 

Main articles: Pharmaceutical drug, Drug class, and Medication

A medication or medicine is a drug taken to cure or ameliorate any symptoms of an illness or medical condition. The use may also be as preventive medicine that has future benefits but does not treat any existing or pre-existing diseases or symptoms. Dispensing of medication is often regulated by governments into three categories—over-the-counter medications, which are available in pharmacies and supermarkets without special restrictions; behind-the-counter medicines, which are dispensed by a pharmacist without needing a doctor's prescription, and prescription only medicines, which must be prescribed by a licensed medical professional, usually a physician.

In the United Kingdom, behind-the-counter medicines are called pharmacy medicines which can only be sold in registered pharmacies, by or under the supervision of a pharmacist. These medications are designated by the letter P on the label. The range of medicines available without a prescription varies from country to country. Medications are typically produced by pharmaceutical companies and are often patented to give the developer exclusive rights to produce them. Those that are not patented (or with expired patents) are called generic drugs since they can be produced by other companies without restrictions or licenses from the patent holder.

Pharmaceutical drugs are usually categorised into drug classes. A group of drugs will share a similar chemical structure, or have the same mechanism of action, the same related mode of action or target the same illness or related illnesses. The Anatomical Therapeutic Chemical Classification System (ATC), the most widely used drug classification system, assigns drugs a unique ATC code, which is an alphanumeric code that assigns it to specific drug classes within the ATC system. Another major classification system is the Biopharmaceutics Classification System. This groups drugs according to their solubility and permeability or absorption properties.

Spiritual and religious use

Main articles: Entheogen and Psychonaut

 

An Amazonian shaman

 

San Pedro, a psychoactive cactus.

Some religions, particularly ethnic religions, are based completely on the use of certain drugs, known as entheogens, which are mostly hallucinogens,—psychedelics, dissociatives, or deliriants. Some drugs used as entheogens include kava which can act as a stimulant, a sedative, a euphoriant and an anesthetic. The roots of the kava plant are used to produce a drink which is consumed throughout the cultures of the Pacific Ocean.

Some shamans from different cultures use entheogens, defined as "generating the divine within" to achieve religious ecstasy. Amazonian shamans use ayahuasca (yagé) a hallucinogenic brew for this purpose. Mazatec shamans have a long and continuous tradition of religious use of Salvia divinorum a psychoactive plant. Its use is to facilitate visionary states of consciousness during spiritual healing sessions.

Silene undulata is regarded by the Xhosa people as a sacred plant and used as an entheogen. Its roots are traditionally used to induce vivid (and according to the Xhosa, prophetic) lucid dreams during the initiation process of shamans, classifying it a naturally occurring oneirogen similar to the more well-known dream herb Calea ternifolia.

Peyote, a small spineless cactus, has been a major source of psychedelic mescaline and has probably been used by Native Americans for at least five thousand years. Most mescaline is now obtained from a few species of columnar cacti in particular from San Pedro and not from the vulnerable peyote.

The entheogenic use of cannabis has also been widely practised for centuries. Rastafari use marijuana (ganja) as a sacrament in their religious ceremonies.

Psychedelic mushrooms (psilocybin mushrooms), commonly called magic mushrooms or shrooms have also long been used as entheogens.

Smart drugs and designer drugs

Main articles: Nootropic, Designer drug, and Psychoactive drug

Nootropics, also commonly referred to as "smart drugs", are drugs that are claimed to improve human cognitive abilities. Nootropics are used to improve memory, concentration, thought, mood, and learning. An increasingly used nootropic among students, also known as a study drug, is methylphenidate branded commonly as Ritalin and used for the treatment of attention deficit hyperactivity disorder (ADHD) and narcolepsy. At high doses methylphenidate can become highly addictive. Serious addiction can lead to psychosis, anxiety and heart problems, and the use of this drug is related to a rise in suicides, and overdoses. Evidence for use outside of student settings is limited but suggests that it is commonplace. Intravenous use of methylphenidate can lead to emphysematous damage to the lungs, known as Ritalin lung.

Other drugs known as designer drugs are produced. An early example of what today would be labelled a 'designer drug' was LSD, which was synthesised from ergot. Other examples include analogs of performance-enhancing drugs such as designer steroids taken to improve physical capabilities and these are sometimes used (legally or not) for this purpose, often by professional athletes. Other designer drugs mimic the effects of psychoactive drugs. Since the late 1990s there has been the identification of many of these synthesised drugs. In Japan and the United Kingdom this has spurred the addition of many designer drugs into a newer class of controlled substances known as a temporary class drug.

Synthetic cannabinoids have been produced for a longer period of time and are used in the designer drug synthetic cannabis.

Recreational drug use

Main article: Recreational drug use

Further information: Prohibition of drugs

 

Cannabis is a commonly used recreational drug.

Recreational drug use is the use of a drug (legal, controlled, or illegal) with the primary intention of altering the state of consciousness through alteration of the central nervous system in order to create positive emotions and feelings. The hallucinogen LSD is a psychoactive drug commonly used as a recreational drug.

Ketamine is a drug used for anesthesia, and is also used as a recreational drug, both in powder and liquid form, for its hallucinogenic and dissociative effects.

Some national laws prohibit the use of different recreational drugs; and medicinal drugs that have the potential for recreational use are often heavily regulated. However, there are many recreational drugs that are legal in many jurisdictions and widely culturally accepted. Cannabis is the most commonly consumed controlled recreational drug in the world (as of 2012). Its use in many countries is illegal but is legally used in several countries usually with the proviso that it can only be used for personal use. It can be used in the leaf form of marijuana (grass), or in the resin form of hashish. Marijuana is a more mild form of cannabis than hashish.

There may be an age restriction on the consumption and purchase of legal recreational drugs. Some recreational drugs that are legal and accepted in many places include alcohol, tobacco, betel nut, and caffeine products, and in some areas of the world the legal use of drugs such as khat is common.

There are a number of legal intoxicants commonly called legal highs that are used recreationally. The most widely used of these is alcohol.

Administration of drugs

Main article: Route of administration

All drugs, can be administered via a number of routes, and many can be administered by more than one.

• Bolus is the administration of a medication, drug or other compound that is given to raise its concentration in blood to an effective level. The administration can be given intravenously, by parenteral,by indovenous, by intramuscular, intrathecal or subcutaneous injection.
• Inhaled, (breathed into the lungs), as an aerosol, inhaler, vape or dry powder (this includes smoking or vaping a substance).
• Injection as a solution, suspension or emulsion either: intramuscular, intravenous, intraperitoneal, intraosseous.
• Insufflation, as a nasal spray or snorting into the nose.
• Orally, as a liquid or solid, that is absorbed through the intestines.
• Rectally as a suppository, that is absorbed by the rectum or colon.
• Sublingually, diffusing into the blood through tissues under the tongue.
• Topically, usually as a cream or ointment. A drug administered in this manner may be given to act locally or systemically.
• Vaginally as a pessary, primarily to treat vaginal infections.

Control of drugs

Main articles: Prohibition of drugs and Drug policy

There are numerous governmental offices in many countries that deal with the control and oversee of drug manufacture and use, and the implementation of various drug laws. The Single Convention on Narcotic Drugs is an international treaty brought about in 1961 to prohibit the use of narcotics save for those used in medical research and treatment. In 1971, a second treaty the Convention on Psychotropic Substances had to be introduced to deal with newer recreational psychoactive and psychedelic drugs.

The legal status of Salvia divinorum varies in many countries and even in states within the United States. Where it is legislated against the degree of prohibition also varies.

The Food and Drug Administration (FDA) in the United States is a federal agency responsible for protecting and promoting public health through the regulation and supervision of food safety, tobacco products, dietary supplements, prescription and over-the-counter medications, vaccines, biopharmaceuticals, blood transfusions, medical devices, electromagnetic radiation emitting devices, cosmetics, animal foods and veterinary drugs.

In India, the Narcotics Control Bureau (abbr. NCB), an Indian federal law enforcement and intelligence agency under the Ministry of Home Affairs, Government of India is tasked with combating drug trafficking and assisting international use of illegal substances under the provisions of Narcotic Drugs and Psychotropic Substances Act.

Identity (social science)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Identity_(social_science) 

Narcissus painting by Caravaggio, depicting Narcissus gazing upon the water after falling in love with his own reflection.

Identity is the qualities, beliefs, personality traits, appearance, and/or expressions that characterize a person or group.

In sociology, emphasis is placed on collective identity, in which an individual's identity is strongly associated with role-behavior or the collection of group memberships that define them. According to Peter Burke, "Identities tell us who we are and they announce to others who we are." Identities subsequently guide behavior, leading "fathers" to behave like "fathers" and "nurses" to act like "nurses."

In psychology, the term "identity" is most commonly used to describe personal identity, or the distinctive qualities or traits that make an individual unique. Identities are strongly associated with self-concept, self-image (one's mental model of oneself), self-esteem, and individuality. Individuals' identities are situated, but also contextual, situationally adaptive and changing. Despite their fluid character, identities often feel as if they are stable ubiquitous categories defining an individual, because of their grounding in the sense of personal identity (the sense of being a continuous and persistent self).

In psychology

Erik Erikson (1902–1994) became one of the earliest psychologists to take an explicit interest in identity. An essential feature of Erikson's theory of psychosocial development was the idea of the ego identity, (often referred to as “the self,”) which is described as an individual's personal sense of continuity. He suggested that people can attain this feeling throughout their lives as they develop and is meant to be an ongoing process. The ego-identity consists of two main features: one's personal characteristics and development, and the culmination of social and cultural factors and roles that impact one's identity. In Erikson's theory, he describes eight distinct stages across the lifespan that are each characterized by a conflict between the inner, personal world and the outer, social world of an individual. Erikson identified the conflict of identity as occurring primarily during adolescence and described potential outcomes that depend on how one deals with this conflict. Those who do not manage a resynthesis of childhood identifications are seen as being in a state of 'identity diffusion' whereas those who retain their given identities unquestioned have 'foreclosed' identities. On some readings of Erikson, the development of a strong ego identity, along with the proper integration into a stable society and culture, lead to a stronger sense of identity in general. Accordingly, a deficiency in either of these factors may increase the chance of an identity crisis or confusion.

The "Neo-Eriksonian" identity status paradigm emerged in 1966, driven largely by the work of James Marcia. This model focuses on the concepts of exploration and commitment. The central idea is that an individual's sense of identity is determined in large part by the degrees to which a person has made certain explorations and the extent to which they have commitments to those explorations or a particular identity. A person may display either relative weakness or strength in terms of both exploration and commitments. When assigned categories, there were four possible results: identity diffusion, identity foreclosure, identity moratorium, and identity achievement. Diffusion is when a person avoids or refuses both exploration and making a commitment. Foreclosure occurs when a person does make a commitment to a particular identity but neglected to explore other options. Identity moratorium is when a person avoids or postpones making a commitment but is still actively exploring their options and different identities. Lastly, identity achievement is when a person has both explored many possibilities and has committed to their identity.

Although the self is distinct from identity, the literature of self-psychology can offer some insight into how identity is maintained. From the vantage point of self-psychology, there are two areas of interest: the processes by which a self is formed (the "I"), and the actual content of the schemata which compose the self-concept (the "Me"). In the latter field, theorists have shown interest in relating the self-concept to self-esteem, the differences between complex and simple ways of organizing self-knowledge, and the links between those organizing principles and the processing of information.

Weinreich's identity variant similarly includes the categories of identity diffusion, foreclosure and crisis, but with a somewhat different emphasis. Here, with respect to identity diffusion for example, an optimal level is interpreted as the norm, as it is unrealistic to expect an individual to resolve all their conflicted identifications with others; therefore we should be alert to individuals with levels which are much higher or lower than the norm – highly diffused individuals are classified as diffused, and those with low levels as foreclosed or defensive. Weinreich applies the identity variant in a framework which also allows for the transition from one to another by way of biographical experiences and resolution of conflicted identifications situated in various contexts – for example, an adolescent going through family break-up may be in one state, whereas later in a stable marriage with a secure professional role may be in another. Hence, though there is continuity, there is also development and change.

Laing's definition of identity closely follows Erikson's, in emphasising the past, present and future components of the experienced self. He also develops the concept of the "metaperspective of self", i.e. the self's perception of the other's view of self, which has been found to be extremely important in clinical contexts such as anorexia nervosa. Harré also conceptualises components of self/identity – the "person" (the unique being I am to myself and others) along with aspects of self (including a totality of attributes including beliefs about one's characteristics including life history), and the personal characteristics displayed to others.

Further information: Self (psychology)

In social psychology

At a general level, self-psychology is compelled to investigate the question of how the personal self relates to the social environment. To the extent that these theories place themselves in the tradition of "psychological" social psychology, they focus on explaining an individual's actions within a group in terms of mental events and states. However, some "sociological" social psychology theories go further by attempting to deal with the issue of identity at both the levels of individual cognition and of collective behaviour.

Collective identity

Main article: Collective identity

Many people gain a sense of positive self-esteem from their identity groups, which furthers a sense of community and belonging. Another issue that researchers have attempted to address is the question of why people engage in discrimination, i.e., why they tend to favour those they consider a part of their "in-group" over those considered to be outsiders. Both questions have been given extensive attention by researchers working in the social identity tradition. For example, in work relating to social identity theory it has been shown that merely crafting cognitive distinction between in- and out-groups can lead to subtle effects on people's evaluations of others.

Different social situations also compel people to attach themselves to different self-identities which may cause some to feel marginalized, switch between different groups and self-identifications, or reinterpret certain identity components. These different selves lead to constructed images dichotomized between what people want to be (the ideal self) and how others see them (the limited self). Educational background and occupational status and roles significantly influence identity formation in this regard.

Identity formation strategies

Another issue of interest in social psychology is related to the notion that there are certain identity formation strategies which a person may use to adapt to the social world. Cote and Levine developed a typology which investigated the different manners of behavior that individuals may have. Their typology includes:

Cote and Levine's identity formation strategy typology

Type	Psychological signs	Personality signs	Social signs
Refuser	Develops cognitive blocks that prevent adoption of adult role-schemas	Engages in childlike behavior	Shows extensive dependency upon others and no meaningful engagement with the community of adults
Drifter	Possesses greater psychological resources than the Refuser (i.e., intelligence, charisma)	Is apathetic toward application of psychological resources	Has no meaningful engagement with or commitment to adult communities
Searcher	Has a sense of dissatisfaction due to high personal and social expectations	Shows disdain for imperfections within the community	Interacts to some degree with role-models, but ultimately these relationships are abandoned
Guardian	Possesses clear personal values and attitudes, but also a deep fear of change	Sense of personal identity is almost exhausted by sense of social identity	Has an extremely rigid sense of social identity and strong identification with adult communities
Resolver	Consciously desires self-growth	Accepts personal skills and competencies and uses them actively	Is responsive to communities that provide opportunity for self-growth

Kenneth Gergen formulated additional classifications, which include the strategic manipulator, the pastiche personality, and the relational self. The strategic manipulator is a person who begins to regard all senses of identity merely as role-playing exercises, and who gradually becomes alienated from their social self. The pastiche personality abandons all aspirations toward a true or "essential" identity, instead viewing social interactions as opportunities to play out, and hence become, the roles they play. Finally, the relational self is a perspective by which persons abandon all sense of exclusive self, and view all sense of identity in terms of social engagement with others. For Gergen, these strategies follow one another in phases, and they are linked to the increase in popularity of postmodern culture and the rise of telecommunications technology.

In social anthropology

See also: Australian Aboriginal identity

Anthropologists have most frequently employed the term identity to refer to this idea of selfhood in a loosely Eriksonian way properties based on the uniqueness and individuality which makes a person distinct from others. Identity became of more interest to anthropologists with the emergence of modern concerns with ethnicity and social movements in the 1970s. This was reinforced by an appreciation, following the trend in sociological thought, of the manner in which the individual is affected by and contributes to the overall social context. At the same time, the Eriksonian approach to identity remained in force, with the result that identity has continued until recently to be used in a largely socio-historical way to refer to qualities of sameness in relation to a person's connection to others and to a particular group of people.

The first favours a primordialist approach which takes the sense of self and belonging to a collective group as a fixed thing, defined by objective criteria such as common ancestry and common biological characteristics. The second, rooted in social constructionist theory, takes the view that identity is formed by a predominantly political choice of certain characteristics. In so doing, it questions the idea that identity is a natural given, characterised by fixed, supposedly objective criteria. Both approaches need to be understood in their respective political and historical contexts, characterised by debate on issues of class, race and ethnicity. While they have been criticized, they continue to exert an influence on approaches to the conceptualisation of identity today.

These different explorations of 'identity' demonstrate how difficult a concept it is to pin down. Since identity is a virtual thing, it is impossible to define it empirically. Discussions of identity use the term with different meanings, from fundamental and abiding sameness, to fluidity, contingency, negotiated and so on. Brubaker and Cooper note a tendency in many scholars to confuse identity as a category of practice and as a category of analysis. Indeed, many scholars demonstrate a tendency to follow their own preconceptions of identity, following more or less the frameworks listed above, rather than taking into account the mechanisms by which the concept is crystallised as reality. In this environment, some analysts, such as Brubaker and Cooper, have suggested doing away with the concept completely. Others, by contrast, have sought to introduce alternative concepts in an attempt to capture the dynamic and fluid qualities of human social self-expression. Stuart Hall for example, suggests treating identity as a process, to take into account the reality of diverse and ever-changing social experience. Some scholars have introduced the idea of identification, whereby identity is perceived as made up of different components that are 'identified' and interpreted by individuals. The construction of an individual sense of self is achieved by personal choices regarding who and what to associate with. Such approaches are liberating in their recognition of the role of the individual in social interaction and the construction of identity.

Anthropologists have contributed to the debate by shifting the focus of research: One of the first challenges for the researcher wishing to carry out empirical research in this area is to identify an appropriate analytical tool. The concept of boundaries is useful here for demonstrating how identity works. In the same way as Barth, in his approach to ethnicity, advocated the critical focus for investigation as being "the ethnic boundary that defines the group rather than the cultural stuff that it encloses", social anthropologists such as Cohen and Bray have shifted the focus of analytical study from identity to the boundaries that are used for purposes of identification. If identity is a kind of virtual site in which the dynamic processes and markers used for identification are made apparent, boundaries provide the framework on which this virtual site is built. They concentrated on how the idea of community belonging is differently constructed by individual members and how individuals within the group conceive ethnic boundaries.

As a non-directive and flexible analytical tool, the concept of boundaries helps both to map and to define the changeability and mutability that are characteristic of people's experiences of the self in society. While identity is a volatile, flexible and abstract 'thing', its manifestations and the ways in which it is exercised are often open to view. Identity is made evident through the use of markers such as language, dress, behaviour and choice of space, whose effect depends on their recognition by other social beings. Markers help to create the boundaries that define similarities or differences between the marker wearer and the marker perceivers, their effectiveness depends on a shared understanding of their meaning. In a social context, misunderstandings can arise due to a misinterpretation of the significance of specific markers. Equally, an individual can use markers of identity to exert influence on other people without necessarily fulfilling all the criteria that an external observer might typically associate with such an abstract identity.

Boundaries can be inclusive or exclusive depending on how they are perceived by other people. An exclusive boundary arises, for example, when a person adopts a marker that imposes restrictions on the behaviour of others. An inclusive boundary is created, by contrast, by the use of a marker with which other people are ready and able to associate. At the same time, however, an inclusive boundary will also impose restrictions on the people it has included by limiting their inclusion within other boundaries. An example of this is the use of a particular language by a newcomer in a room full of people speaking various languages. Some people may understand the language used by this person while others may not. Those who do not understand it might take the newcomer's use of this particular language merely as a neutral sign of identity. But they might also perceive it as imposing an exclusive boundary that is meant to mark them off from the person. On the other hand, those who do understand the newcomer's language could take it as an inclusive boundary, through which the newcomer associates themself with them to the exclusion of the other people present. Equally, however, it is possible that people who do understand the newcomer but who also speak another language may not want to speak the newcomer's language and so see their marker as an imposition and a negative boundary. It is possible that the newcomer is either aware or unaware of this, depending on whether they themself knows other languages or is conscious of the plurilingual quality of the people there and is respectful of it or not.

In philosophy

See also: Personal identity and Identity (philosophy)

Nietzsche called for a rejection of "Soul Atomism" in The Gay Science. Nietzsche supposed that the Soul was an interaction of forces, an ever-changing thing far from the immortal soul posited by both Descartes and the Christian tradition. His "Construction of the Soul" in many ways resembles modern social constructivism.

Heidegger, following Nietzsche, did work on identity. For Heidegger, people only really form an identity after facing death. It's death that allows people to choose from the social constructed meanings in their world, and assemble a finite identity out of seemingly infinite meanings. For Heidegger, most people never escape the "they", a socially constructed identity of "how one ought to be" created mostly to try to escape death through ambiguity.

Ricoeur has introduced the distinction between the ipse identity (selfhood, 'who am I?') and the idem identity (sameness, or a third-person perspective which objectifies identity).

Implications

The implications are multiple as various research traditions are now heavily utilizing the lens of identity to examine phenomena. One implication of identity and of identity construction can be seen in occupational settings. This becomes increasing challenging in stigmatized jobs or "dirty work". Tracy and Trethewey state that "individuals gravitate toward and turn away from particular jobs depending in part, on the extent to which they validate a "preferred organizational self". Some jobs carry different stigmas or acclaims. In her analysis Tracy uses the example of correctional officers trying to shake the stigma of "glorified maids". "The process by which people arrive at justifications of and values for various occupational choices." Among these are workplace satisfaction and overall quality of life. People in these types of jobs are forced to find ways in order to create an identity they can live with. "Crafting a positive sense of self at work is more challenging when one's work is considered "dirty" by societal standards". "In other words, doing taint management is not just about allowing the employee to feel good in that job. "If employees must navigate discourses that question the viability of their work, and/ or experience obstacles in managing taint through transforming dirty work into a badge of honor, it is likely they will find blaming the client to be an efficacious route in affirming their identity".

In any case, the concept that an individual has a unique identity developed relatively recently in history. Factors influencing the emphasis on personal identity may include:

• in the West, the Protestant stress on one's responsibility for one's own soul
• psychology itself, emerging as a distinct field of knowledge and study from the 19th century onwards
• the growth of a sense of privacy since the Renaissance
• specialization of worker roles during the industrial period (as opposed, for example, to the undifferentiated roles of peasants in the feudal system)
• occupation and employment's effect on identity
• increased emphasis on gender identity, including gender dysphoria and transgender issues

Identity changes

An important implication relates to identity change, i.e. the transformation of identity.

Contexts include:

• radical career-change.
• gender identity transition
• national emigration
• adoption